Finite Group Databasebeta

← #10694867

Subgroups — #10694867

Order 768.

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

normal subgroups only

This group has 418 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×#5079C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#70506C211yesmaximal, characteristic
5#70491C211yesmaximal, characteristic
6#69877C211yesmaximal, characteristic
7#69847C211yesmaximal, characteristic
8#9809—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C2×#835C414yes—
14C2×#824—22no—
15#5079—22no—
16#5079—22no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20C3×#343—22no—
21#4546—22no—
22#4482—22no—
23#4479—22no—
24#4454—22no—
25C2×#487—31no—
26#1997D311yescharacteristic
27#1966—31no—
28#1567—31no—
29#1548—31no—
30#1546—31no—
31#1451—31no—
32C22×C3×Q8C2×C412yes—
33C22×Dic6—21no—
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⋊C4—21no—
42C2×C3×C42C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C412yes—
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#835—28no—
56#830—21no—
57#824—44no—
58C4⋊(C3⋊C8)—21no—
59C4⋊(C3⋊C8)—21no—
60C4⋊(C3⋊C8)D412yes—
61C4×C3⋊C8—21no—
62C4×C3⋊C8D412yes—
63C4×C3⋊C8D412yes—
64C2×C4⋊Q8—31no—
65C2×C4⋊Q8C2×D311yescharacteristic
66#487—62no—
67#487—62no—
68#431—31no—
69#423—31no—
70#423C2×D311yescharacteristic
71C2×#103—34no—
72C2×#101—62no—
73C4×M16C2×D311yescharacteristic
74C2×C4×C8—31no—
75#343—22no—
76#335—62no—
77#333—62no—
78#332—62no—
79#322—62no—
80C2×C3×Q8—22no—
81C2×C3×Q8—22no—
82C2×C3×Q8—44no—
83C22×C3×C4—22no—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C22×Dic3—22no—
88C2×Dic6—41no—
89C2×Dic6—41no—
90C2×Dic6—42no—
91C2×Dic6—42no—
92C3×M16—22no—
93C3×M16—22no—
94C3×M16—42no—
95C2×C3×C8—22no—
96C2×C3×C8—41no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C42C2×D411yescharacteristic
102C3×C42C2×D411yescharacteristic
103C3×C42C2×D411yescharacteristic
104C3×C42#4111yescharacteristic
105C4⋊Dic3—22no—
106C4⋊Dic3—22no—
107C4⋊Dic3—41no—
108C4⋊Dic3—41no—
109#208—42no—
110C2×C3⋊C8—22no—
111C2×C3⋊C8—42no—
112C2×C3⋊C8—41no—
113C2×C3⋊C8C22⋊C414yes—
114C22×Q8—61no—
115C22×Q8C4×D312yes—
116C2×M16—21no—
117C2×M16—61no—
118C2×M16#6712yes—
119C22×C8—32no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—61no—
125C4⋊Q8—61no—
126C4⋊Q8—61no—
127C4⋊Q8—61no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—32no—
131C2×C4⋊C4—61no—
132C2×C42C22×D311yescharacteristic
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—32no—
136C4⋊C8—61no—
137C4⋊C8—61no—
138#103—68no—
139#101—124no—
140C22⋊C8—61no—
141#97—21no—
142C4×C8—21no—
143C4×C8—32no—
144C4×C8—32no—
145C4×C8—61no—
146C23×C3C4×D411yescharacteristic
147C3×Q8—44no—
148C3×Q8—82no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—42no—
152C2×C3×C4—82no—
153C2×C3×C4C2×C22⋊C411yescharacteristic
154C2×C3×C4C4×D411yescharacteristic
155C2×C3×C4#12111yescharacteristic
156C2×C3×C4#12111yescharacteristic
157C2×C3×C4#12411yescharacteristic
158C2×C3×C4#12711yescharacteristic
159C2×C3×C4AGL(1,ℤ/8)12yes—
160C2×C3×C4#13712yes—
161C2×Dic3—42no—
162C2×Dic3—44no—
163Dic6—82no—
164Dic6—81no—
165Dic6—81no—
166C3×C8—42no—
167C3×C8—81no—
168C3⋊C8—24no—
169C3⋊C8—42no—
170C3⋊C8—81no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—44no—
174C2×Q8—121no—
175C2×Q8—121no—
176C2×Q8—122no—
177C2×Q8—122no—
178C22×C4—22no—
179C22×C4—62no—
180C22×C4C2×C4×D311yescharacteristic
181C22×C4#23511yescharacteristic
182C22×C4C2×#6711yescharacteristic
183M16—22no—
184M16—22no—
185M16—42no—
186M16—122no—
187C2×C8—22no—
188C2×C8—34no—
189C2×C8—41no—
190C2×C8—62no—
191C2×C8—122no—
192C2×C8—121no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—62no—
198C4⋊C4—62no—
199C4⋊C4—121no—
200C4⋊C4—121no—
201C42D3×D411yescharacteristic
202C42D3×D411yescharacteristic
203C42D3×D411yescharacteristic
204C42#23711yescharacteristic
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#39011yescharacteristic
208C22×C3#44012yes—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—82no—
212C3×C4#41312yes—
213C3×C4#46812yes—
214C3×C4#47312yes—
215C3×C4#49611yescharacteristic
216C3×C4#49711yescharacteristic
217Dic3—82no—
218C23C4×#6711yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—242no—
222Q8—241no—
223Q8—241no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—122no—
229C2×C4—124no—
230C2×C4D3×C22⋊C411yescharacteristic
231C2×C4#67⋊C411yescharacteristic
232C2×C4#88411yescharacteristic
233C2×C4#88511yescharacteristic
234C2×C4#93911yescharacteristic
235C2×C4#94011yescharacteristic
236C2×C4#94212yes—
237C2×C4#94912yes—
238C8—42no—
239C8—64no—
240C8—81no—
241C8—122no—
242C8—241no—
243C2×C3—22no—
244C2×C3#196711yescharacteristic
245C2×C3#197311yescharacteristic
246C2×C3#197911yescharacteristic
247C22—21no—
248C22—21no—
249C22#495711yescharacteristic
250C22#502612yes—
251C4—21no—
252C4—21no—
253C4—82no—
254C4—242no—
255C4#474512yes—
256C4#475512yes—
257C4#504912yes—
258C4#505811yescharacteristic
259C4#508611yescharacteristic
260C3#980911yescharacteristic, complemented, Sylow
261C2—22no—
262C2#6967911yescharacteristic
263C2#7006511yescharacteristic
264C2#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#9809
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
260. C3#9809semidirect

Maximal subgroups

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

subgroupindex
C2×#50792
C3×#19972
#717742
#705062
#704912
#698772
#698472
#98093