Finite Group Databasebeta

← #10829927

Subgroups — #10829927

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5577C211yesmaximal, characteristic, complemented
2#82500C211yesmaximal, characteristic, complemented
3C2×#4843C211yesmaximal, characteristic, complemented
4C2×C4×D24C211yesmaximal, characteristic, complemented
5C3×#1984C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#69772C211yesmaximal, characteristic, complemented
8#68674C212yesmaximal, complemented
9#68673C212yesmaximal, complemented
10#68670C212yesmaximal, complemented
11#68629C212yesmaximal, complemented
12#24184—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—21no—
16C22×D24C2211yescharacteristic
17C22×D24C2212yes—
18#5577—24no—
19#5577—24no—
20C2×#895—21no—
21C2×#895—22no—
22C2×#874C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×D12C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#818C2211yescharacteristic
36#4994—22no—
37#4987—21no—
38#4939C2212yes—
39#4935C2211yescharacteristic
40#4928C2212yes—
41#4843—24no—
42C4×D24—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#550—31no—
47C2×#459—31no—
48C2×C4×D8—31no—
49#2538—31no—
50#2041—32no—
51#2011—32no—
52#1984D311yescharacteristic, complemented
53#1957—31no—
54#1879—32no—
55#1862—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C2×#239—28no—
59C2×#239—44no—
60C2×#239—44no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×D12C2311yescharacteristic
67C22×D12C2311yescharacteristic
68C22×C4×D3—22no—
69C22×C4×D3—21no—
70C22×C4×D3—21no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D412yes—
78C2×C3×C42C2312yes—
79C2×C3×C42C2311yescharacteristic
80#946—44no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—22no—
92C2×D24—22no—
93C2×D24—22no—
94C2×D24—22no—
95C2×D24—24no—
96C2×D24—24no—
97C2×D24—42no—
98#895—28no—
99#895—216no—
100#874—24no—
101#874—24no—
102#874—22no—
103#874D414yes—
104C4×D12—24no—
105C4×D12—22no—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12—24no—
109C4×D12D414yes—
110C3×C4⋊C8—22no—
111C3×C4×C8—22no—
112C3⋊#95—22no—
113#821—22no—
114#821—24no—
115#821—24no—
116#821—22no—
117#821—42no—
118#821—42no—
119#818—22no—
120C22×#41—61no—
121C22×D8—31no—
122C22×D8—32no—
123#550—64no—
124#550—64no—
125C2×#127—31no—
126C2×#127—32no—
127C2×C22⋊Q8—61no—
128C2×#121—61no—
129C2×#121—62no—
130C2×C4×Q8C2×D311yescharacteristic
131C2×C4×D4—31no—
132C2×C4×D4—32no—
133C2×C4×D4—31no—
134#459—64no—
135C4×D8—64no—
136C2×#107—31no—
137C2×C4⋊C8C2×D311yescharacteristic
138C2×#102—32no—
139C2×#102—32no—
140C2×#102—31no—
141C2×#102—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×C4×C8C2×D311yescharacteristic
145#396—62no—
146#390—32no—
147#389—32no—
148#384—31no—
149#380—61no—
150#340—32no—
151#336C2×D312yes—
152#336C2×D312yes—
153C23×D3—21no—
154C23×D3—22no—
155C23×D3—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8D814yescomplemented
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167#239—432no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—22no—
175C2×D12—21no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—22no—
180C2×D12—28no—
181C2×D12—24no—
182C2×D12—22no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—28no—
186C2×D12—24no—
187C2×D12—44no—
188C2×D12—44no—
189C2×D12—42no—
190C2×D12—42no—
191C2×D12—81no—
192C2×D12—81no—
193C2×D12—82no—
194C2×D12—82no—
195C2×D12C2×D414yes—
196C2×D12C2×D412yes—
197C2×D12#4112yescomplemented
198C2×C4×D3—28no—
199C2×C4×D3—24no—
200C2×C4×D3—44no—
201C2×C4×D3—44no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—21no—
213C2×C3×C8—22no—
214C2×C3×C8—21no—
