Finite Group Databasebeta

← #10698421

Subgroups — #10698421

Order 768.

414 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#2607C211yesmaximal, characteristic
2#69534C211yesmaximal, characteristic
3#69442C212yesmaximal
4#67629C211yesmaximal, characteristic
5#67616C212yesmaximal
6#10387—31nomaximal, Sylow
7C2×#926C2211yescharacteristic
8C3×#399—21no—
9C3×#393—21no—
10C3×#391—22no—
11C3×#387—22no—
12C3×#381C2211yescharacteristic
13C3×C4×C22⋊C4C2212yes—
14#5211C2211yescharacteristic
15#5207C2212yes—
16#4527—21no—
17#4527—21no—
18#4523—22no—
19#4523—22no—
20#2607D311yescharacteristic
21#1826—32no—
22#1820—31no—
23#1306—31no—
24#1284—32no—
25C22×C3×Q8—21no—
26C23×C3×C4C2311yescharacteristic
27C2×C3×C4⋊C4—22no—
28C2×C3×C4⋊C4—21no—
29C2×C3×C22⋊C4—22no—
30C2×C3×C22⋊C4D411yescharacteristic
31C2×C3×C22⋊C4D411yescharacteristic
32C2×C3×C42D412yes—
33C2×C3×C42D412yes—
34C2×#217—21no—
35C2×#217—21no—
36#926—22no—
37#926—22no—
38C2×C4⋊Dic3Q812yescomplemented
39C2×C4×Dic3—21no—
40C2×#208—22no—
41C3×#95—21no—
42C3×#95—22no—
43C3×#95—21no—
44C3×#95—22no—
45C3×#95—22no—
46C3⋊#95—22no—
47#830—22no—
48#830—22no—
49C2×#117—31no—
50#407—32no—
51#399—21no—
52#393—21no—
53#391—22no—
54#387—22no—
55#381—31no—
56#381C2×D311yescharacteristic
57C4×C22⋊C4C2×D312yes—
58#342—61no—
59#342—61no—
60#339—62no—
61#339—62no—
62C24×C3C2×D411yescharacteristic
63C2×C3×Q8—44no—
64C2×C3×Q8—81no—
65C22×C3×C4—22no—
66C22×C3×C4—21no—
67C22×C3×C4—22no—
68C22×C3×C4—21no—
69C22×C3×C4—41no—
70C22×C3×C4—41no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×Q811yescharacteristic
73C22×C3×C4#4112yes—
74C22×C3×C4#4111yescharacteristic
75C22×Dic3—22no—
76C22×Dic3—41no—
77C3×C4⋊C4—44no—
78C3×C4⋊C4—81no—
79C3×C22⋊C4—44no—
80C3×C22⋊C4—41no—
81C3×C22⋊C4—41no—
82C3×C42—42no—
83C3×C42—42no—
84#217—41no—
85#217—41no—
86#217—41no—
87#217—41no—
88C4⋊Dic3—24no—
89C4×Dic3—42no—
90#208—42no—
91#208—42no—
92C2×C3⋊C8—42no—
93C22×Q8—21no—
94C23×C4C22×D311yescharacteristic
95C2×M16—62no—
96#117—62no—
97#117—62no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—32no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—61no—
103C2×C22⋊C4—61no—
104C2×C22⋊C4#6711yescharacteristic
105C2×C22⋊C4#6711yescharacteristic
106C2×C42—61no—
107C2×C42#6712yes—
108C2×C42#6712yes—
109C22⋊C8—62no—
110C22⋊C8—62no—
111#95—21no—
112#95—22no—
113#95—21no—
114#95—22no—
115#95—22no—
116#95—62no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—42no—
120C23×C3#12011yescharacteristic
121C23×C3C22⋊Q811yescharacteristic
122C23×C3#12411yescharacteristic
123C3×Q8—84no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—24no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4C22⋊Q812yes—
140C2×C3×C4#12412yes—
141C2×Dic3—42no—
142C2×Dic3—82no—
143C2×Dic3—81no—
144C2×Dic3—82no—
145C3⋊C8—82no—
146C24C2×#6711yescharacteristic
147C2×Q8—44no—
148C2×Q8—81no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—62no—
156C22×C4—121no—
157C22×C4D3×Q811yescharacteristic
158C22×C4#23912yes—
159C22×C4#23911yescharacteristic
160C22×C4C2×#6712yes—
161M16—122no—
162M16—122no—
163C2×C8—122no—
164C4⋊C4—44no—
165C4⋊C4—64no—
166C4⋊C4—81no—
167C22⋊C4—44no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—121no—
171C22⋊C4—121no—
172C22⋊C4—121no—
173C22⋊C4—121no—
174C42—42no—
175C42—42no—
176C42—122no—
177C22×C3—22no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—22no—
181C22×C3—22no—
182C22×C3—81no—
183C22×C3—81no—
184C22×C3#39311yescharacteristic
185C22×C3#45811yescharacteristic
186C22×C3#45811yescharacteristic
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—44no—
190C3×C4—82no—
191C3×C4—81no—
192C3×C4—81no—
193Dic3—82no—
194Dic3—161no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#94611yescharacteristic
199C23#94711yescharacteristic
200C23#95311yescharacteristic
201Q8—84no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—122no—
218C2×C4—242no—
219C2×C4—241no—
220C2×C4—242no—
221C2×C4#94612yes—
222C2×C4#94712yes—
223C8—242no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—41no—
227C2×C3—41no—
228C2×C3#203011yescharacteristic
229C2×C3#203112yes—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—21no—
234C22—21no—
235C22—81no—
236C22—81no—
237C22#523111yescharacteristic
238C22#524311yescharacteristic
239C22#524311yescharacteristic
240C4—22no—
241C4—22no—
242C4—44no—
243C4—81no—
244C4—82no—
245C4—81no—
246C4—242no—
247C4—481no—
248C3#1038711yescharacteristic, complemented, Sylow
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#7184212yes—
254C2#7189411yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2607C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#2607D3
hypocenterC3#10387
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
38. C2×C4⋊Dic3Q8semidirect
248. C3#10387semidirect

Maximal subgroups

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

subgroupindex
C3×#26072
#695342
#694422
#676292
#676162
#103873