Finite Group Databasebeta

← #102173

Subgroups — #102173

Order 512.

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

normal subgroups only

This group has 421 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#33277C211yesmaximal, characteristic
2#9891C211yesmaximal, characteristic, complemented
3#9685C211yesmaximal, characteristic
4#8273C211yesmaximal, characteristic, complemented
5C8×#97C211yesmaximal, 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—
25C2×#322—21no—
26C2×C82—21no—
27#1295—22no—
28#1277—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—
76#322—24no—
77C82—24no—
78C22×Q8—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92#127—41no—
93#127—41no—
94#125—82no—
95#124—42no—
96#124—42no—
97#124—82no—
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—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123#102—42no—
124#102—42no—
125#102—42no—
126#102—42no—
127#97—24no—
128#97—22no—
129#97—42no—
130#97—42no—
131#97SD1614yes—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—41no—
140C4×C8—41no—
141C4×C8—41no—
142#95—41no—
143#95—41no—
144C24—41no—
145C2×Q8—42no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—81no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4—81no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×D411yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12411yescharacteristic
169C22×C4#12711yescharacteristic
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
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—
192C2×C8#10412yes—
193C2×C8#10412yes—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—22no—
219C42—22no—
220C42—41no—
221C42C2×SD1612yes—
222C42#13512yes—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23#39011yescharacteristic
231Q8—162no—
232D4—81no—
233D4—81no—
234D4—81no—
235D4—81no—
236D4—162no—
237D4—162no—
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—21no—
250C2×C4—21no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4#41311yescharacteristic
261C2×C4#41311yescharacteristic
262C2×C4C2×#10411yescharacteristic
263C2×C4C2×#10411yescharacteristic
264C2×C4C2×#10411yescharacteristic
265C2×C4#42111yescharacteristic
266C2×C4C4×SD1612yes—
267C2×C4#44312yes—
268C2×C4#46512yes—
269C2×C4#47212yes—
270C2×C4#49211yescharacteristic
271C2×C4#49511yescharacteristic
272C8—28no—
273C8—22no—
274C8—22no—
275C8—22no—
276C8—44no—
277C8—41no—
278C8—82no—
279C8—82no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22—161no—
284C22—161no—
285C22—161no—
286C22—161no—
287C22#196211yescharacteristic
288C22#196711yescharacteristic
289C22#196811yescharacteristic
290C22#197411yescharacteristic
291C22#197611yescharacteristic
292C22#197811yescharacteristic
293C22#198011yescharacteristic
294C4—22no—
295C4—22no—
296C4—22no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—161no—
302C4—161no—
303C4—161no—
304C4#152411yescharacteristic
305C4#152411yescharacteristic
306C4#152911yescharacteristic
307C4#153211yescharacteristic
308C2—321no—
309C2#968711yescharacteristic
310C2#971711yescharacteristic
311C2#972411yescharacteristic
312C2#976211yescharacteristic
313C2#978911yescharacteristic
314C2#980011yescharacteristic
315C2#984311yescharacteristic
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. C8×#97C2semidirect
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
C8×#972
#76832
#76792