Finite Group Databasebeta

← #10696777

Subgroups — #10696777

Order 768.

442 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 81 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#79320C211yesmaximal, characteristic, complemented
2#79319C211yesmaximal, characteristic
3#78529C211yesmaximal, characteristic
4#70831C211yesmaximal, characteristic, complemented
5#70801C211yesmaximal, characteristic
6#69856C211yesmaximal, characteristic, complemented
7C3×#1291C211yesmaximal, characteristic, complemented
8#10123—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5133—21no—
16#5133—21no—
17#4925C2211yescharacteristic
18#4846—21no—
19#4842—21no—
20#4829—22no—
21#4737—22no—
22#4722—21no—
23#4721—21no—
24#4713—21no—
25#4713—21no—
26#4703—21no—
27#4703—21no—
28#4702—21no—
29#4702—21no—
30#4697—22no—
31#4483—22no—
32#4469—22no—
33#4456—22no—
34#3236—31no—
35#3235—31no—
36#3142—31no—
37#1496—31no—
38#1485—31no—
39#1470—31no—
40#1291D311yescharacteristic, complemented
41C2×#235—21no—
42C2×C3×M16—21no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46#926—21no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
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#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×#97—21no—
72C3×#97—21no—
73C3×#97—21no—
74C3×C4×C8D412yes—
75#830—21no—
76#821—22no—
77#821—22no—
78#821—22no—
79#821—22no—
80#818—22no—
81#818—22no—
82#817—22no—
83#817—22no—
84#816—22no—
85#816—22no—
86#816—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8—21no—
90#533—31no—
91#532—31no—
92C2×#125—31no—
93#501—61no—
94#501—61no—
95#491—61no—
96#491—61no—
97#490—61no—
98#490—61no—
99#488—62no—
100#484—61no—
101#483—61no—
102#482—62no—
103#479—62no—
104#476—61no—
105#474—61no—
106#432C2×D311yescharacteristic
107#432C2×D311yescharacteristic
108#431—31no—
109#417—61no—
110#417—61no—
111C4×M16C2×D311yescharacteristic
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—41no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121#235—44no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C3×M16—42no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158C2×#41—61no—
159C2×M16—21no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163#127—31no—
164#125—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#125—61no—
169#124—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—61no—
173#121—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—61no—
177C4×D4—61no—
178#117—61no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—62no—
186#106—62no—
187#106—62no—
188C4⋊C8—21no—
189C4⋊C8—61no—
190C4⋊C8D1212yes—
191#103—62no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C22⋊C8D1212yes—
202#97—21no—
203#97—21no—
204#97—21no—
205C4×C8—61no—
206C4×C8D1212yes—
207C23×C3#12111yescharacteristic
208C22×D3—41no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218#67—82no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226D12—42no—
227D12—42no—
228D12—42no—
229C4×D3—82no—
230Dic6—42no—
231Dic6—42no—
232Dic6—42no—
233C3×C8—42no—
234C3×C8—42no—
235C3×C8—42no—
236C3⋊C8—81no—
237#41—124no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—121no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—121no—
248C22×C4C2×D1211yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4#23911yescharacteristic
251M16—42no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—121no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42—121no—
276C42C2×D1211yescharacteristic
277C42C2×D1211yescharacteristic
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3#39211yescharacteristic
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4#45412yes—
293Dic3—81no—
294Dic3—81no—
295Dic3—81no—
296Dic3—81no—
297Dic3—81no—
298C23—121no—
299C23#89511yescharacteristic
300Q8—122no—
301Q8—122no—
302Q8—122no—
303D4—122no—
304D4—122no—
305D4—122no—
306D4—242no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—242no—
317C2×C4—241no—
318C2×C4#87411yescharacteristic
319C2×C4#88211yescharacteristic
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#89511yescharacteristic
323C2×C4#89511yescharacteristic
324C8—42no—
325C8—42no—
326C8—42no—
327C8—241no—
328C2×C3—41no—
329C2×C3#200311yescharacteristic
330C2×C3#200911yescharacteristic
331C2×C3#201711yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22#467111yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—241no—
346C4—241no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4#482712yes—
351C3#1012311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—481no—
354C2#6872711yescharacteristic
355C2#6876711yescharacteristic
356C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1291D3
hypocenterC3#10123
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

Complements

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

NH ≅ G/Ntype
1. #79320C2semidirect
4. #70831C2semidirect
6. #69856C2semidirect
7. C3×#1291C2semidirect
40. #1291D3semidirect
351. C3#10123semidirect

Maximal subgroups

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

subgroupindex
#793202
#793192
#785292
#708312
#708012
#698562
C3×#12912
#101233