Finite Group Databasebeta

← #10692865

Subgroups — #10692865

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#79335C211yesmaximal, characteristic
2#73802C211yesmaximal, characteristic
3#70827C211yesmaximal, characteristic
4#70824C211yesmaximal, characteristic, complemented
5#70824C211yesmaximal, characteristic, complemented
6C3×#1285C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9451—31nomaximal, Sylow
9C2×#875C412yes—
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#4987—21no—
16#4987—21no—
17#4929C412yes—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4836—22no—
21#4726—21no—
22D3×C22⋊C8—21no—
23#4706—22no—
24#4674—21no—
25#4674—21no—
26#4481—22no—
27#4481—22no—
28#4468—21no—
29#4468—21no—
30#2973—31no—
31#2547—31no—
32#1558—31no—
33#1558—31no—
34#1466—31no—
35#1285—31no—
36#1285D311yescharacteristic, complemented
37C22×D12C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C22×Dic6C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C42C2×C412yes—
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4D411yescharacteristic
47C2×(C2×D3)⋊C4D411yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×ΓL(1,𝔽25)—22no—
52C2×D3×C8—22no—
53#875—44no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—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⋊Q8—31no—
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⋊C4C2×D311yescharacteristic
96#333—62no—
97#333—62no—
98#323—61no—
99#323—61no—
100C23×D3C22⋊C411yescharacteristic
101C23×D3C22⋊C411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—22no—
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C2×D12—22no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—41no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—42no—
118C2×C4×D3—42no—
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6—42no—
122C2×Dic6—42no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—42no—
126C3×C42—42no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—21no—
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—44no—
136(C2×D3)⋊C4—44no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145ΓL(1,𝔽25)—82no—
146D3×C8—82no—
147C22×Q8—32no—
148C22×D4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C2×M16—62no—
152C22×C8—62no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157#124—124no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C42C4×D312yes—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175C4⋊C8D1212yes—
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—21no—
187#95—62no—
188#95—62no—
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—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220D12—42no—
221D12—42no—
222C4×D3—42no—
223C4×D3—42no—
224C4×D3—81no—
225C4×D3—81no—
226Dic6—82no—
227Dic6—82no—
228C3×C8—42no—
229C3×C8—81no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C24—31no—
233C24—31no—
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8—122no—
237C2×Q8—122no—
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—42no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—124no—
273C22⋊C4—124no—
274C42—42no—
275C42C2×D1211yescharacteristic
276C42#23511yescharacteristic
277C42D3×D411yescharacteristic
278C42#23911yescharacteristic
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—242no—
316Q8—242no—
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×C4C4×D1211yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4#89311yescharacteristic
341C2×C4#89411yescharacteristic
342C2×C4#89511yescharacteristic
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—
366C22#466511yescharacteristic
367C22D3×#10011yescharacteristic
368C22#475111yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—121no—
376C4—121no—
377C4—242no—
378C4#482512yes—
379C3#945111yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2—122no—
383C2—241no—
384C2#6871211yescharacteristic
385C2#6875911yescharacteristic
386C2#6899211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9451
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
4. #70824C2semidirect
5. #70824C2semidirect
6. C3×#1285C2semidirect
7. #67621C2semidirect
36. #1285D3semidirect
379. C3#9451semidirect

Maximal subgroups

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

subgroupindex
#793352
#738022
#708272
#708242
C3×#12852
#676212
#94513