Finite Group Databasebeta

← #10972868

Subgroups — #10972868

Order 768.

2294 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#5479C211yesmaximal, characteristic
3#84066C211yesmaximal, characteristic
4#84065C211yesmaximal, characteristic, complemented
5#82504C214yesmaximal, complemented
6#82502C214yesmaximal, complemented
7#82477C212yesmaximal, complemented
8#82474C212yesmaximal, complemented
9#82456C212yesmaximal, complemented
10#82455C212yesmaximal, complemented
11#82198C214yesmaximal, complemented
12#82180C211yesmaximal, characteristic
13#82178C211yesmaximal, characteristic, complemented
14#82079C214yesmaximal, complemented
15#82060C211yesmaximal, characteristic, complemented
16#35246—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C2×C3×C4×Q8—28no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2212yes—
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2212yes—
25C22×C3×C42C2212yes—
26C22×C3×C42C2211yescharacteristic
27C2×#947C2212yes—
28C2×#944C2212yes—
29C2×#930C2211yescharacteristic
30C2×#929C2212yes—
31C2×C4×#67C2212yes—
32C2×#924C2211yescharacteristic
33#5479—28no—
34C2×#876C2212yes—
35C2×#875C2211yescharacteristic
36C2×C4×D12C2211yescharacteristic
37C2×#872C2212yes—
38C2×#870C2211yescharacteristic
39C2×C4×Dic6C2211yescharacteristic
40#5231C2214yes—
41#5230C2214yes—
42#5229C2212yes—
43#5229C2212yes—
44#5213C2214yes—
45#5213C2214yes—
46#5212C2212yes—
47#5211C2212yes—
48#5210C2214yes—
49#5210C2214yes—
50C4×#217C2211yescharacteristic
51C4×#217C2211yescharacteristic
52#4994C2218yes—
53#4992C2214yes—
54#4989C2218yes—
55#4988C2212yes—
56#4983C2214yes—
57#4982C2218yes—
58#4978C2218yes—
59#4977C2212yes—
60#4975C2212yes—
61#4973C2212yes—
62#4940C2218yes—
63#4940C2218yes—
64#4937C2212yes—
65#4936C2212yes—
66C4×C4⋊Dic3C2212yes—
67#4933C2218yes—
68#4933C2218yes—
69#4932C2212yes—
70C2×#529—31no—
71C22×C4×Q8D311yescharacteristic, complemented
72#2853—34no—
73#2637—34no—
74#2599—34no—
75#2579—34no—
76#2565—32no—
77#2545—32no—
78#2504—31no—
79#2502—31no—
80#2436—31no—
81#2434—31no—
82#2327—32no—
83#2326—32no—
84#2283—31no—
85C22×C3×Q8—28no—
86C22×C3×Q8—24no—
87C22×C3×Q8C2312yes—
88C22×C3×Q8C2312yes—
89C23×C3×C4C2312yes—
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C23×C3×C4C2311yescharacteristic
94C22×#67C2312yes—
95C22×D12C2311yescharacteristic
96C22×C4×D3C2312yes—
97C22×Dic6C2311yescharacteristic
98C3×C4×Q8—28no—
99C3×C4×Q8—28no—
100C3×C4×Q8—28no—
101C3×C4×Q8—28no—
102C2×C3×C4⋊C4—28no—
103C2×C3×C4⋊C4—24no—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C42—28no—
109C2×C3×C42D418yes—
110C2×C3×C42C2314yes—
111C2×C3×C42C2314yes—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×#217C2312yes—
117C2×#217C2312yes—
118C2×#217C2312yes—
119#947—28no—
120#944—28no—
121#930—24no—
122#929—28no—
123C4×#67—28no—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2318yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2318yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2314yes—
133#924—24no—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2318yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2318yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2312yes—
143#876—28no—
144#875—24no—
145C4×D12—24no—
146#872—28no—
147#870—24no—
148C4×Dic6—24no—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2318yes—
153C3⋊#95C2318yes—
154C3⋊#95C2318yes—
155C3⋊#95C2318yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C23×Q8C2×D311yescharacteristic
159C2×C4⋊Q8—32no—
160#529—68no—
161C2×#126—32no—
162C2×#125—31no—
163C2×#124—32no—
164C2×#124—31no—
165C2×#123—32no—
166C2×C22⋊Q8—31no—
167C2×#121—31no—
168C2×C4×Q8—28no—
