Finite Group Databasebeta

← #116918

Subgroups — #116918

Order 512.

596 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 132 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#21959C211yesmaximal, characteristic, complemented
2#21852C211yesmaximal, characteristic
3#17581C212yesmaximal
4C2×#1290C211yesmaximal, characteristic, complemented
5#7952C212yesmaximal, complemented
6C2×#529C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10#2285C2212yes—
11#1985C412yescomplemented
12#1955C412yescomplemented
13#1948—24no—
14#1869—22no—
15#1807—22no—
16#1798—22no—
17#1520—22no—
18#1519—22no—
19#1512—24no—
20#1509—24no—
21#1290—22no—
22#1285—24no—
23C23×C8D412yes—
24#529—22no—
25#529—22no—
26#529—22no—
27#529—22no—
28C2×#126—21no—
29C2×#125Q811yescharacteristic
30C2×#124Q811yescharacteristic
31C2×#123—21no—
32C2×C22⋊Q8D411yescharacteristic
33C2×#121D411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×#117—22no—
38C2×#117—21no—
39C22×C42C2311yescharacteristic
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432D414yes—
46C2×#106—24no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2×C412yes—
52C2×C4⋊C8D412yes—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—24no—
56C2×#97—22no—
57C2×#97C2×C412yes—
58#384—22no—
59#382—22no—
60#380—22no—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4D414yes—
64#340—24no—
65#336—24no—
66#328—28no—
67#327—28no—
68C22×Q8C4⋊C411yescharacteristic
69C22×D4—21no—
70C22×D4C4⋊C411yescharacteristic
71C23×C4—21no—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4#4111yescharacteristic
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—42no—
79C22×C8—82no—
80C22×C8C22⋊C414yes—
81#126—42no—
82#126—42no—
83#125—22no—
84#125—22no—
85#124—22no—
86#124—22no—
87#123—42no—
88#123—42no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91#121—22no—
92#121—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101#117—42no—
102#117—42no—
103#117—84no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C4⋊C411yescharacteristic
112C2×C4⋊C4C4⋊C411yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—42no—
119C2×C42—21no—
120C2×C42—24no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42#4112yes—
127C2×C42#4112yes—
128C2×C42#4111yescharacteristic
129#106—84no—
130C4⋊C8—24no—
131C4⋊C8—24no—
132C4⋊C8—24no—
133C4⋊C8—42no—
134C4⋊C8—42no—
135C22⋊C8—24no—
136C22⋊C8—42no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—84no—
140#97—24no—
141#97—24no—
142#97—42no—
143#95—24no—
144#95—42no—
145C24—21no—
146C24#12111yescharacteristic
147C2×Q8—22no—
148C2×Q8—81no—
149C2×D4—22no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—44no—
153C2×D4—81no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—44no—
165C22×C4—81no—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C2×C4⋊C411yescharacteristic
168C22×C4#11711yescharacteristic
169C22×C4#12011yescharacteristic
170C22×C4#12012yes—
171C22×C4#12112yes—
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4C22⋊Q811yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12312yes—
177C22×C4#12411yescharacteristic
178C22×C4#12612yes—
179C22×C4#12711yescharacteristic
180C2×C8—24no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—82no—
188C2×C8—84no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—84no—
199C4⋊C4—84no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—41no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—41no—
207C22⋊C4—84no—
208C42—41no—
209C42—41no—
210C42—84no—
211C42C4×D412yes—
212C42C4×D412yes—
213C42#12112yes—
214C42#12112yes—
215C42C22⋊Q812yes—
216C42C22⋊Q812yes—
217C42#12312yes—
218C42#12412yes—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—41no—
225C23#38011yescharacteristic
226C23#39211yescharacteristic
227C23#39711yescharacteristic
228C23#47114yes—
229Q8—82no—
230D4—82no—
231D4—82no—
232D4—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4#38512yes—
264C2×C4#38611yescharacteristic
265C2×C4#38812yes—
266C2×C4#39011yescharacteristic
267C2×C4#39311yescharacteristic
268C2×C4#39412yes—
269C2×C4#39612yes—
270C2×C4#39911yescharacteristic
271C8—44no—
272C8—82no—
273C8—84no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—44no—
281C22—84no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#143911yescharacteristic
286C22#181512yes—
287C22#201912yes—
288C22#207912yes—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—164no—
300C4#191111yescharacteristic
301C4#191211yescharacteristic
302C2—22no—
303C2—22no—
304C2—22no—
305C2—161no—
306C2#1107211yescharacteristic
307C2#1110211yescharacteristic
308C2#1110611yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21959C2semidirect
4. C2×#1290C2semidirect
5. #7952C2semidirect
11. #1985C4semidirect
12. #1955C4semidirect

Maximal subgroups

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

subgroupindex
#219592
#218522
#175812
C2×#12902
#79522