Finite Group Databasebeta

← #10694825

Subgroups — #10694825

Order 768.

402 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 113 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×#5093C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic
3#71775C211yesmaximal, characteristic
4#70507C211yesmaximal, characteristic
5#70490C211yesmaximal, characteristic
6#69879C211yesmaximal, characteristic
7#69847C211yesmaximal, characteristic
8#9803—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5093—22no—
14#5093—22no—
15C2×#810C414yes—
16#4925C2211yescharacteristic
17#4924C2211yescharacteristic
18C2×C4×C3⋊C8C2211yescharacteristic
19C3×#343—22no—
20#4554—22no—
21#4547—22no—
22#4481—22no—
23#4480—22no—
24#4454—22no—
25C2×#487—31no—
26#1997—31no—
27#1997D311yescharacteristic
28#1566—31no—
29#1549—31no—
30#1548—31no—
31#1451—31no—
32C22×C3×Q8—21no—
33C22×Dic6C2×C412yes—
34C2×C3×M16—21no—
35C2×C3×M16D412yes—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C42C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×#208—21no—
46C22×C3⋊C8C2×C412yes—
47#869—21no—
48#869—21no—
49#869—21no—
50#869—21no—
51C3×C4⋊C8—21no—
52C3×C4⋊C8—21no—
53C3×#97—21no—
54C3×C4×C8—21no—
55#830—21no—
56#810—28no—
57C4⋊(C3⋊C8)—21no—
58C4⋊(C3⋊C8)—21no—
59C4⋊(C3⋊C8)D412yes—
60C4×C3⋊C8—21no—
61C4×C3⋊C8D412yes—
62C4×C3⋊C8D412yes—
63C2×C4⋊Q8—31no—
64C2×C4⋊Q8C2×D311yescharacteristic
65#487—62no—
66#487—62no—
67#431—31no—
68#423—31no—
69#423C2×D311yescharacteristic
70C2×#103—34no—
71C4×M16C2×D311yescharacteristic
72C2×C4×C8—31no—
73#343—22no—
74#343—62no—
75#334—62no—
76#333—62no—
77#332—62no—
78#322—62no—
79C2×C3×Q8—41no—
80C2×C3×Q8—41no—
81C2×C3×Q8—82no—
82C22×C3×C4—22no—
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4111yescharacteristic
86C22×Dic3—22no—
87C2×Dic6—22no—
88C2×Dic6—22no—
89C2×Dic6—44no—
90C3×M16—22no—
91C3×M16—22no—
92C3×M16—42no—
93C2×C3×C8—22no—
94C2×C3×C8—41no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—41no—
99C3×C42C2×D411yescharacteristic
100C3×C42C2×D411yescharacteristic
101C3×C42C2×D411yescharacteristic
102C3×C42#4111yescharacteristic
103C4⋊Dic3—41no—
104C4⋊Dic3—41no—
105C4⋊Dic3—41no—
106C4⋊Dic3—41no—
107#208—42no—
108C2×C3⋊C8—22no—
109C2×C3⋊C8—42no—
110C2×C3⋊C8—41no—
111C2×C3⋊C8C22⋊C414yes—
112C22×Q8—21no—
113C22×Q8—32no—
114C2×M16—21no—
115C2×M16—61no—
116C2×M16#6712yes—
117C22×C8—32no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—61no—
123C4⋊Q8—61no—
124C4⋊Q8—61no—
125C4⋊Q8—61no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4C4×D312yes—
130C2×C42C22×D311yescharacteristic
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—32no—
134C4⋊C8—61no—
135C4⋊C8—61no—
136#103—68no—
137C22⋊C8—61no—
138#97—21no—
139C4×C8—21no—
140C4×C8—32no—
141C4×C8—32no—
142C4×C8—61no—
143C23×C3C4×D411yescharacteristic
144C3×Q8—82no—
145C3×Q8—82no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—82no—
150C2×C3×C4C2×C22⋊C411yescharacteristic
151C2×C3×C4C4×D411yescharacteristic
152C2×C3×C4#12111yescharacteristic
153C2×C3×C4#12111yescharacteristic
154C2×C3×C4#12411yescharacteristic
155C2×C3×C4#12711yescharacteristic
156C2×C3×C4#13712yes—
157C2×C3×C4#13712yes—
158C2×Dic3—42no—
159C2×Dic3—82no—
160Dic6—44no—
161Dic6—82no—
162C3×C8—42no—
163C3×C8—81no—
164C3⋊C8—24no—
165C3⋊C8—42no—
166C3⋊C8—81no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—62no—
170C2×Q8—62no—
171C2×Q8—82no—
172C2×Q8—124no—
173C22×C4—22no—
174C22×C4—62no—
175C22×C4C2×C4×D311yescharacteristic
176C22×C4#23511yescharacteristic
177C22×C4C2×#6711yescharacteristic
178M16—22no—
179M16—22no—
180M16—42no—
181M16—122no—
182C2×C8—22no—
183C2×C8—34no—
184C2×C8—41no—
185C2×C8—62no—
186C2×C8—122no—
187C2×C8—121no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—121no—
193C4⋊C4—121no—
194C4⋊C4—121no—
195C4⋊C4—121no—
196C42D3×D411yescharacteristic
197C42D3×D411yescharacteristic
198C42D3×D411yescharacteristic
199C42#23711yescharacteristic
200C22×C3—21no—
201C22×C3—21no—
202C22×C3#39011yescharacteristic
203C22×C3#44112yes—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—82no—
207C3×C4#41312yes—
208C3×C4#47012yes—
209C3×C4#47112yes—
210C3×C4#49711yescharacteristic
211C3×C4#49711yescharacteristic
212Dic3—82no—
213C23C4×#6711yescharacteristic
214Q8—82no—
215Q8—82no—
216Q8—124no—
217Q8—242no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—122no—
223C2×C4—242no—
224C2×C4D3×C22⋊C411yescharacteristic
225C2×C4#67⋊C411yescharacteristic
226C2×C4#88411yescharacteristic
227C2×C4#88511yescharacteristic
228C2×C4#93911yescharacteristic
229C2×C4#94011yescharacteristic
230C2×C4#94212yes—
231C2×C4#95112yes—
232C8—42no—
233C8—64no—
234C8—81no—
235C8—122no—
236C8—241no—
237C2×C3—22no—
238C2×C3#196711yescharacteristic
239C2×C3#197111yescharacteristic
240C2×C3#197711yescharacteristic
241C22—21no—
242C22—21no—
243C22#495711yescharacteristic
244C22#503012yes—
245C4—21no—
246C4—21no—
247C4—82no—
248C4—242no—
249C4#474812yes—
250C4#475112yes—
251C4#505212yes—
252C4#506011yescharacteristic
253C4#508611yescharacteristic
254C3#980311yescharacteristic, complemented, Sylow
255C2—22no—
256C2#6967811yescharacteristic
257C2#7007311yescharacteristic
258C2#7009911yescharacteristic
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#9803
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
254. C3#9803semidirect

Maximal subgroups

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

subgroupindex
C2×#50932
C3×#19972
#717752
#705072
#704902
#698792
#698472
#98033