Finite Group Databasebeta

← #121129

Subgroups — #121129

Order 512.

500 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 108 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#8114C211yesmaximal, characteristic
2#8104C211yesmaximal, characteristic
3#8049C211yesmaximal, characteristic
4#8024C211yesmaximal, characteristic
5C2×#1293C211yesmaximal, characteristic
6#7943C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#381C2211yescharacteristic
15#1306—22no—
16#1305—22no—
17#1297—22no—
18#1294—22no—
19#1293C414yescomplemented
20#1284—22no—
21#1282—22no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—21no—
26C23×C8—21no—
27C2×#117—21no—
28C2×#117—21no—
29C22×C22⋊C4—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#407C2×C412yes—
41#407C2×C412yes—
42C2×C22⋊C8—21no—
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⋊C8Q812yes—
54#381—21no—
55#381—21no—
56#381—21no—
57#381—21no—
58#381—21no—
59#381D412yes—
60#380—21no—
61#380—21no—
62#380—21no—
63#380D412yes—
64#379—21no—
65#379—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4D412yes—
70C2×#95—21no—
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—22no—
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—41no—
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—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—82no—
118C2×C42—22no—
119C2×C42—41no—
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—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24C4×Q811yescharacteristic
148C24#12311yescharacteristic
149C24#12311yescharacteristic
150C24#12311yescharacteristic
151C24#12611yescharacteristic
152C24#12611yescharacteristic
153C24#12611yescharacteristic
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—42no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4#11711yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4#12411yescharacteristic
190C22×C4#12411yescharacteristic
191C22×C4#12411yescharacteristic
192C22×C4#12511yescharacteristic
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
202M16—82no—
203M16—82no—
204M16—82no—
205M16—82no—
206M16—82no—
207M16—82no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—44no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C42—82no—
229C42—82no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—88no—
252C23#38711yescharacteristic
253C23#38711yescharacteristic
254C23#38711yescharacteristic
255C23#39711yescharacteristic
256C23#39711yescharacteristic
257C23#39711yescharacteristic
258C23#40111yescharacteristic
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
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—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—164no—
297C2×C4#38311yescharacteristic
298C2×C4#38811yescharacteristic
299C2×C4#38811yescharacteristic
300C2×C4#38811yescharacteristic
301C2×C4#39611yescharacteristic
302C2×C4#40011yescharacteristic
303C2×C4#40011yescharacteristic
304C2×C4#40011yescharacteristic
305C8—82no—
306C8—82no—
307C8—82no—
308C8—82no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
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—44no—
329C22—44no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22#144311yescharacteristic
336C22#208012yes—
337C22#210412yes—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—162no—
347C4—162no—
348C4—162no—
349C2—22no—
350C2—22no—
351C2—41no—
352C2—41no—
353C2—41no—
354C2—41no—
355C2—41no—
356C2#1130511yescharacteristic
357C2#1131111yescharacteristic
358C2#1131811yescharacteristic
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
19. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#81142
#81042
#80492
#80242
C2×#12932
#79432
#79252