Finite Group Databasebeta

← #116906

Subgroups — #116906

Order 512.

584 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 126 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#21960C211yesmaximal, characteristic, complemented
2#21848C211yesmaximal, characteristic
3#17580C212yesmaximal
4#8002C211yesmaximal, characteristic, complemented
5#7858C212yesmaximal, complemented
6C2×#529C2211yescharacteristic
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10#2283C2212yes—
11#1984C412yescomplemented
12#1954C412yescomplemented
13#1949—24no—
14#1869—21no—
15#1869—21no—
16#1808—21no—
17#1807—21no—
18#1798—22no—
19#1521—24no—
20#1511—22no—
21#1510—22no—
22#1509—24no—
23#1292—22no—
24#1274—24no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27#529—22no—
28#529—22no—
29#529—22no—
30#529—22no—
31C2×#126—21no—
32C2×#125Q811yescharacteristic
33C2×#124Q811yescharacteristic
34C2×#123—21no—
35C2×C22⋊Q8D411yescharacteristic
36C2×#121D411yescharacteristic
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4C2×C411yescharacteristic
40C2×#117—21no—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#431—22no—
45#431—22no—
46#431—22no—
47#431D414yes—
48C2×#107—22no—
49C2×#106—22no—
50#423—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2×C412yes—
55C2×C4⋊C8D412yes—
56#407—22no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C4×M16—22no—
61C2×C4×C8C2×C412yes—
62#383—22no—
63#382—22no—
64#381—22no—
65#380—22no—
66C4×C4⋊C4D414yes—
67C4×C22⋊C4—22no—
68#376—22no—
69#340—22no—
70#340—22no—
71#336—22no—
72#336—22no—
73#328—28no—
74#327—28no—
75C22×Q8C4⋊C411yescharacteristic
76C22×D4—21no—
77C22×D4C4⋊C411yescharacteristic
78C23×C4—21no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4#4111yescharacteristic
82C2×M16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—81no—
89C22×C8—81no—
90C22×C8C22⋊C412yes—
91C22×C8C22⋊C412yes—
92#126—42no—
93#126—42no—
94#125—22no—
95#125—22no—
96#124—22no—
97#124—22no—
98#123—42no—
99#123—42no—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102#121—22no—
103#121—22no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—41no—
112#117—42no—
113#117—42no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4C4⋊C411yescharacteristic
120C2×C4⋊C4C4⋊C411yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—42no—
127C2×C42—21no—
128C2×C42—24no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C2×C42#4112yes—
135C2×C42#4112yes—
136C2×C42#4111yescharacteristic
137#107—82no—
138#106—82no—
139C4⋊C8—24no—
140C4⋊C8—24no—
141C4⋊C8—24no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C22⋊C8—24no—
145C22⋊C8—42no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—82no—
149C22⋊C8—82no—
150#97—42no—
151C4×C8—24no—
152C4×C8—24no—
153#95—24no—
154#95—42no—
155#95—42no—
156#95—42no—
157C24—21no—
158C24#12111yescharacteristic
159C2×Q8—22no—
160C2×Q8—42no—
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—44no—
164C2×D4—41no—
165C2×D4—41no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—44no—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4#12012yes—
183C22×C4#12112yes—
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12312yes—
189C22×C4#12411yescharacteristic
190C22×C4#12612yes—
191C22×C4#12711yescharacteristic
192M16—84no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—44no—
203C2×C8—81no—
204C2×C8—81no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—41no—
221C22⋊C4—84no—
222C42—48no—
223C42—41no—
224C42—41no—
225C42C4×D412yes—
226C42C4×D412yes—
227C42#12112yes—
228C42#12112yes—
229C42C22⋊Q812yes—
230C42C22⋊Q812yes—
231C42#12312yes—
232C42#12412yes—
233C23—22no—
234C23—22no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23#38011yescharacteristic
241C23#39211yescharacteristic
242C23#39711yescharacteristic
243Q8—82no—
244D4—82no—
245D4—82no—
246D4—82no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4#38512yes—
277C2×C4#38611yescharacteristic
278C2×C4#38812yes—
279C2×C4#39011yescharacteristic
280C2×C4#39311yescharacteristic
281C2×C4#39412yes—
282C2×C4#39612yes—
283C2×C4#39911yescharacteristic
284C8—42no—
285C8—42no—
286C8—82no—
287C8—84no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—44no—
296C22—84no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#143911yescharacteristic
301C22#181212yes—
302C22#201712yes—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—164no—
314C4#191111yescharacteristic
315C4#191211yescharacteristic
316C2—22no—
317C2—22no—
318C2—41no—
319C2—161no—
320C2#1107211yescharacteristic
321C2#1109811yescharacteristic
322C2#1111111yescharacteristic
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. #21960C2semidirect
4. #8002C2semidirect
5. #7858C2semidirect
11. #1984C4semidirect
12. #1954C4semidirect

Maximal subgroups

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

subgroupindex
#219602
#218482
#175802
#80022
#78582