Finite Group Databasebeta

← #10849326

Subgroups — #10849326

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2#82475C211yesmaximal, characteristic, complemented
3C2×#4842C211yesmaximal, characteristic, complemented
4C2×#4714C211yesmaximal, characteristic, complemented
5C3×#1985C211yesmaximal, characteristic, complemented
6#70458C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic, complemented
8#68684C212yesmaximal, complemented
9#68671C212yesmaximal, complemented
10#68630C212yesmaximal, complemented
11#68629C212yesmaximal
12#25659—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15C2×#944—21no—
16#5585—24no—
17#5585—24no—
18C2×#894—21no—
19C2×#894—21no—
20C2×#893—21no—
21C2×#875C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×#874C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#817C2212yes—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#4981—21no—
38#4981—21no—
39#4973—21no—
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2212yes—
42#4928C2212yes—
43#4842—24no—
44#4714—24no—
45C3×#336C2212yes—
46C3×#336C2212yes—
47#4551C2212yes—
48C2×#559—31no—
49C2×#489—31no—
50C2×#476—31no—
51#2546—31no—
52#2042—32no—
53#2041—32no—
54#1985D311yescharacteristic, complemented
55#1983—31no—
56#1865—32no—
57#1864—32no—
58#1762—31no—
59C22×C3×Q8D411yescharacteristic
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2311yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×C4×D3—21no—
66C22×Dic6—21no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78#947—28no—
79#944—44no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95#894—28no—
96#894—28no—
97#893—28no—
98#875—22no—
99#875—22no—
100#875—22no—
101#875D414yes—
102#874—22no—
103#874—22no—
104C4×D12—22no—
105C4×D12—22no—
106C4×D12—22no—
107C4×D12D414yes—
108#869—22no—
109#869—22no—
110C4×Dic6—22no—
111C4×Dic6—22no—
112C3×C4⋊C8—22no—
113C3×#97—22no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—21no—
117#821—24no—
118#821—24no—
119#821—24no—
120#821—24no—
121#817—24no—
122#816—24no—
123#816—24no—
124#559—64no—
125#559—64no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—61no—
128C2×#127—31no—
129C2×#124—31no—
130C2×#124—31no—
131C2×#124—61no—
132C2×#123—61no—
133C2×#123—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8C2×D311yescharacteristic
136C2×C4×D4—31no—
137C2×C4×D4—31no—
138C2×C4×D4—61no—
139#489—64no—
140#476—64no—
141C2×#106—32no—
142C2×C4⋊C8C2×D311yescharacteristic
143C2×#103—32no—
144C2×#103—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×#97C2×D311yescharacteristic
150#398—61no—
151#398—61no—
152#391—61no—
153#389—32no—
154#389—32no—
155C4×C4⋊C4—31no—
156#340—32no—
157#336C2×D312yes—
158#336C2×D312yes—
159C23×D3—21no—
160C23×D3—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—41no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—24no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—24no—
183C2×D12—42no—
184C2×D12—42no—
185C2×D12—82no—
186C2×D12C2×D412yes—
187C2×D12C2×D412yes—
188C2×C4×D3—24no—
189C2×C4×D3—24no—
190C2×C4×D3—82no—
191C2×C4×D3—82no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—21no—
195C2×Dic6—42no—
196C2×Dic6—42no—
197C2×Dic6—42no—
198C2×Dic6C2×D412yes—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42C2×D412yes—
213C3×C42C2×D412yes—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3—42no—
225C4⋊Dic3C2×D412yes—
226C4⋊Dic3#4114yes—
227Dic3⋊C4—24no—
228Dic3⋊C4—44no—
229Dic3⋊C4—44no—
230C4×Dic3—24no—
231C4×Dic3—42no—
232C22×Q8—31no—
233C22×Q8—61no—
234C22×Q8D1211yescharacteristic
235C22×D4—31no—
236C22×D4—31no—
237C22×D4—61no—
238C22×D4—61no—
239C23×C4—61no—
240C23×C4—61no—
241C22×C8C22×D312yes—
242C22×C8C22×D312yes—
243C4⋊Q8—62no—
244C4⋊Q8—62no—
245C4⋊Q8—124no—
246#127—62no—
247#127—62no—
248#124—34no—
249#124—62no—
250#124—62no—
251#124—68no—
252#124—62no—
253#123—68no—
254#123—68no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—34no—
260C4×D4—62no—
261C4×D4—62no—
262C4×D4—68no—
263C4×D4—62no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4D1211yescharacteristic
272C2×C4⋊C4D1211yescharacteristic
