Finite Group Databasebeta

← #237829

Subgroups — #237829

Order 512.

972 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 194 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#35167C211yesmaximal, characteristic
2#21838C212yesmaximal
3#14289C211yesmaximal, characteristic
4#10343C214yesmaximal
5#10342C214yesmaximal
6#8018C212yesmaximal
7#7912C211yesmaximal, characteristic
8C2×#544—21no—
9C22×#125C2211yescharacteristic
10C2×#432C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2552—22no—
14#2551—22no—
15#2550—24no—
16#2316—21no—
17#2285C2212yes—
18C2×#383C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1992—22no—
21#1992—22no—
22#1989—24no—
23#1824C2214yes—
24#1823C2214yes—
25#1445C2212yes—
26#1445C2212yes—
27#1441—21no—
28#1441—21no—
29#1436—22no—
30#1294C2218yes—
31#1280C2214yes—
32#544—48no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—22no—
35C2×#125—24no—
36C2×#125—22no—
37C2×#125—22no—
38C2×#125—22no—
39C2×#125—24no—
40C2×#125Q814yescomplemented
41C2×C22⋊Q8—24no—
42C2×#117—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51#432—24no—
52#432—24no—
53C2×C4⋊C8—22no—
54C2×C22⋊C8C2314yes—
55C2×#97—22no—
56#398—28no—
57#395—28no—
58#391—24no—
59#389—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#384—22no—
64#384D412yes—
65#384D412yes—
66#383—24no—
67#382—22no—
68#380—24no—
69#380—22no—
70#380C2314yes—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4D414yes—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95C2312yes—
82C2×#95C2312yes—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87C22×Q8—24no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—24no—
97C4⋊Q8—416no—
98C4⋊Q8—84no—
99#125—28no—
100#125—28no—
101#125—416no—
102#125—44no—
103#125—44no—
104C22⋊Q8—48no—
105C22⋊Q8—48no—
106#117—48no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×Q812yes—
135C2×C4⋊C4C2×Q812yes—
136C2×C22⋊C4—24no—
137C2×C42—24no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—82no—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×Q812yes—
144C2×C42#4114yes—
145C2×C42#4112yes—
146C4⋊C8—44no—
147C22⋊C8—24no—
148C22⋊C8—24no—
149#97—44no—
150#95—28no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—42no—
156#95—42no—
157#95—44no—
158#95—44no—
159#95—82no—
160#95—82no—
161C24C22×D411yescharacteristic
162C2×Q8—48no—
163C2×Q8—84no—
164C2×Q8—84no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4#12011yescharacteristic
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4#12412yes—
206C22×C4#12412yes—
207C22×C4AGL(1,ℤ/8)14yes—
208C22×C4#13714yes—
209C22×C4C22×D412yes—
210C22×C4C22×Q811yescharacteristic
211C22×C4C2×#4112yes—
212C22×C4C2×#4111yescharacteristic
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—44no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—48no—
226C4⋊C4—416no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—48no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C22⋊C4—48no—
236C42—44no—
237C42—48no—
238C42—82no—
239C42C22⋊Q814yes—
240C42C22⋊Q814yes—
241C42#12414yes—
242C42#12414yes—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23C2×#12011yescharacteristic
248C23C2×C22⋊Q811yescharacteristic
249C23C2×#12411yescharacteristic
250C23C2×AGL(1,ℤ/8)12yes—
251C23C2×#13712yes—
252Q8—168no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#39314yes—
304C2×C4#39312yes—
305C2×C4#39312yes—
306C2×C4C2×C22⋊Q812yes—
307C2×C4C2×#12412yes—
308C8—84no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22C2×#39311yescharacteristic
318C22#300311yescharacteristic
319C22#300411yescharacteristic
320C22#301911yescharacteristic
321C22#302011yescharacteristic
322C22#302811yescharacteristic
323C22#302811yescharacteristic
324C22#307912yes—
325C22#314112yes—
326C4—24no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—44no—
332C4—44no—
333C4—82no—
334C4—82no—
335C4—84no—
336C4—84no—
337C4—162no—
338C2—22no—
339C2—22no—
340C2#1836311yescharacteristic
341C2#1836411yescharacteristic
342C2#1837311yescharacteristic
343C2#1837411yescharacteristic
344C2#2345211yescharacteristic
345C2#2346211yescharacteristic
346C2#2347011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
40. C2×#125Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218382
#142892
#103432
#103422
#80182
#79122