215C2×C3×C8#4112yescomplemented
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222(C2×D3)⋊C4—28no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—44no—
226(C2×D3)⋊C4—82no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—42no—
233C4⋊Dic3—81no—
234C4⋊Dic3C2×D412yes—
235Dic3⋊C4—44no—
236D24—28no—
237D24—48no—
238C2×#41—68no—
239C2×#41—124no—
240C2×#41—124no—
241C22×Q8D1211yescharacteristic
242C22×D4—31no—
243C22×D4—32no—
244C22×D4—31no—
245C22×D4—62no—
246C22×D4—61no—
247C22×D4—62no—
248C22×D4—61no—
249C23×C4—62no—
250C23×C4—61no—
251C23×C4—61no—
252C2×D8—62no—
253C2×D8—62no—
254C2×D8—64no—
255C2×D8—64no—
256C2×D8—62no—
257C2×D8—122no—
258C22×C8C22×D312yes—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#127—34no—
262#127—64no—
263#127—64no—
264#127—62no—
265C22⋊Q8—124no—
266#121—68no—
267#121—616no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×D4—34no—
271C4×D4—64no—
272C4×D4—64no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4D1211yescharacteristic
283C2×C4⋊C4D1212yes—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C42C22×D312yes—
290C2×C42C22×D311yescharacteristic
291#107—62no—
292C4⋊C8—22no—
293#102—62no—
294#102—62no—
295#102—64no—
296#102—64no—
297#102—122no—
298#102—122no—
299C4×C8—22no—
300#95—62no—
301C23×C3C22×D411yescharacteristic
302C22×D3—24no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—42no—
313C22×D3—41no—
314C22×D3—42no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—42no—
319C22×D3—41no—
320C22×D3—42no—
321C22×D3—41no—
322C22×D3—41no—
323C3×Q8—28no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12012yes—
333C2×C3×C4C2×D814yes—
334C2×C3×C4C2×D812yes—
335C2×C3×C4AGL(1,ℤ/8)12yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351D12—22no—
352D12—22no—
353D12—22no—
354D12—22no—
355D12—24no—
356D12—24no—
357D12—416no—
358D12—42no—
359D12—41no—
360D12—42no—
361D12—41no—
362D12—41no—
363D12—41no—
364D12—48no—
365D12—84no—
366D12—88no—
367C4×D3—416no—
368C4×D3—48no—
369C4×D3—84no—
370C3×C8—22no—
371C3×C8—41no—
372C3×C8—42no—
373C24—61no—
374C24—62no—
375C24—61no—
376#41—1232no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8D2414yescomplemented
380C2×D4—32no—
381C2×D4—34no—
382C2×D4—32no—
383C2×D4—68no—
384C2×D4—68no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—62no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—62no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—62no—
401C2×D4—61no—
402C2×D4—124no—
403C2×D4—124no—
404C2×D4—122no—
405C2×D4—122no—
406C2×D4—241no—
407C2×D4—241no—
408C2×D4—242no—
409C2×D4—242no—
410C22×C4—68no—
411C22×C4—61no—
412C22×C4—64no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4C2×D1212yes—
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1212yes—
425C22×C4C2×D1211yescharacteristic
426C22×C4C23×D311yescharacteristic
427D8—68no—
428D8—128no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—22no—
437C2×C8#23912yescomplemented
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—32no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—124no—
447C4⋊C4—122no—
448C4⋊C4—241no—
449C22⋊C4—68no—
450C22⋊C4—64no—
451C22⋊C4—122no—
452C22⋊C4—124no—
453C22⋊C4—242no—
454C42—22no—
455C42—22no—
456C42D3×D412yes—
457C42D3×D412yes—
458C22×C3C2×#12011yescharacteristic
459C22×C3#54811yescharacteristic
460C22×C3#55011yescharacteristic
461C22×C3C22×D811yescharacteristic
462C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
463C22×C3#57611yescharacteristic
464C22×C3#57711yescharacteristic
465C2×D3—42no—
466C2×D3—42no—
467C2×D3—42no—
468C2×D3—42no—
469C2×D3—44no—
470C2×D3—44no—
