Finite Group Databasebeta

← #10813631

Subgroups — #10813631

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5145C211yesmaximal, characteristic
2#83783C211yesmaximal, characteristic, complemented
3#79081C211yesmaximal, characteristic, complemented
4C2×#4823C211yesmaximal, characteristic
5#78818C211yesmaximal, characteristic
6C3×#1866C211yesmaximal, characteristic, complemented
7#70722C211yesmaximal, characteristic, complemented
8#70693C211yesmaximal, characteristic, complemented
9#70538C211yesmaximal, characteristic, complemented
10#70355C211yesmaximal, characteristic, complemented
11#70292C211yesmaximal, characteristic, complemented
12#68770C211yesmaximal, characteristic, complemented
13#68748C211yesmaximal, characteristic, complemented
14#68694C211yesmaximal, characteristic, complemented
15#68652C211yesmaximal, characteristic, complemented
16#22686—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18Dic3×#41C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#948C2211yescharacteristic
21C2×#907C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5522C2211yescharacteristic
24C2×C3×#108C2211yescharacteristic
25C3×#424C2211yescharacteristic
26C2×C3×#104C2211yescharacteristic
27C3×#416C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5145—24no—
30C2×#825C2211yescharacteristic
31C2×#822C2211yescharacteristic
32C2×#822C2211yescharacteristic
33#5119—21no—
34#5119C2211yescharacteristic
35#5118—21no—
36#5118C2211yescharacteristic
37#5118C2211yescharacteristic
38#5113C2211yescharacteristic
39C2×#819C2211yescharacteristic
40#5102—21no—
41#5102C2211yescharacteristic
42C2×#815C2211yescharacteristic
43C2×Dic3⋊C8—21no—
44C2×Dic3⋊C8C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#4999C2211yescharacteristic
47#4998C2211yescharacteristic
48C4×Dic3⋊C4C2211yescharacteristic
49C2×#805C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4839—21no—
52#4839—21no—
53#4837—21no—
54D3×C4⋊C8—21no—
55#4823—24no—
56#4790C411yescharacteristic, complemented
57#4790C411yescharacteristic, complemented
58#4788C411yescharacteristic, complemented
59#4788C411yescharacteristic, complemented
60#4758C411yescharacteristic, complemented
61#4758C411yescharacteristic, complemented
62#4756C411yescharacteristic, complemented
63#4756C411yescharacteristic, complemented
64C3×#339C2211yescharacteristic
65C3×#336C2211yescharacteristic
66#4555C2211yescharacteristic
67#4550C2211yescharacteristic
68#4532C2211yescharacteristic
69#4523C2211yescharacteristic
70C2×#444—31no—
71C2×#444—31no—
72C4×#135—31no—
73#2906—31no—
74#2906—31no—
75#1946—31no—
76#1946—31no—
77#1866—31no—
78#1866—31no—
79#1866—31no—
80#1866D311yescharacteristic, complemented
81#1848—31no—
82#1806—31no—
83#1806—31no—
84#1770—31no—
85C2×C3×#41C2311yescharacteristic
86C2×#235C2311yescharacteristic
87C3×#131—22no—
88C3×#131—41no—
89C2×C3×M16—21no—
90C2×C3×M16C2311yescharacteristic
91C2×C3×M16C2311yescharacteristic
92C22×C3×C8—21no—
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2311yescharacteristic
95C3×#117C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#950D414yes—
