Finite Group Databasebeta

← #10692863

Subgroups — #10692863

Order 768.

518 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83240C211yesmaximal, characteristic
2#79335C211yesmaximal, characteristic
3C3×#1558C211yesmaximal, characteristic, complemented
4#71777C211yesmaximal, characteristic, complemented
5#70827C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9451—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#947C412yes—
11C2×C4×D12C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5229C412yes—
15#4992—21no—
16#4992—21no—
17#4987—21no—
18#4987—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4836—22no—
23#4726—21no—
24D3×C22⋊C8—21no—
25#4706—22no—
26C3×#333—22no—
27#4546—22no—
28#4468—21no—
29#4468—21no—
30#2973—31no—
31#2547—31no—
32#1558—31no—
33#1558D311yescharacteristic, complemented
34#1466—31no—
35#1285—31no—
36#1285—31no—
37C22×C3×Q8C2×C412yes—
38C22×D12C2×C411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48#947—44no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×(C2×D3)⋊C4D411yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic
55C2×C4×Dic3C2×C412yes—
56C2×ΓL(1,𝔽25)—22no—
57C2×D3×C8—22no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8D412yes—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×#97—21no—
68C3⋊#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71#830—21no—
72#830—21no—
73(C2×D3)⋊C8—22no—
74(C2×D3)⋊C8—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77#803—21no—
78#803—21no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#124—32no—
81C2×C4×D4—31no—
82#435—62no—
83#435—62no—
84#432—31no—
85#432—31no—
86#432C2×D311yescharacteristic
87#432C2×D311yescharacteristic
88#407—61no—
89C2×C22⋊C8—61no—
90#397—61no—
91#397—61no—
92#391—32no—
93#380—61no—
94#380—61no—
95C4×C4⋊C4—31no—
96#333—22no—
97#333—62no—
98#323—61no—
99#323—61no—
100C23×D3C22⋊C411yescharacteristic
101C23×D3C22⋊C411yescharacteristic
102C2×C3×Q8—22no—
103C2×C3×Q8—22no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C22×C3×C4—22no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—22no—
111C22×Dic3C22⋊C411yescharacteristic
112C22×Dic3C22⋊C411yescharacteristic
113C2×D12—22no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—41no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—42no—
122C2×C4×D3—42no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—44no—
138(C2×D3)⋊C4—44no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—42no—
142C4×Dic3—42no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145ΓL(1,𝔽25)—82no—
146D3×C8—82no—
147C22×Q8C4×D312yes—
148C22×D4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C2×M16—62no—
152C22×C8—62no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157#124—124no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C42—32no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175C4⋊C8#6712yes—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—62no—
179C22⋊C8—62no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182#97—21no—
183#97—21no—
184#97—61no—
185#97—61no—
186#95—62no—
187#95—62no—
188#95—61no—
189C23×C3C2×C22⋊C411yescharacteristic
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3#10012yes—
200C3×Q8—82no—
201C3×Q8—82no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222D12—42no—
223D12—42no—
224C4×D3—42no—
225C4×D3—42no—
226C4×D3—81no—
227C4×D3—81no—
228C3×C8—42no—
229C3×C8—81no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C24—31no—
233C24—31no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×D4—62no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C22×C4—22no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—62no—
250C22×C4—122no—
251C22×C4—122no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255M16—242no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—121no—
260C2×C8—242no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—122no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—124no—
273C22⋊C4—124no—
274C42—122no—
275C42#23511yescharacteristic
276C42D3×D411yescharacteristic
277C42#23711yescharacteristic
278C42C2×#6711yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#38611yescharacteristic
282C22×C3C2×#10011yescharacteristic
283C22×C3#41311yescharacteristic
284C2×D3—22no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—81no—
293C2×D3—81no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—82no—
300C3×C4#45512yes—
301Dic3—41no—
302Dic3—41no—
303Dic3—82no—
304C23—32no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—61no—
309C23—61no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23D3×C22⋊C411yescharacteristic
315Q8—82no—
316Q8—82no—
317D4—122no—
318D4—122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—241no—
336C2×C4—241no—
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88311yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C8—42no—
344C8—81no—
345C8—122no—
346C8—241no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3#188511yescharacteristic
350C2×C3#190011yescharacteristic
351C2×C3#190911yescharacteristic
352D3—42no—
353D3—81no—
354C22—21no—
355C22—21no—
356C22—62no—
357C22—122no—
358C22—122no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—241no—
365C22—241no—
366C22D3×#10011yescharacteristic
367C22#474511yescharacteristic
368C22#495611yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—121no—
376C4—121no—
377C4—242no—
378C4#520212yes—
379C3#945111yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2—122no—
383C2—241no—
384C2#7008111yescharacteristic
385C2#7034311yescharacteristic
386C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1558C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1558D3
hypocenterC3#9451
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
3. C3×#1558C2semidirect
4. #71777C2semidirect
6. #67750C2semidirect
7. #67749C2semidirect
33. #1558D3semidirect
379. C3#9451semidirect

Maximal subgroups

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

subgroupindex
#832402
#793352
C3×#15582
#717772
#708272
#677502
#677492
#94513