Finite Group Databasebeta

← #10697920

Subgroups — #10697920

Order 768.

504 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 105 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#79272C211yesmaximal, characteristic
2#79271C211yesmaximal, characteristic
3#73308C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#70778C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic
7C3×#1276C211yesmaximal, characteristic
8#10296—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C43C2211yescharacteristic
15#4936—22no—
16#4929—22no—
17#4929—22no—
18#4924C2211yescharacteristic
19#4851—22no—
20#4850—22no—
21#4722—22no—
22#4721—22no—
23#4682—21no—
24#4681—21no—
25#4681—21no—
26#4680—21no—
27#4523—22no—
28#4486—22no—
29#4482—22no—
30#3230—31no—
31#3229—31no—
32#2862—31no—
33#1994—31no—
34#1549—31no—
35#1546—31no—
36#1276D311yescharacteristic
37C22×Dic6—21no—
38C22×Dic6—21no—
39C2×C3×C42—21no—
40C2×C3×C42—21no—
41C2×C3×C42D412yes—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3D412yes—
47C2×C4⋊Dic3D412yes—
48#924—21no—
49#924—21no—
50C2×#208—21no—
51#870—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8D412yes—
65C3×C4⋊C8D412yes—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70C3⋊#95—28no—
71#818—24no—
72#817—24no—
73#816—24no—
74#816—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8—31no—
80#500—61no—
81#498—61no—
82#487—61no—
83#487—61no—
84#484—62no—
85#483—62no—
86#477—62no—
87#475—62no—
88#431C2×D311yescharacteristic
89#431C2×D311yescharacteristic
90#423—31no—
91#391—62no—
92#391—62no—
93#382—62no—
94C43C2×D311yescharacteristic
95#339—62no—
96#332—62no—
97#332—62no—
98C22×C3×C4—22no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×Dic3—22no—
103C22×Dic3—22no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—44no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×Dic6—42no—
112C2×Dic6—42no—
113C2×C3×C8—22no—
114C2×C3×C8—22no—
115C3×C42—24no—
116C3×C42—22no—
117C3×C42—22no—
118C3×C42—42no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124C4⋊Dic3—22no—
125C4⋊Dic3—22no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—42no—
140C2×C3⋊C8—41no—
141C22×Q8—61no—
142C22×Q8—61no—
143C2×M16—61no—
144C4⋊Q8—31no—
145C4⋊Q8—31no—
146C4⋊Q8—61no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C4×Q8—61no—
158C4×Q8—61no—
159#117—61no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—32no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42D1212yes—
167C2×C42C22×D311yescharacteristic
168#107—64no—
169#106—64no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8D1212yes—
173C4⋊C8D1212yes—
174#103—64no—
175#103—64no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C4×C8—21no—
179C4×C8—21no—
180#95—68no—
181C23×C3#12011yescharacteristic
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—24no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12711yescharacteristic
193C2×C3×C4#13512yes—
194C2×C3×C4#13712yes—
195C2×Dic3—44no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—44no—
201Dic6—42no—
202Dic6—42no—
203Dic6—82no—
204Dic6—82no—
205Dic6—82no—
206Dic6—81no—
207Dic6—81no—
208C3×C8—42no—
209C3×C8—42no—
210C3⋊C8—81no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—124no—
214C2×Q8—121no—
215C2×Q8—121no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—122no—
219C2×Q8—122no—
220C22×C4—22no—
221C22×C4—62no—
222C22×C4—62no—
223C22×C4C2×D1211yescharacteristic
224C22×C4D3×D411yescharacteristic
225C22×C4D3×D411yescharacteristic
226M16—122no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—121no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C22⋊C4—121no—
245C42—24no—
246C42—22no—
247C42—22no—
248C42—42no—
249C42—121no—
250C42C2×D1211yescharacteristic
251C42C2×D1211yescharacteristic
252C42D3×D411yescharacteristic
253C42#23911yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3#39211yescharacteristic
257C22×C3#45212yes—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—44no—
261C3×C4#45412yes—
262C3×C4#45612yes—
263C3×C4#46412yes—
264C3×C4#49511yescharacteristic
265C3×C4#49711yescharacteristic
266Dic3—81no—
267Dic3—82no—
268Dic3—81no—
269Dic3—82no—
270C23#88211yescharacteristic
271Q8—122no—
272Q8—122no—
273Q8—242no—
274Q8—241no—
275Q8—241no—
276Q8—242no—
277Q8—242no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—124no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—124no—
289C2×C4#87411yescharacteristic
290C2×C4#88211yescharacteristic
291C2×C4#88211yescharacteristic
292C2×C4#89511yescharacteristic
293C2×C4#89511yescharacteristic
294C2×C4#89511yescharacteristic
295C2×C4#90412yes—
296C2×C4#90912yes—
297C8—42no—
298C8—42no—
299C8—241no—
300C2×C3—22no—
301C2×C3#200311yescharacteristic
302C2×C3#200511yescharacteristic
303C2×C3#201511yescharacteristic
304C22—21no—
305C22—21no—
306C22#467111yescharacteristic
307C22#473312yes—
308C4—21no—
309C4—21no—
310C4—44no—
311C4—242no—
312C4—242no—
313C4—241no—
314C4—241no—
315C4#469611yescharacteristic
316C4#471511yescharacteristic
317C4#482612yes—
318C4#482712yes—
319C4#484512yes—
320C3#1029611yescharacteristic, complemented, Sylow
321C2—22no—
322C2#6867711yescharacteristic
323C2#6876711yescharacteristic
324C2#6876811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1276D3
hypocenterC3#10296
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
320. C3#10296semidirect

Maximal subgroups

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

subgroupindex
#792722
#792712
#733082
#707802
#707782
#693962
C3×#12762
#102963