Finite Group Databasebeta

← #102188

Subgroups — #102188

Order 512.

350 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 98 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#33278C211yesmaximal, characteristic
2#9891C211yesmaximal, characteristic
3#9890C211yesmaximal, characteristic
4#8274C211yesmaximal, characteristic
5#8132C211yesmaximal, characteristic
6#7683C211yesmaximal, characteristic
7#7682C211yesmaximal, characteristic
8#2862C2211yescharacteristic
9#2859C2211yescharacteristic
10#1990C411yescharacteristic
11#1990C411yescharacteristic
12#1956C411yescharacteristic
13#1956C411yescharacteristic
14#1933—21no—
15#1933—21no—
16#1933—21no—
17#1933—21no—
18#1767C2211yescharacteristic
19#1767C2211yescharacteristic
20#1744C2211yescharacteristic
21C4×#97C2211yescharacteristic
22C42×C8C2211yescharacteristic
23C2×#335—21no—
24#1295—22no—
25#1295—22no—
26#1277—21no—
27#1273—21no—
28#1273—21no—
29#1272—22no—
30#1271—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C4⋊Q8C2×C411yescharacteristic
33C2×#125—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#97—21no—
41C2×#97—21no—
42C2×#97—21no—
43C2×#97D412yes—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#391—21no—
53#391—21no—
54#391—21no—
55#391—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#384C2×C411yescharacteristic
61#384C2×C411yescharacteristic
62#382—21no—
63#382—21no—
64#382—21no—
65C43C2311yescharacteristic
66#340—21no—
67#340—21no—
68#340—21no—
69#340—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336D412yes—
74#335—24no—
75C22×Q8—21no—
76C22×Q8—21no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89#125—82no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113#103—42no—
114#103—42no—
115#103—42no—
116#103—42no—
117#97—22no—
118#97—42no—
119#97—42no—
120#97—42no—
121#97Q1614yes—
122C4×C8—22no—
123C4×C8—41no—
124C4×C8—41no—
125C4×C8—41no—
126C4×C8—41no—
127C4×C8—41no—
128C4×C8—41no—
129C4×C8—41no—
130C4×C8—41no—
131C4×C8—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—42no—
145C2×Q8—81no—
146C2×Q8—81no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C4×D411yescharacteristic
153C22×C4C4×D411yescharacteristic
154C22×C4#12111yescharacteristic
155C22×C4#12111yescharacteristic
156C22×C4#12411yescharacteristic
157C22×C4#12711yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—81no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8#10412yes—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—22no—
204C42—22no—
205C42—41no—
206C42C2×Q1612yes—
207C42#13512yes—
208C23#39011yescharacteristic
209Q8—81no—
210Q8—81no—
211Q8—81no—
212Q8—81no—
213Q8—162no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4#41311yescharacteristic
240C2×C4#41311yescharacteristic
241C2×C4C2×#10411yescharacteristic
242C2×C4#42111yescharacteristic
243C2×C4#42111yescharacteristic
244C2×C4#42111yescharacteristic
245C2×C4C4×Q1612yes—
246C2×C4#44412yes—
247C2×C4#46712yes—
248C2×C4#47112yes—
249C2×C4#49411yescharacteristic
250C2×C4#49511yescharacteristic
251C8—22no—
252C8—44no—
253C8—41no—
254C8—41no—
255C8—41no—
256C8—82no—
257C8—82no—
258C8—82no—
259C22#196311yescharacteristic
260C22#196711yescharacteristic
261C22#196911yescharacteristic
262C22#197511yescharacteristic
263C22#197611yescharacteristic
264C22#197711yescharacteristic
265C22#198011yescharacteristic
266C4—22no—
267C4—22no—
268C4—22no—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—41no—
273C4—161no—
274C4—161no—
275C4—161no—
276C4—161no—
277C4#152911yescharacteristic
278C4#153211yescharacteristic
279C4#153511yescharacteristic
280C4#153511yescharacteristic
281C2#968811yescharacteristic
282C2#971811yescharacteristic
283C2#972811yescharacteristic
284C2#976611yescharacteristic
285C2#979011yescharacteristic
286C2#979511yescharacteristic
287C2#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

Maximal subgroups

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

subgroupindex
#332782
#98912
#98902
#82742
#81322
#76832
#76822