Finite Group Databasebeta

← #10691840

Subgroups — #10691840

Order 768.

410 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 109 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#83224C211yesmaximal, characteristic
2#78480C211yesmaximal, characteristic
3C3×#1993C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic
6#67660C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9288—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5102—21no—
14#5101—21no—
15#5093C412yescomplemented
16#5091—22no—
17#5078C412yescomplemented
18#4970—22no—
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4701—22no—
23#4701—22no—
24C3×#340—21no—
25C3×#340—21no—
26#4546—22no—
27#4452—22no—
28#3230—31no—
29#2961—31no—
30#1993D311yescharacteristic
31#1548—31no—
32#1518—31no—
33#1286—31no—
34#1275—31no—
35C22×C3×Q8—21no—
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8C2×C412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924D412yes—
49C2×#208—21no—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×#97—21no—
56C3×#97—21no—
57C3×#95—21no—
58C3×#95—21no—
59#830—21no—
60#830—21no—
61#815—22no—
62Dic3⋊C8—24no—
63C8×Dic3—22no—
64#810—24no—
65#807—24no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)D412yes—
70C4×C3⋊C8—21no—
71C4×C3⋊C8C2×C412yes—
72#533—31no—
73C2×C4⋊Q8C2×D311yescharacteristic
74#500—32no—
75#487—32no—
76#484—62no—
77#477—62no—
78#446—62no—
79#446—62no—
80#431—31no—
81#431—31no—
82#424—61no—
83#423—31no—
84#405—61no—
85C2×#97C2×D311yescharacteristic
86#382C2×D311yescharacteristic
87#340—21no—
88#340—21no—
89#333—62no—
90#328—62no—
91C2×C3×Q8—22no—
92C2×C3×Q8—41no—
93C2×C3×Q8—82no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C2×Dic6—21no—
100C2×Dic6C22⋊C412yes—
101C2×C3×C8—21no—
102C2×C3×C8—21no—
103C2×C3×C8—21no—
104C2×C3×C8—21no—
105C2×C3×C8—21no—
106C2×C3×C8Q1612yes—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C42C22×C411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117#217—22no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C22⋊C412yes—
122Dic3⋊C4—22no—
123Dic3⋊C4—22no—
124C4×Dic3—22no—
125#208—41no—
126#208—41no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—41no—
130C22×Q8—21no—
131C2×M16—61no—
132C22×C8#6711yescharacteristic
133C22×C8#6711yescharacteristic
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8C4×D312yes—
140#125—61no—
141C22⋊Q8—32no—
142C22⋊Q8—61no—
143C4×Q8—32no—
144C4×Q8—61no—
145#117—61no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C42C22×D311yescharacteristic
150#107—64no—
151C4⋊C8—32no—
152C4⋊C8—64no—
153C4⋊C8—61no—
154C4⋊C8—61no—
155C4⋊C8—61no—
156#103—64no—
157C22⋊C8—61no—
158C22⋊C8—61no—
159#97—21no—
160#97—21no—
161#97—62no—
162C4×C8—32no—
163C4×C8—62no—
164C4×C8—61no—
165#95—21no—
166#95—21no—
167C23×C3#12111yescharacteristic
168C3×Q8—82no—
169C3×Q8—82no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—42no—
173C2×C3×C4—82no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4C2×C22⋊C411yescharacteristic
178C2×C3×C4C4×D411yescharacteristic
179C2×C3×C4C4×D411yescharacteristic
180C2×C3×C4#12011yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12311yescharacteristic
183C2×C3×C4C2×Q1611yescharacteristic
184C2×C3×C4#13511yescharacteristic
185C2×C3×C4#13711yescharacteristic
186C2×C3×C4#13711yescharacteristic
187C2×Dic3—22no—
188C2×Dic3—22no—
189Dic6—42no—
190Dic6—81no—
191C3×C8—22no—
192C3×C8—41no—
193C3⋊C8—42no—
194C3⋊C8—42no—
195C3⋊C8—81no—
196C2×Q8—22no—
197C2×Q8—32no—
198C2×Q8—41no—
199C2×Q8—61no—
200C2×Q8—82no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4D3×D411yescharacteristic
204C22×C4#23711yescharacteristic
205C22×C4C2×#6711yescharacteristic
206M16—121no—
207M16—121no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—121no—
216C2×C8#21612yes—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—32no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—61no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C22⋊C4—62no—
230C42—62no—
231C42C2×C4×D311yescharacteristic
232C42#23511yescharacteristic
233C42D3×D411yescharacteristic
234C42D3×D411yescharacteristic
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#38611yescharacteristic
238C22×C3#46211yescharacteristic
239C22×C3#46411yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—82no—
245C3×C4—161no—
246C3×C4C4×Q1611yescharacteristic
247C3×C4#44111yescharacteristic
248C3×C4#45111yescharacteristic
249C3×C4#45211yescharacteristic
250Dic3—42no—
251Dic3—42no—
252C23#93811yescharacteristic
253Q8—82no—
254Q8—82no—
255Q8—122no—
256Q8—241no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4D3×C22⋊C411yescharacteristic
267C2×C4#67⋊C411yescharacteristic
268C2×C4#88311yescharacteristic
269C2×C4#88411yescharacteristic
270C2×C4C4×#6711yescharacteristic
271C2×C4#93711yescharacteristic
272C2×C4#94211yescharacteristic
273C2×C4C2×#21611yescharacteristic
274C2×C4#95011yescharacteristic
275C2×C4#95111yescharacteristic
276C8—22no—
277C8—41no—
278C8—122no—
279C8—122no—
280C8—241no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3#187811yescharacteristic
284C2×C3#190311yescharacteristic
285C2×C3#190911yescharacteristic
286C22—21no—
287C22—21no—
288C22#495611yescharacteristic
289C22#505411yescharacteristic
290C22#509011yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—82no—
296C4—122no—
297C4—122no—
298C4—161no—
299C4C4×#21611yescharacteristic
300C4#503011yescharacteristic
301C4#504711yescharacteristic
302C4#505011yescharacteristic
303C3#928811yescharacteristic, complemented, Sylow
304C2—21no—
305C2—21no—
306C2#6969011yescharacteristic
307C2#7032911yescharacteristic
308C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1993D3
hypocenterC3#9288
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
15. #5093C4semidirect
17. #5078C4semidirect
303. C3#9288semidirect

Maximal subgroups

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

subgroupindex
#832242
#784802
C3×#19932
#717742
#695462
#676602
#676312
#92883