Finite Group Databasebeta

← #129313

Subgroups — #129313

Order 512.

994 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C22×#435C211yesmaximal, characteristic
2#12679C212yesmaximal, complemented
3C2×#1322C212yesmaximal, complemented
4C2×#1314C212yesmaximal, complemented
5C23×M16C411yescharacteristic
6C24×C8C2211yescharacteristic
7C22×C4×D4C411yescharacteristic
8C2×#435—24no—
9C2×#435—24no—
10C2×#435—24no—
11C22×C4⋊C8C2211yescharacteristic
12C22×C22⋊C8C411yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×#97C411yescharacteristic
15C2×C22⋊C16—24no—
16C2×C22⋊C16C2214yes—
17#1322—24no—
18#1314—24no—
19C23×D4C811yescharacteristic
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C22×M16—22no—
23C22×M16—22no—
24C22×M16—22no—
25C22×M16—44no—
26C23×C8—22no—
27C23×C8—22no—
28C23×C8—44no—
29C23×C8C2×C411yescharacteristic
30C23×C8C2×C411yescharacteristic
31C23×C8C2×C411yescharacteristic
32C23×C8D414yes—
33C23×C8C2311yescharacteristic
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C22×C4⋊C4C811yescharacteristic
41C22×C22⋊C4C811yescharacteristic
42C22×C22⋊C4C811yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C16—24no—
45#435—232no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8D414yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8D414yes—
55C2×#97—22no—
56C2×#97—22no—
57C2×#97—22no—
58C22⋊C16—24no—
59C22⋊C16—24no—
60C22⋊C16—48no—
61C25C2×C811yescharacteristic
62C25C2×C811yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—44no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4C22⋊C412yes—
80C23×C4C22⋊C412yes—
81C23×C4C2×C811yescharacteristic
82C23×C4C2×C811yescharacteristic
83C23×C4C2×C811yescharacteristic
84C23×C4C2×C811yescharacteristic
85C23×C4M1612yes—
86C23×C4M1612yes—
87C23×C4C22×C411yescharacteristic
88C2×M16—28no—
89C2×M16—48no—
90C2×M16—48no—
91C2×M16—44no—
92C2×M16—44no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—24no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—28no—
101C22×C8—28no—
102C22×C8—44no—
103C22×C8—42no—
104C22×C8—48no—
105C22×C8—48no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8C22⋊C412yes—
109C22×C8C22⋊C412yes—
110C22×C8C22⋊C412yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4×D4—28no—
115C4×D4—28no—
116C4×D4—28no—
117C4×D4—28no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C22⋊C412yes—
132C2×C42M1612yes—
133C2×C16—48no—
134C2×C16—44no—
135C4⋊C8—28no—
136C22⋊C8—28no—
137C22⋊C8—28no—
138#97—28no—
139C24—24no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—24no—
145C24—24no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24C22⋊C812yes—
150C24C22⋊C812yes—
151C24C22×C811yescharacteristic
152C2×D4—28no—
153C2×D4—48no—
154C2×D4—44no—
155C2×D4—48no—
156C2×D4—44no—
157C22×C4—22no—
158C22×C4—22no—
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—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4C22⋊C812yes—
198C22×C4C22⋊C812yes—
199C22×C4C22⋊C812yes—
200C22×C4C22⋊C812yes—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×M1611yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4C2×M1611yescharacteristic
207M16—416no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—416no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8#9914yes—
225C2×C8#12014yes—
226C4⋊C4—28no—
227C22⋊C4—28no—
228C22⋊C4—28no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—22no—
233C16—84no—
234C23—22no—
235C23—22no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—22no—
240C23—24no—
241C23—22no—
242C23—22no—
243C23—24no—
244C23—416no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—88no—
258C23#34914yes—
259C23C2×C22⋊C811yescharacteristic
260C23C2×C22⋊C811yescharacteristic
261C23C2×C22⋊C811yescharacteristic
262C23#50414yes—
263D4—416no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
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—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4#37911yescharacteristic
298C2×C4#40711yescharacteristic
299C2×C4#40711yescharacteristic
300C2×C4#40711yescharacteristic
301C2×C4C2×#9912yes—
302C2×C4#41312yes—
303C8—24no—
304C8—24no—
305C8—42no—
306C8—42no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—28no—
312C22—24no—
313C22—28no—
314C22—48no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—44no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—84no—
323C22—84no—
324C22—84no—
325C22#145912yes—
326C22#177911yescharacteristic
327C22#211312yes—
328C22#211312yes—
329C22#211312yes—
330C22C2×#34912yes—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4#146512yes—
338C4#178111yescharacteristic
339C4#178511yescharacteristic
340C2—22no—
341C2—22no—
342C2—44no—
343C2—82no—
344C2#830111yescharacteristic
345C2#1146714yes—
346C2#1268011yescharacteristic
347C2#1268111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12679C2semidirect
3. C2×#1322C2semidirect
4. C2×#1314C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
#126792
C2×#13222
C2×#13142