Finite Group Databasebeta

← #125472

Subgroups — #125472

Order 512.

536 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 80 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#15195C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7122C214yesmaximal, complemented
5C2×#384C411yescharacteristic
6C2×#380—21no—
7C2×C4×C22⋊C4C2211yescharacteristic
8#1779—21no—
9#1749—21no—
10C2×#323—21no—
11#1442—21no—
12#1442C2212yes—
13#1439—21no—
14#1439C412yes—
15C4⋊#95—21no—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C412yes—
25C22×C22⋊C4—22no—
26C22×C22⋊C4C2×C411yescharacteristic
27C22×C22⋊C4D411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C22×C42D411yescharacteristic
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—41no—
33C2×C22⋊C8—41no—
34C2×#97—41no—
35#384—41no—
36#384—41no—
37#380—44no—
38C4×C22⋊C4—41no—
39C4×C22⋊C4—41no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95C2×C412yes—
45C2×#95D411yescharacteristic
46C2×#95C2311yescharacteristic
47#336—41no—
48#323—24no—
49C25C22⋊C411yescharacteristic
50C23×C4—22no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—41no—
56C23×C4—42no—
57C23×C4—42no—
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C22×C8—42no—
65C22×C8—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—82no—
76C2×C4⋊C4—82no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—82no—
85C2×C22⋊C4—82no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—42no—
91C2×C42—42no—
92C22⋊C8—24no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96C22⋊C8—82no—
97C22⋊C8—82no—
98#97—82no—
99#95—24no—
100#95—41no—
101#95—41no—
102#95—42no—
103#95—44no—
104#95—82no—
105#95—82no—
106C24—21no—
107C24—21no—
108C24—21no—
109C24—21no—
110C24—21no—
111C24—21no—
112C24—41no—
113C24—41no—
114C24—41no—
115C24—41no—
116C24#10112yes—
117C24C2×C22⋊C411yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—48no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
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—84no—
165C22×C4#9911yescharacteristic
166C22×C4#9911yescharacteristic
167C22×C4#10412yes—
168C22×C4#13112yes—
169C22×C4AGL(1,ℤ/8)12yes—
170C22×C4#13712yes—
171C2×C8—44no—
172C2×C8—82no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8—82no—
177C4⋊C4—84no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C22⋊C4—44no—
183C22⋊C4—44no—
184C22⋊C4—88no—
185C42—42no—
186C42—42no—
187C42—42no—
188C42—81no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23#40811yescharacteristic
217C23C2×#9911yescharacteristic
218C23C2×#10111yescharacteristic
219C23#41811yescharacteristic
220C23#41911yescharacteristic
221C23C2×#10411yescharacteristic
222C23#42111yescharacteristic
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#35212yes—
271C2×C4#35412yes—
272C8—84no—
273C8—162no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22#147211yescharacteristic
298C22#148411yescharacteristic
299C22#149711yescharacteristic
300C22#150811yescharacteristic
301C22#151011yescharacteristic
302C22#153111yescharacteristic
303C22#155611yescharacteristic
304C22C2×#35211yescharacteristic
305C22#212211yescharacteristic
306C22C2×#35411yescharacteristic
307C22#213411yescharacteristic
308C4—22no—
309C4—22no—
310C4—42no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—81no—
315C4—162no—
316C4—162no—
317C2—21no—
318C2—21no—
319C2—21no—
320C2—21no—
321C2—82no—
322C2#770811yescharacteristic
323C2#1156911yescharacteristic
324C2#1157611yescharacteristic
325C2#1178711yescharacteristic
326C2#1179011yescharacteristic
327C2#1179711yescharacteristic
328C2#1180611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#151952
#87002
#81522
#71222