Finite Group Databasebeta

← #496409

Subgroups — #496409

Order 512.

866 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 198 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#35333C211yesmaximal, characteristic
2#20728C212yesmaximal
3#13303C218yesmaximal, complemented
4#12863C214yesmaximal
5C22×#125C411yescharacteristic
6C2×C4×#41C2211yescharacteristic
7#2920—22no—
8#2851—22no—
9#2629—22no—
10#2579—22no—
11#2579—22no—
12#2566C414yes—
13#2565C2214yes—
14#2317C412yes—
15#2311C2212yes—
16#2285C411yescharacteristic
17#1822—22no—
18#1801C2214yes—
19#1800—22no—
20#1785C2214yes—
21C4×#101C2214yes—
22C4×#100—22no—
23#1394C22116yes—
24C22×#41C2311yescharacteristic
25C2×#125—21no—
26C2×#125—21no—
27C2×#125—41no—
28C2×#125—42no—
29C2×#125—41no—
30C2×#125C2×C414yes—
31C4×#41—41no—
32C4×#41—42no—
33C4×#41—41no—
34C2×C4×Q8C2312yes—
35C2×C4×D4C2×C412yes—
36C2×C4×D4D412yes—
37C2×C4×D4D412yes—
38C2×#117C2×C411yescharacteristic
39C2×#117C2×C412yes—
40C2×#117D412yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C412yes—
43C22×C42C2×C411yescharacteristic
44C22×C42D412yes—
45#432—42no—
46#432—42no—
47#431—42no—
48#431—42no—
49#423—24no—
50C2×#101C2318yes—
51C2×#100—24no—
52#407—24no—
53C4×M16—24no—
54#398—24no—
55#398—24no—
56#396—24no—
57#396—24no—
58#391C2314yes—
59#390C2×C414yes—
60#389—22no—
61#389—22no—
62#387—24no—
63#387—24no—
64#386—22no—
65#386—22no—
66#385—24no—
67#385—24no—
68#385—22no—
69#385—22no—
70#384—21no—
71#384C2×C414yes—
72#382—21no—
73#382C2×C414yes—
74#380—22no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4C2×C414yes—
78#343—24no—
79#343—24no—
80C2×#41—24no—
81C2×#41—41no—
82C2×#41—41no—
83C2×#41—42no—
84C22×Q8C2411yescharacteristic
85C22×D4C22×C411yescharacteristic
86C22×D4C2×D412yes—
87C23×C4—22no—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D412yes—
93C2×M16—28no—
94#125—44no—
95#125—44no—
96#125—42no—
97#125—42no—
98#125—41no—
99#125—42no—
100#125—41no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—42no—
109#117—41no—
110#117—41no—
111#117—42no—
112#117—42no—
113#117—44no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C22×C414yes—
126C2×C4⋊C4C22×C412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4C22⋊C411yescharacteristic
133C2×C22⋊C4C22⋊C411yescharacteristic
134C2×C22⋊C4C22⋊C412yes—
135C2×C22⋊C4C22⋊C412yes—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C22⋊C412yes—
144C2×C42C22⋊C411yescharacteristic
145C2×C42C22⋊C411yescharacteristic
146C2×C42C22×C412yes—
147C2×C42C22×C414yes—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C4⋊C8—44no—
151C4⋊C8—44no—
152#101—216no—
153#100—48no—
154C22⋊C8—44no—
155C22⋊C8—44no—
156#97—44no—
157C4×C8—44no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163C24C2×C22⋊C412yes—
164C24C2×C22⋊C411yescharacteristic
165#41—48no—
166#41—48no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q82−1+414yes—
170C2×D4—24no—
171C2×D4—22no—
172C2×D4—24no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—41no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—84no—
196C22×C4—82no—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C23×C411yescharacteristic
203C22×C4C22×D412yes—
204M16—48no—
205M16—48no—
206C2×C8—48no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—44no—
225C22⋊C4—84no—
226C22⋊C4—84no—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—21no—
234C42—21no—
235C42—22no—
236C42—22no—
237C42—44no—
238C42#9911yescharacteristic
239C42#9912yes—
240C42#9911yescharacteristic
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23#41012yes—
250C23C22×C22⋊C411yescharacteristic
251Q8—44no—
252D4—48no—
253D4—44no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—88no—
284C2×C4C2×#9911yescharacteristic
285C2×C4C2×#9911yescharacteristic
286C2×C4C2×#9912yes—
287C2×C4C2×#9912yes—
288C2×C4#51912yes—
289C2×C4#53614yes—
290C8—88no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—44no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—82no—
301C22#231011yescharacteristic
302C22C22×#9911yescharacteristic
303C22C2×#41011yescharacteristic
304C4—24no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—42no—
311C4—42no—
312C4—42no—
313C4—84no—
314C4—84no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2—82no—
319C2#2182511yescharacteristic
320C2#3336912yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13303C2semidirect

Maximal subgroups

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

subgroupindex
#353332
#207282
#133032
#128632