Finite Group Databasebeta

← #10697464

Subgroups — #10697464

Order 768.

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

normal subgroups only

This group has 462 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#79313C211yesmaximal, characteristic
2#79272C211yesmaximal, characteristic
3#78479C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic
5#70793C211yesmaximal, characteristic
6#69828C211yesmaximal, characteristic
7C3×#1281C211yesmaximal, characteristic
8#10224—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15#5106—22no—
16#4936C2211yescharacteristic
17#4850—22no—
18#4849—22no—
19#4739—22no—
20#4722—22no—
21#4702—21no—
22#4702—21no—
23#4702—21no—
24#4702—21no—
25#4699—21no—
26#4682—21no—
27#4681—21no—
28#4681—21no—
29#4551—22no—
30#4467—22no—
31#4467—22no—
32#3230—31no—
33#3209—31no—
34#3162—31no—
35#1993—31no—
36#1518—31no—
37#1518—31no—
38#1281D311yescharacteristic
39C22×Dic6—21no—
40C22×C3×C8D412yes—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43#926—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47#924—21no—
48#924—21no—
49#924—21no—
50#924—21no—
51C2×Dic12—24no—
52#870—21no—
53#870—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869D411yescharacteristic
58#869D411yescharacteristic
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D412yes—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8D412yes—
69C3×C22⋊C8D412yes—
70C3×#97—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76#818—24no—
77#818—22no—
78#818—22no—
79#817—22no—
80#817—22no—
81#816—24no—
82#816—24no—
83#533—31no—
84#533—31no—
85C2×C4⋊Q8—31no—
86#501—61no—
87#501—61no—
88#501—61no—
89#501—61no—
90#500—61no—
91#494—61no—
92#487—61no—
93#487—61no—
94#484—62no—
95#477—62no—
96#462—62no—
97#451—62no—
98#431C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100#427—62no—
101C2×#97C2×D311yescharacteristic
102#382—31no—
103#340—62no—
104#328—62no—
105#328—62no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—41no—
110C22×Dic3—41no—
111C2×Dic6—21no—
112C2×Dic6—22no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C2×C3×C8Q1614yescomplemented
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127#217—41no—
128#217—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—42no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148Dic12—84no—
149C22×Q8—61no—
150C2×Q16—64no—
151C22×C8D1212yes—
152C4⋊Q8—32no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160#125—61no—
161#125—61no—
162C22⋊Q8—61no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—61no—
171#117—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#107—62no—
178#107—64no—
179#106—62no—
180#106—62no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183#103—64no—
184#103—64no—
185C22⋊C8D1212yes—
186C22⋊C8D1212yes—
187#97—21no—
188#97—21no—
189C4×C8—21no—
190C4×C8—21no—
191#95—61no—
192#95—61no—
193C23×C3#12711yescharacteristic
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4C2×Q1612yes—
203C2×C3×C4#13712yes—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—82no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213Dic6—42no—
214Dic6—44no—
215Dic6—42no—
216Dic6—82no—
217Dic6—81no—
218Dic6—81no—
219C3×C8—24no—
220C3×C8—42no—
221C3×C8—42no—
222C2×Q8—61no—
223C2×Q8—62no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8—121no—
228C2×Q8—242no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4C2×D1211yescharacteristic
232C22×C4C2×D1211yescharacteristic
233C22×C4C2×D1211yescharacteristic
234Q16—244no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8Dic1214yescomplemented
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—62no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—122no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—121no—
260C42—121no—
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C42D3×D411yescharacteristic
264C42#23911yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39211yescharacteristic
268C22×C3#49411yescharacteristic
269C22×C3#49711yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4#45112yes—
275C3×C4#46212yes—
276Dic3—82no—
277Dic3—81no—
278Dic3—81no—
279Dic3—81no—
280Dic3—81no—
281Dic3—161no—
282C23#87411yescharacteristic
283Q8—122no—
284Q8—124no—
285Q8—122no—
286Q8—241no—
287Q8—241no—
288Q8—242no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—122no—
296C2×C4—242no—
297C2×C4—241no—
298C2×C4—241no—
299C2×C4—241no—
300C2×C4#88211yescharacteristic
301C2×C4#88211yescharacteristic
302C2×C4#88211yescharacteristic
303C2×C4#89511yescharacteristic
304C2×C4#89511yescharacteristic
305C2×C4#89511yescharacteristic
306C2×C4C2×Dic1212yes—
307C2×C4#90912yes—
308C8—24no—
309C8—42no—
310C8—42no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#200811yescharacteristic
314C2×C3#200911yescharacteristic
315C2×C3#201611yescharacteristic
316C22—21no—
317C22—21no—
318C22#467111yescharacteristic
319C22#469911yescharacteristic
320C22#471511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—242no—
326C4—241no—
327C4—241no—
328C4—241no—
329C4—241no—
330C4—481no—
331C4#473912yes—
332C4#484912yes—
333C3#1022411yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#6868811yescharacteristic
337C2#6902811yescharacteristic
338C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1281D3
hypocenterC3#10224
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
122. C2×C3×C8Q16semidirect
239. C2×C8Dic12semidirect
333. C3#10224semidirect

Maximal subgroups

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

subgroupindex
#793132
#792722
#784792
#707932
#698282
C3×#12812
#102243