273C2×C4⋊C4D1211yescharacteristic
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—61no—
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284C2×C42C22×D311yescharacteristic
285#106—64no—
286C4⋊C8—22no—
287#103—64no—
288#103—64no—
289#102—64no—
290#102—64no—
291#102—64no—
292#102—64no—
293#97—22no—
294#95—61no—
295#95—62no—
296#95—62no—
297C23×C3C22×D411yescharacteristic
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C3×Q8—44no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4AGL(1,ℤ/8)12yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4112yes—
327C2×C3×C42−1+412yes—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—24no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338D12—22no—
339D12—22no—
340D12—24no—
341D12—48no—
342D12—41no—
343D12—41no—
344D12—42no—
345D12—84no—
346C4×D3—84no—
347C4×D3—84no—
348Dic6—24no—
349Dic6—41no—
350Dic6—41no—
351Dic6—84no—
352C3×C8—42no—
353C3×C8—42no—
354C24—61no—
355C24—61no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—32no—
359C2×Q8—41no—
360C2×Q8—61no—
361C2×Q8—62no—
362C2×Q8—62no—
363C2×Q8—122no—
364C2×Q8—122no—
365C2×Q8—122no—
366C2×D4—32no—
367C2×D4—32no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—242no—
379C22×C4—64no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—242no—
386C22×C4—242no—
387C22×C4C2×D1211yescharacteristic
388C22×C4C2×D1211yescharacteristic
389C22×C4C2×D1211yescharacteristic
390C22×C4C2×D1211yescharacteristic
391C22×C4C2×D1211yescharacteristic
392C22×C4C2×D1211yescharacteristic
393C22×C4C23×D311yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—32no—
404C4⋊C4—34no—
405C4⋊C4—64no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—124no—
411C4⋊C4—122no—
412C4⋊C4—124no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—64no—
424C42—122no—
425C42D3×D412yes—
426C42D3×D412yes—
427C22×C3C2×#12011yescharacteristic
428C22×C3#54812yes—
429C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C3#57511yescharacteristic
431C22×C3#57712yes—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4#53211yescharacteristic
449C3×C4#53611yescharacteristic
450C3×C4#54712yes—
451Dic3—44no—
452Dic3—42no—
453Dic3—42no—
454Dic3—82no—
455C23—62no—
456C23—62no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23C22×D1211yescharacteristic
466Q8—44no—
467Q8—64no—
468Q8—121no—
469Q8—121no—
470Q8—244no—
471D4—62no—
472D4—62no—
473D4—64no—
474D4—128no—
475D4—121no—
476D4—121no—
477D4—122no—
478D4—244no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—64no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—244no—
502C2×C4—244no—
503C2×C4#88212yes—
504C2×C4#88212yes—
505C2×C4#90812yes—
506C2×C4C2×D3×D411yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4#100712yes—
509C2×C4D3×#4112yes—
510C8—42no—
511C8—42no—
512C2×C3#243211yescharacteristic
513C2×C3#300611yescharacteristic
514C2×C3#300911yescharacteristic
515C2×C3#337112yes—
516C2×C3#337312yes—
517D3—82no—
518D3—82no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—242no—
526C22—242no—
527C22—242no—
528C22—242no—
529C22C2×#88211yescharacteristic
530C22#557612yes—
531C22C2×#90811yescharacteristic
532C22#574811yescharacteristic
533C22#575112yes—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—122no—
541C4—124no—
542C4—122no—
543C4—242no—
544C4#563112yes—
545C4#566811yescharacteristic
546C4#572411yescharacteristic
547C3#2565911yescharacteristic, complemented, Sylow
548C2—242no—
549C2—242no—
550C2#7649811yescharacteristic
551C2#7863911yescharacteristic
552C2#7867111yescharacteristic
553C2#7997312yes—
554C2#7998212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1985D3
hypocenterC3#25659
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #82475C2semidirect
3. C2×#4842C2semidirect
4. C2×#4714C2semidirect
5. C3×#1985C2semidirect
6. #70458C2semidirect
7. #69812C2semidirect
8. #68684C2semidirect
9. #68671C2semidirect
10. #68630C2semidirect
54. #1985D3semidirect
547. C3#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#824752
C2×#48422
C2×#47142
C3×#19852
#704582
#698122
#686842
#686712
#686302
#686292
#256593