Finite Group Databasebeta

← #10695084

Subgroups — #10695084

Order 768.

422 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 121 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#83344C211yesmaximal, characteristic
2C3×#1994C211yesmaximal, characteristic
3#71826C211yesmaximal, characteristic
4#70501C211yesmaximal, characteristic
5#70491C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic
7#69388C211yesmaximal, characteristic
8#9848—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C43C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5092C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5051—22no—
18C4×#216—22no—
19#4924C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C3×#339—22no—
23#4535—22no—
24#4524—22no—
25#4523—22no—
26#4484—22no—
27#4479—22no—
28#3170—31no—
29#1994—31no—
30#1994D311yescharacteristic
31#1744—31no—
32#1567—31no—
33#1493—31no—
34#1493—31no—
35C2×C3×M16—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C42—21no—
45C2×C3×C42—21no—
46C2×C3×C42D412yes—
47C2×C3×C42C2311yescharacteristic
48C2×#216—22no—
49#926—21no—
50#924—21no—
51#924—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8D412yes—
55#870—21no—
56#869C2×C411yescharacteristic
57#869C2×C411yescharacteristic
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62#835—22no—
63#810—22no—
64#807—22no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4×C3⋊C8—21no—
68C4×C3⋊C8C2×C411yescharacteristic
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8D412yes—
71C4×C3⋊C8D412yes—
72#533—31no—
73#533C2×D311yescharacteristic
74#500—31no—
75#494—31no—
76#487—31no—
77#487—31no—
78#467—62no—
79C4×Q16—62no—
80#423—31no—
81#423C2×D311yescharacteristic
82C2×C4×C8—31no—
83C2×C4×C8—31no—
84C43C2×D311yescharacteristic
85#339—22no—
86#339—62no—
87#336—62no—
88#335—62no—
89Q8⋊C8—62no—
90Q8⋊C8—62no—
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C22×C3×C4—22no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C2×Dic6—21no—
98C2×Dic6—21no—
99C3×M16—42no—
100C2×C3×C8—41no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—41no—
108C3×C42—22no—
109C3×C42—42no—
110C3×C42—42no—
111C3×C42—41no—
112C3×C42Q1614yescomplemented
113C3×C42C22×C411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117#217—41no—
118#216—44no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—41no—
126#208—42no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—42no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8C22⋊C412yes—
132C2×C3⋊C8C22⋊C412yes—
133C2×Q16—62no—
134C2×M16—21no—
135C2×M16—61no—
136C22×C8—32no—
137C22×C8—61no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8C4×D311yescharacteristic
141C4⋊Q8C4×D311yescharacteristic
142#125—21no—
143#125—61no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—61no—
151C4×Q8—61no—
152#117—21no—
153#117—61no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42#6712yes—
157C2×C42C22×D311yescharacteristic
158#107—62no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#103—62no—
164#103—62no—
165C4×C8—31no—
166C4×C8—31no—
167C4×C8—32no—
168C4×C8—32no—
169C4×C8—61no—
170C23×C3#12111yescharacteristic
171C3×Q8—42no—
172C3×Q8—42no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4C2×C22⋊C411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4#12111yescharacteristic
184C2×C3×C4#12411yescharacteristic
185C2×C3×C4#12711yescharacteristic
186C2×C3×C4C2×Q1612yes—
187C2×C3×C4#13512yes—
188C2×Dic3—41no—
189C2×Dic3—41no—
190Dic6—42no—
191Dic6—42no—
192C3×C8—81no—
193C3⋊C8—22no—
194C3⋊C8—22no—
195C3⋊C8—42no—
196C3⋊C8—81no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—61no—
200C2×Q8—61no—
201C22×C4—22no—
202C22×C4D3×D411yescharacteristic
203C22×C4#23711yescharacteristic
204C22×C4C2×#6711yescharacteristic
205Q16—124no—
206M16—42no—
207M16—122no—
208C2×C8—32no—
209C2×C8—32no—
210C2×C8—41no—
211C2×C8—62no—
212C2×C8—122no—
213C2×C8—122no—
214C2×C8—121no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C22⋊C4—41no—
228C22⋊C4—121no—
229C42—22no—
230C42—42no—
231C42—42no—
232C42—41no—
233C42—121no—
234C42#21614yescomplemented
235C42C2×C4×D311yescharacteristic
236C42#23511yescharacteristic
237C42D3×D411yescharacteristic
238C42D3×D411yescharacteristic
239C22×C3—21no—
240C22×C3—21no—
241C22×C3#39011yescharacteristic
242C22×C3#46712yes—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—44no—
246C3×C4—81no—
247C3×C4—81no—
248C3×C4#42112yes—
249C3×C4C4×Q1612yes—
250C3×C4#44412yes—
251C3×C4#49411yescharacteristic
252C3×C4#49511yescharacteristic
253Dic3—81no—
254Dic3—81no—
255C23#93911yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—122no—
259Q8—122no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4D3×C22⋊C411yescharacteristic
270C2×C4#67⋊C411yescharacteristic
271C2×C4#88411yescharacteristic
272C2×C4#88511yescharacteristic
273C2×C4C4×#6711yescharacteristic
274C2×C4#94011yescharacteristic
275C2×C4C2×#21612yes—
276C2×C4#95012yes—
277C8—62no—
278C8—62no—
279C8—81no—
280C8—122no—
281C8—241no—
282C2×C3—22no—
283C2×C3#196311yescharacteristic
284C2×C3#196911yescharacteristic
285C2×C3#197511yescharacteristic
286C22—21no—
287C22—21no—
288C22#495711yescharacteristic
289C22#505112yes—
290C4—21no—
291C4—21no—
292C4—44no—
293C4—81no—
294C4—81no—
295C4—241no—
296C4—241no—
297C4#482112yes—
298C4#482312yes—
299C4C4×#21612yes—
300C4#505911yescharacteristic
301C4#509211yescharacteristic
302C3#984811yescharacteristic, complemented, Sylow
303C2—22no—
304C2#6967711yescharacteristic
305C2#7035711yescharacteristic
306C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1994D3
hypocenterC3#9848
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
15. #5079C4semidirect
16. #5078C4semidirect
112. C3×C42Q16semidirect
234. C42#216semidirect
302. C3#9848semidirect

Maximal subgroups

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

subgroupindex
#833442
C3×#19942
#718262
#705012
#704912
#693962
#693882
#98483