Finite Group Databasebeta

← #10778348

Subgroups — #10778348

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83213C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic, complemented
3#79303C211yesmaximal, characteristic
4#79041C211yesmaximal, characteristic, complemented
5C3×#1549C211yesmaximal, characteristic, complemented
6#70745C211yesmaximal, characteristic, complemented
7#70502C211yesmaximal, characteristic, complemented
8#68454C214yesmaximal, complemented
9#68446C214yesmaximal, complemented
10#19198—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#907—21no—
13#5477C2211yescharacteristic
14C2×#874C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#423C2211yescharacteristic
17#5227—22no—
18#5129—21no—
19#5088—21no—
20#5088—21no—
21#5088C2212yes—
22#5088C2212yes—
23#5085—21no—
24#5083C2214yescomplemented
25#5079—21no—
26C2×#809—21no—
27C2×#809—21no—
28#4925C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4843C2214yes—
31#4838—21no—
32#4838C2212yescomplemented
33#4834—21no—
34#4834C2212yescomplemented
35#4732—22no—
36#4695—21no—
37#4694—21no—
38C3×#332C2214yescomplemented
39#4485C2214yes—
40#4480C2212yescomplemented
41#4479C2212yescomplemented
42#3206—31no—
43#3205—31no—
44C2×#486—31no—
45#2931—31no—
46#1701—34no—
47#1681—34no—
48#1564—31no—
49#1549D311yescharacteristic, complemented
50#1545—31no—
51C22×C3×Q8—21no—
52C2×#235—21no—
53C22×D12—21no—
54C22×D12—21no—
55C22×D12—21no—
56C2×C3×M16—21no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8C2312yes—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C2×#215—24no—
65#930—21no—
66#930D412yes—
67C22×C3⋊C8D411yescharacteristic
68C22×C3⋊C8D411yescharacteristic
69#907—44no—
70C2×D24—24no—
71C2×ΓL(1,𝔽25)—22no—
72C2×D3×C8—22no—
73#875—21no—
74#874—21no—
75#874—21no—
76#874—21no—
77#874D411yescharacteristic
78#874C2312yes—
79C4×D12—21no—
80C4×D12C2312yes—
81#869D411yescharacteristic
82C3×C4⋊C8—21no—
83C3×C4⋊C8C2312yes—
84C3×C4⋊C8C2312yes—
85C3×C22⋊C8—21no—
86C3×C4×C8—21no—
87#835—24no—
88#830—21no—
89#821—24no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92#809—22no—
93#809—22no—
94#809—22no—
95#809—22no—
96C4⋊(C3⋊C8)C2312yes—
97C4×C3⋊C8—21no—
98C4×C3⋊C8D411yescharacteristic
99C4×C3⋊C8D411yescharacteristic
100C4×C3⋊C8C2311yescharacteristic
101C4×C3⋊C8C2311yescharacteristic
102C2×#131—61no—
103#532—31no—
104C2×C4⋊Q8C2×D311yescharacteristic
105C2×#127—31no—
106#493—61no—
107#492—61no—
108#487—61no—
109#486—32no—
110#486—32no—
111#486—61no—
112#486—61no—
113#486—61no—
114#461—34no—
115#459—34no—
116#448—62no—
117#447—62no—
118#436—32no—
119#436—61no—
120C8×D4—32no—
121C8×D4—61no—
122#431—31no—
123#431C2×D311yescharacteristic
124#423C2×D311yescharacteristic
125C2×#102—61no—
126C2×#102—61no—
127#408—61no—
128C2×C4×C8—31no—
129#335—32no—
130#334—32no—
131#332C2×D314yescomplemented
132#331—34no—
133C23×D3—41no—
134C2×C3×Q8—22no—
135C2×C3×Q8—41no—
136C2×C3×Q8—82no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—22no—
142#235—42no—
143#235—42no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—42no—
150C2×D12—42no—
151C2×D12—41no—
152C2×D12—41no—
153C2×D12—41no—
154C2×D12—82no—
155C2×D12—82no—
156C2×D12C2×D412yes—
157C2×C4×D3—22no—
158C2×Dic6—21no—
159C3×M16—42no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—41no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4111yescharacteristic
169C3×C42C2411yescharacteristic
170#215—84no—
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173C4⋊Dic3—21no—
174C4⋊Dic3C2×D412yes—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8C2×D411yescharacteristic
179C2×C3⋊C8C2×D411yescharacteristic
180C2×C3⋊C8C2×D411yescharacteristic
181C2×C3⋊C8C2×D411yescharacteristic
182D24—84no—
183ΓL(1,𝔽25)—44no—
184D3×C8—44no—
185C2×#41—61no—
186C22×Q8—21no—
187C22×D4—61no—
188C22×D4—61no—
189C22×D4—61no—
190C2×SD16—64no—
191C2×D8—64no—
192#131—124no—
193C2×M16—21no—
194C2×M16—62no—
195C22×C8—31no—
196C22×C8—31no—
197C22×C8—62no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—31no—
202C4⋊Q8C22×D312yes—
203#127—31no—
204#127—32no—
205#127—61no—
206#127—61no—
207#127—61no—
208#124—61no—
209#121—32no—
210#121—61no—
211C4×D4—32no—
212C4×D4—61no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C42C22×D311yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—32no—
218C4⋊C8C22×D312yes—
219C4⋊C8C22×D312yes—
