Finite Group Databasebeta

← #128012

Subgroups — #128012

Order 512.

1508 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1508 conjugacy classes of subgroups — too many to draw. Only its 150 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#20591C212yesmaximal
2#13851C211yesmaximal, characteristic
3#7308C214yesmaximal, complemented
4C23×#41C2211yescharacteristic
5C2×#413—22no—
6C2×#413C414yescomplemented
7C2×#407—22no—
8C2×C4×C22⋊C4—22no—
9C2×C4×C22⋊C4C412yes—
10#1785—22no—
11#1785C414yescomplemented
12#1784C2214yes—
13#1783—22no—
14#1446C2212yes—
15#1438—22no—
16#1438C412yes—
17C22×#41—42no—
18C22×#41—42no—
19C22×#41—48no—
20C22×#41C2×C412yes—
21C22×#41C2×C412yes—
22C22×#41D411yescharacteristic
23C22×#41D411yescharacteristic
24C22×#41Q811yescharacteristic
25C22×#41Q811yescharacteristic
26C23×Q8C2311yescharacteristic
27C23×D4C2×C411yescharacteristic
28C23×D4D412yes—
29C24×C4C2×C411yescharacteristic
30C24×C4D412yes—
31C22×M16—24no—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—24no—
34C22×C42—22no—
35#413—48no—
36#413—48no—
37C2×#101—24no—
38C2×#101C2×C418yes—
39C2×#100—24no—
40C2×#100—24no—
41#407—24no—
42#407—24no—
43#407—44no—
44C2×C22⋊C8—44no—
45C4×C22⋊C4—48no—
46C4×C22⋊C4—42no—
47C4×C22⋊C4—42no—
48C2×#95—24no—
49C2×#95C2×C414yes—
50C25C22⋊C412yes—
51C25C22⋊C411yescharacteristic
52C2×#41—28no—
53C2×#41—28no—
54C2×#41—42no—
55C2×#41—44no—
56C2×#41—42no—
57C2×#41—44no—
58C2×#41—41no—
59C2×#41—41no—
60C2×#41—42no—
61C2×#41—42no—
62C2×#41—41no—
63C2×#41—44no—
64C2×#41—41no—
65C2×#41—44no—
66C2×#41—44no—
67C2×#41—44no—
68C2×#41—44no—
69C2×#41—432no—
70C2×#41—432no—
71C22×Q8—42no—
72C22×Q8—44no—
73C22×Q8C22×C412yes—
74C22×Q8C2×D411yescharacteristic
75C22×Q8C2×Q811yescharacteristic
76C22×D4—44no—
77C22×D4—42no—
78C22×D4—44no—
79C22×D4—44no—
80C22×D4—44no—
81C22×D4C4212yes—
82C22×D4C22⋊C412yes—
83C22×D4C22⋊C412yes—
84C22×D4C4⋊C411yescharacteristic
85C22×D4C4⋊C412yes—
86C22×D4C4⋊C411yescharacteristic
87C22×D4C4⋊C412yes—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—28no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4C4211yescharacteristic
97C23×C4C4211yescharacteristic
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C4⋊C411yescharacteristic
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C412yes—
104C23×C4C4⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D412yes—
107C2×M16—28no—
108C2×M16—44no—
109C2×M16—48no—
110C2×M16—48no—
111C22×C8—44no—
112C2×C22⋊C4—416no—
113C2×C22⋊C4—48no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—44no—
116C2×C42—48no—
117C2×C42—42no—
118C2×C42—42no—
119#101—48no—
120#101—48no—
121#100—48no—
122#100—48no—
123C22⋊C8—48no—
124C22⋊C8—48no—
125#95—44no—
126#95—48no—
127C24—24no—
128C24—22no—
129C24—24no—
130C24—42no—
131C24—42no—
132C24—48no—
133C24—44no—
134C24#9511yescharacteristic
135C24#9511yescharacteristic
136C24#9512yes—
137C24#9512yes—
138C24C2×C22⋊C411yescharacteristic
139#41—48no—
140#41—48no—
141#41—416no—
142#41—416no—
143#41—416no—
144#41—432no—
145#41—416no—
146#41—48no—
147#41—48no—
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—416no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—432no—
176C2×D4—416no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—416no—
208C22×C4—416no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—48no—
214C22×C4—44no—
215C22×C4#9511yescharacteristic
216C22×C4#9511yescharacteristic
217C22×C4#9912yes—
218C22×C4#9912yes—
219C22×C4#9918yes—
220C22×C4C2×C4211yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C4⋊C411yescharacteristic
223C22×C4C2×C4⋊C412yes—
224M16—48no—
225M16—48no—
226M16—88no—
227C2×C8—48no—
228C2×C8—88no—
229C22⋊C4—432no—
230C22⋊C4—416no—
231C42—84no—
232C42—84no—
233C23—24no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—48no—
245C23—44no—
246C23—44no—
247C23—48no—
248C23—44no—
249C23—44no—
250C23—48no—
251C23—42no—
252C23—42no—
253C23—48no—
254C23—44no—
255C23—84no—
256C23—84no—
257C23—84no—
258C23—84no—
259C23—84no—
260C23#34214yes—
261C23C2×#9511yescharacteristic
262C23C2×#9912yes—
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—48no—
267D4—48no—
268D4—44no—
269D4—48no—
270D4—48no—
271D4—48no—
272D4—44no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—416no—
300C2×C4—48no—
301C2×C4—84no—
302C2×C4—88no—
303C2×C4—84no—
304C2×C4#34212yes—
305C2×C4#34211yescharacteristic
306C2×C4#34214yes—
307C2×C4#34211yescharacteristic
308C2×C4C2×#9914yes—
309C8—88no—
310C22—24no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—24no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—42no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—82no—
326C22—816no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—88no—
331C22C2×#34212yes—
332C22C2×#34211yescharacteristic
333C22#213214yes—
334C4—24no—
335C4—22no—
336C4—41no—
337C4—42no—
338C4—41no—
339C4—88no—
340C4#130412yes—
341C4#130412yes—
342C2—24no—
343C2—22no—
344C2—44no—
345C2—82no—
346C2—81no—
347C2—81no—
348C2C2×#130411yescharacteristic
349C2#1258512yes—
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
socleC22C2×#342
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7308C2semidirect
6. C2×#413C4semidirect
11. #1785C4semidirect

Maximal subgroups

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

subgroupindex
#205912
#138512
#73082