Finite Group Databasebeta

← #10831745

Subgroups — #10831745

Order 768.

1248 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5727C211yesmaximal, characteristic
2C2×#4875C211yesmaximal, characteristic, complemented
3C2×#4889C211yesmaximal, characteristic
4#76076C211yesmaximal, characteristic, complemented
5C3×#2057C211yesmaximal, characteristic, complemented
6#71794C211yesmaximal, characteristic, complemented
7#70638C211yesmaximal, characteristic
8#70617C211yesmaximal, characteristic, complemented
9#69625C211yesmaximal, characteristic
10#68910C211yesmaximal, characteristic, complemented
11#68896C211yesmaximal, characteristic
12#68893C211yesmaximal, characteristic
13#68806C211yesmaximal, characteristic, complemented
14#68672C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#24342—31nomaximal, Sylow
17C22×D3×Q8—21no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×#946—21no—
20C22×#204C2211yescharacteristic
21#5727—24no—
22#5727—24no—
23C2×#896—21no—
24C2×#896—21no—
25C2×#893C2211yescharacteristic
26C2×#893C2211yescharacteristic
27C2×#888C2211yescharacteristic
28C2×#888C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#107C2211yescharacteristic
33C2×C3×#107C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4980—21no—
40#4979—21no—
41#4977C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4929C2211yescharacteristic
50#4889—24no—
51#4875—24no—
52#4658—21no—
53#4652—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×D4×Q8—31no—
65C2×#477—31no—
66C2×#440—31no—
67#2581—31no—
68#2057—31no—
69#2057D311yescharacteristic, complemented
70#2024—31no—
71#2024—31no—
72#1993—31no—
73#1985—31no—
74#1877—31no—
75#1877—31no—
76#1862—31no—
77#1862—31no—
78#1804—31no—
79C22×C3×Q8—21no—
80C2×D3×Q8—28no—
81C2×D3×Q8—44no—
82C2×D3×Q8—44no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×C4×D3—21no—
87C22×Dic6—21no—
88C22×Dic6—21no—
89C22×Dic6D411yescharacteristic
90C22×Dic6C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×C4⋊Q8—22no—
94C3×C4⋊Q8—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102#946—28no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×#204—22no—
121C2×#204—22no—
122C2×#204Q814yescomplemented
123#896—28no—
124#896—28no—
125#893—22no—
126#893—22no—
127#893—22no—
128#893D414yes—
129#888—22no—
130#888—22no—
131#888—22no—
132#888D414yes—
133#887—22no—
134#887—22no—
135C4×D12—22no—
136C4×D12—22no—
137C4×Dic6—22no—
138C4×Dic6—22no—
139C3×#107—22no—
140C3×#107—22no—
141C3×#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#821—22no—
150#816—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#810—22no—
154#810—22no—
155#809—22no—
156#809—22no—
157#807—22no—
158C23×Q8—61no—
159C22×SD16—31no—
160D4×Q8—64no—
161D4×Q8—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—31no—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×C22⋊Q8—61no—
166C2×C22⋊Q8—61no—
167C2×C22⋊Q8—61no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×D4—31no—
171C2×C4×D4—31no—
172C2×C4×D4—31no—
173#477—64no—
174#440—64no—
175C2×#107—31no—
176C2×#107C2×D311yescharacteristic
177C2×#107C2×D311yescharacteristic
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#97—31no—
186#400—61no—
187#395—61no—
188#391—31no—
189#391—31no—
190#389—31no—
191#389C2×D311yescharacteristic
192#387—61no—
193#387—61no—
194#382—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199#340C2×D311yescharacteristic
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C2×C3×Q8—24no—
207C2×C3×Q8—42no—
208C2×C3×Q8—42no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223D3×Q8—432no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12C2×D412yes—
