Finite Group Databasebeta

← #128049

Subgroups — #128049

Order 512.

966 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 138 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#20722C212yesmaximal
2#13847C211yesmaximal, characteristic
3#7307C214yesmaximal
4C22×C4×D4C2211yescharacteristic
5#2280—22no—
6C2×C4×C4⋊C4C411yescharacteristic, complemented
7C2×C4×C22⋊C4—22no—
8C2×#376C411yescharacteristic, complemented
9#1819—22no—
10#1799—22no—
11#1799C414yescomplemented
12#1781C2214yes—
13C4×#99—22no—
14C4×#99C414yescomplemented
15#1445C412yescomplemented
16#1439—22no—
17#1438C2212yes—
18C23×D4C2311yescharacteristic
19C24×C4D412yes—
20C2×C4×D4—41no—
21C2×C4×D4—41no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4C2×C412yes—
25C2×C4×D4C2×C412yes—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4D411yescharacteristic
28C2×C4×D4Q811yescharacteristic
29C2×C4×D4Q811yescharacteristic
30C2×#117—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C22⋊C4—24no—
34C22×C22⋊C4D412yes—
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C2×#99—24no—
39C2×#99C2×C418yes—
40#381—42no—
41#380—24no—
42#380—42no—
43#379—24no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C4×C22⋊C4—24no—
47C4×C22⋊C4—48no—
48C4×C22⋊C4—42no—
49#376—41no—
50#376—41no—
51C2×#95—21no—
52C2×#95—24no—
53C2×#95C2×C412yes—
54C2×#95C2×C412yes—
55#342—24no—
56#342—24no—
57C25C2×D412yes—
58C22×D4—41no—
59C22×D4—41no—
60C22×D4—44no—
61C22×D4C22×C412yes—
62C22×D4C2×D411yescharacteristic
63C22×D4C2×Q811yescharacteristic
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—44no—
71C23×C4—44no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C412yes—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C4⋊C412yes—
78C23×C4C4⋊C412yes—
79C23×C4C22×C411yescharacteristic
80C4×D4—42no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—41no—
92#117—84no—
93#117—84no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4C4212yes—
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C22⋊C4—28no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—48no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—48no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4C22⋊C412yes—
111C2×C22⋊C4C22⋊C412yes—
112C2×C22⋊C4C4⋊C412yes—
113C2×C22⋊C4C4⋊C412yes—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—44no—
123C2×C42—44no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42C4211yescharacteristic
127C2×C42C4211yescharacteristic
128C2×C42C4⋊C411yescharacteristic
129C2×C42C4⋊C411yescharacteristic
130#99—48no—
131#99—48no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—48no—
136#95—44no—
137#95—44no—
138C24—24no—
139C24—22no—
140C24—22no—
141C24—42no—
142C24—42no—
143C24—44no—
144C24—42no—
145C24C2×C22⋊C411yescharacteristic
146C24C2×C22⋊C412yes—
147C24C2×C4⋊C412yes—
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—42no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—48no—
156C2×D4—48no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—48no—
194C22×C4—84no—
195C22×C4—84no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4#9511yescharacteristic
199C22×C4#9511yescharacteristic
200C22×C4#9511yescharacteristic
201C22×C4#9511yescharacteristic
202C22×C4#9512yes—
203C22×C4#9512yes—
204C22×C4#9911yescharacteristic
205C22×C4#9911yescharacteristic
206C22×C4#9911yescharacteristic
207C22×C4#9911yescharacteristic
208C22×C4C2×C4211yescharacteristic
209C22×C4C2×C4⋊C411yescharacteristic
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—48no—
214C4⋊C4—88no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—432no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—88no—
223C22⋊C4—84no—
224C22⋊C4—84no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—44no—
234C42—44no—
235C42—82no—
236C42—88no—
237C42—82no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—44no—
248C23—44no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—48no—
254C23—48no—
255C23—42no—
256C23—42no—
257C23—44no—
258C23—82no—
259C23—82no—
260C23—84no—
261C23—84no—
262C23C2×#9511yescharacteristic
263C23C2×#9911yescharacteristic
264C23C2×#9911yescharacteristic
265C23#41014yes—
266D4—44no—
267D4—48no—
268D4—44no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—88no—
313C2×C4—88no—
314C2×C4—88no—
315C2×C4#34211yescharacteristic
316C2×C4#34211yescharacteristic
317C2×C4#34211yescharacteristic
318C2×C4#34211yescharacteristic
319C2×C4#35312yes—
320C2×C4#35412yes—
321C22—24no—
322C22—24no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—24no—
330C22—44no—
331C22—44no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22—84no—
339C22—88no—
340C22—84no—
341C22—82no—
342C22C2×#34211yescharacteristic
343C22#174012yes—
344C22C2×#35311yescharacteristic
345C22C2×#35411yescharacteristic
346C22#212612yes—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—84no—
356C4—84no—
357C4—168no—
358C4#139312yes—
359C4#139312yes—
360C2—21no—
361C2—22no—
362C2—22no—
363C2—21no—
364C2—44no—
365C2—82no—
366C2#810011yescharacteristic
367C2C2×#139311yescharacteristic
368C2#1258111yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC23×D4C23
socleC22C2×#342
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C4×C4⋊C4C4semidirect
8. C2×#376C4semidirect
11. #1799C4semidirect
14. C4×#99C4semidirect
15. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#138472
#73072