Finite Group Databasebeta

← #213929

Subgroups — #213929

Order 512.

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

normal subgroups only

This group has 921 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#36958C211yesmaximal, characteristic
2#21757C212yesmaximal
3#9513C214yesmaximal, complemented
4#9513C214yesmaximal, complemented
5#8755C212yesmaximal, complemented
6#8554C211yesmaximal, characteristic, complemented
7#8554C211yesmaximal, characteristic, complemented
8C2×#537C2211yescharacteristic
9C22×#125C2211yescharacteristic
10#2879C2214yes—
11#2854C412yes—
12#2720—24no—
13#2711—24no—
14#2618C412yes—
15#2591C412yes—
16#2501C412yes—
17#2406C2212yes—
18#2406C2212yes—
19#2313C2211yescharacteristic
20#1901—24no—
21#1900—24no—
22#1821C2212yes—
23#1821C2212yes—
24#1785C2214yes—
25#1463C2212yes—
26#1463C2212yes—
27#1463C2212yes—
28#1463C2212yes—
29#1306C2218yes—
30C22×#41C2312yes—
31C22×M16—24no—
32#537—42no—
33#537—42no—
34C2×#125—24no—
35C2×#125—42no—
36C2×#125—42no—
37C2×#123C2×C414yes—
38C2×#123C2×C414yes—
39C2×C22⋊Q8D414yes—
40C2×#121C2×C412yes—
41C2×#121C2×C412yes—
42C2×#121D414yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×#101—24no—
51C2×#100—24no—
52#408—44no—
53#408—44no—
54#407—24no—
55#407C2314yes—
56#407C2314yes—
57#400—22no—
58#399—24no—
59#398—22no—
60#396—24no—
61#395—22no—
62#395—22no—
63#394—22no—
64#394—22no—
65#394—22no—
66#394—22no—
67#390—22no—
68#390—22no—
69#389—22no—
70#388—24no—
71#388—24no—
72#387—24no—
73#386C2×C414yes—
74#386C2×C414yes—
75#385—24no—
76#385—24no—
77#384—22no—
78#381C2312yes—
79#380C2×C412yes—
80C4×C22⋊C4C2×C412yes—
81#343—24no—
82#343—24no—
83#324—42no—
84#324—42no—
85#324—42no—
86#324—42no—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—84no—
90C22×Q8C2×D412yes—
91C22×D4C22⋊C412yes—
92C22×D4C22⋊C412yes—
93C22×D4C22×C414yes—
94C22×D4C2×D412yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D414yes—
102C23×C4C2411yescharacteristic
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—44no—
106C2×M16—84no—
107C2×M16—84no—
108C22×C8—44no—
109#125—48no—
110#125—48no—
111#123—84no—
112#123—84no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115#121—44no—
116#121—44no—
117#121—42no—
118#121—42no—
119#121—42no—
120#121—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C2×D414yes—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4C22⋊C414yes—
149C2×C22⋊C4C22⋊C414yes—
150C2×C22⋊C4C22×C414yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—41no—
154C2×C42—41no—
155#101—84no—
156#100—84no—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—48no—
162#95—22no—
163#95—22no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168C24C2×C22⋊C414yes—
169C24C23×C411yescharacteristic
170#41—84no—
171#41—84no—
172#41—84no—
173#41—84no—
174C2×Q8—24no—
175C2×Q8—44no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—82no—
185C2×D4—82no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C22×D412yes—
218C22×C4H(5,𝔽2)12yes—
219C22×C4H(5,𝔽2)12yes—
220M16—44no—
221M16—44no—
222M16—44no—
223M16—44no—
224M16—88no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—88no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—84no—
238C4⋊C4—84no—
239C4⋊C4—82no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—48no—
243C22⋊C4—44no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C22⋊C4—84no—
247C42—44no—
248C42—82no—
249C23—22no—
250C23—24no—
251C23—24no—
252C23—24no—
253C23—41no—
254C23—41no—
255C23—44no—
256C23—44no—
257C23C22×C22⋊C411yescharacteristic
258C23#51811yescharacteristic
259C23#51811yescharacteristic
260Q8—84no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—162no—
265D4—162no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4#53414yes—
307C8—84no—
308C8—84no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—84no—
322C22—84no—
323C22#230811yescharacteristic
324C22#288312yes—
325C4—24no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—44no—
330C4—84no—
331C4—81no—
332C4—81no—
333C4—81no—
334C2—22no—
335C2—41no—
336C2—41no—
337C2—84no—
338C2#2175011yescharacteristic
339C2#2177911yescharacteristic
340C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#537C22
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
3. #9513C2semidirect
4. #9513C2semidirect
5. #8755C2semidirect
6. #8554C2semidirect
7. #8554C2semidirect

Maximal subgroups

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

subgroupindex
#369582
#217572
#95132
#87552
#85542