Finite Group Databasebeta

← #10697476

Subgroups — #10697476

Order 768.

488 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 97 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#79271C211yesmaximal, characteristic, complemented
2#79268C211yesmaximal, characteristic
3#78497C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic
6#69822C211yesmaximal, characteristic, complemented
7C3×#1281C211yesmaximal, characteristic, complemented
8#10226—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15#5113—22no—
16C2×#816—22no—
17#4936C2211yescharacteristic
18#4851—22no—
19#4846—22no—
20#4736—22no—
21#4721—22no—
22#4714—21no—
23#4714—21no—
24#4713—21no—
25#4703—21no—
26#4694—21no—
27#4682—21no—
28#4681—21no—
29#4680—21no—
30#4470—22no—
31#4467—22no—
32#3229—31no—
33#3226—31no—
34#3135—31no—
35#1960—31no—
36#1518—31no—
37#1517—31no—
38#1281D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6—21no—
41C22×C3×C8D412yes—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930—21no—
45#926—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51#875—21no—
52#874D411yescharacteristic
53C4×D12—21no—
54C4×D12—21no—
55#870—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D412yes—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8D412yes—
68C3×C22⋊C8D412yes—
69C3×#97—21no—
70C3×#97—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75#821—24no—
76#821—22no—
77#821—22no—
78#818—24no—
79#817—24no—
80#816—24no—
81#816—22no—
82#816—22no—
83#816—44no—
84#533—31no—
85#532—31no—
86C2×C4⋊Q8—31no—
87#500—61no—
88#498—61no—
89#491—61no—
90#490—61no—
91#489—61no—
92#489—61no—
93#487—61no—
94#486—61no—
95#483—62no—
96#480—62no—
97#475—62no—
98#474—62no—
99#431C2×D311yescharacteristic
100#431C2×D311yescharacteristic
101#416—62no—
102C2×#103—62no—
103C2×#97C2×D311yescharacteristic
104#382—31no—
105#328—62no—
106#327—62no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—41no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—42no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152C2×#41—61no—
153C22×Q8—61no—
154C22×C8D1212yes—
155C4⋊Q8—32no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162#127—31no—
163#125—61no—
164#124—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167#121—61no—
168#121—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—61no—
172C4×D4—61no—
173#117—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179#106—64no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182#103—62no—
183#103—62no—
184#103—64no—
185#103—124no—
186#102—62no—
187#102—62no—
188#102—64no—
189C22⋊C8D1212yes—
190C22⋊C8D1212yes—
191#97—21no—
192#97—21no—
193C4×C8—21no—
194C4×C8—21no—
195#95—61no—
196#95—61no—
197C23×C3#12711yescharacteristic
198C22×D3—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#13512yes—
208C2×C3×C4AGL(1,ℤ/8)12yes—
209#67—81no—
210#67—81no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220D12—42no—
221D12—42no—
222D12—81no—
223C4×D3—82no—
224Dic6—42no—
225Dic6—42no—
226Dic6—44no—
227Dic6—82no—
228Dic6—81no—
229C3×C8—42no—
230C3×C8—42no—
231C3×C8—42no—
232#41—122no—
233#41—122no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—62no—
237C2×Q8—61no—
238C2×Q8—121no—
239C2×Q8—122no—
240C2×Q8—122no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—121no—
245C22×C4—121no—
246C22×C4—121no—
247C22×C4—121no—
248C22×C4C2×D1211yescharacteristic
249C22×C4C2×D1211yescharacteristic
250C22×C4C2×D1211yescharacteristic
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—122no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—121no—
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C42#23911yescharacteristic
277C22×C3—21no—
278C22×C3—21no—
279C22×C3#39211yescharacteristic
280C22×C3#49511yescharacteristic
281C22×C3#49611yescharacteristic
282C2×D3—81no—
283C2×D3—81no—
284C2×D3—81no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4#44912yes—
290C3×C4#46312yes—
291Dic3—81no—
292Dic3—81no—
293Dic3—82no—
294Dic3—81no—
295Dic3—161no—
296C23—121no—
297C23#87411yescharacteristic
298Q8—122no—
299Q8—122no—
300Q8—124no—
301Q8—241no—
302Q8—242no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307D4—241no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4—242no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88211yescharacteristic
323C2×C4#89511yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4#89511yescharacteristic
326C2×C4#90412yes—
327C2×C4#90812yes—
328C8—42no—
329C8—42no—
330C8—42no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#200911yescharacteristic
334C2×C3#200911yescharacteristic
335C2×C3#201211yescharacteristic
336D3—161no—
337C22—21no—
338C22—21no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#467111yescharacteristic
343C22#469611yescharacteristic
344C22#471211yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—241no—
350C4—242no—
351C4—241no—
352C4—241no—
353C4—481no—
354C4#473412yes—
355C4#484412yes—
356C3#1022611yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—481no—
360C2#6867511yescharacteristic
361C2#6903111yescharacteristic
362C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1281D3
hypocenterC3#10226
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
1. #79271C2semidirect
4. #70793C2semidirect
6. #69822C2semidirect
7. C3×#1281C2semidirect
38. #1281D3semidirect
356. C3#10226semidirect

Maximal subgroups

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

subgroupindex
#792712
#792682
#784972
#707932
#707902
#698222
C3×#12812
#102263