Finite Group Databasebeta

← #10692892

Subgroups — #10692892

Order 768.

432 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2568C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71776C212yesmaximal
4#71625C211yesmaximal, characteristic
5#67621C212yesmaximal
6#9456—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#400—21no—
10C3×#400—21no—
11C3×#391—21no—
12C3×#391—21no—
13C3×#391C412yes—
14C3×#382C412yes—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5024—24no—
17#4926—21no—
18#4926C2212yes—
19#4926C2212yes—
20#4925—21no—
21#4547—24no—
22#4528—21no—
23#4528—21no—
24#2983—31no—
25#2568D311yescharacteristic
26#1556—32no—
27#1467—31no—
28#1285—32no—
29C22×C3×Q8—21no—
30C22×C3×Q8C2×C411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C42C2×C412yes—
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45C3×#95—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—21no—
50#830—22no—
51#830—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)D414yes—
56#803—22no—
57#803—22no—
58#803—22no—
59C4×C3⋊C8—22no—
60C2×C4⋊Q8C2×D311yescharacteristic
61C2×C4×Q8C2×D311yescharacteristic
62#446—64no—
63#432—32no—
64#432—32no—
65#432—61no—
66#431—61no—
67#400—21no—
68#400—21no—
69#391—21no—
70#391—21no—
71#391Dic312yes—
72#382Dic312yes—
73C4×C4⋊C4C2×D311yescharacteristic
74#332—64no—
75#324—61no—
76#324—61no—
77C2×C3×Q8—22no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8—44no—
81C2×C3×Q8—41no—
82C2×C3×Q8—41no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—24no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—42no—
98C3×C42—21no—
99C3×C42—21no—
100C3×C42—21no—
101C3×C42—21no—
102C3×C42—42no—
103C3×C42C2×D412yes—
104C3×C42#4112yes—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—42no—
107C22×Q8—21no—
108C22×Q8C2×Dic311yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2×Dic312yes—
116C2×C4⋊C4C2×Dic311yescharacteristic
117C2×C4⋊C4#6711yescharacteristic
118C2×C4⋊C4#6711yescharacteristic
119C2×C42C2×Dic312yes—
120C2×C42#6711yescharacteristic
121C2×C42#6711yescharacteristic
122C2×C42C22×D311yescharacteristic
123C4⋊C8—34no—
124C4⋊C8—62no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127C22⋊C8—62no—
128C22⋊C8—62no—
129#97—62no—
130#97—62no—
131#97—62no—
132C4×C8—62no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95—22no—
137#95—21no—
138C23×C3C2×C22⋊C411yescharacteristic
139C3×Q8—44no—
140C3×Q8—82no—
141C3×Q8—82no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—44no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4#10112yes—
161C2×C3×C4C4×D412yes—
162C2×C3×C4#12011yescharacteristic
163C2×C3×C4#12112yes—
164C2×C3×C4#12311yescharacteristic
165C3⋊C8—44no—
166C3⋊C8—82no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—41no—
170C2×Q8—44no—
171C2×Q8—41no—
172C2×Q8—41no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4#21711yescharacteristic
176C22×C4#21711yescharacteristic
177C22×C4#21711yescharacteristic
178C22×C4#21711yescharacteristic
179C22×C4C22×Dic311yescharacteristic
180C22×C4C2×#6711yescharacteristic
181C22×C4C2×#6711yescharacteristic
182C2×C8—64no—
183C2×C8—122no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—42no—
195C42D3×D412yes—
196C42#23712yes—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3#38611yescharacteristic
200C22×C3C2×#10111yescharacteristic
201C22×C3#41311yescharacteristic
202C3×C4—21no—
203C3×C4—22no—
204C3×C4—22no—
205C3×C4—22no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4—41no—
209C3×C4—82no—
210C3×C4—82no—
211C3×C4#45412yes—
212C23C2×#21711yescharacteristic
213Q8—44no—
214Q8—82no—
215Q8—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4#83612yes—
235C2×C4D4×Dic312yes—
236C2×C4#93511yescharacteristic
237C2×C4#93711yescharacteristic
238C2×C4#93912yes—
239C8—124no—
240C8—242no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3#188811yescharacteristic
244C2×C3#190111yescharacteristic
245C2×C3#190911yescharacteristic
246C22—21no—
247C22—21no—
248C22#522011yescharacteristic
249C22C2×#83611yescharacteristic
250C22#523711yescharacteristic
251C4—22no—
252C4—22no—
253C4—21no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—82no—
259C4—82no—
260C4#506212yes—
261C3#945611yescharacteristic, complemented, Sylow
262C2—21no—
263C2—21no—
264C2#7169811yescharacteristic
265C2#7172411yescharacteristic
266C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2568C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2568D3
hypocenterC3#9456
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
261. C3#9456semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
#824362
#717762
#716252
#676212
#94563