Finite Group Databasebeta

← #10804150

Subgroups — #10804150

Order 768.

1294 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5648C211yesmaximal, characteristic
2#82230C211yesmaximal, characteristic, complemented
3#74053C211yesmaximal, characteristic, complemented
4C3×#1899C211yesmaximal, characteristic, complemented
5#71705C211yesmaximal, characteristic, complemented
6#71595C211yesmaximal, characteristic, complemented
7#70029C212yesmaximal, complemented
8#70027C212yesmaximal
9#69535C211yesmaximal, characteristic, complemented
10#68954C212yesmaximal, complemented
11#68953C212yesmaximal, complemented
12#21751—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×#123C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#935C2211yescharacteristic
18C2×#929C2211yescharacteristic
19C2×#926C2211yescharacteristic
20#5648—24no—
21#5648—22no—
22#5648—22no—
23C2×#896C412yescomplemented
24C2×#888C412yescomplemented
25C2×#886C2211yescharacteristic
26C2×#885C412yescomplemented
27C2×#878C412yescomplemented
28C2×#877C2211yescharacteristic
29C2×#877C2212yes—
30C2×C3×#99C2212yes—
31C3×#381C2211yescharacteristic
32C2×#834C2212yes—
33#5211C2211yescharacteristic
34#4973—22no—
35C2×#806C2212yes—
36C2×#806C2212yes—
37#4943C2212yes—
38#4943C2212yes—
39#4943C2212yes—
40#4929—22no—
41#4646—24no—
42C3×#342C2212yes—
43#4527C2212yes—
44#4527C2212yes—
45#4526C2212yes—
46C2×#536—31no—
47#2310—31no—
48#2310—31no—
49#1899—32no—
50#1899—31no—
51#1899—32no—
52#1899—31no—
53#1899—31no—
54#1899D311yescharacteristic, complemented
55#1820—32no—
56#1782—32no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67C2×C411yescharacteristic
60C22×#67C2311yescharacteristic
61C23×Dic3C2×C411yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C2×#237—22no—
65C2×#237—22no—
66C2×#237—42no—
67C2×#237—42no—
68C22×C4×D3C2×C411yescharacteristic
69C22×Dic6—42no—
70C22×Dic6—42no—
71C22×Dic6C2×C411yescharacteristic
72C22×Dic6C2×C412yes—
73C22×Dic6C2×C412yes—
74C22×Dic6D414yes—
75C22×Dic6D412yes—
76C22×Dic6D412yes—
77C3×#123—22no—
78C3×#123—21no—
79C3×#123—21no—
80C3×#117—42no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2×C412yes—
83C2×C3×C22⋊C4C2×C412yes—
84C2×C3×C22⋊C4C2312yes—
85C2×C3×C22⋊C4C2312yes—
86C2×C3×C22⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×#217C2×C412yes—
89C2×#217C2312yes—
90C2×#217C2312yes—
91C2×#217C2311yescharacteristic
92#935—22no—
93#935—22no—
94#929—22no—
95#929—21no—
96#929—21no—
97C2×(C2×D3)⋊C4C2×C412yes—
98C2×(C2×D3)⋊C4C2×C412yes—
99#926—42no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2×C412yes—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4C2×C412yes—
104C2×Dic3⋊C4C2×C412yes—
105C2×Dic3⋊C4C2×C412yes—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3C2×C412yes—
109#896—24no—
110#896—24no—
111#888—24no—
112#888—24no—
113#886—22no—
114#886—22no—
115#885—24no—
116#885—24no—
117#878—24no—
118#878—24no—
119#877—22no—
120#877—22no—
121#877—42no—
122#877—42no—
123C3×#99—42no—
124C3×#95—22no—
125#834—42no—
126C3⋊#95—22no—
127C3⋊#95—24no—
128C3⋊#95—24no—
129C3⋊#95—24no—
130#806—42no—
131#806—42no—
132C22×#41—31no—
133C23×Q8—31no—
134#536—64no—
135#536—62no—
136#536—62no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—31no—
140C2×#123—31no—
141C2×#123—31no—
142C2×#123C2×D311yescharacteristic
143C2×C22⋊Q8—32no—
144C2×C22⋊Q8—32no—
145C2×#117—31no—
146C2×#117—31no—
147C2×#117—32no—
148C2×#117C2×D311yescharacteristic
149C2×#99—32no—
150C2×#99—32no—
151C2×#99—32no—
152C2×#99C2×D312yes—
153#391—64no—
154#391—62no—
155#391—62no—
156#381—31no—
157#381—32no—
158#381—32no—
159#381—32no—
160#381C2×D311yescharacteristic
161#342—32no—
162#342—32no—
