Finite Group Databasebeta

← #116604

Subgroups — #116604

Order 512.

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

normal subgroups only

This group has 670 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#21374C211yesmaximal, characteristic
2#20748C212yesmaximal
3#17819C211yesmaximal, characteristic, complemented
4#8074C212yesmaximal, complemented
5#7980C211yesmaximal, characteristic, complemented
6C2×C4×#41C2211yescharacteristic
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C2×C4×M16C2211yescharacteristic
10#2283C2212yes—
11#1985C412yescomplemented
12#1955C412yescomplemented
13#1870—22no—
14#1826—22no—
15#1806—24no—
16#1789—22no—
17#1759—24no—
18C4×C22⋊C8—22no—
19#1574—24no—
20#1568—22no—
21#1300—24no—
22#1289—22no—
23C22×#41D411yescharacteristic
24C22×M16—21no—
25C4×#41—22no—
26C4×#41—22no—
27C4×#41—22no—
28C4×#41—22no—
29C2×C4×Q8C2×C411yescharacteristic
30C2×C4×Q8Q811yescharacteristic
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33C2×C4×D4C2×C411yescharacteristic
34C2×C4×D4Q811yescharacteristic
35C2×#117—21no—
36C2×#117D411yescharacteristic
37C22×C42—21no—
38C22×C42C2311yescharacteristic
39#432—22no—
40#432—22no—
41#431—22no—
42#431—22no—
43#431—22no—
44#431—22no—
45#423D414yes—
46C2×C4⋊C8C2×C412yes—
47C2×#104—28no—
48#407—22no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8D412yes—
51C4×M16—22no—
52C4×M16—22no—
53C4×M16—22no—
54C2×#97C2×C412yes—
55C2×C4×C8—21no—
56C2×C4×C8D412yes—
57#383—22no—
58#382—22no—
59#381—22no—
60#380—22no—
61C4×C4⋊C4—22no—
62C4×C22⋊C4—22no—
63#376D414yes—
64#339—28no—
65#336—24no—
66#335—24no—
67#334—24no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—44no—
71C22×Q8C4⋊C411yescharacteristic
72C22×D4—21no—
73C22×D4C4⋊C411yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4#4111yescharacteristic
80C2×M16—24no—
81C2×M16—44no—
82C2×M16—42no—
83C22×C8—22no—
84C22×C8C22⋊C414yes—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—42no—
94C4×D4—42no—
95C4×D4—42no—
96C4×D4—42no—
97#117—22no—
98#117—22no—
99#117—42no—
100#117—42no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C4⋊C411yescharacteristic
105C2×C4⋊C4C4⋊C411yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—42no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—24no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42#4112yes—
124C2×C42#4112yes—
125C2×C42#4111yescharacteristic
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—24no—
129#104—416no—
130C22⋊C8—24no—
131C22⋊C8—48no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#97—24no—
135#97—42no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—42no—
139C4×C8—41no—
140C4×C8—41no—
141#95—24no—
142#95—24no—
143#95—42no—
144#95—42no—
145C24—21no—
146C24#12111yescharacteristic
147#41—28no—
148#41—44no—
149#41—44no—
150#41—44no—
151C2×Q8—22no—
152C2×Q8—42no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—44no—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C2×C4⋊C411yescharacteristic
177C22×C4#11711yescharacteristic
178C22×C4#12012yes—
179C22×C4#12112yes—
180C22×C4#12111yescharacteristic
181C22×C4#12112yes—
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4#12312yes—
185C22×C4#12312yes—
186C22×C4#12412yes—
187M16—48no—
188M16—48no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—44no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—84no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—84no—
204C42—48no—
205C42—41no—
206C42—41no—
207C42—41no—
208C42—41no—
209C42—41no—
210C42—41no—
211C42C4×D412yes—
212C42C4×D412yes—
213C42#12011yescharacteristic
214C42#12112yes—
215C42C22⋊Q812yes—
216C42C22⋊Q812yes—
217C42#12311yescharacteristic
218C42#12411yescharacteristic
219C42#12612yes—
220C42#12711yescharacteristic
221C23—22no—
222C23—22no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—41no—
229C23#38011yescharacteristic
230C23#39412yes—
231Q8—24no—
232Q8—42no—
233D4—24no—
234D4—42no—
235D4—48no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4#38512yes—
265C2×C4#38611yescharacteristic
266C2×C4#38812yes—
267C2×C4#39011yescharacteristic
268C2×C4#39211yescharacteristic
269C2×C4#39311yescharacteristic
270C2×C4#39612yes—
271C2×C4#39711yescharacteristic
272C2×C4#39911yescharacteristic
273C8—44no—
274C8—44no—
275C8—84no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—44no—
282C22—42no—
283C22—82no—
284C22—82no—
285C22—81no—
286C22—81no—
287C22#143911yescharacteristic
288C22#181412yes—
289C4—21no—
290C4—22no—
291C4—21no—
292C4—22no—
293C4—21no—
294C4—21no—
295C4—22no—
296C4—42no—
297C4—81no—
298C4—81no—
299C4—164no—
300C4#188412yes—
301C4#201012yes—
302C2—22no—
303C2—42no—
304C2—82no—
305C2#1105611yescharacteristic
306C2#1110011yescharacteristic
307C2#1110811yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17819C2semidirect
4. #8074C2semidirect
5. #7980C2semidirect
11. #1985C4semidirect
12. #1955C4semidirect

Maximal subgroups

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

subgroupindex
#213742
#207482
#178192
#80742
#79802