Finite Group Databasebeta

← #125343

Subgroups — #125343

Order 512.

584 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 98 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
2C4×#323C211yesmaximal, characteristic
3#7911C211yesmaximal, characteristic
4#7122C214yesmaximal, complemented
5C2×#407—21no—
6C2×#384C411yescharacteristic
7C2×#380—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9C4×C22⋊C8—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×M16—41no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—22no—
27C22×C22⋊C4C2×C411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31#407—24no—
32#407—42no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C4×C8—41no—
37#384—41no—
38#384—41no—
39#380—42no—
40#380—42no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95C2×C412yes—
48C2×#95D411yescharacteristic
49C2×#95C2311yescharacteristic
50#336—41no—
51#323—24no—
52C25C22⋊C411yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—42no—
59C23×C4—42no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4D812yes—
64C23×C4SD1612yes—
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C2×M16—44no—
69C2×M16—82no—
70C2×M16—82no—
71C22×C8—42no—
72C22×C8—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—82no—
83C2×C4⋊C4—82no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—82no—
92C2×C22⋊C4—82no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—42no—
98C2×C42—42no—
99C22⋊C8—24no—
100C22⋊C8—24no—
101C22⋊C8—44no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C4×C8—82no—
105#95—24no—
106#95—41no—
107#95—41no—
108#95—42no—
109#95—42no—
110#95—42no—
111#95—82no—
112#95—82no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—21no—
119C24—41no—
120C24—41no—
121C24—41no—
122C24—41no—
123C24#10212yes—
124C24C2×C22⋊C411yescharacteristic
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
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—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—84no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4#9911yescharacteristic
173C22×C4#9911yescharacteristic
174C22×C4#10112yes—
175C22×C4#10212yes—
176C22×C4#10412yes—
177C22×C4#10412yes—
178C22×C4#13112yes—
179C22×C4C2×D811yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4#13712yes—
182M16—88no—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—88no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—81no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23#40811yescharacteristic
229C23C2×#9911yescharacteristic
230C23C2×#10111yescharacteristic
231C23C2×#10211yescharacteristic
232C23#41911yescharacteristic
233C23C2×#10411yescharacteristic
234C23C2×#10411yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4#32712yes—
285C2×C4#32712yes—
286C2×C4#35212yes—
287C2×C4#35412yes—
288C8—84no—
289C8—84no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#147211yescharacteristic
314C22#148511yescharacteristic
315C22#149211yescharacteristic
316C22C2×#32711yescharacteristic
317C22#150811yescharacteristic
318C22#152611yescharacteristic
319C22#154611yescharacteristic
320C22C2×#35211yescharacteristic
321C22#212211yescharacteristic
322C22C2×#35411yescharacteristic
323C22#213411yescharacteristic
324C4—22no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—162no—
332C4—162no—
333C4#134414yes—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—21no—
338C2—82no—
339C2#768511yescharacteristic
340C2#1156911yescharacteristic
341C2#1157911yescharacteristic
342C2C2×#134411yescharacteristic
343C2#1169911yescharacteristic
344C2#1178711yescharacteristic
345C2#1179011yescharacteristic
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
C4×#3232
#79112
#71222