Finite Group Databasebeta

← #10829231

Subgroups — #10829231

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×#5084C211yesmaximal, characteristic, complemented
3C2×#4711C211yesmaximal, characteristic, complemented
4#74239C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69837C211yesmaximal, characteristic, complemented
7#69556C211yesmaximal, characteristic, complemented
8#68886C212yesmaximal, complemented
9#68880C212yesmaximal, complemented
10#68875C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#24128—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946—21no—
16C22×D24C2211yescharacteristic
17#5726—24no—
18#5726—24no—
19C2×#895C2212yes—
20C2×#895C2212yes—
21C2×#894—22no—
22C2×D3×C4⋊C4C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×(C2×D3)⋊C8C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2211yescharacteristic
33#5084—24no—
34#4987C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#808C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4711—24no—
40#4674—21no—
41#4671—21no—
42#4665—22no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#476—31no—
48C2×#442—31no—
49#2535—31no—
50#2034—32no—
51#1999—32no—
52#1993D311yescharacteristic, complemented
53#1955—31no—
54#1874—32no—
55#1853—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—48no—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12D411yescharacteristic
63C22×D12C2311yescharacteristic
64C22×D12C2311yescharacteristic
65C22×C4×D3—22no—
66C22×C4×D3—21no—
67C22×C4×D3—21no—
68C22×C4×D3C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D414yescomplemented
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C42C2311yescharacteristic
77#946—28no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4D412yes—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—22no—
88C22×C3⋊C8C2312yes—
89C2×D24—22no—
90C2×D24—22no—
91C2×D24—22no—
92#895—24no—
93#895—24no—
94#895—24no—
95#895—42no—
96#894—216no—
97D3×C4⋊C4—24no—
98D3×C4⋊C4—24no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12—22no—
104C4×D12D414yescomplemented
105C3×#97—22no—
106C3×#95—21no—
107C3×#95—21no—
108C3⋊#95—22no—
109#821—22no—
110#821—22no—
111(C2×D3)⋊C8—24no—
112(C2×D3)⋊C8—42no—
113#817—22no—
114#809—24no—
115#809—42no—
116#808—24no—
117C4⋊(C3⋊C8)—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×D4—31no—
130C22×C4⋊C4—32no—
131#476—64no—
132#442—64no—
133C2×#106—31no—
134C2×#106—32no—
135C2×C4⋊C8—31no—
136C2×#102—31no—
137C2×#102—31no—
138C2×#102—32no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97C2×D311yescharacteristic
143#397—61no—
144#392—61no—
145#386—62no—
146#382C2×D311yescharacteristic
147#380—32no—
148#340—32no—
149#340C2×D312yes—
150#340C2×D312yes—
151C23×D3—21no—
152C23×D3—21no—
153C23×D3C2×D412yes—
154C2×C3×Q8—24no—
155C2×C3×Q8—44no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3C2×D412yes—
164#239—432no—
165C2×D12—24no—
166C2×D12—22no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—28no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—44no—
179C2×D12—44no—
180C2×D12—44no—
181C2×D12—44no—
182C2×D12—82no—
183C2×D12C2×D412yes—
184C2×D12#4112yescomplemented
185C2×C4×D3—28no—
186C2×C4×D3—24no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×C4×D3—42no—
197C2×C4×D3—82no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8#4112yescomplemented
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4C2×D414yes—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—42no—
216(C2×D3)⋊C4—82no—
217C4⋊Dic3—22no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3C2×D412yes—
222Dic3⋊C4—28no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226D24—28no—
227C2×#41—68no—
228C2×#41—128no—
229C22×Q8—21no—
230C22×D4—31no—
231C22×D4—31no—
232C22×D4—31no—
233C22×D4—62no—
234C22×D4—62no—
235C23×C4—32no—
236C23×C4—62no—
237C23×C4—61no—
238C23×C4—61no—
239C2×D8—62no—
240C2×D8—62no—
241C2×D8—62no—
242C22×C8—32no—
243C22×C8C22×D311yescharacteristic
244C22×C8C22×D311yescharacteristic
245C4⋊Q8—22no—
246C4⋊Q8D1214yescomplemented
247#123—616no—
248C22⋊Q8—68no—
249#121—64no—
250#121—64no—
251#121—64no—
252#121—122no—
253C4×D4—34no—
254C4×D4—62no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—62no—
258C4×D4—62no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—32no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—62no—
274C2×C42C22×D311yescharacteristic
275#106—62no—
276#106—64no—
277C4⋊C8—62no—
278#102—62no—
279#102—62no—
280#102—64no—
281#102—122no—
282C22⋊C8—64no—
283C22⋊C8—122no—
284#97—22no—
285#95—21no—
