Finite Group Databasebeta

← #213927

Subgroups — #213927

Order 512.

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

normal subgroups only

This group has 962 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#37187C211yesmaximal, characteristic
2#21802C211yesmaximal, characteristic, complemented
3#21756C211yesmaximal, characteristic, complemented
4#9512C214yesmaximal, complemented
5#9511C214yesmaximal, complemented
6#8754C211yesmaximal, characteristic, complemented
7#8753C211yesmaximal, characteristic, complemented
8#8552C211yesmaximal, characteristic, complemented
9#8552C211yesmaximal, characteristic, complemented
10C2×#536C2211yescharacteristic
11C22×#124C2211yescharacteristic
12#2921C2212yes—
13#2879C2212yes—
14#2853C412yes—
15#2733—22no—
16#2731—22no—
17#2730—24no—
18#2613C412yes—
19#2482C2212yes—
20#2411C414yes—
21#2404C2212yes—
22#2310C2211yescharacteristic
23#1967—22no—
24#1966—22no—
25#1901—24no—
26#1822C2212yes—
27#1785C2212yes—
28#1784C2211yescharacteristic
29#1784C2211yescharacteristic
30#1784C2211yescharacteristic
31#1784C2211yescharacteristic
32#1462C2214yes—
33#1462C2214yes—
34#1306C2214yes—
35#1306C2214yes—
36C22×#41C2311yescharacteristic
37C23×Q8C2311yescharacteristic
38C23×D4C2×C411yescharacteristic
39C22×M16—22no—
40#536—44no—
41C2×C4⋊Q8D412yes—
42C2×#124—22no—
43C2×#124—22no—
44C2×#124—44no—
45C2×#124D412yes—
46C2×#123C2×C412yes—
47C2×#123C2×C412yes—
48C2×C22⋊Q8D414yes—
49C2×#120C2×C414yes—
50C2×#117C2×C412yes—
51C2×#117C2311yescharacteristic
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊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#399—22no—
71#397—22no—
72#396—24no—
73#394—24no—
74#394—24no—
75#393—22no—
76#392—22no—
77#392—24no—
78#391—22no—
79#391—22no—
80#390—24no—
81#388—24no—
82#388—24no—
83#388—24no—
84#387—24no—
85#386—24no—
86#386C2×C414yes—
87#385C2×C414yes—
88#381C2×C412yes—
89#381C2312yes—
90#379C2×C414yes—
91#343—24no—
92#323—44no—
93#323—44no—
94C25C22×C412yes—
95C2×#41—24no—
96C2×#41—82no—
97C22×Q8—24no—
98C22×Q8—82no—
99C22×Q8C2×D412yes—
100C22×Q8C2×D412yes—
101C22×Q8C2×D411yescharacteristic
102C22×D4—24no—
103C22×D4—22no—
104C22×D4—42no—
105C22×D4—82no—
106C22×D4C22×C412yes—
107C22×D4C2×D411yescharacteristic
108C23×C4C22×C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2411yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—42no—
118C2×M16—84no—
119C22×C8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—42no—
122#124—42no—
123#124—42no—
124#124—48no—
125#124—44no—
126#124—44no—
127#123—44no—
128#123—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131#120—84no—
132#117—24no—
133#117—82no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4C22⋊C414yes—
137C2×C4⋊C4C22⋊C414yes—
138C2×C4⋊C4C2×D412yes—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4C22⋊C414yes—
151C2×C22⋊C4C22⋊C414yes—
152C2×C22⋊C4C22×C412yes—
153C2×C22⋊C4C22×C414yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—42no—
157C2×C42C2×D412yes—
158C4⋊C8—44no—
159#101—82no—
160#101—82no—
161#101—82no—
162#101—82no—
163#101—82no—
164#100—82no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—24no—
169C22⋊C8—44no—
170#97—42no—
171C4×C8—42no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177C24—22no—
178C24—22no—
179C24—22no—
180C24—44no—
181C24—82no—
182C24—82no—
183C24C2×C22⋊C412yes—
184C24C23×C411yescharacteristic
185#41—84no—
186#41—84no—
187C2×Q8—24no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—88no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—88no—
205C2×D4—84no—
206C2×D4—84no—
207C2×D4—84no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C414yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4H(5,𝔽2)14yes—
228M16—42no—
229M16—42no—
230M16—42no—
231M16—42no—
232M16—44no—
233M16—44no—
234M16—84no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—84no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—82no—
244C4⋊C4—84no—
245C22⋊C4—24no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—48no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—84no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C22⋊C4—84no—
256C42—22no—
257C42—22no—
258C42—44no—
259C42—82no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—42no—
264C23—44no—
265C23—42no—
266C23—44no—
267C23—42no—
268C23—44no—
269C23—44no—
270C23—42no—
271C23—44no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—82no—
276C23—168no—
277C23C22×C22⋊C411yescharacteristic
278C23#51812yes—
279Q8—84no—
280Q8—82no—
281Q8—82no—
282Q8—82no—
283D4—82no—
284D4—84no—
285D4—88no—
286D4—168no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#53412yes—
313C2×C4#53412yes—
314C8—82no—
315C8—82no—
316C8—82no—
317C8—82no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—84no—
326C22—84no—
327C22—84no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—164no—
332C22—164no—
333C22#230811yescharacteristic
334C22#288312yes—
335C4—22no—
336C4—22no—
337C4—44no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—84no—
342C4—82no—
343C2—22no—
344C2—42no—
345C2—82no—
346C2—162no—
347C2#2174911yescharacteristic
348C2#2178112yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radical#37187C2
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
2. #21802C2semidirect
3. #21756C2semidirect
4. #9512C2semidirect
5. #9511C2semidirect
6. #8754C2semidirect
7. #8753C2semidirect
8. #8552C2semidirect
9. #8552C2semidirect

Maximal subgroups

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

subgroupindex
#371872
#218022
#217562
#95122
#95112
#87542
#87532
#85522