Finite Group Databasebeta

← #124779

Subgroups — #124779

Order 512.

512 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C4×#398C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7123C212yesmaximal
5#7104C212yesmaximal
6C2×#398C411yescharacteristic
7C2×C4×C4⋊C4—21no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1817—21no—
11#1749—21no—
12C2×#324—21no—
13#1445C2211yescharacteristic
14#1444C2211yescharacteristic
15#1442—21no—
16#1441—21no—
17#1441C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—41no—
37#398—22no—
38#398—22no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—44no—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95D411yescharacteristic
55C2×#95C2311yescharacteristic
56#336—41no—
57#336—41no—
58#336—41no—
59#324—24no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C22×C8—42no—
72C22×C8—42no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4M1612yes—
92C2×C42—24no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—41no—
108C2×C42—41no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#97—82no—
114#95—24no—
115#95—21no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95M1612yes—
127C24C2×C22⋊C411yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4#9911yescharacteristic
189C22×C4#9911yescharacteristic
190C22×C4#10412yes—
191C22×C4C2×M1611yescharacteristic
192C22×C4C2×M1611yescharacteristic
193C22×C4#13512yes—
194C22×C4AGL(1,ℤ/8)12yes—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—48no—
207C4⋊C4—42no—
208C4⋊C4—48no—
209C4⋊C4—42no—
210C42—42no—
211C42—42no—
212C42—48no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—81no—
217C42—82no—
218C42—82no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23#40711yescharacteristic
224C23C2×#9911yescharacteristic
225C23#41311yescharacteristic
226C23#41611yescharacteristic
227C23#41811yescharacteristic
228C23C2×#10411yescharacteristic
229C23#42111yescharacteristic
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
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—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#35212yes—
299C2×C4#35612yes—
300C8—84no—
301C8—162no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22#146511yescharacteristic
315C22#148811yescharacteristic
316C22#149011yescharacteristic
317C22#150111yescharacteristic
318C22#151011yescharacteristic
319C22#153211yescharacteristic
320C22#155311yescharacteristic
321C22C2×#35211yescharacteristic
322C22#212011yescharacteristic
323C22#212611yescharacteristic
324C22C2×#35611yescharacteristic
325C4—22no—
326C4—22no—
327C4—44no—
328C4—41no—
329C4—41no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—82no—
334C4—81no—
335C4—82no—
336C4—82no—
337C4—81no—
338C2—21no—
339C2—21no—
340C2—21no—
341C2—21no—
342C2#762011yescharacteristic
343C2#1152611yescharacteristic
344C2#1153211yescharacteristic
345C2#1172711yescharacteristic
346C2#1173611yescharacteristic
347C2#1179811yescharacteristic
348C2#1180311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C4×#3982
#87012
#81532
#71232
#71042