Finite Group Databasebeta

← #129705

Subgroups — #129705

Order 512.

599 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2994C211yesmaximal, characteristic
2#11639C212yesmaximal, complemented
3#11634C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic, complemented
5#11459C211yesmaximal, characteristic, complemented
6C2×#529C411yescharacteristic
7#2994—24no—
8#2994—24no—
9C2×#446C411yescharacteristic
10C2×C8×Q8C2211yescharacteristic
11C2×#436C2211yescharacteristic
12C2×#435—21no—
13C2×C8×D4—21no—
14C2×C8×D4C411yescharacteristic
15C2×#433—21no—
16C2×#431C2211yescharacteristic
17C2×#424—21no—
18C22×C4⋊C8C411yescharacteristic
19#2163C2212yes—
20#2162C2212yes—
21Q8⋊C16—24no—
22#1330—24no—
23#1322—21no—
24#1322—21no—
25#1314—21no—
26#1314—21no—
27C22×M16—21no—
28C23×C8—21no—
29C23×C8—21no—
30#529—22no—
31#529—22no—
32#529—22no—
33#529—22no—
34C2×#126—21no—
35C2×#125C811yescharacteristic
36C2×#124C811yescharacteristic
37C2×#123—21no—
38C2×C22⋊Q8C811yescharacteristic
39C2×#121C811yescharacteristic
40C2×C4×Q8C2×C411yescharacteristic
41C2×C4×D4—21no—
42C2×C4×D4C2×C411yescharacteristic
43C2×#117—21no—
44C22×C42C2×C411yescharacteristic
45#446—22no—
46#446—22no—
47C8×Q8—22no—
48C8×Q8D414yes—
49#436—22no—
50#436D414yes—
51#435—28no—
52C8×D4—28no—
53C8×D4—22no—
54C8×D4—22no—
55#433—28no—
56#431—22no—
57#431—22no—
58#424—28no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2×C411yescharacteristic
65C2×C4⋊C8C2×C411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8D411yescharacteristic
71C2×C22⋊C8D411yescharacteristic
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C4⋊C16—22no—
77C4⋊C16—22no—
78C22⋊C16—22no—
79C22⋊C16—22no—
80#346—22no—
81C4×C16—22no—
82C22×Q8C2×C811yescharacteristic
83C22×D4—21no—
84C22×D4C2×C811yescharacteristic
85C23×C4—21no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C2×C811yescharacteristic
88C23×C4M1611yescharacteristic
89C2×M16—24no—
90C2×M16—82no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—24no—
94C22×C8—24no—
95C22×C8—82no—
96C22×C8—82no—
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101#126—24no—
102#126—24no—
103#125—22no—
104#125—22no—
105#124—22no—
106#124—22no—
107#123—24no—
108#123—24no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111#121—22no—
112#121—22no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8C22⋊C412yes—
116C4×Q8M1612yes—
117C4×D4—24no—
118C4×D4—24no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4C22⋊C412yes—
122C4×D4M1612yes—
123#117—24no—
124#117—24no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C2×C811yescharacteristic
130C2×C4⋊C4C2×C811yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42C22⋊C411yescharacteristic
142C2×C42C2×C811yescharacteristic
143C2×C42M1611yescharacteristic
144C2×C42C22×C411yescharacteristic
145C2×C16—42no—
146C2×C16—42no—
147C4⋊C8—24no—
148C4⋊C8—24no—
149C4⋊C8—24no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8C22⋊C412yes—
155C4⋊C8C22⋊C412yes—
156C22⋊C8—24no—
157C22⋊C8—24no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160#97—24no—
161C4×C8—24no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8C2×D412yes—
165C4×C8C2×D412yes—
166C24—21no—
167C24C22⋊C811yescharacteristic
168C2×Q8—21no—
169C2×Q8—42no—
170C2×Q8C22⋊C812yes—
171C2×D4—24no—
172C2×D4—21no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4C22⋊C812yes—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4C22⋊C811yescharacteristic
196C22×C4C22⋊C811yescharacteristic
197C22×C4C22⋊C811yescharacteristic
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C22×C811yescharacteristic
200C22×C4C2×M1611yescharacteristic
201M16—84no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—84no—
213C2×C8—84no—
214C2×C8#12012yes—
215C2×C8#12012yes—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4C22⋊C812yes—
224C4⋊C4C22⋊C812yes—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42C2×C22⋊C411yescharacteristic
240C42C2×C22⋊C411yescharacteristic
241C42C2×M1611yescharacteristic
242C42C2×M1611yescharacteristic
243C16—82no—
244C16—82no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23#35012yes—
255C23C2×C22⋊C811yescharacteristic
256Q8—24no—
257Q8—42no—
258D4—42no—
259D4—48no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#37911yescharacteristic
291C2×C4C2×C22⋊C811yescharacteristic
292C2×C4C2×C22⋊C811yescharacteristic
293C2×C4#40711yescharacteristic
294C2×C4#40711yescharacteristic
295C2×C4#40711yescharacteristic
296C8—22no—
297C8—22no—
298C8—42no—
299C8—42no—
300C8—41no—
301C8—41no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—82no—
313C22—82no—
314C22#177911yescharacteristic
315C22C2×#35011yescharacteristic
316C22#211711yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—42no—
323C4—41no—
324C4—42no—
325C4—41no—
326C4—42no—
327C4#179511yescharacteristic
328C4#179611yescharacteristic
329C4#211512yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—82no—
334C2#832411yescharacteristic
335C2#1268211yescharacteristic
336C2#1268311yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11639C2semidirect
3. #11634C2semidirect
4. #11460C2semidirect
5. #11459C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#116392
#116342
#114602
#114592