Finite Group Databasebeta

← #237948

Subgroups — #237948

Order 512.

1087 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C2×#2862C211yesmaximal, characteristic, complemented
2#21415C211yesmaximal, characteristic
3#17588C212yesmaximal
4#10363C214yesmaximal, complemented
5#10362C214yesmaximal, complemented
6#7959C212yesmaximal, complemented
7#7867C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9C22×C4⋊Q8—21no—
10#2920C2212yes—
11#2920C2211yescharacteristic
12#2862—24no—
13#2862—22no—
14#2862C2214yes—
15C2×#391—24no—
16C2×#382—21no—
17C2×#382C2212yes—
18C2×C43C2211yescharacteristic
19#1994C2212yes—
20#1962C2212yes—
21#1896—21no—
22#1896—21no—
23C4×#104—22no—
24#1518C2214yes—
25#1517C2214yes—
26#1512—24no—
27#1504—24no—
28#1286C2218yes—
29#1276C2214yes—
30C23×Q8—22no—
31#533Q812yescomplemented
32#533C2311yescharacteristic
33#532Q812yescomplemented
34#532C2311yescharacteristic
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—48no—
38#529—22no—
39#529—22no—
40#529—22no—
41#525—21no—
42#525—22no—
43C4×#41—21no—
44C4×#41—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C42—22no—
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42C2311yescharacteristic
51#432—22no—
52#432—22no—
53#432—22no—
54#431—22no—
55#431—22no—
56#431C2314yes—
57#423—22no—
58#423C2312yes—
59C2×#104—22no—
60C2×#104—22no—
61#407—22no—
62#407—21no—
63C4×M16—22no—
64C4×M16—21no—
65#391—216no—
66#391—216no—
67#382—24no—
68#382—22no—
69#382—22no—
70#382—24no—
71#382D414yes—
72#382C2314yes—
73C2×#95—24no—
74C2×#95—22no—
75C2×#95D414yes—
76C43—22no—
77C43D412yes—
78C43C2312yes—
79#339—22no—
80#339—22no—
81#328—28no—
82#327—28no—
83C2×#41—22no—
84C2×#41—21no—
85C22×Q8—28no—
86C22×Q8—48no—
87C22×Q8—416no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—24no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C2×M16—22no—
94C2×M16—42no—
95C4⋊Q8—416no—
96C4⋊Q8—48no—
97C4⋊Q8—48no—
98C4⋊Q8C2×Q812yescomplemented
99C4⋊Q8C2×Q811yescharacteristic, complemented
100#127C2×Q811yescharacteristic, complemented
101#126—42no—
102#125—22no—
103#125—21no—
104#124—22no—
105#124—21no—
106#123—41no—
107#123—42no—
108C22⋊Q8—22no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111#121—22no—
112#121—22no—
113#121—22no—
114#120—41no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124#117—22no—
125#117—21no—
126#117—41no—
127#117—42no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—48no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—48no—
140C2×C42—41no—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×Q812yes—
146C2×C42#4112yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C2×C42C2411yescharacteristic
150C4⋊C8—24no—
151C4⋊C8—24no—
152C4⋊C8—42no—
153#104—44no—
154#104—44no—
155C22⋊C8—24no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158#97—42no—
159#97—41no—
160C4×C8—24no—
161C4×C8—41no—
162#95—216no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95C2×D418yes—
167C24#12011yescharacteristic
168#41—42no—
169#41—42no—
170#41—84no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—416no—
174C2×Q8—432no—
175C2×Q8—832no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—41no—
179C2×D4—42no—
180C22×C4—24no—
181C22×C4—28no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—48no—
191C22×C4—416no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4#12011yescharacteristic
195C22×C4#12012yes—
196C22×C4C22⋊Q812yes—
197C22×C4C22⋊Q812yes—
198C22×C4C22⋊Q812yes—
199C22×C4#12412yes—
200C22×C4#12412yes—
201C22×C4#12412yes—
202C22×C4C22×D412yes—
203C22×C4C22×D411yescharacteristic
204M16—42no—
205M16—42no—
206M16—84no—
207C2×C8—44no—
208C2×C8—42no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—48no—
213C4⋊C4—416no—
214C4⋊C4—42no—
215C4⋊C4—48no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C42—24no—
224C42—24no—
225C42—22no—
226C42—22no—
227C42—48no—
228C42—48no—
229C42—41no—
230C42—42no—
231C42C22⋊Q812yes—
232C42C22⋊Q812yes—
233C42C22⋊Q812yes—
234C42#12411yescharacteristic
235C42#12412yes—
236C42#12411yescharacteristic
237C42#12412yes—
238C42AGL(1,ℤ/8)14yes—
239C42#13714yes—
240C42C22×Q811yescharacteristic
241C42C2×#4111yescharacteristic
242C42C2×#4112yes—
243C23—21no—
244C23—21no—
245C23—24no—
246C23—41no—
247C23#39312yes—
248C23C2×#12011yescharacteristic
249Q8—42no—
250Q8—82no—
251Q8—832no—
252D4—42no—
253D4—82no—
254D4—81no—
255D4—81no—
256D4—84no—
257C2×C4—28no—
258C2×C4—216no—
259C2×C4—21no—
260C2×C4—24no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—416no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—48no—
277C2×C4—84no—
278C2×C4—816no—
279C2×C4—82no—
280C2×C4#39312yes—
281C2×C4#39312yes—
282C2×C4#39312yes—
283C2×C4C2×C22⋊Q812yes—
284C2×C4C2×C22⋊Q811yescharacteristic
285C2×C4C2×#12412yes—
286C2×C4C2×#12411yescharacteristic
287C2×C4C2×AGL(1,ℤ/8)12yes—
288C2×C4C2×#13712yes—
289C8—84no—
290C8—82no—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—81no—
297C22—82no—
298C22#203114yes—
299C22C2×#39311yescharacteristic
300C22#300311yescharacteristic
301C22#300411yescharacteristic
302C4—28no—
303C4—22no—
304C4—22no—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—44no—
309C4—88no—
310C4—81no—
311C4—82no—
312C4#307912yes—
313C4#314112yes—
314C2—21no—
315C2—21no—
316C2—24no—
317C2—161no—
318C2#2345211yescharacteristic
319C2C2×#203112yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2862C2semidirect
4. #10363C2semidirect
5. #10362C2semidirect
6. #7959C2semidirect
7. #7867C2semidirect
31. #533Q8semidirect
33. #532Q8semidirect
98. C4⋊Q8C2×Q8semidirect
99. C4⋊Q8C2×Q8semidirect
100. #127C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
#214152
#175882
#103632
#103622
#79592
#78672