Finite Group Databasebeta

← #10876056

Subgroups — #10876056

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82461C211yesmaximal, characteristic, complemented
2#74172C211yesmaximal, characteristic
3C3×#1805C211yesmaximal, characteristic, complemented
4#71633C211yesmaximal, characteristic
5#70483C211yesmaximal, characteristic
6#70448C211yesmaximal, characteristic, complemented
7#70240C211yesmaximal, characteristic
8#70240C211yesmaximal, characteristic
9#70238C211yesmaximal, characteristic, complemented
10#70220C211yesmaximal, characteristic, complemented
11#69542C211yesmaximal, characteristic, complemented
12#68903C211yesmaximal, characteristic
13#68856C211yesmaximal, characteristic
14#68846C211yesmaximal, characteristic, complemented
15#68818C211yesmaximal, characteristic, complemented
16#27577—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#815C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4984C2211yescharacteristic
33#4977—21no—
34#4977C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4972—21no—
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39#4939C2211yescharacteristic
40C4×Dic3⋊C4—21no—
41C4×Dic3⋊C4C2211yescharacteristic
42#4928—21no—
43#4928C2211yescharacteristic
44#4675—21no—
45#4674—21no—
46#4670—21no—
47#4669—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2351—31no—
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2065—31no—
66#1936—31no—
67#1920—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1865—31no—
73#1805D311yescharacteristic, complemented
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87Q8×Dic3—22no—
88Q8×Dic3—22no—
89Q8×Dic3—22no—
90Q8×Dic3—41no—
91C2×#215—22no—
92C2×#215—22no—
93C2×#215—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C3×C4⋊C8—41no—
118C3×#103—22no—
119C3×#103—22no—
120C3×#95—21no—
121C3×#95—21no—
122#835—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—22no—
132#821—41no—
133#817—22no—
134#815—41no—
135#809—41no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×Q8C2×D311yescharacteristic
140C2×#106—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#103—31no—
143C2×#103C2×D311yescharacteristic
144C2×#103C2×D311yescharacteristic
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97—31no—
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#389—61no—
158#389—61no—
159#388—61no—
160#388—61no—
161#382—31no—
162#382—61no—
163#382C2×D311yescharacteristic
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8#4112yes—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—42no—
222#215—44no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—81no—
226(C2×D3)⋊C4—81no—
227(C2×D3)⋊C4—81no—
228(C2×D3)⋊C4—81no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—42no—
236C4⋊Dic3—81no—
237C4⋊Dic3#4112yes—
238Dic3⋊C4—41no—
239Dic3⋊C4—41no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242C4×Dic3—24no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—42no—
248C4×Dic3—81no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C22×Q8C22×D311yescharacteristic
256C22×D4—31no—
257C2×SD16—62no—
258C2×SD16—62no—
259C2×SD16—62no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4×Q8—22no—
265C4×Q8—41no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×Q8—121no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—21no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297#106—62no—
298C4⋊C8—41no—
299#103—22no—
300#103—22no—
301#103—62no—
302#102—62no—
303#102—121no—
304#102—121no—
305#97—121no—
306#95—21no—
307#95—21no—
308#95—31no—
309#95—31no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8#12614yes—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—22no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—41no—
374D12—41no—
375D12—41no—
376D12—41no—
377C3×C8—41no—
378C3×C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—61no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8#23512yes—
386C2×Q8#23712yes—
387C2×Q8#23912yes—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—124no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—32no—
428C4⋊C4—42no—
429C4⋊C4—64no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—121no—
436C4⋊C4—121no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—241no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—241no—
444C22⋊C4—241no—
445C22⋊C4—241no—
446C22⋊C4—241no—
447C42—21no—
448C42—21no—
449C42—42no—
450C42—64no—
451C42—61no—
452C42—61no—
453C42—61no—
454C42—61no—
455C42—122no—
456C42—241no—
457C22×C3#52511yescharacteristic
458C22×C3#54611yescharacteristic
459C22×C3#54811yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57511yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57711yescharacteristic
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C3×C4—24no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4C2×#12611yescharacteristic
478C3×C4#53911yescharacteristic
479C3×C4#55211yescharacteristic
480C3×C4#55311yescharacteristic
481Dic3—41no—
482Dic3—42no—
483Dic3—41no—
484Dic3—41no—
485Dic3—41no—
486Dic3—81no—
487Dic3—81no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23C2×D3×D411yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—22no—
499Q8#89814yes—
500D4—121no—
501D4—121no—
502D4—121no—
503D4—121no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4C2×#23511yescharacteristic
544C2×C4C2×#23711yescharacteristic
545C2×C4#100411yescharacteristic
546C2×C4C2×#23911yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4#101011yescharacteristic
549C8—41no—
550C8—41no—
551C8—121no—
552C8—121no—
553C2×C3#248111yescharacteristic
554C2×C3#311111yescharacteristic
555C2×C3#312611yescharacteristic
556C2×C3#332111yescharacteristic
557C2×C3#332611yescharacteristic
558C2×C3#337111yescharacteristic
559C2×C3#337111yescharacteristic
560D3—161no—
561C22—241no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22#549011yescharacteristic
569C22#559811yescharacteristic
570C22#562711yescharacteristic
571C22#576811yescharacteristic
572C22#577011yescharacteristic
573C22#577711yescharacteristic
574C22#577811yescharacteristic
575C4—24no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—121no—
582C4—121no—
583C4—122no—
584C4—121no—
585C4—121no—
586C4—241no—
587C4—241no—
588C4C2×#89811yescharacteristic
589C4#553111yescharacteristic
590C4#560111yescharacteristic
591C4#564511yescharacteristic
592C3#2757711yescharacteristic, complemented, Sylow
593C2—481no—
594C2#7391011yescharacteristic
595C2#7886011yescharacteristic
596C2#7897011yescharacteristic
597C2#8058311yescharacteristic
598C2#8058411yescharacteristic
599C2#8120611yescharacteristic
600C2#8121011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1805D3
hypocenterC3#27577
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. #82461C2semidirect
3. C3×#1805C2semidirect
6. #70448C2semidirect
9. #70238C2semidirect
10. #70220C2semidirect
11. #69542C2semidirect
14. #68846C2semidirect
15. #68818C2semidirect
73. #1805D3semidirect
592. C3#27577semidirect

Maximal subgroups

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

subgroupindex
#824612
#741722
C3×#18052
#716332
#704832
#704482
#702402
#702382
#702202
#695422
#689032
#688562
#688462
#688182
#275773