163#342—32no—
164#342C2×D312yes—
165C24×C3C22×C411yescharacteristic
166C24×C3C2411yescharacteristic
167C23×D3C22×C411yescharacteristic
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C22×C412yes—
178C22×C3×C4C22×C412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C2×#67—21no—
183C2×#67—21no—
184C2×#67—22no—
185C2×#67—22no—
186C2×#67—22no—
187C2×#67—42no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—42no—
196C22×Dic3C22×C411yescharacteristic
197C22×Dic3C22×C411yescharacteristic
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C412yes—
200C22×Dic3C22×C412yes—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D412yes—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205#237—44no—
206#237—44no—
207#237—48no—
208C2×C4×D3—22no—
209C2×C4×D3—42no—
210C2×Dic6—22no—
211C2×Dic6—44no—
212C2×Dic6—44no—
213C2×Dic6—44no—
214C2×Dic6—44no—
215C2×Dic6—42no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C2×Dic6—42no—
219C2×Dic6—42no—
220C2×Dic6C22⋊C414yes—
221C2×Dic6C22⋊C414yes—
222C2×Dic6C22⋊C414yes—
223C2×Dic6C22⋊C414yes—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—42no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C42—42no—
236#217—21no—
237#217—21no—
238#217—24no—
239#217—22no—
240#217—22no—
241#217—22no—
242#217—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—24no—
246C4⋊Dic3—24no—
247C4⋊Dic3—42no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—24no—
251Dic3⋊C4—24no—
252Dic3⋊C4—44no—
253C4×Dic3—24no—
254C4×Dic3—44no—
255C4×Dic3—42no—
256C2×#41—62no—
257C2×#41—62no—
258C2×#41—122no—
259C2×#41—122no—
260C22×Q8—32no—
261C22×Q8—32no—
262C22×Q8—31no—
263C22×Q8—32no—
264C22×Q8—32no—
265C22×Q8—34no—
266C22×Q8—122no—
267C22×Q8—122no—
268C22×D4—31no—
269C22×D4—31no—
270C22×D4C22×D311yescharacteristic
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4C22×D311yescharacteristic
276C4⋊Q8—64no—
277C4⋊Q8—64no—
278#124—64no—
279#124—64no—
280#123—22no—
281#123—21no—
282#123—21no—
283#123—62no—
284#123—62no—
285#123—62no—
286#123—62no—
287#123—62no—
288#123—61no—
289#123—61no—
290C22⋊Q8—64no—
291C22⋊Q8—64no—
292C22⋊Q8—64no—
293C22⋊Q8—64no—
294#117—42no—
295#117—62no—
296#117—62no—
297#117—122no—
298#117—122no—
299#117—122no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C4×D312yes—
308C2×C22⋊C4—32no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—32no—
311C2×C22⋊C4—32no—
312C2×C22⋊C4—32no—
313C2×C22⋊C4—32no—
314C2×C22⋊C4C4×D312yes—
315C2×C22⋊C4C22×D312yes—
316C2×C22⋊C4C22×D312yes—
317C2×C22⋊C4C22×D311yescharacteristic
318C2×C42—21no—
319C2×C42—32no—
320C2×C42—62no—
321C2×C42—61no—
322#99—42no—
323#99—122no—
324#99—122no—
325#99—122no—
326#95—22no—
327#95—64no—
328#95—64no—
329#95—64no—
330#95—62no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3C23×C411yescharacteristic
336C23×C3C22×D411yescharacteristic
337C23×C3C22×D411yescharacteristic
338C23×C3H(5,𝔽2)12yes—
339C23×C3H(5,𝔽2)12yes—
340C22×D3—21no—
341C22×D3—22no—
342C22×D3—42no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—42no—
346C2×C3×C4—24no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4C2×C22⋊C412yes—
361C2×C3×C4C2×C22⋊C412yes—
362#67—44no—
363#67—41no—
364#67—41no—
365#67—42no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—24no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—44no—
380C2×Dic3—42no—
381C2×Dic3—44no—
382C2×Dic3—44no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3C2×C22⋊C412yes—
386C2×Dic3C2×C22⋊C412yes—
387C2×Dic3C2×C22⋊C414yes—
388C4×D3—44no—
389Dic6—44no—
390Dic6—44no—
391Dic6—44no—
392Dic6—44no—
393Dic6—44no—
394C24—31no—
395C24C2×C4×D311yescharacteristic
396C24C23×D311yescharacteristic