220#103—64no—
221#102—62no—
222#102—62no—
223#102—62no—
224#102—62no—
225#102—64no—
226C22⋊C8—21no—
227C22⋊C8—62no—
228C22⋊C8—62no—
229C22⋊C8—61no—
230C4×C8—21no—
231C4×C8—31no—
232C4×C8—31no—
233C4×C8—31no—
234C4×C8—31no—
235C4×C8—61no—
236C23×C3#12111yescharacteristic
237C22×D3—22no—
238C22×D3—42no—
239C22×D3—82no—
240C22×D3—82no—
241C22×D3—82no—
242C3×Q8—44no—
243C3×Q8—82no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—42no—
247C2×C3×C4—82no—
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#13512yes—
252C2×C3×C4AGL(1,ℤ/8)12yes—
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C22×D411yescharacteristic
255C2×C3×C4C2×#4111yescharacteristic
256#67—44no—
257C2×Dic3—22no—
258D12—24no—
259D12—42no—
260D12—42no—
261D12—44no—
262D12—42no—
263D12—42no—
264D12—82no—
265D12—82no—
266D12—82no—
267D12—82no—
268C4×D3—42no—
269C4×D3—42no—
270Dic6—42no—
271C3×C8—42no—
272C3×C8—42no—
273C3⋊C8—21no—
274C3⋊C8—21no—
275C3⋊C8—21no—
276C3⋊C8—21no—
277C3⋊C8—42no—
278C24—121no—
279#41—122no—
280#41—122no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—61no—
284C2×Q8—82no—
285C2×D4—32no—
286C2×D4—62no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—242no—
298C2×D4—242no—
299C22×C4—41no—
300C22×C4—62no—
301C22×C4#23511yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304SD16—244no—
305D8—244no—
306M16—42no—
307M16—124no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—31no—
311C2×C8—31no—
312C2×C8—31no—
313C2×C8—31no—
314C2×C8—62no—
315C2×C8—62no—
316C2×C8—62no—
317C2×C8—124no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—32no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—61no—
324C22⋊C4—62no—
325C22⋊C4—62no—
326C42#23511yescharacteristic
327C42D3×D411yescharacteristic
328C42D3×D411yescharacteristic
329C42C23×D311yescharacteristic
330C22×C3—21no—
331C22×C3—21no—
332C22×C3#46312yes—
333C22×C3C2×#12111yescharacteristic
334C2×D3—42no—
335C2×D3—42no—
336C2×D3—42no—
337C2×D3—82no—
338C2×D3—82no—
339C2×D3—82no—
340C2×D3—164no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—82no—
344C3×C4#46312yes—
345C3×C4#47112yes—
346C3×C4C2×#13511yescharacteristic
347C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
348C3×C4#57612yes—
349Dic3—42no—
350C23—62no—
351C23—122no—
352C23—242no—
353C23—242no—
354C23—242no—
355C23#88411yescharacteristic
356Q8—44no—
357Q8—82no—
358Q8—122no—
359D4—64no—
360D4—124no—
361D4—122no—
362D4—122no—
363D4—124no—
364D4—122no—
365D4—122no—
366D4—242no—
367D4—242no—
368D4—242no—
369D4—242no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—62no—
374C2×C4—82no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4#88411yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4#91412yes—
381C2×C4#91912yes—
382C2×C4C2×#23511yescharacteristic
383C2×C4C2×D3×D411yescharacteristic
384C2×C4C2×D3×D411yescharacteristic
385C8—42no—
386C8—42no—
387C8—61no—
388C8—61no—
389C8—61no—
390C8—61no—
391C8—122no—
392C2×C3—22no—
393C2×C3C2×#46311yescharacteristic
394C2×C3#305811yescharacteristic
395C2×C3C2×#47111yescharacteristic
396D3—82no—
397D3—162no—
398C22—21no—
399C22—21no—
400C22—122no—
401C22—122no—
402C22—122no—
403C22—242no—
404C22—242no—
405C22—242no—
406C22—484no—
407C22#481912yes—
408C22C2×#88411yescharacteristic
409C4—21no—
410C4—21no—
411C4—82no—
412C4—122no—
413C4#475412yes—
414C4#481912yes—
415C4C2×#91411yescharacteristic
416C4C2×#91911yescharacteristic
417C4#577412yes—
418C3#1919811yescharacteristic, complemented, Sylow
419C2—22no—
420C2—242no—
421C2—482no—
422C2C2×#475411yescharacteristic
423C2C2×#481911yescharacteristic
424C2#7893211yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1549C2
socleC22×C3C2×#121
prime radical#79303C2
hypercenter#1549D3
hypocenterC3#19198
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #83213C2semidirect
2. C2×#5088C2semidirect
4. #79041C2semidirect
5. C3×#1549C2semidirect
6. #70745C2semidirect
7. #70502C2semidirect
8. #68454C2semidirect
9. #68446C2semidirect
24. #5083C22semidirect
32. #4838C22semidirect
34. #4834C22semidirect
38. C3×#332C22semidirect
40. #4480C22semidirect
41. #4479C22semidirect
49. #1549D3semidirect
131. #332C2×D3semidirect
418. C3#19198semidirect

Maximal subgroups

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

subgroupindex
#832132
C2×#50882
#793032
#790412
C3×#15492
#707452
#705022
#684542
#684462
#191983