Finite Group Databasebeta

← #10694682

Subgroups — #10694682

Order 768.

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

normal subgroups only

This group has 390 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×#5078C211yesmaximal, characteristic
2C3×#1549C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#70742C211yesmaximal, characteristic
6#70516C211yesmaximal, characteristic
7#70507C211yesmaximal, characteristic
8#9778—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5078—21no—
14#5078—21no—
15#5078C414yes—
16C2×#807—22no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20C3×#332—22no—
21#4558—22no—
22#4546—22no—
23#4482—22no—
24#4481—22no—
25#4455—21no—
26#4454—21no—
27C2×#500—31no—
28#1837—31no—
29#1549—31no—
30#1549D311yescharacteristic
31#1548—31no—
32#1548—31no—
33#1452—31no—
34C22×C3×Q8—21no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×#208—21no—
47C22×C3⋊C8D412yes—
48#869—21no—
49#869—21no—
50#869—21no—
51#869C2×C412yes—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8D412yes—
55C3×C22⋊C8—21no—
56C3×C4×C8—21no—
57#830—21no—
58#807—24no—
59#807—24no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)D412yes—
63C4×C3⋊C8—21no—
64C4×C3⋊C8C2×C412yes—
65C4×C3⋊C8D411yescharacteristic
66C4×C3⋊C8D411yescharacteristic
67C2×C4⋊Q8—31no—
68C2×C4⋊Q8C2×D311yescharacteristic
69#500—34no—
70#500—61no—
71#500—61no—
72#431—31no—
73#431C2×D311yescharacteristic
74C2×#107—62no—
75#423—31no—
76#423C2×D311yescharacteristic
77C2×C4×C8—31no—
78#341—62no—
79#333—62no—
80#333—62no—
81#332—22no—
82#332—62no—
83#322—61no—
84#322—61no—
85C2×C3×Q8—22no—
86C2×C3×Q8—41no—
87C2×C3×Q8—82no—
88C22×C3×C4—41no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4111yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—41no—
93C2×Dic6—22no—
94C2×Dic6—41no—
95C2×Dic6—82no—
96C3×M16—42no—
97C2×C3×C8—22no—
98C2×C3×C8—41no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C42C22×C411yescharacteristic
104C3×C42C2×D411yescharacteristic
105C3×C42C2×D411yescharacteristic
106C3×C42C2×D411yescharacteristic
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109C4⋊Dic3—41no—
110C4⋊Dic3—41no—
111#208—42no—
112C2×C3⋊C8—22no—
113C2×C3⋊C8—22no—
114C2×C3⋊C8—22no—
115C2×C3⋊C8—41no—
116C2×C3⋊C8C22⋊C412yes—
117C2×C3⋊C8C22⋊C412yes—
118C22×Q8—21no—
119C22×Q8—61no—
120C2×M16—21no—
121C2×M16—61no—
122C22×C8—32no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—32no—
127C4⋊Q8—61no—
128C4⋊Q8—61no—
129C4⋊Q8—61no—
130C4⋊Q8C4×D312yes—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C42C22×D311yescharacteristic
136#107—64no—
137#107—64no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—32no—
141C4⋊C8—61no—
142C4⋊C8—61no—
143C4⋊C8#6712yes—
144C22⋊C8—21no—
145C22⋊C8—61no—
146C4×C8—21no—
147C4×C8—32no—
148C4×C8—31no—
149C4×C8—31no—
150C4×C8—61no—
151C23×C3#12411yescharacteristic
152C3×Q8—82no—
153C3×Q8—82no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—42no—
157C2×C3×C4—82no—
158C2×C3×C4C2×C22⋊C411yescharacteristic
159C2×C3×C4C4×D411yescharacteristic
160C2×C3×C4C4×D411yescharacteristic
161C2×C3×C4#12111yescharacteristic
162C2×C3×C4#12111yescharacteristic
163C2×C3×C4#12711yescharacteristic
164C2×C3×C4#13712yes—
165C2×C3×C4#13712yes—
166C2×Dic3—42no—
167C2×Dic3—82no—
168Dic6—82no—
169Dic6—82no—
170C3×C8—42no—
171C3×C8—81no—
172C3⋊C8—22no—
173C3⋊C8—22no—
174C3⋊C8—42no—
175C3⋊C8—81no—
176C2×Q8—22no—
177C2×Q8—41no—
178C2×Q8—62no—
179C2×Q8—82no—
180C2×Q8—121no—
181C2×Q8—242no—
182C22×C4—41no—
183C22×C4—121no—
184C22×C4#23511yescharacteristic
185C22×C4D3×D411yescharacteristic
186C22×C4#23711yescharacteristic
187M16—42no—
188M16—122no—
189C2×C8—22no—
190C2×C8—32no—
191C2×C8—32no—
192C2×C8—41no—
193C2×C8—62no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—121no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—62no—
202C4⋊C4—62no—
203C4⋊C4—121no—
204C4⋊C4—121no—
205C42C2×C4×D311yescharacteristic
206C42D3×D411yescharacteristic
207C42D3×D411yescharacteristic
208C42C2×#6711yescharacteristic
209C22×C3—21no—
210C22×C3—21no—
211C22×C3#39011yescharacteristic
212C22×C3#49112yes—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—82no—
216C3×C4#41912yes—
217C3×C4#47212yes—
218C3×C4#47312yes—
219C3×C4#49711yescharacteristic
220C3×C4#49711yescharacteristic
221Dic3—82no—
222C23#88511yescharacteristic
223Q8—82no—
224Q8—82no—
225Q8—242no—
226Q8—242no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—122no—
232C2×C4—242no—
233C2×C4D3×C22⋊C411yescharacteristic
234C2×C4#67⋊C411yescharacteristic
235C2×C4#88411yescharacteristic
236C2×C4#91512yes—
237C2×C4#91812yes—
238C2×C4C4×#6711yescharacteristic
239C2×C4#93911yescharacteristic
240C2×C4#94011yescharacteristic
241C8—42no—
242C8—62no—
243C8—62no—
244C8—81no—
245C8—122no—
246C8—241no—
247C2×C3—22no—
248C2×C3#196011yescharacteristic
249C2×C3#197811yescharacteristic
250C2×C3#197911yescharacteristic
251C22—21no—
252C22—21no—
253C22#479712yes—
254C22#495711yescharacteristic
255C4—21no—
256C4—21no—
257C4—82no—
258C4—242no—
259C4#475512yes—
260C4#480212yes—
261C4#516711yescharacteristic
262C4#519011yescharacteristic
263C4#519912yes—
264C3#977811yescharacteristic, complemented, Sylow
265C2—22no—
266C2#7010811yescharacteristic
267C2#7022811yescharacteristic
268C2#7031611yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenter#1549D3
hypocenterC3#9778
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
264. C3#9778semidirect

Maximal subgroups

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

subgroupindex
C2×#50782
C3×#15492
#717742
#707802
#707422
#705162
#705072
#97783