Finite Group Databasebeta

← #124806

Subgroups — #124806

Order 512.

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

normal subgroups only

This group has 604 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×#395C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7123C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C2×#395C411yescharacteristic
8C2×C4×C4⋊C4—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1817—21no—
11#1749—21no—
12C2×#324—21no—
13#1446C2211yescharacteristic
14#1444—21no—
15#1444C2211yescharacteristic
16#1441—21no—
17#1441C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C2×C4×Q8—44no—
25C2×C4×Q8—41no—
26C2×C4×Q8—41no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42—21no—
34C22×C42—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×#97—41no—
41#395—22no—
42#395—22no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—44no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95C2311yescharacteristic
57#336—41no—
58#336—41no—
59#336—41no—
60#324—24no—
61C22×Q8—21no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8M1612yes—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—42no—
79C22×C8—42no—
80C4×Q8—44no—
81C4×Q8—44no—
82C4×Q8—44no—
83C4×Q8—44no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C42—24no—
103C2×C42—24no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—22no—
110C2×C42—21no—
111C2×C42—41no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—42no—
118C2×C42—42no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#97—82no—
124#95—24no—
125#95—21no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95M1612yes—
136C24C2×C22⋊C411yescharacteristic
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—48no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
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—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4#9911yescharacteristic
207C22×C4#9911yescharacteristic
208C22×C4#10412yes—
209C22×C4C2×M1611yescharacteristic
210C22×C4C2×M1611yescharacteristic
211C22×C4#13512yes—
212C22×C4#13712yes—
213C2×C8—44no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C4⋊C4—44no—
220C4⋊C4—48no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C4⋊C4—42no—
227C42—42no—
228C42—42no—
229C42—48no—
230C42—44no—
231C42—42no—
232C42—42no—
233C42—44no—
234C42—42no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23#40711yescharacteristic
241C23C2×#9911yescharacteristic
242C23#41311yescharacteristic
243C23#41611yescharacteristic
244C23#41911yescharacteristic
245C23C2×#10411yescharacteristic
246C23#42111yescharacteristic
247Q8—44no—
248Q8—44no—
249Q8—44no—
250Q8—44no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4AΣL(1,𝔽16)12yes—
320C2×C4#35612yes—
321C8—84no—
322C8—162no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22#146511yescharacteristic
336C22#148811yescharacteristic
337C22#149111yescharacteristic
338C22#150111yescharacteristic
339C22#151111yescharacteristic
340C22#153211yescharacteristic
341C22#155411yescharacteristic
342C22C2×AΣL(1,𝔽16)11yescharacteristic
343C22#212011yescharacteristic
344C22#212611yescharacteristic
345C22C2×#35611yescharacteristic
346C4—22no—
347C4—22no—
348C4—44no—
349C4—44no—
350C4—41no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—81no—
357C4—81no—
358C4—82no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2#762311yescharacteristic
364C2#1152511yescharacteristic
365C2#1153211yescharacteristic
366C2#1172511yescharacteristic
367C2#1173611yescharacteristic
368C2#1181011yescharacteristic
369C2#1181511yescharacteristic
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×#3952
#87012
#81532
#71312
#71232