Finite Group Databasebeta

← #10896411

Subgroups — #10896411

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73391C211yesmaximal, characteristic, complemented
2#73387C211yesmaximal, characteristic
3C3×#1832C211yesmaximal, characteristic, complemented
4#70458C211yesmaximal, characteristic
5#70458C211yesmaximal, characteristic
6#70454C211yesmaximal, characteristic, complemented
7#70453C211yesmaximal, characteristic, complemented
8#68693C211yesmaximal, characteristic, complemented
9#68689C211yesmaximal, characteristic, complemented
10#68684C211yesmaximal, characteristic, complemented
11#68683C211yesmaximal, characteristic, complemented
12#68680C211yesmaximal, characteristic
13#68678C211yesmaximal, characteristic
14#68671C211yesmaximal, characteristic
15#68670C211yesmaximal, characteristic
16#29267—31nomaximal, Sylow
17C2×#875C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C3×#376C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4939C2211yescharacteristic
37#4936C2211yescharacteristic
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4929C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4670—21no—
45#4670—21no—
46#4668—21no—
47#4668—21no—
48#4654—21no—
49#4654—21no—
50#4646—21no—
51#4646—21no—
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2331—31no—
61#2326—31no—
62#2097—31no—
63#2092—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983—31no—
68#1948—31no—
69#1947—31no—
70#1865—31no—
71#1864—31no—
72#1863—31no—
73#1862—31no—
74#1832D311yescharacteristic, complemented
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×Dic6—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106#875—22no—
107#875Q814yescomplemented
108C4×D12—41no—
109C4×D12—41no—
110#869—22no—
111#869—22no—
112C4×Dic6—41no—
113C4×Dic6—41no—
114C3×C4⋊C8—22no—
115C3×C4⋊C8—22no—
116C3×#95D411yescharacteristic
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#821—22no—
129#821—22no—
130#821—41no—
131#821—41no—
132#818—41no—
133#818—41no—
134#817—41no—
135#817—41no—
136#816—22no—
137#816—22no—
138#816—41no—
139#816—41no—
140C2×C4⋊Q8—31no—
141C2×#124—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158#391—31no—
159#391—61no—
160#391—61no—
161#390—31no—
162#389—31no—
163#389—31no—
164#388—61no—
165#388—61no—
166#387—61no—
167#387—61no—
168#385—61no—
169#385—61no—
170#384—31no—
171#382—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
174#376C2×D311yescharacteristic
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#336C2×D311yescharacteristic
180#336C2×D311yescharacteristic
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C23×D3—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12C2×Q812yes—
204C2×C4×D3—42no—
205C2×C4×D3—81no—
206C2×C4×D3—81no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—24no—
213C2×Dic6—81no—
214C2×Dic6—81no—
215C2×Dic6C2×Q812yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42C2×Q812yes—
233C3×C42#4112yes—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239C4⋊Dic3—24no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—81no—
251C4⋊Dic3—81no—
252C4⋊Dic3#4112yes—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256Dic3⋊C4—81no—
257C4×Dic3—42no—
258C4×Dic3—81no—
259C4×Dic3—81no—
260C22×Q8—31no—
261C22×Q8—61no—
262C22×D4—31no—
263C23×C4—61no—
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C4⋊Q8—62no—
269C4⋊Q8—62no—
270#124—34no—
271#124—62no—
272C4×Q8—121no—
273C4×Q8—121no—
274C4×D4—121no—
275C4×D4—121no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—61no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—121no—
300#107—121no—
301#106—121no—
302#106—121no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305#103—62no—
306#103—62no—
307#103—121no—
308#103—121no—
309#102—62no—
310#102—62no—
311#102—121no—
312#102—121no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95D1211yescharacteristic
322#95D1211yescharacteristic
323#95D1211yescharacteristic
324#95D1211yescharacteristic
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4C22⋊Q812yes—
350C2×C3×C4C22⋊Q812yes—
351C2×C3×C4C22×Q811yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378D12—22no—
379D12—22no—
380D12—41no—
381D12—41no—
382C4×D3—84no—
383Dic6—22no—
384Dic6—22no—
385Dic6—41no—
386Dic6—41no—
387Dic6—84no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C24—61no—
393C2×Q8—32no—
394C2×Q8—64no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—241no—
401C2×Q8—241no—
402C2×D4—32no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C4⋊C4—32no—
438C4⋊C4—32no—
439C4⋊C4—64no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—122no—
465C42—241no—
466C42—241no—
467C42#23712yes—
468C42D3×Q812yes—
469C22×C3C2×C22⋊Q811yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#53511yescharacteristic
472C22×C3#57611yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57711yescharacteristic
475C22×C3#57811yescharacteristic
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—81no—
480C2×D3—81no—
481C2×D3—81no—
482C2×D3—81no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4#55411yescharacteristic
490C3×C4#55411yescharacteristic
491C3×C4#55911yescharacteristic
492C3×C4#55911yescharacteristic
493Dic3—42no—
494Dic3—42no—
495Dic3—42no—
496Dic3—81no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500C23—62no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23C22×D1211yescharacteristic
508Q8—62no—
509Q8—62no—
510Q8—121no—
511Q8—121no—
512Q8—244no—
513D4—62no—
514D4—62no—
515D4—121no—
516D4—121no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—244no—
555C2×C4#89712yes—
556C2×C4#89712yes—
557C2×C4C2×#23711yescharacteristic
558C2×C4C2×D3×Q811yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C2×C4#100911yescharacteristic
562C2×C4#100911yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C2×C3#258011yescharacteristic
568C2×C3#308311yescharacteristic
569C2×C3#308411yescharacteristic
570C2×C3#311811yescharacteristic
571C2×C3#312111yescharacteristic
572C2×C3#322411yescharacteristic
573C2×C3#322511yescharacteristic
574D3—82no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22—241no—
582C22C2×#89711yescharacteristic
583C22#552411yescharacteristic
584C22#552511yescharacteristic
585C22#575111yescharacteristic
586C22#575111yescharacteristic
587C22#575211yescharacteristic
588C22#575311yescharacteristic
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—122no—
596C4—122no—
597C4—122no—
598C4—241no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#568111yescharacteristic
603C4#568111yescharacteristic
604C4#569011yescharacteristic
605C4#569011yescharacteristic
606C3#2926711yescharacteristic, complemented, Sylow
607C2—242no—
608C2#7419011yescharacteristic
609C2#7909111yescharacteristic
610C2#7909211yescharacteristic
611C2#7938311yescharacteristic
612C2#7939211yescharacteristic
613C2#7939311yescharacteristic
614C2#7939411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1832C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1832D3
hypocenterC3#29267
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1832

Complements

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

NH ≅ G/Ntype
1. #73391C2semidirect
3. C3×#1832C2semidirect
6. #70454C2semidirect
7. #70453C2semidirect
8. #68693C2semidirect
9. #68689C2semidirect
10. #68684C2semidirect
11. #68683C2semidirect
74. #1832D3semidirect
107. #875Q8semidirect
606. C3#29267semidirect

Maximal subgroups

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

subgroupindex
#733912
#733872
C3×#18322
#704582
#704542
#704532
#686932
#686892
#686842
#686832
#686802
#686782
#686712
#686702
#292673