397#41—124no—
398#41—124no—
399#41—128no—
400C2×Q8—34no—
401C2×Q8—34no—
402C2×Q8—34no—
403C2×Q8—34no—
404C2×Q8—62no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8—124no—
408C2×Q8—124no—
409C2×Q8—122no—
410C2×Q8—122no—
411C2×Q8—124no—
412C2×Q8—124no—
413C2×Q8—122no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—62no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—122no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—31no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—31no—
436C22×C4—32no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4C2×C4×D311yescharacteristic
448C22×C4C2×C4×D312yes—
449C22×C4C2×C4×D312yes—
450C22×C4D3×D412yes—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—64no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—122no—
462C4⋊C4—124no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C42—42no—
482C42—64no—
483C42—122no—
484C42—124no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—82no—
493C22×C3C22×C22⋊C411yescharacteristic
494C22×C3#51812yes—
495C22×C3#53411yescharacteristic
496C22×C3#53411yescharacteristic
497C22×C3#53412yes—
498C2×D3—41no—
499C2×D3—42no—
500C2×D3—82no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—41no—
504C3×C4—42no—
505C3×C4—82no—
506C3×C4—82no—
507Dic3—22no—
508Dic3—22no—
509Dic3—24no—
510Dic3—41no—
511Dic3—42no—
512Dic3—82no—
513Dic3—82no—
514C23—21no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23—61no—
519C23—62no—
520C23—122no—
521C23C22×C4×D311yescharacteristic
522C23C2×D3×D411yescharacteristic
523C23C2×D3×D411yescharacteristic
524C23#100412yes—
525C23#100912yes—
526Q8—124no—
527Q8—124no—
528Q8—124no—
529Q8—124no—
530Q8—124no—
531D4—41no—
532D4—41no—
533D4—42no—
534D4—124no—
535D4—121no—
536D4—121no—
537D4—122no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—32no—
548C2×C4—34no—
549C2×C4—32no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—64no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—124no—
569C2×C4—124no—
570C2×C4—122no—
571C2×C4—124no—
572C2×C4—124no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4D3×C22⋊C412yes—
576C2×C4D3×C22⋊C412yes—
577C2×C3—21no—
578C2×C3—22no—
579C2×C3—21no—
580C2×C3—22no—
581C2×C3—81no—
582C2×C3#230811yescharacteristic
583C2×C3#288412yes—
584D3—81no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—41no—
592C22—82no—
593C22—122no—
594C22—121no—
595C22—242no—
596C22C2×D3×C22⋊C411yescharacteristic
597C22#554512yes—
598C22#558911yescharacteristic
599C22#560512yes—
600C22#565311yescharacteristic
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—62no—
606C4—62no—
607C4—64no—
608C4—82no—
609C4—82no—
610C4—122no—
611C4—121no—
612C4—242no—
613C4—242no—
614C3#2175111yescharacteristic, complemented, Sylow
615C2—21no—
616C2—22no—
617C2—21no—
618C2—22no—
619C2—81no—
620C2—241no—
621C2#7487411yescharacteristic
622C2#7868912yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1899C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#237C22
cosocleC24×C3C24
hypercenter#1899D3
hypocenterC3#21751
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1899

Complements

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

NH ≅ G/Ntype
2. #82230C2semidirect
3. #74053C2semidirect
4. C3×#1899C2semidirect
5. #71705C2semidirect
6. #71595C2semidirect
7. #70029C2semidirect
9. #69535C2semidirect
10. #68954C2semidirect
11. #68953C2semidirect
23. C2×#896C4semidirect
24. C2×#888C4semidirect
26. C2×#885C4semidirect
27. C2×#878C4semidirect
54. #1899D3semidirect
614. C3#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#56482
#822302
#740532
C3×#18992
#717052
#715952
#700292
#700272
#695352
#689542
#689532
#217513