169C2×C4×Q8—31no—
170C2×C4×Q8C2×D312yes—
171C2×C4×Q8C2×D312yes—
172C2×C4×Q8C2×D312yes—
173C2×C4×Q8C2×D312yes—
174C2×C4×D4—32no—
175C2×C4×D4—31no—
176C2×#117—32no—
177C22×C4⋊C4C2×D312yes—
178C22×C4⋊C4C2×D311yescharacteristic
179C22×C42C2×D312yes—
180C22×C42C2×D311yescharacteristic
181#398—38no—
182#398—34no—
183#397—34no—
184#396—38no—
185#395—34no—
186#393—34no—
187#391—32no—
188#391—32no—
189#391—32no—
190#390—32no—
191#388—38no—
192#388—38no—
193#387—38no—
194#387—38no—
195#387—38no—
196#386—34no—
197#386—34no—
198#385—34no—
199#385—34no—
200#385—38no—
201#384—32no—
202#383—32no—
203#382—32no—
204#382—32no—
205#381—32no—
206#380—32no—
207C4×C4⋊C4—32no—
208C4×C22⋊C4—31no—
209C4×C22⋊C4—31no—
210#376—32no—
211C24×C3C2411yescharacteristic
212C23×D3C2412yes—
213C2×C3×Q8—232no—
214C2×C3×Q8—24no—
215C2×C3×Q8—24no—
216C2×C3×Q8—28no—
217C2×C3×Q8—24no—
218C2×C3×Q8#4118yes—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—28no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—42no—
225C22×C3×C4—44no—
226C22×C3×C4C2×D414yes—
227C22×C3×C4C2×D414yes—
228C22×C3×C4C2×D414yes—
229C22×C3×C4C2414yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2414yes—
236C22×C3×C4C2411yescharacteristic
237C2×#67—24no—
238C2×#67—28no—
239C22×Dic3C2412yes—
240C22×Dic3C2414yes—
241C22×Dic3C2414yes—
242C22×Dic3C2412yes—
243C22×Dic3C2412yes—
244C2×D12—22no—
245C2×D12—24no—
246C2×C4×D3—24no—
247C2×C4×D3—44no—
248C2×Dic6—22no—
249C2×Dic6—24no—
250C3×C4⋊C4—28no—
251C3×C4⋊C4—28no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C42—28no—
255C3×C42—28no—
256C3×C42C2×D414yes—
257C3×C42C2×D414yes—
258C3×C42C2×D414yes—
259C3×C42C2×D414yes—
260#217—24no—
261#217—24no—
262#217—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—216no—
268(C2×D3)⋊C4—48no—
269C4⋊Dic3—22no—
270C4⋊Dic3—22no—
271C4⋊Dic3—44no—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—216no—
277Dic3⋊C4—48no—
278C4×Dic3—24no—
279C4×Dic3—24no—
280C4×Dic3—24no—
281C22×Q8—28no—
282C22×Q8—24no—
283C22×Q8—31no—
284C22×Q8C22×D312yes—
285C22×Q8C22×D312yes—
286C22×D4—32no—
287C22×D4—31no—
288C23×C4—32no—
289C23×C4C22×D312yes—
290C23×C4C22×D311yescharacteristic
291C23×C4C22×D311yescharacteristic
292C23×C4C22×D312yes—
293C23×C4C22×D311yescharacteristic
294C4⋊Q8—68no—
295#126—68no—
296#125—64no—
297#124—68no—
298#124—64no—
299#123—68no—
300C22⋊Q8—64no—
301#121—64no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—28no—
305C4×Q8—28no—
306C4×Q8—64no—
307C4×D4—68no—
308C4×D4—64no—
309#117—68no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—38no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—38no—
321C2×C4⋊C4C22×D312yes—
322C2×C4⋊C4C22×D314yes—
323C2×C4⋊C4C22×D314yes—
324C2×C4⋊C4C22×D312yes—
325C2×C22⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—32no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—32no—
331C2×C22⋊C4—32no—
332C2×C22⋊C4—38no—
333C2×C22⋊C4—38no—
334C2×C42—28no—
335C2×C42—32no—
336C2×C42—32no—
337C2×C42—32no—
338C2×C42#6718yes—
339C2×C42C22×D314yes—
340C2×C42C22×D314yes—
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D311yescharacteristic
345#95—34no—
346#95—34no—
347#95—32no—
348#95—32no—
349#95—34no—
350#95—38no—
351#95—38no—
352#95—38no—
353#95—38no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3C22×D412yes—
357C23×C32−1+414yes—
358C23×C3C2511yescharacteristic
359C22×D3—22no—
360C22×D3—24no—
361C22×D3—24no—
362C22×D3—24no—
363C3×Q8—216no—
364C3×Q8—216no—
365C2×C3×C4—216no—
366C2×C3×C4—28no—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—24no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—28no—
375C2×C3×C4—24no—
376C2×C3×C4—28no—
