Finite Group Databasebeta

← #102189

Subgroups — #102189

Order 512.

398 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 102 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
4#8132C211yesmaximal, 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—
19#1295—24no—
20#1277—21no—
21#1273—21no—
22#1273—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—22no—
37C2×#97—21no—
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—
58C22×Q8—22no—
59C22×Q8—22no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—42no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C4⋊Q8—42no—
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C4⋊Q8—42no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42#4111yescharacteristic
87C2×C42#4111yescharacteristic
88#107—42no—
89#107—42no—
90C4⋊C8—24no—
91C4⋊C8—42no—
92#103—44no—
93#103—44no—
94#97—22no—
95#97—42no—
96#97—44no—
97#97Q1614yescomplemented
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—41no—
102C4×C8—41no—
103C4×C8—41no—
104C4×C8—41no—
105C4×C8—41no—
106C4×C8—41no—
107C4×C8—41no—
108#95—42no—
109#95—42no—
110#95—42no—
111#95—42no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—44no—
119C2×Q8—82no—
120C2×Q8—82no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C4×D411yescharacteristic
125C22×C4C4×D411yescharacteristic
126C22×C4#12111yescharacteristic
127C22×C4#12111yescharacteristic
128C22×C4#12411yescharacteristic
129C22×C4#12711yescharacteristic
130Q16—84no—
131Q16—84no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—24no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—44no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8#10012yes—
151C2×C8#10012yes—
152C2×C8#10412yes—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—22no—
165C42—22no—
166C42—41no—
167C42C2×Q1612yes—
168C42#13512yes—
169C23#39011yescharacteristic
170Q8—82no—
171Q8—82no—
172Q8—82no—
173Q8—82no—
174Q8—164no—
175C2×C4—21no—
176C2×C4—21no—
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—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4C2×#10011yescharacteristic
195C2×C4C2×#10011yescharacteristic
196C2×C4C2×#10411yescharacteristic
197C2×C4#42111yescharacteristic
198C2×C4#42111yescharacteristic
199C2×C4#42111yescharacteristic
200C2×C4C4×Q1612yes—
201C2×C4#44412yes—
202C2×C4#46712yes—
203C2×C4#47312yes—
204C2×C4#49411yescharacteristic
205C2×C4#49511yescharacteristic
206C8—22no—
207C8—22no—
208C8—22no—
209C8—44no—
210C8—41no—
211C8—82no—
212C8—84no—
213C22#196311yescharacteristic
214C22#196511yescharacteristic
215C22#196911yescharacteristic
216C22#197511yescharacteristic
217C22#197611yescharacteristic
218C22#197911yescharacteristic
219C22#198011yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—22no—
224C4—21no—
225C4—21no—
226C4—41no—
227C4—162no—
228C4—162no—
229C4#152711yescharacteristic
230C4#153011yescharacteristic
231C4#153311yescharacteristic
232C4#153611yescharacteristic
233C2#968911yescharacteristic
234C2#971811yescharacteristic
235C2#972711yescharacteristic
236C2#976511yescharacteristic
237C2#979111yescharacteristic
238C2#979811yescharacteristic
239C2#984811yescharacteristic
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
97. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#332422
#98902
#82742
#81322
#76822