228C2×D12C2×Q812yes—
229C2×D12#4112yes—
230C2×C4×D3—24no—
231C2×C4×D3—24no—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×Dic6—22no—
240C2×Dic6—22no—
241C2×Dic6—22no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—24no—
247C2×Dic6—21no—
248C2×Dic6—24no—
249C2×Dic6—42no—
250C2×Dic6—42no—
251C2×Dic6—42no—
252C2×Dic6—42no—
253C2×Dic6C2×Q812yescomplemented
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8C2×Q812yescomplemented
260C3×C4⋊C4—24no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—42no—
268C3×C4⋊C4C2×D412yes—
269C3×C4⋊C4C2×D412yes—
270C3×C42—21no—
271C3×C42—21no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3—22no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3C2×D412yes—
280Dic3⋊C4—24no—
281Dic3⋊C4—22no—
282Dic3⋊C4—22no—
283Dic3⋊C4—24no—
284Dic3⋊C4—42no—
285Dic3⋊C4—42no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—42no—
289C4×Dic3C2×D412yes—
290C4×Dic3C2×D412yes—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297#204—28no—
298C22×Q8—21no—
299C22×Q8—31no—
300C22×Q8—31no—
301C22×Q8—61no—
302C22×Q8—61no—
303C22×Q8—68no—
304C22×Q8—124no—
305C22×Q8—124no—
306C22×D4—31no—
307C23×C4—61no—
308C23×C4—61no—
309C23×C4—61no—
310C2×SD16—34no—
311C2×SD16—62no—
312C2×SD16—62no—
313C22×C8—31no—
314C22×C8—31no—
315C22×C8C22×D311yescharacteristic
316C22×C8C22×D311yescharacteristic
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319C4⋊Q8—34no—
320C4⋊Q8—62no—
321C4⋊Q8—62no—
322C4⋊Q8—62no—
323C22⋊Q8—68no—
324C22⋊Q8—68no—
325C22⋊Q8—68no—
326C4×Q8—62no—
327C4×Q8—62no—
328C4×Q8—62no—
329C4×Q8—62no—
330C4×D4—34no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4—62no—
334C4×D4—62no—
335C4×D4—62no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C4⋊C4C22×D311yescharacteristic
349C2×C4⋊C4C22×D311yescharacteristic
350C2×C4⋊C4C22×D311yescharacteristic
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—61no—
357C2×C42C22×D311yescharacteristic
358#107—22no—
359#107—22no—
360#107—62no—
361C4⋊C8—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365#102—62no—
366#102—62no—
367#102—62no—
368#97—62no—
369#95—21no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3C22×D411yescharacteristic
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C3×Q8—48no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4AGL(1,ℤ/8)12yes—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×Q811yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3#12012yes—
429C2×Dic3#12012yes—
430C2×Dic3AGL(1,ℤ/8)12yes—
431D12—22no—
432D12—22no—
433D12—22no—
434D12C22⋊Q814yes—
435C4×D3—48no—
436C4×D3—48no—
437C4×D3—48no—
438Dic6—28no—
439Dic6—22no—
440Dic6—22no—
441Dic6—48no—
442Dic6—48no—
443Dic6—41no—
444Dic6—41no—
445C3×C8—22no—
446C3×C8—41no—
447C3⋊C8—41no—
448C3⋊C8—41no—
449C24—61no—
450C2×Q8—24no—
451C2×Q8—32no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—62no—
455C2×Q8—62no—
456C2×Q8—64no—
457C2×Q8—64no—
458C2×Q8—62no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—122no—
465C2×Q8—122no—
466C2×Q8—122no—
467C2×Q8—122no—
468C2×Q8—1232no—
469C2×D4—32no—
470C2×D4—32no—
471C2×D4—32no—
472C2×D4—61no—
473C2×D4—61no—
474C2×D4—61no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—64no—
484C22×C4—64no—
485C22×C4—64no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4D3×D411yescharacteristic
496C22×C4D3×D411yescharacteristic
497C22×C4C23×D311yescharacteristic
498SD16—68no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8D3×Q812yescomplemented
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—32no—
519C4⋊C4—42no—
