Finite Group Databasebeta

← #10845324

Subgroups — #10845324

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82778C211yesmaximal, characteristic
2C2×#4791C211yesmaximal, characteristic, complemented
3C2×#67⋊C8C211yesmaximal, characteristic
4#73937C211yesmaximal, characteristic
5C3×#2035C211yesmaximal, characteristic, complemented
6#71696C211yesmaximal, characteristic, complemented
7#70577C211yesmaximal, characteristic, complemented
8#70563C211yesmaximal, characteristic, complemented
9#70274C211yesmaximal, characteristic, complemented
10#70270C211yesmaximal, characteristic, complemented
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69940C211yesmaximal, characteristic, complemented
13#69511C211yesmaximal, characteristic, complemented
14#68891C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#25341—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C22×D3×C8C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#885—21no—
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4994—21no—
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4976—21no—
43#4975C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4957—21no—
48#4952—21no—
49#4948—21no—
50#4791—24no—
51#67⋊C8—24no—
52#4675—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#475—31no—
63C2×C8×D4—31no—
64#2547—31no—
65#2538—31no—
66#2077—31no—
67#2077—31no—
68#2035—31no—
69#2035D311yescharacteristic, complemented
70#1990—31no—
71#1948—31no—
72#1892—31no—
73#1892—31no—
74#1807—31no—
75#1807—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×#67D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217D411yescharacteristic
95#946—22no—
96#946—41no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99C4×#67—22no—
100C4×#67—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8—22no—
120C22×C3⋊C8—22no—
121C22×C3⋊C8—41no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C2×D3×C8—22no—
125C2×D3×C8—22no—
126C2×D3×C8—41no—
127#888—22no—
128#888—22no—
129#885—44no—
130C3×#106—41no—
131C3×#103—22no—
132C3×#103—22no—
133C3×C22⋊C8—22no—
134C3×#95—21no—
135#835—22no—
136#835—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143(C2×D3)⋊C8—22no—
144#817—22no—
145Dic3⋊C8D414yes—
146C8×Dic3—22no—
147#808—22no—
148#808—41no—
149C23×C8—31no—
150C23×C8—31no—
151C2×C4⋊Q8—31no—
152C2×#124—61no—
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8C2×D311yescharacteristic
155C2×C4×Q8—31no—
156C2×C4×D4—31no—
157#475—64no—
158C8×D4—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170C2×C4×C8—31no—
171#397—61no—
172#397—61no—
173#396—61no—
174#396—61no—
175#391—61no—
176#390—61no—
177#389—61no—
178#384—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194C24×C3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—22no—
215C2×#67—22no—
216C2×#67SD1614yes—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×Dic6—42no—
231C2×Dic6—42no—
232C2×Dic6—42no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D412yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248#217—22no—
249#217—42no—
250#217—81no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265Dic3⋊C4—22no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—81no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3C2×D412yes—
273C4×Dic3#4112yes—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—44no—
279C2×C3⋊C8C2×D412yes—
280C2×C3⋊C8#4112yes—
281D3×C8—44no—
282C22×Q8—61no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—31no—
285C23×C4—31no—
286C23×C4C22×D311yescharacteristic
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8—62no—
293C22×C8—121no—
294C22×C8—121no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299#124—124no—
300C22⋊Q8—22no—
301C22⋊Q8—41no—
302C22⋊Q8—62no—
303C22⋊Q8—121no—
304C4×Q8—62no—
305C4×Q8—62no—
306C4×D4—62no—
307C4×D4—62no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42—61no—
335#106—41no—
336#106—62no—
337#106—62no—
338#106—121no—
339C4⋊C8—34no—
340#103—22no—
341#103—22no—
342#103—62no—
343#103—62no—
344C22⋊C8—22no—
345C22⋊C8—62no—
346C4×C8—62no—
347#95—21no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3C2×SD1612yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3C2×SD1612yes—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—41no—
372C3×Q8—41no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398#67—28no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3#12112yes—
417C2×Dic3#12112yes—
418C2×Dic3C2×SD1612yes—
419C2×Dic3#13512yes—
420C4×D3—41no—
421C4×D3—41no—
422C4×D3—41no—
423C4×D3—41no—
424Dic6—84no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C24—31no—
430C24D3×D411yescharacteristic
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—122no—
437C2×Q8—122no—
438C2×Q8—122no—
439C2×Q8#23712yes—
440C2×D4—34no—
441C2×D4—62no—
442C2×D4—62no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—32no—
474C2×C8—32no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—124no—
480C2×C8—124no—
481C2×C8D3×D412yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—62no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—241no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—62no—
506C22⋊C4—62no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—241no—
512C22⋊C4—241no—
513C42—32no—
514C42—32no—
515C42—122no—
516C42—122no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#54611yescharacteristic
526C22×C3#54711yescharacteristic
527C22×C3C22×SD1611yescharacteristic
528C22×C3C2×#13511yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57811yescharacteristic
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—22no—
534C2×D3—41no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—41no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—22no—
548Dic3—41no—
549Dic3—81no—
550Dic3—81no—
551Dic3—81no—
552Dic3—81no—
553Dic3#46514yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—32no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23D3×SD1612yes—
568C23C2×D3×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—244no—
574D4—68no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4#100411yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4#101011yescharacteristic
625C8—22no—
626C8—41no—
627C8—62no—
628C8—62no—
629C2×C3—42no—
630C2×C3#242711yescharacteristic
631C2×C3C2×#46511yescharacteristic
632C2×C3#306311yescharacteristic
633C2×C3#329811yescharacteristic
634C2×C3#330411yescharacteristic
635C2×C3#333511yescharacteristic
636C2×C3#334211yescharacteristic
637D3—42no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—62no—
646C22—62no—
647C22—62no—
648C22—121no—
649C22—121no—
650C22—121no—
651C22—121no—
652C22#559111yescharacteristic
653C22D3×#12111yescharacteristic
654C22#565511yescharacteristic
655C22C2×D3×SD1611yescharacteristic
656C22D3×#13511yescharacteristic
657C22#577811yescharacteristic
658C22#577911yescharacteristic
659C4—34no—
660C4—41no—
661C4—42no—
662C4—62no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—121no—
667C4—241no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#559811yescharacteristic
672C4#561011yescharacteristic
673C4#564911yescharacteristic
674C4#566911yescharacteristic
675C3#2534111yescharacteristic, complemented, Sylow
676C2—42no—
677C2—122no—
678C2#7690211yescharacteristic
679C2#8004911yescharacteristic
680C2#8024911yescharacteristic
681C2D3×#46511yescharacteristic
682C2#8090711yescharacteristic
683C2#8143511yescharacteristic
684C2#8153311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2035C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#2035D3
hypocenterC3#25341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
2. C2×#4791C2semidirect
5. C3×#2035C2semidirect
6. #71696C2semidirect
7. #70577C2semidirect
8. #70563C2semidirect
9. #70274C2semidirect
10. #70270C2semidirect
11. Dic3×#103C2semidirect
12. #69940C2semidirect
13. #69511C2semidirect
14. #68891C2semidirect
15. #68803C2semidirect
69. #2035D3semidirect
675. C3#25341semidirect

Maximal subgroups

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

subgroupindex
#827782
C2×#47912
C2×#67⋊C82
#739372
C3×#20352
#716962
#705772
#705632
#702742
#702702
Dic3×#1032
#699402
#695112
#688912
#688032
#253413