Finite Group Databasebeta

← #124349

Subgroups — #124349

Order 512.

419 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 86 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#21419C211yesmaximal, characteristic
2#8731C211yesmaximal, characteristic, complemented
3#7866C211yesmaximal, characteristic, complemented
4#7134C212yesmaximal, complemented
5#7106C212yesmaximal, complemented
6#3455C411yescharacteristic
7#2921C411yescharacteristic
8C2×#431—21no—
9C4×#117C2211yescharacteristic
10#1995C411yescharacteristic
11#1994C2211yescharacteristic
12#1963C411yescharacteristic
13#1962C2211yescharacteristic
14#1897—21no—
15#1896—21no—
16#1759—21no—
17C4×#104—21no—
18#1458—21no—
19#1276—21no—
20#1276C2212yes—
21#1276C2212yes—
22#1274—21no—
23#542—22no—
24#541C812yescomplemented
25#540C812yescomplemented
26#539—41no—
27#538—41no—
28#533C2×C411yescharacteristic
29#532C2×C411yescharacteristic
30#529—42no—
31#525—21no—
32C4×#41—21no—
33C2×#117C2×C411yescharacteristic
34C2×#117D411yescharacteristic
35C22×C42D411yescharacteristic
36#433—42no—
37#431—22no—
38#431—22no—
39#431—22no—
40#431—41no—
41#431—41no—
42#423C2×C411yescharacteristic
43#423C2×C411yescharacteristic
44C2×C4⋊C8—22no—
45C2×#104—21no—
46C2×#104—21no—
47C2×#104—21no—
48C2×#104—21no—
49#407—21no—
50C2×C22⋊C8—41no—
51C4×M16—21no—
52C2×C4×C8—41no—
53C4×C4⋊C4—41no—
54C4×C22⋊C4—41no—
55#376C2×C411yescharacteristic
56#376D411yescharacteristic
57C43D411yescharacteristic
58C43C2311yescharacteristic
59#339—21no—
60#339—21no—
61#339—21no—
62#339—21no—
63#324—41no—
64#323—41no—
65C2×#41—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×M16—21no—
69C2×M16—21no—
70C22×C8—42no—
71C4⋊Q8—21no—
72C4⋊Q8C2×C811yescharacteristic, complemented
73#127C2×C811yescharacteristic, complemented
74#126—42no—
75#126—42no—
76#125—42no—
77#125C2×C812yescomplemented
78#124—21no—
79#124—42no—
80#123—42no—
81#123—41no—
82#123—42no—
83C22⋊Q8—22no—
84C22⋊Q8—42no—
85C22⋊Q8—41no—
86#121—22no—
87#121—42no—
88#121—41no—
89#120—41no—
90C4×Q8—22no—
91C4×Q8—41no—
92C4×D4—22no—
93C4×D4—41no—
94C4×D4—42no—
95C4×D4—42no—
96#117—22no—
97#117—22no—
98#117—21no—
99#117—41no—
100#117—41no—
101#117C2×C812yes—
102#117M1612yes—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22⋊C411yescharacteristic
116C2×C42C22⋊C411yescharacteristic
117C2×C42C22⋊C411yescharacteristic
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C4⋊C8—24no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126#104—42no—
127#104—41no—
128#104—41no—
129#104—42no—
130#104—41no—
131#104—41no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136#97—41no—
137C4×C8—42no—
138C4×C8—41no—
139C4×C8—42no—
140C4×C8—42no—
141#95—41no—
142#95—41no—
143C24#9911yescharacteristic
144#41—41no—
145#41—41no—
146#41—41no—
147#41—41no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—42no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—82no—
171C22×C4C22⋊C812yes—
172C22×C4#9911yescharacteristic
173C22×C4#10012yes—
174C22×C4C2×C22⋊C411yescharacteristic
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—44no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—84no—
183C2×C8—82no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—41no—
192C4⋊C4—42no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—42no—
213C42—41no—
214C42—82no—
215C42—82no—
216C42C22⋊C812yes—
217C42#10411yescharacteristic
218C42#10411yescharacteristic
219C42C22×C811yescharacteristic
220C42C2×M1611yescharacteristic
221C42#13512yes—
222C42AGL(1,ℤ/8)11yescharacteristic
223C42#13711yescharacteristic
224C23—21no—
225C23—21no—
226C23—41no—
227C23—81no—
228C23#32312yes—
229C23C2×#9911yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—161no—
233D4—41no—
234D4—41no—
235D4—81no—
236D4—81no—
237D4—161no—
238D4—162no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—162no—
269C2×C4#32312yes—
270C2×C4C2×C22⋊C811yescharacteristic
271C2×C4C2×#10011yescharacteristic
272C2×C4#41611yescharacteristic
273C2×C4#41711yescharacteristic
274C2×C4C2×#10411yescharacteristic
275C2×C4#42111yescharacteristic
276C8—84no—
277C8—81no—
278C8—81no—
279C8—162no—
280C22—22no—
281C22—41no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—82no—
287C22—81no—
288C22C2×#32311yescharacteristic
289C22#150411yescharacteristic
290C22#151611yescharacteristic
291C4—22no—
292C4—22no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—82no—
302C4—81no—
303C4#147611yescharacteristic
304C4#147711yescharacteristic
305C4#153011yescharacteristic
306C4#155111yescharacteristic
307C2—21no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—161no—
312C2#757811yescharacteristic
subgroupG/H
centerC4#1551
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC2#7578
prime radical#3455C4
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8731C2semidirect
3. #7866C2semidirect
4. #7134C2semidirect
5. #7106C2semidirect
24. #541C8semidirect
25. #540C8semidirect
72. C4⋊Q8C2×C8semidirect
73. #127C2×C8semidirect
77. #125C2×C8semidirect

Maximal subgroups

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

subgroupindex
#214192
#87312
#78662
#71342
#71062