520C4⋊C4—62no—
521C4⋊C4—62no—
522C4⋊C4—62no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—122no—
529C4⋊C4—122no—
530C4⋊C4D3×D412yes—
531C4⋊C4D3×D412yes—
532C22⋊C4—64no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C42—21no—
536C42—21no—
537C42—32no—
538C42—32no—
539C42—61no—
540C42—61no—
541C42—122no—
542C22×C3C2×#12011yescharacteristic
543C22×C3#54811yescharacteristic
544C22×C3D4×Q811yescharacteristic
545C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
546C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
547C22×C3#57811yescharacteristic
548C22×C3#57811yescharacteristic
549C2×D3—24no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C3×C4—22no—
557C3×C4—42no—
558C3×C4—42no—
559C3×C4—41no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4C2×C22⋊Q811yescharacteristic
563C3×C4#53611yescharacteristic
564C3×C4#54811yescharacteristic
565C3×C4D4×Q811yescharacteristic
566Dic3—22no—
567Dic3—22no—
568Dic3—22no—
569Dic3—42no—
570Dic3—42no—
571Dic3—41no—
572Dic3—81no—
573C23—62no—
574C23—62no—
575C23—62no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23C2×D3×D411yescharacteristic
581Q8—48no—
582Q8—68no—
583Q8—62no—
584Q8—62no—
585Q8—128no—
586Q8—128no—
587Q8—121no—
588Q8—121no—
589D4—34no—
590D4—62no—
591D4—62no—
592D4—62no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—62no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—128no—
625C2×C4—128no—
626C2×C4—128no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4#91112yes—
635C2×C4C2×D3×D411yescharacteristic
636C2×C4C2×D3×D411yescharacteristic
637C2×C4C2×D3×Q811yescharacteristic
638C2×C4#100711yescharacteristic
639C2×C4#100711yescharacteristic
640C2×C4D3×#4111yescharacteristic
641C8—22no—
642C8—41no—
643C8—121no—
644C8—121no—
645C2×C3#240911yescharacteristic
646C2×C3#300311yescharacteristic
647C2×C3#301111yescharacteristic
648C2×C3#336711yescharacteristic
649C2×C3#336711yescharacteristic
650C2×C3#338611yescharacteristic
651C2×C3#338611yescharacteristic
652D3—44no—
653C22—64no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22—122no—
659C22—122no—
660C22D3×#12011yescharacteristic
661C22#563311yescharacteristic
662C22#572711yescharacteristic
663C22C2×#91111yescharacteristic
664C22#576611yescharacteristic
665C22D3×AGL(1,ℤ/8)11yescharacteristic
666C22#577911yescharacteristic
667C4—22no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—62no—
672C4—62no—
673C4—62no—
674C4—81no—
675C4—81no—
676C4—121no—
677C4—122no—
678C4—122no—
679C4—241no—
680C4D3×C22⋊Q811yescharacteristic
681C4#563311yescharacteristic
682C4#572411yescharacteristic
683C4#572711yescharacteristic
684C3#2434211yescharacteristic, complemented, Sylow
685C2—124no—
686C2#7717211yescharacteristic
687C2#8000211yescharacteristic
688C2#8018511yescharacteristic
689C2#8099311yescharacteristic
690C2#8110011yescharacteristic
691C2#8195511yescharacteristic
692C2#8196611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2057D3
hypocenterC3#24342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
2. C2×#4875C2semidirect
4. #76076C2semidirect
5. C3×#2057C2semidirect
6. #71794C2semidirect
8. #70617C2semidirect
10. #68910C2semidirect
13. #68806C2semidirect
15. #68620C2semidirect
69. #2057D3semidirect
122. C2×#204Q8semidirect
253. C2×Dic6C2×Q8semidirect
259. C2×C3×C8C2×Q8semidirect
510. C2×C8D3×Q8semidirect
684. C3#24342semidirect

Maximal subgroups

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

subgroupindex
C2×#57272
C2×#48752
C2×#48892
#760762
C3×#20572
#717942
#706382
#706172
#696252
#689102
#688962
#688932
#688062
#686722
#686202
#243423