Finite Group Databasebeta

← #102181

Subgroups — #102181

Order 512.

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

normal subgroups only

This group has 389 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#33277C211yesmaximal, characteristic
2#9891C211yesmaximal, characteristic, complemented
3#9685C211yesmaximal, characteristic
4#8273C211yesmaximal, characteristic, complemented
5#8132C211yesmaximal, characteristic, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7679C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9#2858C2211yescharacteristic
10#1990C411yescharacteristic
11#1990C411yescharacteristic
12#1957C411yescharacteristic
13#1957C411yescharacteristic
14#1934—21no—
15#1934—21no—
16#1934—21no—
17#1934—21no—
18#1767C2211yescharacteristic
19#1767C2211yescharacteristic
20#1744C2211yescharacteristic
21C4×#97C2211yescharacteristic
22C42×C8C2211yescharacteristic
23C2×#334—21no—
24C2×#329—22no—
25#1295—22no—
26#1277—21no—
27#1273—21no—
28#1273—21no—
29#1271—22no—
30#1270—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×#127C2×C411yescharacteristic
33C2×#125—21no—
34C2×#124—21no—
35C2×#124—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×#102—21no—
42C2×#97—21no—
43C2×#97—21no—
44C2×#97—21no—
45C2×#97D412yes—
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#390—21no—
55#390—21no—
56#390—21no—
57#390—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
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#334—24no—
75#329—44no—
76C22×Q8—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C4⋊Q8—41no—
89C4⋊Q8—41no—
90#127—41no—
91#127—41no—
92#125—82no—
93#124—42no—
94#124—42no—
95#124—82no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121#102—42no—
122#102—42no—
123#102—42no—
124#102—42no—
125#97—22no—
126#97—42no—
127#97—42no—
128#97—42no—
129#97SD1614yes—
130C4×C8—22no—
131C4×C8—41no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8—41no—
136C4×C8—41no—
137C4×C8—41no—
138C4×C8—41no—
139C4×C8—41no—
140#95—41no—
141#95—41no—
142C24—41no—
143C2×Q8—42no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—81no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4—81no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C4×D411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4#12411yescharacteristic
167C22×C4#12711yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—24no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8#10412yes—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42—41no—
219C42C2×SD1612yes—
220C42#13512yes—
221C23—81no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23#39011yescharacteristic
229Q8—162no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—81no—
234D4—162no—
235D4—162no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4#41311yescharacteristic
259C2×C4#41311yescharacteristic
260C2×C4C2×#10411yescharacteristic
261C2×C4#42111yescharacteristic
262C2×C4#42111yescharacteristic
263C2×C4#42111yescharacteristic
264C2×C4C4×SD1612yes—
265C2×C4#44412yes—
266C2×C4#46512yes—
267C2×C4#47212yes—
268C2×C4#49211yescharacteristic
269C2×C4#49511yescharacteristic
270C8—22no—
271C8—44no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—82no—
276C8—82no—
277C8—82no—
278C22—161no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22—161no—
284C22—161no—
285C22#196311yescharacteristic
286C22#196711yescharacteristic
287C22#196811yescharacteristic
288C22#197511yescharacteristic
289C22#197611yescharacteristic
290C22#197811yescharacteristic
291C22#198011yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—21no—
297C4—21no—
298C4—41no—
299C4—161no—
300C4—161no—
301C4—161no—
302C4#152911yescharacteristic
303C4#153211yescharacteristic
304C4#153511yescharacteristic
305C4#153511yescharacteristic
306C2—321no—
307C2#968811yescharacteristic
308C2#971711yescharacteristic
309C2#972611yescharacteristic
310C2#976411yescharacteristic
311C2#979011yescharacteristic
312C2#980011yescharacteristic
313C2#984611yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
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
2. #9891C2semidirect
4. #8273C2semidirect
5. #8132C2semidirect
6. #7683C2semidirect

Maximal subgroups

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

subgroupindex
#332772
#98912
#96852
#82732
#81322
#76832
#76792