471C2×D3—42no—
472C2×D3—44no—
473C2×D3—42no—
474C2×D3—81no—
475C2×D3—82no—
476C2×D3—82no—
477C2×D3—81no—
478C2×D3—81no—
479C2×D3—82no—
480C2×D3—81no—
481C2×D3—82no—
482C2×D3—81no—
483C2×D3—81no—
484C2×D3—81no—
485C2×D3—81no—
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4#44714yes—
490C3×C4#52511yescharacteristic
491C3×C4#53711yescharacteristic
492C3×C4#54611yescharacteristic
493C3×C4#54711yescharacteristic
494Dic3—42no—
495Dic3—82no—
496Dic3—81no—
497C23—64no—
498C23—62no—
499C23—62no—
500C23—121no—
501C23—122no—
502C23—121no—
503C23—121no—
504C23—122no—
505C23—121no—
506C23—121no—
507C23—122no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—122no—
512C23—121no—
513C23—122no—
514C23—121no—
515C23—122no—
516C23—121no—
517C23—121no—
518C23C22×D1211yescharacteristic
519Q8—28no—
520D4—62no—
521D4—64no—
522D4—64no—
523D4—62no—
524D4—62no—
525D4—62no—
526D4—1216no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—128no—
531D4—122no—
532D4—121no—
533D4—122no—
534D4—244no—
535D4—248no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—62no—
544C2×C4—1216no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—128no—
548C2×C4—122no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—244no—
555C2×C4#88212yes—
556C2×C4#88212yes—
557C2×C4C2×D2414yes—
558C2×C4C2×D2412yes—
559C2×C4#90812yes—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4C2×#23911yescharacteristic
563C2×C4#100711yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C2×C4#100911yescharacteristic
566C8—22no—
567C8—42no—
568C8—41no—
569C2×C3#240711yescharacteristic
570C2×C3C2×#44711yescharacteristic
571C2×C3#301211yescharacteristic
572C2×C3#335811yescharacteristic
573C2×C3#336611yescharacteristic
574C2×C3#339111yescharacteristic
575C2×C3#339711yescharacteristic
576D3—82no—
577D3—82no—
578D3—84no—
579C22—124no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—124no—
584C22—122no—
585C22—124no—
586C22—122no—
587C22—122no—
588C22—241no—
589C22—241no—
590C22—242no—
591C22—241no—
592C22—242no—
593C22—241no—
594C22—242no—
595C22—241no—
596C22—241no—
597C22—241no—
598C22—242no—
599C22—241no—
600C22C2×#88211yescharacteristic
601C22#557611yescharacteristic
602C22#557711yescharacteristic
603C22C22×D2411yescharacteristic
604C22C2×#90811yescharacteristic
605C22#575111yescharacteristic
606C22#575211yescharacteristic
607C4—24no—
608C4—22no—
609C4—42no—
610C4—122no—
611C4—241no—
612C4—242no—
613C4#473214yes—
614C4#562711yescharacteristic
615C4#562911yescharacteristic
616C4#567611yescharacteristic
617C4#572611yescharacteristic
618C3#2418411yescharacteristic, complemented, Sylow
619C2—244no—
620C2—242no—
621C2—242no—
622C2#7661811yescharacteristic
623C2C2×#473211yescharacteristic
624C2#7867211yescharacteristic
625C2#7991011yescharacteristic
626C2#7992411yescharacteristic
627C2#7995011yescharacteristic
628C2#7996911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
hypercenter#1984D3
hypocenterC3#24184
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
1. C2×#5577C2semidirect
2. #82500C2semidirect
3. C2×#4843C2semidirect
4. C2×C4×D24C2semidirect
5. C3×#1984C2semidirect
6. #70479C2semidirect
7. #69772C2semidirect
8. #68674C2semidirect
9. #68673C2semidirect
10. #68670C2semidirect
11. #68629C2semidirect
52. #1984D3semidirect
158. C2×C3×Q8D8semidirect
197. C2×D12#41semidirect
215. C2×C3×C8#41semidirect
379. C2×Q8D24semidirect
437. C2×C8#239semidirect
618. C3#24184semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
#825002
C2×#48432
C2×C4×D242
C3×#19842
#704792
#697722
#686742
#686732
#686702
#686292
#241843