98#950Q814yes—
99#948—22no—
100#948—41no—
101Q8×Dic3C2×C411yescharacteristic
102Q8×Dic3C2×C411yescharacteristic
103C2×#216C2×C411yescharacteristic
104C2×#215C2×C411yescharacteristic
105D4×Dic3—21no—
106D4×Dic3—21no—
107D4×Dic3C2×C411yescharacteristic
108D4×Dic3C2×C411yescharacteristic
109C2×#214C2×C411yescharacteristic
110C2×#213C2×C411yescharacteristic
111#926—21no—
112#926C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×#208—21no—
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#907—22no—
124#907—41no—
125#901—22no—
126#901—41no—
127C2×ΓL(1,𝔽25)—21no—
128C2×ΓL(1,𝔽25)—21no—
129C2×D3×C8—21no—
130C2×D3×C8—21no—
131#893C2×C411yescharacteristic
132#893C2×C411yescharacteristic
133#887C2×C411yescharacteristic
134#887C2×C411yescharacteristic
135#67⋊C4—21no—
136#67⋊C4—21no—
137#872—21no—
138C42×D3—21no—
139C3×#108—41no—
140C3×C4⋊C8—21no—
141C3×C4⋊C8—21no—
142C3×#104—22no—
143C3×#103C2×C411yescharacteristic
144C3×#103C2×C411yescharacteristic
145C3×#102C2×C411yescharacteristic
146C3×#102C2×C411yescharacteristic
147#837—22no—
148C3⋊#95—21no—
149#825—22no—
150#822—22no—
151#822—22no—
152#821C2×C411yescharacteristic
153#821C2×C411yescharacteristic
154#819—41no—
155#816C2×C411yescharacteristic
156#816C2×C411yescharacteristic
157#815—22no—
158Dic3⋊C8—21no—
159Dic3⋊C8—21no—
160Dic3⋊C8—21no—
161Dic3⋊C8—21no—
162Dic3⋊C8—21no—
163Dic3⋊C8—21no—
164Dic3⋊C8—21no—
165Dic3⋊C8—21no—
166Dic3⋊C8D411yescharacteristic
167Dic3⋊C8D411yescharacteristic
168Dic3⋊C8D411yescharacteristic
169Dic3⋊C8D411yescharacteristic
170C8×Dic3C2×C411yescharacteristic
171C8×Dic3C2×C411yescharacteristic
172C8×Dic3C2×C411yescharacteristic
173C8×Dic3C2×C411yescharacteristic
174#808C2×C411yescharacteristic
175#808C2×C411yescharacteristic
176#807C2×C411yescharacteristic
177#807C2×C411yescharacteristic
178#805—22no—
179C4⋊(C3⋊C8)—21no—
180C4⋊(C3⋊C8)—21no—
181#803—22no—
182C2×#135—31no—
183C2×#131—31no—
184C2×#131—31no—
185C2×#131—31no—
186C2×#131C2×D311yescharacteristic
187C4×#41—31no—
188C4×#41—31no—
189#444—64no—
190#444—64no—
191C4×Q16—31no—
192C4×Q16—31no—
193C4×SD16—31no—
194C4×SD16—31no—
195C4×SD16—31no—
196C4×SD16—31no—
197C4×D8—31no—
198C4×D8—31no—
199C2×#108—31no—
200C2×#108—31no—
201C2×#108—31no—
202C2×#108C2×D311yescharacteristic
203#427—31no—
204#424—31no—
205#424—31no—
206#424—31no—
207#424—61no—
208#424—61no—
209#424—61no—
210#424—61no—
211#424C2×D311yescharacteristic
212#423—31no—
213#423—31no—
214#423—61no—
215#423—61no—
216C2×C4⋊C8—31no—
217C2×C4⋊C8—61no—
218C2×C4⋊C8—61no—
219C2×#104—31no—
220C2×#104—31no—
221C2×#104—31no—
222C2×#104C2×D311yescharacteristic
223#416—31no—
224#416C2×D311yescharacteristic
225C2×#97—31no—
226C2×#97—31no—
227C2×C4×C8—31no—
228C4×C4⋊C4—31no—
229#339—31no—
230#339—31no—
231#339—31no—
232#339C2×D311yescharacteristic
233#336—31no—
234#336C2×D311yescharacteristic
235C3×#41—21no—
236C3×#41—21no—
237C3×#41C2×D412yes—
238C3×#41C2×Q812yes—
239C2×C3×Q8C22×C411yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4C22×C411yescharacteristic
242C22×C3×C4—21no—
243C22×C3×C4—21no—
244C22×C3×C4C2411yescharacteristic
245C2×#67—21no—
