Finite Group Databasebeta

← #108641

Subgroups — #108641

Order 512.

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

normal subgroups only

This group has 633 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#20525C211yesmaximal, characteristic
2#8107C212yesmaximal
3#8019C212yesmaximal
4#8008C211yesmaximal, characteristic
5C2×#1289C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9C2×#380C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1753—22no—
12#1752C412yescomplemented
13C8×C22⋊C4C412yescomplemented
14#1749—21no—
15C4×C22⋊C8—21no—
16#1305C418yescomplemented
17#1294—22no—
18#1294—22no—
19#1293—22no—
20#1289—21no—
21#1289—21no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—22no—
25C23×C8C2×C412yes—
26C22×C4⋊C4—22no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4—22no—
29C22×C42—21no—
30#407—22no—
31#407—22no—
32#407—22no—
33#407—22no—
34#407C2×C414yes—
35#407C2×C414yes—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8C2×C412yes—
42C2×C22⋊C8D411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×C22⋊C8D412yes—
45C2×#97—22no—
46C2×C4×C8—22no—
47#380—22no—
48#380—22no—
49#380—22no—
50#380C2×C414yes—
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4C2×C411yescharacteristic
54C4×C22⋊C4C2×C411yescharacteristic
55C4×C22⋊C4Q811yescharacteristic
56C4×C22⋊C4Q811yescharacteristic
57C2×#95—22no—
58C2×#95D411yescharacteristic
59C2×#95D411yescharacteristic
60#336—22no—
61#336—22no—
62C25#4111yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—42no—
66C23×C4—82no—
67C23×C4C22×C411yescharacteristic
68C23×C4C22×C412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×Q811yescharacteristic
73C23×C4#4112yes—
74C23×C4#4112yes—
75C23×C4#4111yescharacteristic
76C23×C4#4112yes—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—82no—
81C2×M16—84no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—42no—
86C22×C8—82no—
87C22×C8C22⋊C412yes—
88C22×C8C22⋊C412yes—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—84no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—82no—
100C2×C22⋊C4—82no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—82no—
106C22⋊C8—22no—
107C22⋊C8—24no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110C22⋊C8—24no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8C4214yes—
116#97—82no—
117C4×C8—82no—
118#95—21no—
119#95—21no—
120#95—24no—
121#95—42no—
122#95C4⋊C411yescharacteristic
123#95C4⋊C411yescharacteristic
124#95C4⋊C411yescharacteristic
125#95C4⋊C411yescharacteristic
126C24—22no—
127C24—21no—
128C24—21no—
129C24—42no—
130C24—81no—
131C24#11711yescharacteristic
132C24#11712yes—
133C24#12111yescharacteristic
134C24C22⋊Q811yescharacteristic
135C24#12612yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—44no—
151C22×C4—82no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—82no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—84no—
161C22×C4C2×C4211yescharacteristic
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C2×C4⋊C411yescharacteristic
164C22×C4#11711yescharacteristic
165C22×C4#11711yescharacteristic
166C22×C4#11711yescharacteristic
167C22×C4#11712yes—
168C22×C4C4×D412yes—
169C22×C4C4×D412yes—
170C22×C4C4×D412yes—
171C22×C4C4×Q812yes—
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4#12312yes—
175C22×C4#12312yes—
176C22×C4#12412yes—
177C22×C4#12512yes—
178C22×C4#12612yes—
179M16—84no—
180M16—84no—
181M16—84no—
182C2×C8—24no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—44no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—84no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—84no—
202C42—42no—
203C42—42no—
204C42—81no—
205C42—81no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—41no—
223C23—42no—
224C23—84no—
225C23—82no—
226C23—82no—
227C23#37611yescharacteristic
228C23#38011yescharacteristic
229C23#38311yescharacteristic
230C23#38712yes—
231C23#38812yes—
232C23#47112yes—
233C23#47211yescharacteristic
234C23#47311yescharacteristic
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—168no—
272C2×C4C4×C22⋊C411yescharacteristic
273C2×C4C4×C4⋊C411yescharacteristic
274C2×C4#38111yescharacteristic
275C2×C4#38211yescharacteristic
276C2×C4#38712yes—
277C2×C4#38812yes—
278C8—44no—
279C8—42no—
280C8—42no—
281C8—84no—
282C22—21no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—21no—
291C22—22no—
292C22—21no—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—44no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22—82no—
305C22#143411yescharacteristic
306C22#181511yescharacteristic
307C22#181611yescharacteristic
308C22#184112yes—
309C22#193012yes—
310C22#193112yes—
311C4—21no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—164no—
321C4#182812yes—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—22no—
326C2—22no—
327C2—41no—
328C2—42no—
329C2#1080511yescharacteristic
330C2#1082711yescharacteristic
331C2#1082911yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1752C4semidirect
13. C8×C22⋊C4C4semidirect
16. #1305C4semidirect

Maximal subgroups

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

subgroupindex
#205252
#81072
#80192
#80082
C2×#12892