Finite Group Databasebeta

← #10694798

Subgroups — #10694798

Order 768.

474 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#5092C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#70507C211yesmaximal, characteristic
5#70491C211yesmaximal, characteristic
6#69879C211yesmaximal, characteristic
7M16×C3⋊C8C211yesmaximal, characteristic
8#9798—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216C414yes—
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5092—21no—
15#5092—21no—
16#5092—22no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20C3×#343—22no—
21#4554—22no—
22#4546—22no—
23#4481—22no—
24#4479—22no—
25#4455—21no—
26C8×C3⋊C8—21no—
27C2×#494—31no—
28#1997—31no—
29#1997D311yescharacteristic
30#1567—31no—
31#1548—31no—
32#1548—31no—
33C8×M16—31no—
34C22×C3×Q8C2×C412yes—
35C22×Dic6C2×C412yes—
36C2×C3×M16—21no—
37C2×C3×M16D412yes—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42C2311yescharacteristic
45C2×#216—24no—
46C2×#216—24no—
47C2×#216—48no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×#208—21no—
51C22×C3⋊C8C2×C412yes—
52#869—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×#97—21no—
59C3×C4×C8—21no—
60#830—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)D412yes—
64C4×C3⋊C8—21no—
65C4×C3⋊C8D411yescharacteristic
66C4×C3⋊C8D411yescharacteristic
67C4×C3⋊C8D412yes—
68C22×Q16—34no—
69C2×C4⋊Q8—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#494—61no—
72#494—61no—
73#494—62no—
74#431—31no—
75#423—31no—
76#423C2×D311yescharacteristic
77C4×M16C2×D311yescharacteristic
78C2×C4×C8—31no—
79#343—22no—
80#343—62no—
81#335—62no—
82#333—62no—
83#333—62no—
84#322—61no—
85C82—61no—
86C2×C3×Q8—22no—
87C2×C3×Q8—22no—
88C2×C3×Q8—44no—
89C22×C3×C4—22no—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4#4111yescharacteristic
93C22×Dic3—22no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—44no—
97C3×M16—22no—
98C3×M16—42no—
99C3×M16Q1614yescomplemented
100C2×C3×C8—22no—
101C2×C3×C8—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C42C2×D411yescharacteristic
107C3×C42C2×D411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42#4111yescharacteristic
110#216—48no—
111#216—48no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115C4⋊Dic3—41no—
116#208—42no—
117C2×C3⋊C8—24no—
118C2×C3⋊C8—22no—
119C2×C3⋊C8—41no—
120C2×C3⋊C8C22⋊C412yes—
121C2×C3⋊C8C22⋊C412yes—
122C22×Q8—32no—
123C22×Q8C4×D312yes—
124C2×Q16—64no—
125C2×Q16—64no—
126C2×Q16—128no—
127C2×M16—21no—
128C2×M16—61no—
129C2×M16#6712yes—
130C22×C8—32no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—61no—
136C4⋊Q8—61no—
137C4⋊Q8—61no—
138C4⋊Q8—61no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C42C22×D311yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—32no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149C22⋊C8—61no—
150#97—21no—
151C4×C8—21no—
152C4×C8—31no—
153C4×C8—31no—
154C4×C8—32no—
155C4×C8—61no—
156C23×C3C4×D411yescharacteristic
157C3×Q8—44no—
158C3×Q8—82no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—42no—
162C2×C3×C4—82no—
163C2×C3×C4C2×C22⋊C411yescharacteristic
164C2×C3×C4C4×D411yescharacteristic
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12111yescharacteristic
167C2×C3×C4#12411yescharacteristic
168C2×C3×C4#12711yescharacteristic
169C2×C3×C4C2×Q1612yes—
170C2×C3×C4#13512yes—
171C2×Dic3—42no—
172C2×Dic3—82no—
173Dic6—44no—
174Dic6—82no—
175C3×C8—24no—
176C3×C8—81no—
177C3⋊C8—22no—
178C3⋊C8—42no—
179C3⋊C8—81no—
180C3⋊C8#10014yes—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—44no—
184C2×Q8—62no—
185C2×Q8—62no—
186C2×Q8—124no—
187C22×C4—22no—
188C22×C4—62no—
189C22×C4C2×C4×D311yescharacteristic
190C22×C4#23511yescharacteristic
191C22×C4C2×#6711yescharacteristic
192Q16—128no—
193Q16—128no—
194M16—22no—
195M16—42no—
196M16—122no—
197M16#21614yescomplemented
198C2×C8—22no—
199C2×C8—32no—
200C2×C8—32no—
201C2×C8—41no—
202C2×C8—64no—
203C2×C8—62no—
204C2×C8—121no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—121no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C42D3×D411yescharacteristic
214C42D3×D411yescharacteristic
215C42D3×D411yescharacteristic
216C42#23711yescharacteristic
217C22×C3—21no—
218C22×C3—21no—
219C22×C3#39011yescharacteristic
220C22×C3C4×Q1612yes—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—82no—
224C3×C4C2×#10012yes—
225C3×C4#46712yes—
226C3×C4#47312yes—
227C3×C4#49411yescharacteristic
228C3×C4#49511yescharacteristic
229Dic3—82no—
230C23C4×#6711yescharacteristic
231Q8—44no—
232Q8—82no—
233Q8—124no—
234Q8—242no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—122no—
240C2×C4—242no—
241C2×C4D3×C22⋊C411yescharacteristic
242C2×C4#67⋊C411yescharacteristic
243C2×C4#88411yescharacteristic
244C2×C4#88511yescharacteristic
245C2×C4#93911yescharacteristic
246C2×C4#94011yescharacteristic
247C2×C4C2×#21612yes—
248C2×C4#95012yes—
249C8—24no—
250C8—34no—
251C8—62no—
252C8—81no—
253C8—122no—
254C8—241no—
255C2×C3—22no—
256C2×C3#196511yescharacteristic
257C2×C3#196911yescharacteristic
258C2×C3#197911yescharacteristic
259C22—21no—
260C22—21no—
261C22#495711yescharacteristic
262C22C4×#21612yes—
263C4—21no—
264C4—21no—
265C4—82no—
266C4—242no—
267C4D3×#10012yes—
268C4#475512yes—
269C4#505112yes—
270C4#505911yescharacteristic
271C4#509211yescharacteristic
272C3#979811yescharacteristic, complemented, Sylow
273C2—22no—
274C2#6967711yescharacteristic
275C2#7006411yescharacteristic
276C2#7010811yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1997D3
hypocenterC3#9798
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
99. C3×M16Q16semidirect
197. M16#216semidirect
272. C3#9798semidirect

Maximal subgroups

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

subgroupindex
C2×#50922
C3×#19972
#717742
#705072
#704912
#698792
M16×C3⋊C82
#97983