Finite Group Databasebeta

← #102178

Subgroups — #102178

Order 512.

430 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 110 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#33242C211yesmaximal, characteristic
2#9890C212yesmaximal
3#8274C211yesmaximal, characteristic
4C8×#97C211yesmaximal, characteristic
5#7682C212yesmaximal
6C2×#500C411yescharacteristic
7C2×#494C411yescharacteristic
8C2×#487C412yes—
9#2862C2212yes—
10#1969—21no—
11#1969—21no—
12#1969—21no—
13#1969—21no—
14#1767C2212yes—
15#1744C2211yescharacteristic
16C4×#97C2211yescharacteristic
17C42×C8C2211yescharacteristic
18C2×#335—21no—
19C2×#322—21no—
20C2×C82—21no—
21#1295—24no—
22#1277—21no—
23#1272—24no—
24C22×Q16—21no—
25C22×Q16—21no—
26C2×C4⋊Q8C2×C412yes—
27C2×C4⋊Q8C2×C412yes—
28#500—42no—
29#494—42no—
30#487—44no—
31C2×#107—21no—
32C2×#107—21no—
33C2×C4⋊C8—22no—
34C2×#103—22no—
35C2×#103—22no—
36C2×#97—21no—
37C2×#97—22no—
38C2×#97D412yes—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—22no—
48#391—22no—
49#391—22no—
50#391—22no—
51#382—22no—
52#382—22no—
53C43C2311yescharacteristic
54#336—22no—
55#336—21no—
56#336D412yes—
57#335—24no—
58#322—24no—
59C82—24no—
60C22×Q8—22no—
61C22×Q8—22no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—42no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C4⋊Q8—42no—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C42C22×C411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D411yescharacteristic
87C2×C42C2×D411yescharacteristic
88C2×C42#4111yescharacteristic
89C2×C42#4111yescharacteristic
90#107—42no—
91#107—42no—
92C4⋊C8—24no—
93C4⋊C8—42no—
94#103—44no—
95#103—44no—
96#97—24no—
97#97—22no—
98#97—44no—
99#97Q1614yescomplemented
100C4×C8—22no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—41no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—44no—
121C2×Q8—82no—
122C2×Q8—82no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C4×D411yescharacteristic
127C22×C4C4×D411yescharacteristic
128C22×C4#12111yescharacteristic
129C22×C4#12111yescharacteristic
130C22×C4#12411yescharacteristic
131C22×C4#12711yescharacteristic
132Q16—84no—
133Q16—84no—
134C2×C8—24no—
135C2×C8—24no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—24no—
141C2×C8—24no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—44no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8#10012yes—
151C2×C8#10012yes—
152C2×C8#10412yes—
153C2×C8#10412yes—
154C2×C8#10412yes—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—22no—
167C42—22no—
168C42—41no—
169C42C2×Q1612yes—
170C42#13512yes—
171C23#39011yescharacteristic
172Q8—82no—
173Q8—82no—
174Q8—82no—
175Q8—82no—
176Q8—164no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4C2×#10011yescharacteristic
197C2×C4C2×#10011yescharacteristic
198C2×C4C2×#10411yescharacteristic
199C2×C4C2×#10411yescharacteristic
200C2×C4C2×#10411yescharacteristic
201C2×C4#42111yescharacteristic
202C2×C4C4×Q1612yes—
203C2×C4#44312yes—
204C2×C4#46712yes—
205C2×C4#47312yes—
206C2×C4#49411yescharacteristic
207C2×C4#49511yescharacteristic
208C8—28no—
209C8—22no—
210C8—22no—
211C8—44no—
212C8—41no—
213C8—84no—
214C8#32914yes—
215C22#196211yescharacteristic
216C22#196511yescharacteristic
217C22#196911yescharacteristic
218C22#197411yescharacteristic
219C22#197611yescharacteristic
220C22#197911yescharacteristic
221C22#198011yescharacteristic
222C4—22no—
223C4—22no—
224C4—22no—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—41no—
229C4—162no—
230C4—162no—
231C4C2×#32911yescharacteristic
232C4#152511yescharacteristic
233C4#152711yescharacteristic
234C4#153011yescharacteristic
235C2#968511yescharacteristic
236C2#971811yescharacteristic
237C2#972511yescharacteristic
238C2#976311yescharacteristic
239C2#978711yescharacteristic
240C2#979811yescharacteristic
241C2#984511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
99. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#332422
#98902
#82742
C8×#972
#76822