Finite Group Databasebeta

← #116899

Subgroups — #116899

Order 512.

633 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 136 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#21969C211yesmaximal, characteristic, complemented
2#21847C211yesmaximal, characteristic
3#17450C212yesmaximal
4#7988C211yesmaximal, characteristic, complemented
5#7965C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9C2×#431C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11#1984C412yes—
12#1954C412yes—
13#1950—24no—
14#1869—22no—
15#1810—22no—
16#1797—22no—
17#1489—24no—
18#1488—24no—
19#1479—22no—
20#1478—22no—
21#1290—22no—
22#1287—24no—
23C22×#41Q811yescharacteristic
24C23×C8D412yes—
25C4×#41—22no—
26C4×#41—22no—
27C4×#41—22no—
28C4×#41—22no—
29C2×C4×Q8C2×C411yescharacteristic
30C2×C4×Q8D411yescharacteristic
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33C2×C4×D4C2×C411yescharacteristic
34C2×C4×D4D411yescharacteristic
35C2×#117—21no—
36C2×#117Q811yescharacteristic
37C22×C42—21no—
38C22×C42C2311yescharacteristic
39#432—22no—
40#432—22no—
41#432—22no—
42#432D414yes—
43#431—22no—
44#431—22no—
45#431—22no—
46#431—22no—
47#431—22no—
48#431—22no—
49C2×#108—24no—
50#423—22no—
51#423—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yes—
56C2×C4⋊C8D412yes—
57#407—22no—
58#407—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—24no—
61C4×M16—22no—
62C4×M16D414yes—
63C2×C4×C8C2×C412yes—
64#341—24no—
65#336—24no—
66Q8⋊C8—28no—
67D4⋊C8—28no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—42no—
73C22×Q8C4⋊C411yescharacteristic
74C22×D4—21no—
75C22×D4C4⋊C411yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×Q811yescharacteristic
81C23×C4#4111yescharacteristic
82C2×M16—24no—
83C2×M16—42no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—82no—
88C22×C8C22⋊C414yes—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—42no—
100C4×D4—41no—
101C4×D4—42no—
102C4×D4—41no—
103#117—22no—
104#117—22no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C4⋊C4C4⋊C411yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4112yes—
125C2×C42#4112yes—
126C2×C42#4111yescharacteristic
127#108—84no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C4⋊C8—42no—
135C22⋊C8—24no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—42no—
141C22⋊C8—84no—
142#97—24no—
143#97—24no—
144C4×C8—24no—
145C4×C8—24no—
146C4×C8—42no—
147C24—21no—
148C24C22⋊Q811yescharacteristic
149#41—44no—
150#41—44no—
151#41—44no—
152#41—44no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8#10812yescomplemented
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4#10812yescomplemented
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—81no—
182C22×C4—82no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4#11711yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C22⋊Q812yes—
190C22×C4C22⋊Q812yes—
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12312yes—
194C22×C4#12412yes—
195C22×C4#12411yescharacteristic
196C22×C4#12711yescharacteristic
197M16—44no—
198M16—44no—
199M16—84no—
200C2×C8—24no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—44no—
207C2×C8—44no—
208C2×C8—82no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C42—41no—
220C42—41no—
221C42—42no—
222C42—42no—
223C42C4×D412yes—
224C42C4×D412yes—
225C42#12012yes—
226C42#12112yes—
227C42#12112yes—
228C42#12112yes—
229C42#12312yes—
230C42#12612yes—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—41no—
240C23#38011yescharacteristic
241C23#39311yescharacteristic
242C23#39911yescharacteristic
243Q8—24no—
244Q8—42no—
245D4—24no—
246D4—44no—
247D4—44no—
248D4—42no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—48no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4#38512yes—
281C2×C4#38611yescharacteristic
282C2×C4#38812yes—
283C2×C4#39011yescharacteristic
284C2×C4#39211yescharacteristic
285C2×C4#39412yes—
286C2×C4#39612yes—
287C2×C4#39711yescharacteristic
288C2×C4C2×#10811yescharacteristic
289C2×C4C2×#10811yescharacteristic
290C2×C4#43012yes—
291C8—44no—
292C8—82no—
293C8—84no—
294C8—84no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—42no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22#143911yescharacteristic
311C22#181011yescharacteristic
312C22#181111yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4#185512yes—
323C4#185512yes—
324C4#191111yescharacteristic
325C4#191211yescharacteristic
326C2—21no—
327C2—21no—
328C2—41no—
329C2—41no—
330C2—82no—
331C2#1107211yescharacteristic
332C2#1109711yescharacteristic
333C2#1111011yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41Q8
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. #21969C2semidirect
4. #7988C2semidirect
5. #7965C2semidirect
156. C2×Q8#108semidirect
164. C2×D4#108semidirect

Maximal subgroups

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

subgroupindex
#219692
#218472
#174502
#79882
#79652