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3#4111yescharacteristic
250C22×Dic3#4111yescharacteristic
251#235—21no—
252#235—21no—
253#235C2×D412yes—
254#235C2×Q812yes—
255C2×D12C22×C411yescharacteristic
256C2×C4×D3—21no—
257C2×C4×D3—21no—
258C2×C4×D3—21no—
259C2×Dic6C22×C411yescharacteristic
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C3×M16—42no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C2×C3×C8—42no—
272C2×C3×C8C22×C411yescharacteristic
273C2×C3×C8C22×C411yescharacteristic
274C2×C3×C8C2×D411yescharacteristic
275C2×C3×C8C2×D411yescharacteristic
276C3×C4⋊C4C22×C411yescharacteristic
277C3×C4⋊C4C22×C411yescharacteristic
278C3×C22⋊C4—21no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—21no—
282#217—21no—
283#217—21no—
284#217—21no—
285#216C4⋊C414yescomplemented
286#215C4⋊C414yescomplemented
287#214C4⋊C414yescomplemented
288#213C4⋊C414yescomplemented
289(C2×D3)⋊C4—21no—
290(C2×D3)⋊C4—21no—
291C4⋊Dic3—21no—
292C4⋊Dic3—21no—
293C4⋊Dic3C22×C411yescharacteristic
294C4⋊Dic3C22×C411yescharacteristic
295Dic3⋊C4—21no—
296Dic3⋊C4—21no—
297Dic3⋊C4—42no—
298C4×Dic3—21no—
299C4×Dic3—21no—
300C4×Dic3—21no—
301C4×Dic3—21no—
302C4×Dic3—21no—
303C4×Dic3—21no—
304C4×Dic3—21no—
305C4×Dic3C22×C411yescharacteristic
306C4×Dic3C22×C411yescharacteristic
307C4×Dic3C2×D411yescharacteristic
308C4×Dic3C2×D411yescharacteristic
309#208—21no—
310#208—21no—
311#208—21no—
312#208—21no—
313#208—42no—
314C2×C3⋊C8—21no—
315C2×C3⋊C8—21no—
316C2×C3⋊C8—21no—
317C2×C3⋊C8—21no—
318C2×C3⋊C8—42no—
319C2×C3⋊C8C22×C411yescharacteristic
320C2×C3⋊C8C22×C411yescharacteristic
321C2×C3⋊C8C2×D411yescharacteristic
322C2×C3⋊C8C2×D411yescharacteristic
323C2×C3⋊C8C2×D412yes—
324C2×C3⋊C8C2×Q812yes—
325ΓL(1,𝔽25)—42no—
326ΓL(1,𝔽25)—42no—
327D3×C8—42no—
328D3×C8—42no—
329C2×#41—31no—
330C2×#41C22×D311yescharacteristic
331#135—34no—
332#135—34no—
333C2×Q16—31no—
334C2×SD16—31no—
335C2×SD16—31no—
336C2×D8—31no—
337#131—22no—
338#131—41no—
339#131—62no—
340#131—62no—
341#131—62no—
342#131—121no—
343#131—121no—
344#131—121no—
345C2×M16—21no—
346C2×M16—31no—
347C2×M16—31no—
348C2×M16—61no—
349C2×M16—61no—
350C2×M16—61no—
351C2×M16C22×D311yescharacteristic
352C2×M16C22×D311yescharacteristic
353C22×C8—21no—
354C22×C8—31no—
355C22×C8—31no—
356C22×C8—61no—
357C22×C8—61no—
358C22×C8—61no—
359C22×C8C22×D311yescharacteristic
360C22×C8C22×D311yescharacteristic
361C4×Q8—31no—
362C4×Q8—31no—
363C4×Q8—31no—
364C4×Q8—31no—
365C4×D4—31no—
366C4×D4—31no—
367C4×D4—31no—
368C4×D4—31no—
369C4×D4—61no—
370C4×D4—61no—
371C4×D4—61no—
372C4×D4—61no—
373#117—31no—
374#117—61no—
375#117—61no—
376#117C22×D311yescharacteristic
377C2×C4⋊C4—61no—
378C2×C42—31no—
379C2×C42—31no—
380C2×C42—61no—
381C2×C42—61no—
382C2×C42C22×D311yescharacteristic
383#108—41no—
384#108—62no—
385#108—62no—
386#108—121no—
387#107—31no—
388#107—31no—
389#106—31no—
390#106—31no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393C4⋊C8—31no—
394C4⋊C8—31no—
395C4⋊C8—31no—
396C4⋊C8—31no—
397C4⋊C8—61no—
398C4⋊C8—61no—
399C4⋊C8—61no—
400C4⋊C8—61no—
401C4⋊C8—61no—
402C4⋊C8—61no—
403C4⋊C8—61no—
404C4⋊C8—61no—
