Finite Group Databasebeta

← #10696567

Subgroups — #10696567

Order 768.

496 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 85 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#79318C211yesmaximal, characteristic
2#78528C211yesmaximal, characteristic
3#73410C211yesmaximal, characteristic, complemented
4#70822C211yesmaximal, characteristic, complemented
5#69863C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic
7C3×#1300C211yesmaximal, characteristic, complemented
8#10093—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#5133—22no—
16#4936—21no—
17#4935—21no—
18#4926C2211yescharacteristic
19#4847—22no—
20#4736—21no—
21#4735—21no—
22#4714—22no—
23#4713—21no—
24#4702—21no—
25#4694—21no—
26#4681—21no—
27#4661—22no—
28#4656—22no—
29#4486—22no—
30#4465—22no—
31#4451—22no—
32#3234—31no—
33#3141—31no—
34#2660—31no—
35#1556—31no—
36#1528—31no—
37#1505—31no—
38#1300D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—24no—
48C2×C4⋊Dic3D412yes—
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53#875—21no—
54#874D411yescharacteristic
55C4×D12—21no—
56C4×D12—21no—
57#870—21no—
58#870—21no—
59#870—21no—
60#870—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D411yescharacteristic
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71C3×#95—21no—
72C3×#95D412yes—
73C3⋊#95—24no—
74#830—21no—
75#821—24no—
76#821—22no—
77#821—22no—
78#818—22no—
79#817—22no—
80#816—24no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#532—31no—
84C2×C4⋊Q8—31no—
85C2×#125—31no—
86#501—61no—
87#490—61no—
88#489—62no—
89#487—61no—
90#486—61no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432—31no—
95#432C2×D311yescharacteristic
96#417—62no—
97C4×M16C2×D311yescharacteristic
98#398—62no—
99#395—62no—
100#384—61no—
101#382—61no—
102#376C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#327—62no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—41no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—21no—
114#235—44no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C3×M16—42no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C3×C42—24no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—22no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—84no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C2×C3⋊C8—41no—
149C2×#41—61no—
150C22×Q8—61no—
151C2×M16—21no—
152C4⋊Q8—31no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157#127—31no—
158#125—61no—
159#125—61no—
160#125—61no—
161#125—61no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—64no—
174C2×C42—21no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#106—62no—
178C4⋊C8—61no—
179C4⋊C8D1212yes—
180#103—64no—
181#102—64no—
182#102—62no—
183#102—62no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186#97—21no—
187#97—61no—
188#97D1212yes—
189C4×C8—21no—
190#95—21no—
191#95—64no—
192#95D1212yes—
193C23×C3#12011yescharacteristic
194C22×D3—41no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12711yescharacteristic
207#67—82no—
208C2×Dic3—42no—
209C2×Dic3—44no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215D12—42no—
216D12—42no—
217D12—81no—
218C4×D3—82no—
219Dic6—42no—
220Dic6—82no—
221Dic6—82no—
222C3×C8—42no—
223C3×C8—42no—
224C3⋊C8—81no—
225#41—124no—
226C2×Q8—61no—
227C2×Q8—121no—
228C2×Q8—121no—
229C2×Q8—242no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—121no—
234C22×C4—22no—
235C22×C4—61no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242M16—42no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—121no—
246C4⋊C4—62no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—244no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C42—24no—
264C42C2×D1211yescharacteristic
265C42C2×D1211yescharacteristic
266C42D3×D411yescharacteristic
267C42#23911yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C22×C3#45412yes—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—44no—
281C3×C4#45612yes—
282Dic3—81no—
283Dic3—81no—
284Dic3—82no—
285Dic3—81no—
286C23—121no—
287C23#88211yescharacteristic
288Q8—122no—
289Q8—242no—
290Q8—242no—
291D4—122no—
292D4—122no—
293D4—241no—
294D4—242no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—122no—
302C2×C4—124no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—242no—
309C2×C4#87411yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88211yescharacteristic
312C2×C4#89511yescharacteristic
313C2×C4#89511yescharacteristic
314C2×C4#89511yescharacteristic
315C8—42no—
316C8—42no—
317C8—241no—
318C2×C3—22no—
319C2×C3#200311yescharacteristic
320C2×C3#200511yescharacteristic
321C2×C3#200811yescharacteristic
322D3—161no—
323C22—21no—
324C22—21no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#467111yescharacteristic
329C22#482712yes—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—44no—
336C4—241no—
337C4—241no—
338C4—242no—
339C4—241no—
340C4#482612yes—
341C3#1009311yescharacteristic, complemented, Sylow
342C2—22no—
343C2—481no—
344C2#6876711yescharacteristic
345C2#6876811yescharacteristic
346C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1300D3
hypocenterC3#10093
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #73410C2semidirect
4. #70822C2semidirect
5. #69863C2semidirect
7. C3×#1300C2semidirect
38. #1300D3semidirect
341. C3#10093semidirect

Maximal subgroups

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

subgroupindex
#793182
#785282
#734102
#708222
#698632
#694322
C3×#13002
#100933