Finite Group Databasebeta

← #10698260

Subgroups — #10698260

Order 768.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#74218C211yesmaximal, characteristic
2C3×#1960C211yesmaximal, characteristic
3#71788C212yesmaximal
4#67659C212yesmaximal
5#67630C211yesmaximal, characteristic
6#10359—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#103—21no—
9C2×C3×#103—21no—
10C2×C3×#97C2211yescharacteristic
11C3×#382C2211yescharacteristic
12#4986—21no—
13#4979—21no—
14#4936C2212yes—
15C2×#803C2212yes—
16#4660—22no—
17#4656—22no—
18#4529—22no—
19#4529—22no—
20#2844—31no—
21#1993—32no—
22#1960D311yescharacteristic
23#1295—32no—
24#1295—31no—
25C22×C3×Q8—21no—
26C22×Dic6—21no—
27C22×C3×C8—21no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—21no—
30C3×C4⋊Q8Q812yescomplemented
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3D412yes—
37C2×C4⋊Dic3D412yes—
38C2×Dic3⋊C4—22no—
39C22×C3⋊C8—22no—
40C3×#103—42no—
41C3×#103—42no—
42C3×#97—22no—
43C3×#95—22no—
44C3⋊#95—22no—
45C3⋊#95—21no—
46C3⋊#95—21no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49#803—22no—
50#803—22no—
51C2×C4⋊Q8C2×D311yescharacteristic
52C2×#103—21no—
53C2×#103—21no—
54C2×#97—32no—
55C2×#97C2×D311yescharacteristic
56#400—62no—
57#400—61no—
58#395—62no—
59#395—61no—
60#382—32no—
61#382C2×D311yescharacteristic
62#340—62no—
63#340—62no—
64C2×C3×Q8—22no—
65C2×C3×Q8—41no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8—41no—
69C2×C3×Q8—41no—
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—41no—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—22no—
76C22×Dic3—22no—
77C2×Dic6—81no—
78C2×Dic6—82no—
79C2×C3×C8—22no—
80C2×C3×C8—41no—
81C2×C3×C8—41no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—41no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—42no—
86C3×C42C2×Q811yescharacteristic
87C3×C42#4111yescharacteristic
88C3×C42#4112yes—
89C4⋊Dic3—42no—
90C4⋊Dic3—42no—
91C4⋊Dic3—81no—
92Dic3⋊C4—82no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8—42no—
95C2×C3⋊C8—42no—
96C22×Q8—21no—
97C22×Q8—61no—
98C22×C8—21no—
99C22×C8—62no—
100C4⋊Q8—21no—
101C4⋊Q8—22no—
102C4⋊Q8Dic612yescomplemented
103C2×C4⋊C4—22no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4#6711yescharacteristic
109C2×C4⋊C4#6711yescharacteristic
110C2×C42C22×D311yescharacteristic
111#103—42no—
112#103—42no—
113#97—22no—
114#97—62no—
115#97—62no—
116#95—22no—
117#95—62no—
118#95—61no—
119#95—61no—
120#95—62no—
121#95—62no—
122C23×C3#12011yescharacteristic
123C3×Q8—42no—
124C3×Q8—42no—
125C3×Q8—81no—
126C3×Q8—81no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—42no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—81no—
137C2×C3×C4—81no—
138C2×C3×C4C22⋊Q812yes—
139C2×C3×C4C22⋊Q811yescharacteristic
140C2×C3×C4#12412yes—
141C2×C3×C4#12411yescharacteristic
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148Dic6—84no—
149C3×C8—42no—
150C3⋊C8—44no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—242no—
158C2×Q8—241no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4D3×D412yes—
165C22×C4C2×#6711yescharacteristic
166C2×C8—22no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—64no—
170C2×C8—122no—
171C2×C8—122no—
172C4⋊C4—24no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—122no—
177C4⋊C4—122no—
178C4⋊C4—242no—
179C4⋊C4—241no—
180C42C2×Dic611yescharacteristic
181C42#23511yescharacteristic
182C42#23712yes—
183C22×C3#39311yescharacteristic
184C22×C3#45311yescharacteristic
185C22×C3#45411yescharacteristic
186C22×C3#45512yes—
187C22×C3#45612yes—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—81no—
193C3×C4—81no—
194C3×C4—82no—
195Dic3—82no—
196Dic3—82no—
197C23#93711yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—81no—
201Q8—81no—
202Q8—244no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4#87812yes—
221C2×C4#88512yes—
222C2×C4#92411yescharacteristic
223C2×C4#93611yescharacteristic
224C8—42no—
225C8—124no—
226C2×C3#165911yescharacteristic
227C2×C3#166011yescharacteristic
228C2×C3#166412yes—
229C2×C3#202711yescharacteristic
230C2×C3#202812yes—
231C22#494711yescharacteristic
232C22#506211yescharacteristic
233C22#507711yescharacteristic
234C22#520212yes—
235C22#520512yes—
236C4—21no—
237C4—21no—
238C4—22no—
239C4—22no—
240C4—82no—
241C4—81no—
242C4—81no—
243C4—242no—
244C4—242no—
245C3#1035911yescharacteristic, complemented, Sylow
246C2#6964311yescharacteristic
247C2#7033312yes—
248C2#7125312yes—
249C2#7132911yescharacteristic
250C2#7133011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socle, prime radicalC23×C3#120
hypercenter#1960D3
hypocenterC3#10359
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
30. C3×C4⋊Q8Q8semidirect
102. C4⋊Q8Dic6semidirect
245. C3#10359semidirect

Maximal subgroups

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

subgroupindex
#742182
C3×#19602
#717882
#676592
#676302
#103593