Finite Group Databasebeta

← #121210

Subgroups — #121210

Order 512.

483 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 110 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#8115C211yesmaximal, characteristic
2#8057C211yesmaximal, characteristic
3C2×#1298C211yesmaximal, characteristic
4#8009C211yesmaximal, characteristic
5C2×#1293C211yesmaximal, characteristic
6#7927C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2280C2211yescharacteristic
15#1306—22no—
16#1298—22no—
17#1298C414yescomplemented
18#1293—22no—
19#1293—22no—
20#1282—22no—
21#1282—22no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—21no—
26C22×M16—21no—
27C23×C8—21no—
28C23×C8—21no—
29C2×#117—21no—
30#407—21no—
31#407—21no—
32#407—21no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407C2×C412yes—
45#407D412yes—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8C2×C412yes—
63C2×C22⋊C8C2×C412yes—
64C2×C22⋊C8D412yes—
65C2×C22⋊C8D412yes—
66#381—21no—
67#380—21no—
68#380—21no—
69#379Q812yes—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C25C2×Q811yescharacteristic
73C23×C4—82no—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4#4111yescharacteristic
79C23×C4#4111yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—82no—
97C2×M16—82no—
98C2×M16—82no—
99C2×M16—82no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—82no—
111C22×C8—82no—
112#117—82no—
113C2×C4⋊C4—41no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C42—41no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142#95—41no—
143#95—41no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24C4×Q811yescharacteristic
148C24C22⋊Q811yescharacteristic
149C24C22⋊Q811yescharacteristic
150C24C22⋊Q811yescharacteristic
151C24#12511yescharacteristic
152C24#12511yescharacteristic
153C24#12511yescharacteristic
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4#11711yescharacteristic
170C22×C4#11711yescharacteristic
171C22×C4#11711yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12311yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12411yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4#12611yescharacteristic
188C22×C4#12611yescharacteristic
189C22×C4#12611yescharacteristic
190C22×C4#12611yescharacteristic
191C22×C4#12611yescharacteristic
192C22×C4#12611yescharacteristic
193C22×C4#12611yescharacteristic
194C22×C4#12611yescharacteristic
195C22×C4#12611yescharacteristic
196C22×C4#12611yescharacteristic
197M16—82no—
198M16—82no—
199M16—82no—
200M16—82no—
201M16—82no—
202M16—82no—
203M16—82no—
204M16—82no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C4⋊C4—82no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C42—82no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—88no—
248C23#38711yescharacteristic
249C23#38711yescharacteristic
250C23#38711yescharacteristic
251C23#39611yescharacteristic
252C23#40011yescharacteristic
253C23#40011yescharacteristic
254C23#40011yescharacteristic
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#38311yescharacteristic
287C2×C4#38811yescharacteristic
288C2×C4#38811yescharacteristic
289C2×C4#38811yescharacteristic
290C2×C4#39711yescharacteristic
291C2×C4#39711yescharacteristic
292C2×C4#39711yescharacteristic
293C2×C4#40111yescharacteristic
294C8—82no—
295C8—82no—
296C8—82no—
297C8—82no—
298C8—82no—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—44no—
320C22—44no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22#144311yescharacteristic
327C22#210312yes—
328C22#210312yes—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—162no—
337C4#210412yes—
338C2—22no—
339C2—22no—
340C2—41no—
341C2—41no—
342C2—41no—
343C2—41no—
344C2—41no—
345C2#1130611yescharacteristic
346C2#1130611yescharacteristic
347C2#1131811yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ 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
17. #1298C4semidirect

Maximal subgroups

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

subgroupindex
#81152
#80572
C2×#12982
#80092
C2×#12932
#79272
#79252