Finite Group Databasebeta

← #10829230

Subgroups — #10829230

Order 768.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5726C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70581C212yesmaximal, complemented
7#70579C212yesmaximal, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70471C211yesmaximal, characteristic, complemented
10#70204C212yesmaximal, complemented
11#70200C212yesmaximal, complemented
12#24128—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946—21no—
17C2×#938C2212yes—
18C22×#213C2211yescharacteristic
19#5726—24no—
20#5726—24no—
21C2×#895C2212yes—
22C2×#894—22no—
23C2×D3×C4⋊C4C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C3×#476—24no—
27C2×C3×#106C2212yes—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×#832C2211yescharacteristic
31#5213—22no—
32C2×#821C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5014—24no—
36#4987C2212yes—
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4960—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#476D311yescharacteristic, complemented
48C2×#442—31no—
49#2535—31no—
50#2034—32no—
51#1999—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1853—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C2×#239—28no—
62C2×#239—48no—
63C22×D12—22no—
64C22×D12D411yescharacteristic
65C22×D12C2311yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3—21no—
68C22×C4×D3C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yescomplemented
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80#946—28no—
81#938—24no—
82#938—42no—
83C2×#213—22no—
84C2×#213—22no—
85C2×#213—22no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4D412yes—
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—22no—
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96#895—24no—
97#895—24no—
98#894—216no—
99D3×C4⋊C4—24no—
100D3×C4⋊C4—24no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12—22no—
104C4×D12D414yescomplemented
105C3×#106—24no—
106C3×C4⋊C8—22no—
107C3×#102—24no—
108#832—22no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112#821—42no—
113(C2×D3)⋊C8—24no—
114(C2×D3)⋊C8—42no—
115#809—22no—
116#808—22no—
117#803—22no—
118C22×#41—61no—
119C22×D8—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#123—62no—
124C2×C22⋊Q8—61no—
125C2×#121—32no—
126C2×#121—32no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#476—24no—
132#442—64no—
133C2×#106—31no—
134C2×#106C2×D312yes—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#102—31no—
137C2×#102—31no—
138C2×#102—32no—
139C2×#102C2×D312yes—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97—31no—
143#397—61no—
144#392—61no—
145#386—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C23×D3—21no—
153C23×D3C2×D412yes—
154C2×C3×Q8—24no—
155C2×C3×Q8—44no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4#4112yescomplemented
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—42no—
165C22×C3×C4—82no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—44no—
170C2×#67—44no—
171C2×#67—82no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3C2×D412yes—
175#239—432no—
176C2×D12—24no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—28no—
183C2×D12—21no—
184C2×D12—44no—
185C2×D12—44no—
186C2×D12C2×D412yes—
187C2×C4×D3—28no—
188C2×C4×D3—24no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—44no—
196C2×C4×D3—44no—
197C2×C4×D3—44no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4C2×D414yes—
206C3×C22⋊C4—42no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209#217—44no—
210#213—28no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—82no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—28no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8#4112yescomplemented
227C2×#41—68no—
228C2×#41—128no—
229C22×Q8—21no—
230C22×D4—31no—
231C22×D4—31no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4C22×D311yescharacteristic
235C23×C4—21no—
236C23×C4—32no—
237C23×C4—62no—
238C23×C4—61no—
239C2×D8—62no—
240C2×D8—62no—
241C2×D8—62no—
242C22×C8—31no—
243C22×C8—31no—
244C22×C8C22×D312yes—
245C4⋊Q8—22no—
246C4⋊Q8#6714yescomplemented
247#123—616no—
248C22⋊Q8—68no—
249#121—64no—
250#121—64no—
251#121—64no—
252#121—122no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—34no—
256C4×D4—62no—
257C4×D4—62no—
258C4×D4—62no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—32no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C42C22×D311yescharacteristic
275#106—24no—
276#106—62no—
277C4⋊C8—22no—
278#102—24no—
279#102—62no—
280#102—62no—
281#102—122no—
282C22⋊C8—64no—
283C22⋊C8—122no—
284#97—62no—
285#95—62no—
286#95—61no—
287#95—61no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3AGL(1,ℤ/8)14yes—