405C4⋊C8—61no—
406C4⋊C8—61no—
407#104—22no—
408#104—62no—
409#104—62no—
410#104—62no—
411#103—31no—
412#103—31no—
413#103C4×D311yescharacteristic
414#103C4×D311yescharacteristic
415#102—31no—
416#102—31no—
417#102C4×D311yescharacteristic
418#102C4×D311yescharacteristic
419#97—62no—
420#97—62no—
421C4×C8—31no—
422C4×C8—31no—
423C4×C8—31no—
424C4×C8—31no—
425#95—61no—
426C23×C3—21no—
427C23×C3C2×#4111yescharacteristic
428C22×D3—21no—
429C3×Q8—21no—
430C3×Q8C2×C4⋊C412yes—
431C3×D4—22no—
432C3×D4—22no—
433C3×D4—21no—
434C3×D4C2×C4⋊C412yes—
435C2×C3×C4—21no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—21no—
440C2×C3×C4—41no—
441C2×C3×C4—41no—
442C2×C3×C4—41no—
443C2×C3×C4C23×C411yescharacteristic
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446C2×C3×C4C22×Q811yescharacteristic
447C2×C3×C4H(5,𝔽2)11yescharacteristic
448C2×C3×C42−1+411yescharacteristic
449#67—22no—
450#67—22no—
451C2×Dic3—21no—
452C2×Dic3—22no—
453C2×Dic3—21no—
454C2×Dic3—21no—
455C2×Dic3—41no—
456C2×Dic3—42no—
457C2×Dic3—41no—
458C2×Dic3—41no—
459C2×Dic3—41no—
460C2×Dic3—81no—
461C2×Dic3C4×D411yescharacteristic
462C2×Dic3C4×D411yescharacteristic
463C2×Dic3C4×D411yescharacteristic
464C2×Dic3C4×D411yescharacteristic
465D12—21no—
466D12C2×C4⋊C412yes—
467C4×D3—22no—
468C4×D3—22no—
469C4×D3—42no—
470C4×D3—81no—
471Dic6—21no—
472Dic6C2×C4⋊C412yes—
473C3×C8—21no—
474C3×C8—21no—
475C3×C8—21no—
476C3×C8—21no—
477C3×C8—41no—
478C3×C8—41no—
479C3⋊C8—21no—
480C3⋊C8—21no—
481C3⋊C8—41no—
482C3⋊C8—41no—
483C3⋊C8C2×C4⋊C412yes—
484C3⋊C8C2×C4⋊C412yes—
485#41—21no—
486#41—21no—
487#41—32no—
488#41—32no—
489#41—61no—
490#41—61no—
491#41D3×D412yes—
492#41D3×Q812yes—
493C2×Q8—31no—
494C2×Q8C2×C4×D311yescharacteristic
495C2×D4—21no—
496C2×D4—31no—
497C2×D4—61no—
498C2×D4C2×C4×D311yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—61no—
504C22×C4—61no—
505C22×C4—61no—
506C22×C4—61no—
507C22×C4—61no—
508C22×C4—61no—
509C22×C4C23×D311yescharacteristic
510Q16—34no—
511SD16—34no—
512SD16—34no—
513D8—34no—
514M16—21no—
515M16—21no—
516M16—21no—
517M16—21no—
518M16—42no—
519M16—61no—
520M16—61no—
521M16—61no—
522M16—61no—
523M16—122no—
524M16—122no—
525M16—122no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—31no—
533C2×C8—31no—
534C2×C8—32no—
535C2×C8—32no—
536C2×C8—31no—
537C2×C8—31no—
538C2×C8—42no—
539C2×C8—61no—
540C2×C8—61no—
541C2×C8—61no—
542C2×C8—61no—
543C2×C8—122no—
544C2×C8—122no—
545C2×C8—122no—
546C2×C8C2×C4×D311yescharacteristic
547C2×C8C2×C4×D311yescharacteristic
548C2×C8D3×D411yescharacteristic
549C2×C8D3×D411yescharacteristic
550C4⋊C4—31no—
551C4⋊C4—31no—
552C4⋊C4—61no—
553C4⋊C4—61no—
554C4⋊C4—61no—
555C4⋊C4—61no—
556C4⋊C4—122no—
557C4⋊C4C2×C4×D311yescharacteristic
558C4⋊C4C2×C4×D311yescharacteristic
559C22⋊C4—21no—
560C22⋊C4—61no—
561C22⋊C4—61no—
562C22⋊C4—61no—
563C22⋊C4—61no—
564C22⋊C4—61no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—31no—
569C42—31no—
570C42—31no—
571C42—31no—
572C42—61no—
573C42—61no—
574C42—61no—
575C42—61no—
576C42—61no—
577C42—61no—
578C42—61no—
579C22×C3—21no—
580C22×C3—21no—
581C22×C3—22no—
582C22×C3—41no—
