Finite Group Databasebeta

← #10692671

Subgroups — #10692671

Order 768.

380 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#3185C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic
3#71837C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic
5#71823C211yesmaximal, characteristic
6#67670C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic
8#9416—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C3×#498C411yescharacteristic, complemented
12C3×#494C411yescharacteristic, complemented
13C3×#487C411yescharacteristic, complemented
14C3×#487C411yescharacteristic, complemented
15C3×#470—22no—
16C3×#441—22no—
17C3×C4×M16C2211yescharacteristic
18#5024—22no—
19#5024—22no—
20#5000—22no—
21#4937C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25#4543—22no—
26#4538—22no—
27#4538—22no—
28#3185D311yescharacteristic
29#2987—31no—
30#1537—31no—
31#1515—31no—
32#1515—31no—
33#1291—31no—
34#1288—31no—
35C2×C3×Q16—22no—
36C2×C3×M16—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8D412yes—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8D412yes—
51C3×#117—21no—
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54C3×#106—22no—
55C3×#103—22no—
56C3×#103—22no—
57C3×#97—21no—
58C3×C4×C8C2×C411yescharacteristic
59C3×C4×C8C2×C411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63#830—21no—
64#830—21no—
65#830D412yes—
66C4⋊(C3⋊C8)—24no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70#803—22no—
71#803—21no—
72#803—21no—
73#803D412yes—
74C4×C3⋊C8—22no—
75#533C2×D311yescharacteristic
76#533C2×D311yescharacteristic
77#498Dic311yescharacteristic, complemented
78#494Dic311yescharacteristic, complemented
79#487Dic311yescharacteristic, complemented
80#487Dic311yescharacteristic, complemented
81#470—22no—
82#446—62no—
83#446—62no—
84#441—22no—
85#433—62no—
86#432—31no—
87#432—31no—
88#432—31no—
89C4×M16C2×D311yescharacteristic
90#383—31no—
91#330—62no—
92#328—62no—
93#328—62no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8C22⋊C412yes—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—41no—
102C3×Q16—44no—
103C3×M16—42no—
104C2×C3×C8—21no—
105C2×C3×C8—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4C22⋊C412yes—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—41no—
120C3×C42—22no—
121C3×C42—41no—
122C3×C42C22×C411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C2×C3⋊C8—22no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—41no—
129C2×Q16—22no—
130C2×M16—21no—
131C4⋊Q8C2×Dic311yescharacteristic
132C4⋊Q8C2×Dic311yescharacteristic
133C4⋊Q8C2×Dic311yescharacteristic
134C4⋊Q8C2×Dic311yescharacteristic
135#125—21no—
136#125—21no—
137C22⋊Q8—21no—
138C22⋊Q8—21no—
139C22⋊Q8—21no—
140C22⋊Q8#6712yes—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8#6712yes—
145#117—21no—
146#117—21no—
147C2×C42C22×D311yescharacteristic
148#106—22no—
149C4⋊C8—64no—
150C4⋊C8—61no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153#103—22no—
154#103—22no—
155C22⋊C8—32no—
156C22⋊C8—61no—
157C22⋊C8—61no—
158#97—21no—
159#97—32no—
160#97—62no—
161#97—61no—
162#97—61no—
163C4×C8—62no—
164C4×C8C2×Dic311yescharacteristic
165C4×C8C2×Dic311yescharacteristic
166#95—61no—
167#95—61no—
168#95—61no—
169C23×C3#12111yescharacteristic
170C3×Q8—42no—
171C3×Q8—42no—
172C3×Q8—81no—
173C3×Q8—81no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4C2×C22⋊C411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12311yescharacteristic
187C2×Dic3—81no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190C3×C8—22no—
191C3×C8—41no—
192C3⋊C8—42no—
193C3⋊C8—42no—
194C3⋊C8—81no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8#21712yes—
199C22×C4—121no—
200C22×C4D3×D411yescharacteristic
201C22×C4#23711yescharacteristic
202C22×C4C2×#6711yescharacteristic
203Q16—44no—
204M16—42no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—121no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4#21712yes—
222C22⋊C4—22no—
223C22⋊C4—41no—
224C42—22no—
225C42—41no—
226C42D3×D411yescharacteristic
227C42#23711yescharacteristic
228C42C22×Dic311yescharacteristic
229C42C2×#6711yescharacteristic
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3#38611yescharacteristic
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—42no—
241C3×C4—42no—
242C3×C4—81no—
243C3×C4—81no—
244Dic3—161no—
245C23#93911yescharacteristic
246Q8—42no—
247Q8—42no—
248Q8—81no—
249Q8—81no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—241no—
258C2×C4—241no—
259C2×C4—241no—
260C2×C4D4×Dic311yescharacteristic
261C2×C4D4×Dic311yescharacteristic
262C2×C4#93511yescharacteristic
263C2×C4#93711yescharacteristic
264C2×C4#93911yescharacteristic
265C2×C4C2×#21711yescharacteristic
266C8—22no—
267C8—41no—
268C8—122no—
269C8—122no—
270C8—241no—
271C2×C3—41no—
272C2×C3#188411yescharacteristic
273C2×C3#190211yescharacteristic
274C2×C3#190511yescharacteristic
275C22—21no—
276C22—21no—
277C22—21no—
278C22#522011yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—481no—
290C3#941611yescharacteristic, complemented, Sylow
291C2—41no—
292C2#7181711yescharacteristic
293C2#7210111yescharacteristic
294C2#7210211yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3185C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#3185D3
hypocenterC3#9416
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
11. C3×#498C4semidirect
12. C3×#494C4semidirect
13. C3×#487C4semidirect
14. C3×#487C4semidirect
77. #498Dic3semidirect
78. #494Dic3semidirect
79. #487Dic3semidirect
80. #487Dic3semidirect
290. C3#9416semidirect

Maximal subgroups

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

subgroupindex
C3×#31852
#833332
#718372
#718232
#676702
#676662
#94163