377C2×C3×C4—44no—
378C2×C3×C4—48no—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D412yes—
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C2×#4118yes—
386C2×C3×C4C2×#4114yes—
387#67—48no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3—24no—
400D12—44no—
401C4×D3—48no—
402Dic6—44no—
403C24—32no—
404C24C23×D311yescharacteristic
405C2×Q8—232no—
406C2×Q8—28no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—64no—
411C2×Q8—62no—
412C2×Q8#23518yes—
413C2×D4—68no—
414C2×D4—64no—
415C2×D4—64no—
416C2×D4—62no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—64no—
430C22×C4—124no—
431C22×C4C2×#6714yes—
432C22×C4C2×#6714yes—
433C22×C4C2×#6714yes—
434C22×C4C23×D314yes—
435C22×C4C23×D314yes—
436C22×C4C23×D311yescharacteristic
437C22×C4C23×D311yescharacteristic
438C22×C4C23×D311yescharacteristic
439C22×C4C23×D311yescharacteristic
440C22×C4C23×D311yescharacteristic
441C22×C4C23×D311yescharacteristic
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—616no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—62no—
450C4⋊C4—64no—
451C4⋊C4—62no—
452C4⋊C4—64no—
453C4⋊C4—128no—
454C4⋊C4—124no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C22⋊C4—616no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—128no—
464C42—28no—
465C42—28no—
466C42—64no—
467C42—64no—
468C42—64no—
469C42C2×#6714yes—
470C42C2×#6714yes—
471C42C2×#6714yes—
472C42C2×#6714yes—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—28no—
476C22×C3#53614yes—
477C22×C3C23×D411yescharacteristic
478C22×C3C22×#4112yes—
479C22×C3C2×2−1+412yes—
480C22×C3#58512yes—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—48no—
485C3×C4—22no—
486C3×C4—28no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—44no—
492C3×C4#52514yes—
493Dic3—42no—
494Dic3—42no—
495Dic3—42no—
496Dic3—44no—
497Dic3—44no—
498C23—22no—
499C23—22no—
500C23—64no—
501C23—64no—
502C23—64no—
503C23—62no—
504C23#100714yes—
505C23C22×#6712yes—
506C23C24×D311yescharacteristic
507Q8—216no—
508Q8—216no—
509Q8—124no—
510D4—128no—
511D4—124no—
512C2×C4—216no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—128no—
539C2×C4C2×#23518yes—
540C2×C4C2×#23514yes—
541C2×C4C22×#6712yes—
542C2×C4C22×#6712yes—
543C2×C4C22×#6712yes—
544C2×C4C22×#6712yes—
545C2×C4C22×#6712yes—
546C2×C4C22×#6712yes—
547C2×C3—24no—
548C2×C3C2×#52511yescharacteristic
549C2×C3C2×#53611yescharacteristic
550C2×C3#344911yescharacteristic
551C2×C3#351614yes—
552D3—82no—
553C22—28no—
554C22—22no—
555C22—22no—
556C22—128no—
557C22—122no—
558C22—122no—
559C22—122no—
560C22#581614yes—
561C22C22×#23512yes—
562C22C2×#100712yes—
563C22#596412yes—
564C22C23×#6711yescharacteristic
565C4—28no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—44no—
572C4—124no—
573C4—124no—
574C4—122no—
575C4—122no—
576C4—122no—
577C4#579114yes—
578C3#3524611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—242no—
581C2#8582014yes—
582C2C2×#579111yescharacteristic
583C2C2×#581611yescharacteristic
584C2#8726811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC22×C3×C4×Q8C2
prime radicalC2×#930C22
hypercenterC22×C4×Q8D3
hypocenterC3#35246
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Q8C2semidirect
4. #84065C2semidirect
5. #82504C2semidirect
6. #82502C2semidirect
7. #82477C2semidirect
8. #82474C2semidirect
9. #82456C2semidirect
10. #82455C2semidirect
11. #82198C2semidirect
13. #82178C2semidirect
14. #82079C2semidirect
15. #82060C2semidirect
71. C22×C4×Q8D3semidirect
578. C3#35246semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×#54792
#840662
#840652
#825042
#825022
#824772
#824742
#824562
#824552
#821982
#821802
#821782
#820792
#820602
#352463