583C22×C3—41no—
584C22×C3C2×C4×D411yescharacteristic
585C22×C3#54711yescharacteristic
586C22×C3#55411yescharacteristic
587C2×D3—22no—
588C2×D3—41no—
589C2×D3—41no—
590C3×C4—22no—
591C3×C4—41no—
592C3×C4—41no—
593C3×C4—81no—
594C3×C4C22×C4⋊C411yescharacteristic
595C3×C4#52011yescharacteristic
596C3×C4D4211yescharacteristic
597C3×C4D4×Q811yescharacteristic
598Dic3—22no—
599Dic3—21no—
600Dic3—21no—
601Dic3—21no—
602Dic3—41no—
603Dic3—41no—
604Dic3—81no—
605Dic3#44412yescomplemented
606C23—21no—
607C23—61no—
608C23D3×#4111yescharacteristic
609Q8—21no—
610Q8—32no—
611Q8—61no—
612Q8D3×C4⋊C412yes—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—32no—
617D4—62no—
618D4—62no—
619D4—61no—
620D4D3×C4⋊C412yes—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—31no—
627C2×C4—31no—
628C2×C4—31no—
629C2×C4—31no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—62no—
634C2×C4—61no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—241no—
646C2×C4—241no—
647C2×C4C22×C4×D311yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C2×C4#100411yescharacteristic
651C2×C4C2×D3×Q811yescharacteristic
652C2×C4#101011yescharacteristic
653C8—21no—
654C8—21no—
655C8—21no—
656C8—21no—
657C8—32no—
658C8—32no—
659C8—41no—
660C8—41no—
661C8—61no—
662C8—61no—
663C8—121no—
664C8—121no—
665C2×C3—21no—
666C2×C3—21no—
667C2×C3—42no—
668C2×C3D4×C4⋊C411yescharacteristic
669C2×C3C2×#44411yescharacteristic
670C2×C3#295711yescharacteristic
671D3—42no—
672C22—21no—
673C22—21no—
674C22—22no—
675C22—41no—
676C22—41no—
677C22—62no—
678C22—121no—
679C22—121no—
680C22C4×D3×D411yescharacteristic
681C22#563611yescharacteristic
682C22#565511yescharacteristic
683C4—22no—
684C4—32no—
685C4—41no—
686C4—41no—
687C4—61no—
688C4—61no—
689C4—61no—
690C4—62no—
691C4—81no—
692C4—121no—
693C4—121no—
694C4—241no—
695C4C2×D3×C4⋊C411yescharacteristic
696C4#554611yescharacteristic
697C4#559011yescharacteristic
698C4#563211yescharacteristic
699C3#2268611yescharacteristic, complemented, Sylow
700C2—21no—
701C2—21no—
702C2—42no—
703C2—122no—
704C2#7439311yescharacteristic
705C2#7989011yescharacteristic
706C2D3×#44411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1866C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1866D3
hypocenterC3#22686
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #83783C2semidirect
3. #79081C2semidirect
6. C3×#1866C2semidirect
7. #70722C2semidirect
8. #70693C2semidirect
9. #70538C2semidirect
10. #70355C2semidirect
11. #70292C2semidirect
12. #68770C2semidirect
13. #68748C2semidirect
14. #68694C2semidirect
15. #68652C2semidirect
56. #4790C4semidirect
57. #4790C4semidirect
58. #4788C4semidirect
59. #4788C4semidirect
60. #4758C4semidirect
61. #4758C4semidirect
62. #4756C4semidirect
63. #4756C4semidirect
80. #1866D3semidirect
285. #216C4⋊C4semidirect
286. #215C4⋊C4semidirect
287. #214C4⋊C4semidirect
288. #213C4⋊C4semidirect
605. Dic3#444semidirect
699. C3#22686semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#51452
#837832
#790812
C2×#48232
#788182
C3×#18662
#707222
#706932
#705382
#703552
#702922
#687702
#687482
#686942
#686522
#226863