Finite Group Databasebeta

← #122190

Subgroups — #122190

Order 512.

870 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 142 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#35090C211yesmaximal, characteristic
2#10738C212yesmaximal
3#8114C211yesmaximal, characteristic
4#8114C211yesmaximal, characteristic
5C2×#1306C212yesmaximal
6C2×#543C411yescharacteristic
7C22×#117C2211yescharacteristic
8C2×#407C2212yes—
9C2×#407C2212yes—
10#2855—22no—
11#2627—22no—
12#2438—24no—
13#2438—24no—
14#2315C411yescharacteristic
15#2315C411yescharacteristic
16#2315C411yescharacteristic
17C2×#381C2211yescharacteristic
18C2×#381C2211yescharacteristic
19C2×#344—24no—
20#1306—24no—
21#1306—22no—
22#1306—22no—
23C24×C4C2311yescharacteristic
24C22×M16—22no—
25C22×M16—22no—
26#543—44no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×#117—22no—
31C2×#117—44no—
32C2×#117D414yes—
33C22×C4⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C22⋊C4C2×C411yescharacteristic
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C42—21no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407Q814yescomplemented
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8Q814yescomplemented
53#400—24no—
54#398—24no—
55#398—24no—
56#395—24no—
57#391—24no—
58#389—24no—
59#387—24no—
60#387—24no—
61#385—22no—
62#385—22no—
63#385—22no—
64#385—22no—
65#385—22no—
66#385—22no—
67#381—21no—
68#381—21no—
69#381—22no—
70#381D412yes—
71#381D412yes—
72#381D414yes—
73#380—22no—
74#380—22no—
75#380—22no—
76C2×#95—22no—
77C2×#95—22no—
78#344—48no—
79C25C22×C411yescharacteristic
80C22×Q8—22no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×Q812yes—
94C23×C4C2×Q812yes—
95C23×C4#4112yes—
96C23×C4#4112yes—
97C2×M16—24no—
98C2×M16—44no—
99C2×M16—44no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—44no—
104C2×M16—44no—
105C22×C8—24no—
106C22×C8—42no—
107C22⋊Q8—48no—
108C22⋊Q8—44no—
109C22⋊Q8—48no—
110C22⋊Q8—44no—
111#117—28no—
112#117—44no—
113#117—48no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—44no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4C22⋊C412yes—
135C2×C22⋊C4C22⋊C412yes—
136C2×C22⋊C4C22⋊C412yes—
137C2×C22⋊C4C22⋊C412yes—
138C2×C22⋊C4C22⋊C412yes—
139C2×C22⋊C4C22⋊C412yes—
140C2×C42—24no—
141C2×C42—44no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157C24—22no—
158C24—22no—
159C24—82no—
160C24C2×C22⋊C411yescharacteristic
161C24C2×C22⋊C411yescharacteristic
162C24C2×C22⋊C411yescharacteristic
163C24C4×Q812yes—
164C24C4×Q812yes—
165C2×Q8—44no—
166C2×Q8—84no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4#12012yes—
198C22×C4#12012yes—
199C22×C4C22⋊Q812yes—
200C22×C4C22⋊Q812yes—
201C22×C4C22⋊Q812yes—
202C22×C4C22⋊Q812yes—
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4#12412yes—
206C22×C4#12411yescharacteristic
207C22×C4#12412yes—
208C22×C4#12411yescharacteristic
209C22×C4C4⋊Q812yes—
210C22×C4C4⋊Q811yescharacteristic
211C22×C4C4⋊Q812yes—
212C22×C4C4⋊Q811yescharacteristic
213M16—48no—
214M16—84no—
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219C2×C8—48no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—84no—
223C4⋊C4—48no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—24no—
231C22⋊C4—22no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4#9914yes—
243C42—48no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—24no—
249C23—24no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—84no—
256C23—84no—
257C23#37911yescharacteristic
258C23#39112yes—
259C23#39111yescharacteristic
260C23#39111yescharacteristic
261C23#39112yes—
262C23C2×#9912yes—
263C23#41012yes—
264Q8—164no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—162no—
298C2×C4—162no—
299C2×C4—162no—
300C2×C4—162no—
301C2×C4#39312yes—
302C2×C4#39312yes—
303C2×C4#39512yes—
304C2×C4#39512yes—
305C8—84no—
306C8—84no—
307C22—22no—
308C22—24no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—24no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—84no—
322C22—84no—
323C22#144611yescharacteristic
324C22#178111yescharacteristic
325C22#178211yescharacteristic
326C22#199614yes—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—44no—
332C4—82no—
333C4—82no—
334C4—82no—
335C4—82no—
336C4—82no—
337C2—21no—
338C2—21no—
339C2—24no—
340C2—41no—
341C2—41no—
342C2—42no—
343C2#1138111yescharacteristic
344C2#1139512yes—
subgroupG/H
center, socleC22#1446
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
48. #407Q8semidirect
52. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#350902
#107382
#81142
C2×#13062