Finite Group Databasebeta

← #121257

Subgroups — #121257

Order 512.

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

normal subgroups only

This group has 529 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#8108C211yesmaximal, characteristic
2#8012C211yesmaximal, characteristic
3C2×#1293C211yesmaximal, characteristic
4#7946C211yesmaximal, characteristic
5#7942C211yesmaximal, characteristic
6#7929C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1305—22no—
16#1294—22no—
17#1293—22no—
18#1284—22no—
19#1284—22no—
20#1282—22no—
21#1282C414yescomplemented
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—21no—
26C23×C8—21no—
27C2×#117—21no—
28C2×#117—21no—
29C22×C22⋊C4—21no—
30C22×C42—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#407D412yes—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
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⋊C8C2×C412yes—
53C2×C22⋊C8C2×C412yes—
54C2×C22⋊C8Q812yes—
55#381—21no—
56#381—21no—
57#380—21no—
58#380—21no—
59#380—21no—
60#380D412yes—
61#379—21no—
62#379—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C412yes—
70C4×C22⋊C4D412yes—
71C2×#95—21no—
72C25#4111yescharacteristic
73C23×C4—41no—
74C23×C4—82no—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q811yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—82no—
96C2×M16—82no—
97C2×M16—82no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—82no—
105#117—82no—
106#117—82no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—82no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—41no—
143#95—41no—
144#95—41no—
145C24—42no—
146C24—42no—
147C24—81no—
148C24#11711yescharacteristic
149C24#12311yescharacteristic
150C24#12311yescharacteristic
151C24#12411yescharacteristic
152C24#12511yescharacteristic
153C24#12611yescharacteristic
154C24#12611yescharacteristic
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4#11711yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4C4×Q811yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12411yescharacteristic
192C22×C4#12411yescharacteristic
193C22×C4#12511yescharacteristic
194C22×C4#12511yescharacteristic
195C22×C4#12611yescharacteristic
196C22×C4#12611yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4#12611yescharacteristic
199C22×C4#12611yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4#12611yescharacteristic
202C22×C4#12611yescharacteristic
203M16—82no—
204M16—82no—
205M16—82no—
206M16—82no—
207M16—82no—
208M16—82no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C22⋊C4—24no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—82no—
231C42—82no—
232C42—82no—
233C42—82no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—88no—
256C23#38311yescharacteristic
257C23#38711yescharacteristic
258C23#38811yescharacteristic
259C23#39711yescharacteristic
260C23#39711yescharacteristic
261C23#40011yescharacteristic
262C23#40011yescharacteristic
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
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—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—88no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#38711yescharacteristic
302C2×C4#38711yescharacteristic
303C2×C4#38811yescharacteristic
304C2×C4#38811yescharacteristic
305C2×C4#39611yescharacteristic
306C2×C4#39711yescharacteristic
307C2×C4#40011yescharacteristic
308C2×C4#40111yescharacteristic
309C8—82no—
310C8—82no—
311C8—82no—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—44no—
333C22—44no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22#144311yescharacteristic
340C22#184112yes—
341C22#208012yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—84no—
350C4—162no—
351C4—162no—
352C4#210412yes—
353C2—22no—
354C2—22no—
355C2—41no—
356C2—41no—
357C2—41no—
358C2—41no—
359C2—41no—
360C2#1130011yescharacteristic
361C2#1131811yescharacteristic
362C2#1132011yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
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
21. #1282C4semidirect

Maximal subgroups

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

subgroupindex
#81082
#80122
C2×#12932
#79462
#79422
#79292
#79252