Finite Group Databasebeta

← #10695153

Subgroups — #10695153

Order 768.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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#83339C211yesmaximal, characteristic
2C3×#1493C211yesmaximal, characteristic
3#71826C211yesmaximal, characteristic
4#70518C211yesmaximal, characteristic
5#70501C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7C2×#4454C211yesmaximal, characteristic
8#9861—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#822—22no—
14#5080C411yescharacteristic
15#5080C411yescharacteristic
16#5078C411yescharacteristic
17#5078C411yescharacteristic
18#5065—22no—
19#4998—22no—
20#4924C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×Q8⋊C8—22no—
24#4535—22no—
25#4484—22no—
26#4456—22no—
27#4454—22no—
28#3160—31no—
29#1848—31no—
30#1493—31no—
31#1493—31no—
32#1493—31no—
33#1493D311yescharacteristic
34C2×#322—31no—
35C2×C3×M16—21no—
36C22×C3×C8—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C22×C3⋊C8D412yes—
52#870—21no—
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4×C8—21no—
60C3×C4×C8D412yes—
61#822—44no—
62#808—22no—
63#808—22no—
64#807—22no—
65#807—22no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4×C3⋊C8—21no—
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8D412yes—
72C4×C3⋊C8D412yes—
73#533—31no—
74#533C2×D311yescharacteristic
75#500—31no—
76#500—31no—
77#499—62no—
78#498—31no—
79#498—31no—
80C2×#108—62no—
81#427—62no—
82#423—31no—
83#423C2×D311yescharacteristic
84C2×C4×C8—31no—
85C2×C4×C8—31no—
86C2×C4×C8C2×D311yescharacteristic
87Q8⋊C8—22no—
88Q8⋊C8—62no—
89Q8⋊C8—62no—
90Q8⋊C8—62no—
91#322—62no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C2×Dic6—21no—
98C2×Dic6—21no—
99C3×M16—42no—
100C2×C3×C8—22no—
101C2×C3×C8—42no—
102C2×C3×C8—41no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C42—41no—
111C3×C42C22×C411yescharacteristic
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115#217—41no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122C4×Dic3—41no—
123#208—42no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—42no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8C22⋊C412yes—
128C2×C3⋊C8C22⋊C412yes—
129C2×C3⋊C8SD1614yes—
130C2×C3⋊C8Q1614yes—
131C2×M16—21no—
132C2×M16—61no—
133C22×C8—21no—
134C22×C8—32no—
135C22×C8—61no—
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8C4×D311yescharacteristic
139C4⋊Q8C4×D311yescharacteristic
140#125—21no—
141#125—61no—
142C22⋊Q8—21no—
143C22⋊Q8—21no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—61no—
149C4×Q8—61no—
150#117—21no—
151#117—61no—
152C2×C42C22×D311yescharacteristic
153#108—124no—
154#107—62no—
155#107—62no—
156#106—62no—
157#106—62no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4×C8—21no—
163C4×C8—31no—
164C4×C8—31no—
165C4×C8—32no—
166C4×C8—32no—
167C4×C8—61no—
168C4×C8#6712yes—
169C23×C3#12411yescharacteristic
170C3×Q8—42no—
171C3×Q8—42no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4C2×C22⋊C411yescharacteristic
177C2×C3×C4C4×D411yescharacteristic
178C2×C3×C4C4×D411yescharacteristic
179C2×C3×C4#12111yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12711yescharacteristic
182C2×C3×C4C2×SD1612yes—
183C2×C3×C4C2×Q1612yes—
184C2×Dic3—41no—
185C2×Dic3—41no—
186Dic6—42no—
187Dic6—42no—
188C3×C8—42no—
189C3×C8—81no—
190C3⋊C8—42no—
191C3⋊C8—81no—
192C3⋊C8#10314yes—
193C3⋊C8#10314yes—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—61no—
197C2×Q8—61no—
198C22×C4#23511yescharacteristic
199C22×C4D3×D411yescharacteristic
200C22×C4#23711yescharacteristic
201M16—42no—
202M16—122no—
203C2×C8—22no—
204C2×C8—32no—
205C2×C8—32no—
206C2×C8—34no—
207C2×C8—34no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—62no—
211C2×C8—122no—
212C2×C8—121no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C22⋊C4—41no—
226C22⋊C4—121no—
227C42—41no—
228C42—121no—
229C42C2×C4×D311yescharacteristic
230C42D3×D411yescharacteristic
231C42D3×D411yescharacteristic
232C42C2×#6711yescharacteristic
233C22×C3—21no—
234C22×C3—21no—
235C22×C3#39011yescharacteristic
236C22×C3#48712yes—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—81no—
240C3×C4—81no—
241C3×C4C2×#10312yes—
242C3×C4#44412yes—
243C3×C4#44412yes—
244C3×C4#49211yescharacteristic
245C3×C4#49411yescharacteristic
246Dic3—81no—
247Dic3—81no—
248C23#88511yescharacteristic
249Q8—42no—
250Q8—42no—
251Q8—122no—
252Q8—122no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—121no—
258C2×C4—121no—
259C2×C4D3×C22⋊C411yescharacteristic
260C2×C4#67⋊C411yescharacteristic
261C2×C4#88411yescharacteristic
262C2×C4D3×SD1612yes—
263C2×C4D3×Q1612yes—
264C2×C4C4×#6711yescharacteristic
265C2×C4#93911yescharacteristic
266C2×C4#94011yescharacteristic
267C8—34no—
268C8—34no—
269C8—42no—
270C8—81no—
271C8—122no—
272C8—241no—
273C2×C3—22no—
274C2×C3#195611yescharacteristic
275C2×C3#197511yescharacteristic
276C2×C3#197511yescharacteristic
277C22—21no—
278C22—21no—
279C22#479212yes—
280C22#495711yescharacteristic
281C4—21no—
282C4—21no—
283C4—81no—
284C4—81no—
285C4—241no—
286C4—241no—
287C4D3×#10312yes—
288C4#482312yes—
289C4#514512yes—
290C4#517511yescharacteristic
291C4#518311yescharacteristic
292C3#986111yescharacteristic, complemented, Sylow
293C2—22no—
294C2#7000011yescharacteristic
295C2#7022511yescharacteristic
296C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenter#1493D3
hypocenterC3#9861
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
292. C3#9861semidirect

Maximal subgroups

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

subgroupindex
#833392
C3×#14932
#718262
#705182
#705012
#697382
C2×#44542
#98613