286#95—21no—
287#95—62no—
288C23×C3C22×D411yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3AGL(1,ℤ/8)14yes—
306C3×Q8—48no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322C2×C3×C42−1+411yescharacteristic
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330D12—28no—
331D12—22no—
332D12—22no—
333D12—22no—
334D12—22no—
335D12—416no—
336D12—41no—
337D12—41no—
338D12—41no—
339D12—41no—
340D12—88no—
341C4×D3—48no—
342C4×D3—416no—
343C4×D3—42no—
344C4×D3—42no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—84no—
348C3×C8—22no—
349C3×C8—41no—
350C3⋊C8—42no—
351C24—32no—
352C24—61no—
353C24—61no—
354#41—1232no—
355C2×Q8—24no—
356C2×Q8—44no—
357C2×D4—32no—
358C2×D4—32no—
359C2×D4—64no—
360C2×D4—68no—
361C2×D4—62no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4—242no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—32no—
380C22×C4—68no—
381C22×C4—61no—
382C22×C4—64no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4—124no—
393C22×C4—122no—
394C22×C4—242no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—68no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8#23912yescomplemented
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—32no—
413C4⋊C4—68no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4C2×D1214yes—
419C22⋊C4—68no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—124no—
423C22⋊C4—122no—
424C22⋊C4—242no—
425C42C2×D1212yes—
426C42D3×D412yes—
427C22×C3C2×#12011yescharacteristic
428C22×C3#54611yescharacteristic
429C22×C3#54711yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)12yes—
431C22×C3#57712yes—
432C2×D3—24no—
433C2×D3—24no—
434C2×D3—24no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—82no—
445C2×D3—82no—
446C2×D3—81no—
447C2×D3—81no—
448C2×D3—82no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—44no—
452C3×C4—82no—
453C3×C4#52511yescharacteristic
454C3×C4#53711yescharacteristic
455C3×C4#54811yescharacteristic
456C3×C4#55011yescharacteristic
457Dic3—42no—
458Dic3—42no—
459Dic3—81no—
460C23—34no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—121no—
470C23—121no—
471C23—121no—
472C23—121no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23C2×D3×D411yescharacteristic
478Q8—48no—
479D4—68no—
480D4—62no—
481D4—62no—
482D4—62no—
483D4—62no—
484D4—1216no—
485D4—121no—
486D4—121no—
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—42no—
497C2×C4—42no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—128no—
503C2×C4—1216no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—121no—
510C2×C4—122no—
511C2×C4—244no—
512C2×C4#88212yes—
513C2×C4#88212yes—
514C2×C4C22×D1211yescharacteristic
515C2×C4C2×D3×D411yescharacteristic
516C2×C4C2×#23911yescharacteristic
517C2×C4#100711yescharacteristic
518C2×C4D3×#4111yescharacteristic
519C2×C4#100911yescharacteristic
520C8—22no—
521C8—41no—
522C8—122no—
523C2×C3#240711yescharacteristic
524C2×C3#300311yescharacteristic
525C2×C3#301011yescharacteristic
526C2×C3#330612yes—
527C2×C3#334512yes—
528D3—44no—
529D3—82no—
530D3—82no—
531C22—64no—
532C22—64no—
533C22—64no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—124no—
539C22—122no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—242no—
544C22—242no—
545C22—242no—
546C22—241no—
547C22—241no—
548C22C2×#88211yescharacteristic
549C22#567611yescharacteristic
550C22#572611yescharacteristic
551C22D3×AGL(1,ℤ/8)12yes—
552C22#577812yes—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—82no—
557C4—122no—
558C4—122no—
559C4—241no—
560C4#557611yescharacteristic
561C4#557711yescharacteristic
562C4#562711yescharacteristic
563C4#562911yescharacteristic
564C3#2412811yescharacteristic, complemented, Sylow
565C2—124no—
566C2—242no—
567C2—242no—
568C2#7661811yescharacteristic
569C2#7892611yescharacteristic
570C2#7896011yescharacteristic
571C2#8169612yes—
572C2#8199212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
hypercenter#1993D3
hypocenterC3#24128
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5726C2semidirect
2. C2×#5084C2semidirect
3. C2×#4711C2semidirect
4. #74239C2semidirect
5. C3×#1993C2semidirect
6. #69837C2semidirect
7. #69556C2semidirect
8. #68886C2semidirect
9. #68880C2semidirect
10. #68875C2semidirect
11. #68823C2semidirect
52. #1993D3semidirect
72. C3×C4⋊Q8D4semidirect
104. C4×D12D4semidirect
184. C2×D12#41semidirect
203. C2×C3×C8#41semidirect
246. C4⋊Q8D12semidirect
407. C2×C8#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×#50842
C2×#47112
#742392
C3×#19932
#698372
#695562
#688862
#688802
#688752
#688232
#241283