Finite Group Databasebeta

← #213926

Subgroups — #213926

Order 512.

906 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 186 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#37258C211yesmaximal, characteristic
2#21802C211yesmaximal, characteristic
3#21756C211yesmaximal, characteristic
4#9514C214yesmaximal, complemented
5#9513C214yesmaximal
6#8754C211yesmaximal, characteristic, complemented
7#8753C211yesmaximal, characteristic, complemented
8#8551C211yesmaximal, characteristic, complemented
9#8551C211yesmaximal, characteristic, complemented
10C2×#536C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#2921C2212yes—
13#2879C2212yes—
14#2853C412yes—
15#2734—22no—
16#2731—22no—
17#2729—24no—
18#2614C412yes—
19#2484C2212yes—
20#2413C414yes—
21#2406C2212yes—
22#2310C2211yescharacteristic
23#1967—22no—
24#1966—22no—
25#1901—24no—
26#1822C2212yes—
27#1785C2212yes—
28#1784C2211yescharacteristic
29#1784C2211yescharacteristic
30#1784C2211yescharacteristic
31#1784C2211yescharacteristic
32#1463C2214yes—
33#1463C2214yes—
34#1306C2214yes—
35#1306C2214yes—
36C22×#41C2311yescharacteristic
37C23×Q8C2×C411yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16—22no—
40#536—44no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—44no—
44C2×C4⋊Q8D412yes—
45C2×#124D412yes—
46C2×#123C2×C414yes—
47C2×#123C2×C412yes—
48C2×#123C2×C412yes—
49C2×C22⋊Q8D414yes—
50C2×#117C2×C412yes—
51C2×#117C2311yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2×C411yescharacteristic
55#433—44no—
56#423—22no—
57C2×#101—22no—
58C2×#101—22no—
59C2×#101—22no—
60C2×#101—22no—
61C2×#101—22no—
62C2×#100—22no—
63#408—44no—
64#407—22no—
65#407C2312yes—
66#407C2312yes—
67#407C2312yes—
68#407C2312yes—
69C4×M16—22no—
70#400—22no—
71#398—24no—
72#398—22no—
73#396—24no—
74#395—22no—
75#395—24no—
76#394—22no—
77#393—24no—
78#391—22no—
79#391—22no—
80#389—24no—
81#388—24no—
82#388—24no—
83#387—24no—
84#387—24no—
85#386C2×C414yes—
86#385—24no—
87#385C2×C414yes—
88#381C2×C412yes—
89#381C2×C414yes—
90#381C2312yes—
91#343—24no—
92#324—44no—
93#324—44no—
94C2×#41—24no—
95C2×#41—82no—
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—42no—
99C22×Q8—82no—
100C22×Q8—82no—
101C22×Q8C2×D412yes—
102C22×Q8C2×D412yes—
103C22×Q8C2×D411yescharacteristic
104C22×D4C22×C412yes—
105C22×D4C2×D411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—42no—
117C2×M16—84no—
118C22×C8—42no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—48no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124#124—42no—
125#124—42no—
126#123—44no—
127#123—44no—
128#123—84no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131#117—24no—
132#117—82no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22⋊C414yes—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4C22⋊C414yes—
149C2×C22⋊C4C22⋊C414yes—
150C2×C22⋊C4C22×C412yes—
151C2×C22⋊C4C22×C414yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—42no—
155C2×C42C2×D412yes—
156C4⋊C8—44no—
157#101—82no—
158#101—82no—
159#101—82no—
160#101—82no—
161#101—82no—
162#100—82no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—44no—
168#97—42no—
169C4×C8—42no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177C24C2×C22⋊C412yes—
178C24C23×C411yescharacteristic
179#41—84no—
180#41—84no—
181C2×Q8—24no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—88no—
194C2×Q8—88no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—44no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C414yes—
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4H(5,𝔽2)14yes—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227M16—44no—
228M16—44no—
229M16—84no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—84no—
235C4⋊C4—24no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—82no—
243C4⋊C4—84no—
244C22⋊C4—24no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—48no—
248C22⋊C4—44no—
249C22⋊C4—84no—
250C42—22no—
251C42—22no—
252C42—44no—
253C42—82no—
254C23—22no—
255C23—22no—
256C23—24no—
257C23—42no—
258C23—44no—
259C23C22×C22⋊C411yescharacteristic
260C23#51812yes—
261Q8—84no—
262Q8—82no—
263Q8—82no—
264Q8—82no—
265Q8—88no—
266D4—82no—
267D4—84no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#53412yes—
301C2×C4#53412yes—
302C8—82no—
303C8—82no—
304C8—82no—
305C8—82no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—84no—
314C22#230811yescharacteristic
315C22#288312yes—
316C4—22no—
317C4—22no—
318C4—44no—
319C4—44no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—84no—
324C4—82no—
325C2—22no—
326C2—42no—
327C2—82no—
328C2#2174911yescharacteristic
329C2#2178012yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9514C2semidirect
6. #8754C2semidirect
7. #8753C2semidirect
8. #8551C2semidirect
9. #8551C2semidirect

Maximal subgroups

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

subgroupindex
#372582
#218022
#217562
#95142
#95132
#87542
#87532
#85512