Finite Group Databasebeta

← #125332

Subgroups — #125332

Order 512.

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

normal subgroups only

This group has 604 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#15150C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7911C211yesmaximal, characteristic
4#7104C214yesmaximal, complemented
5C2×#407—21no—
6C2×#397—21no—
7C2×#397C412yes—
8C2×#384C411yescharacteristic
9C2×#380—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#323—21no—
13#1445—21no—
14#1445C2212yes—
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#397—44no—
38#397—44no—
39#384—41no—
40#384—41no—
41#380—42no—
42#380—42no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C412yes—
50C2×#95D411yescharacteristic
51C2×#95C2311yescharacteristic
52#336—41no—
53#323—24no—
54C25C22⋊C411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4D812yes—
66C23×C4SD1612yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C2×M16—44no—
71C2×M16—82no—
72C2×M16—82no—
73C22×C8—42no—
74C22×C8—42no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—44no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—44no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—42no—
102C2×C42—42no—
103C22⋊C8—24no—
104C22⋊C8—24no—
105C22⋊C8—44no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C4×C8—82no—
109#95—41no—
110#95—41no—
111#95—42no—
112#95—42no—
113#95—44no—
114#95—42no—
115#95—42no—
116#95—44no—
117C24—21no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24#10212yes—
128C24C2×C22⋊C411yescharacteristic
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
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—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—84no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4#9911yescharacteristic
177C22×C4#9911yescharacteristic
178C22×C4#10012yes—
179C22×C4#10212yes—
180C22×C4#10412yes—
181C22×C4#10412yes—
182C22×C4#13112yes—
183C22×C4C2×D811yescharacteristic
184C22×C4C2×SD1611yescharacteristic
185C22×C4AGL(1,ℤ/8)12yes—
186M16—88no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—82no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C4⋊C4—84no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—88no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—81no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23#40811yescharacteristic
233C23C2×#9911yescharacteristic
234C23C2×#10011yescharacteristic
235C23C2×#10211yescharacteristic
236C23#41811yescharacteristic
237C23C2×#10411yescharacteristic
238C23C2×#10411yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
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—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#32712yes—
289C2×C4#32712yes—
290C2×C4#35212yes—
291C2×C4#35412yes—
292C8—84no—
293C8—84no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—44no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#147111yescharacteristic
318C22#148411yescharacteristic
319C22#149211yescharacteristic
320C22C2×#32711yescharacteristic
321C22#150711yescharacteristic
322C22#152511yescharacteristic
323C22#154511yescharacteristic
324C22C2×#35211yescharacteristic
325C22#212311yescharacteristic
326C22C2×#35411yescharacteristic
327C22#213511yescharacteristic
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—162no—
336C4—162no—
337C4#134414yes—
338C2—21no—
339C2—21no—
340C2—21no—
341C2—21no—
342C2—82no—
343C2#768411yescharacteristic
344C2#1156211yescharacteristic
345C2#1156411yescharacteristic
346C2C2×#134411yescharacteristic
347C2#1170011yescharacteristic
348C2#1177711yescharacteristic
349C2#1178011yescharacteristic
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. #7104C2semidirect

Maximal subgroups

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

subgroupindex
#151502
C4×#3232
#79112
#71042