306C3×Q8—48no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—41no—
310C3×D4—41no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—84no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12012yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4H(5,𝔽2)11yescharacteristic
326C2×C3×C42−1+411yescharacteristic
327#67—88no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336D12—28no—
337D12—22no—
338D12—22no—
339D12—416no—
340D12—41no—
341D12—41no—
342C4×D3—48no—
343C4×D3—416no—
344C4×D3—42no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—42no—
348C3×C8—42no—
349C3⋊C8—22no—
350C3⋊C8—41no—
351C24—21no—
352C24—32no—
353C24—61no—
354#41—1232no—
355C2×Q8—24no—
356C2×Q8—44no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—32no—
363C2×D4—64no—
364C2×D4—68no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—124no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—242no—
376C2×D4#23912yescomplemented
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—32no—
380C22×C4—42no—
381C22×C4—68no—
382C22×C4—61no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—82no—
392C22×C4—124no—
393C22×C4—124no—
394C22×C4—124no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—68no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—32no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—32no—
413C4⋊C4—68no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4C2×#6714yes—
419C22⋊C4—42no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—124no—
424C22⋊C4—242no—
425C42D3×D412yes—
426C42C2×#6712yes—
427C22×C3—42no—
428C22×C3—44no—
429C22×C3—82no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54611yescharacteristic
434C22×C3#54711yescharacteristic
435C22×C3C2×AGL(1,ℤ/8)12yes—
436C22×C3#57712yes—
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—82no—
448C2×D3—82no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—44no—
452C3×C4—81no—
453C3×C4#52511yescharacteristic
454C3×C4#53711yescharacteristic
455C3×C4#54811yescharacteristic
456C3×C4#55011yescharacteristic
457Dic3—42no—
458Dic3—42no—
459Dic3—82no—
460C23—22no—
461C23—34no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23—62no—
471C23—62no—
472C23—62no—
473C23—122no—
474C23—122no—
475C23—121no—
476C23—121no—
477C23C2×D3×D411yescharacteristic
478Q8—48no—
479D4—22no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—68no—
484D4—62no—
485D4—62no—
486D4—1216no—
487D4—121no—
488D4—121no—
489D4—248no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—84no—
502C2×C4—128no—
503C2×C4—1216no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4#93712yes—
513C2×C4#93712yes—
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×#23911yescharacteristic
516C2×C4#100711yescharacteristic
517C2×C4D3×#4111yescharacteristic
518C2×C4#100911yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—42no—
521C8—62no—
522C8—121no—
523C2×C3—82no—
524C2×C3#240711yescharacteristic
525C2×C3#300311yescharacteristic
526C2×C3#301011yescharacteristic
527C2×C3#330612yes—
528C2×C3#334512yes—
529D3—44no—
530D3—82no—
531C22—42no—
532C22—44no—
533C22—64no—
534C22—64no—
535C22—64no—
536C22—81no—
537C22—82no—
538C22—81no—
539C22—122no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—122no—
546C22—242no—
547C22—242no—
548C22#563011yescharacteristic
549C22#563111yescharacteristic
550C22#576212yes—
551C22D3×AGL(1,ℤ/8)12yes—
552C22C2×#93711yescharacteristic
553C4—22no—
554C4—22no—
555C4—44no—
556C4—81no—
557C4—122no—
558C4—122no—
559C4—242no—
560C4#562711yescharacteristic
561C4#562911yescharacteristic
562C4#581711yescharacteristic
563C4#583411yescharacteristic
564C3#2412811yescharacteristic, complemented, Sylow
565C2—82no—
566C2—124no—
567C2—242no—
568C2#8107712yes—
569C2#8108312yes—
570C2#8281211yescharacteristic
571C2#8386011yescharacteristic
572C2#8394411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#476C2
socleC23×C3C22×D4
hypercenterC2×#476D3
hypocenterC3#24128
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#5726C2semidirect
2. C2×C3×#476C2semidirect
3. #82416C2semidirect
4. C2×#5014C2semidirect
5. #71788C2semidirect
6. #70581C2semidirect
7. #70579C2semidirect
8. #70480C2semidirect
9. #70471C2semidirect
10. #70204C2semidirect
11. #70200C2semidirect
47. C2×#476D3semidirect
71. C3×C4⋊Q8D4semidirect
104. C4×D12D4semidirect
161. C2×C3×D4#41semidirect
226. C2×C3⋊C8#41semidirect
246. C4⋊Q8#67semidirect
376. C2×D4#239semidirect
564. C3#24128semidirect

Maximal subgroups

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

subgroupindex
C2×#57262
C2×C3×#4762
#824162
C2×#50142
#717882
#705812
#705792
#704802
#704712
#702042
#702002
#241283