Finite Group Databasebeta

← #96965

Subgroups — #96965

Order 512.

1464 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 146 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#34257C211yesmaximal, characteristic
2#11380C211yesmaximal, characteristic, complemented
3#8303C211yesmaximal, characteristic, complemented
4#7578C214yesmaximal, complemented
5C22×2−1+4C411yescharacteristic
6C2×#519C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2341—24no—
9Q8×C22⋊C4—24no—
10#2311—22no—
11#2311C412yes—
12#2297—22no—
13#2297C2212yes—
14#1817C2212yes—
15C2×#336—22no—
16C2×#336C2212yes—
17C2×#324—22no—
18#1272—24no—
19#1270—24no—
20C2×2−1+4—24no—
21C2×2−1+4—22no—
22C2×2−1+4—44no—
23C2×2−1+4—44no—
24C2×2−1+4C814yescomplemented
25C22×#41—22no—
26C22×#41—22no—
27C22×#41C2×C412yes—
28C23×Q8—22no—
29C23×Q8C2×C411yescharacteristic
30C23×C8—22no—
31#519—44no—
32C2×C4×Q8—24no—
33C2×C4×Q8—24no—
34C2×C4×Q8—44no—
35C2×#117—22no—
36C2×#117C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—22no—
42#391—24no—
43#391—24no—
44#391—22no—
45#391D414yes—
46#390—24no—
47#390—24no—
48#390—22no—
49#390D414yes—
50#386—24no—
51#386—22no—
52#386—22no—
53#380—28no—
54#380—24no—
55C4×C22⋊C4—28no—
56C4×C22⋊C4—24no—
57#336—24no—
58#336—24no—
59#336—44no—
60#324—44no—
612−1+4—416no—
622−1+4—432no—
632−1+4—416no—
64C2×#41—28no—
65C2×#41—28no—
66C2×#41—28no—
67C2×#41—28no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—44no—
71C2×#41—44no—
72C2×#41—44no—
73C2×#41—44no—
74C2×#41—48no—
75C2×#41—48no—
76C2×#41—44no—
77C2×#41—44no—
78C2×#41C2×C814yes—
79C2×#41M1614yes—
80C22×Q8—22no—
81C22×Q8—28no—
82C22×Q8—28no—
83C22×Q8—24no—
84C22×Q8—24no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—48no—
88C22×Q8—48no—
89C22×Q8C22⋊C412yes—
90C22×Q8C2×C812yes—
91C22×Q8M1612yes—
92C22×D4—22no—
93C22×D4—22no—
94C22×D4C22⋊C412yes—
95C23×C4—24no—
96C23×C4—22no—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—24no—
103C22×C8—42no—
104C22×C8—44no—
105C22×C8—44no—
106C22×C8—42no—
107C22×C8—42no—
108C4×Q8—416no—
109C4×Q8—416no—
110#117—44no—
111#117—44no—
112#117—44no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C22⋊C414yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—24no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—22no—
127C2×C42—24no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42C2×D414yes—
131C22⋊C8—44no—
132#95—28no—
133#95—24no—
134#95—24no—
135C24—22no—
136C24C2×C22⋊C411yescharacteristic
137#41—416no—
138#41—48no—
139#41—416no—
140#41—416no—
141#41—48no—
142#41—48no—
143#41—48no—
144#41—432no—
145#41—432no—
146#41—416no—
147C2×Q8—28no—
148C2×Q8—28no—
149C2×Q8—216no—
150C2×Q8—28no—
151C2×Q8—28no—
152C2×Q8—28no—
153C2×Q8—24no—
154C2×Q8—24no—
155C2×Q8—22no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—432no—
161C2×Q8—432no—
162C2×Q8C22⋊C814yes—
163C2×D4—28no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—22no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4C22⋊C814yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—216no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—48no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4C22⋊C814yes—
199C22×C4#9914yes—
200C22×C4#10014yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4#12012yes—
203C22×C4#12012yes—
204C22×C4C22×C811yescharacteristic
205C22×C4C2×M1612yes—
206C22×C4C2×M1611yescharacteristic
207C2×C8—48no—
208C2×C8—44no—
209C2×C8—84no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—84no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—84no—
222C22⋊C4—88no—
223C42—44no—
224C42—48no—
225C42—48no—
226C42—44no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—44no—
231C23—44no—
232C23—42no—
233C23—42no—
234C23#37911yescharacteristic
235C23C2×C22⋊C811yescharacteristic
236C23#40711yescharacteristic
237C23C2×#9912yes—
238C23C2×#10012yes—
239Q8—216no—
240Q8—216no—
241Q8—44no—
242Q8—416no—
243Q8—44no—
244D4—44no—
245D4—416no—
246D4—44no—
247D4—48no—
248D4—48no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—432no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#32318yes—
282C2×C4C2×C22⋊C812yes—
283C2×C4#40712yes—
284C2×C4#45412yes—
285C2×C4#45612yes—
286C8—84no—
287C8—162no—
288C22—22no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—44no—
294C22—48no—
295C22—82no—
296C22—84no—
297C22—82no—
298C22C2×#32312yes—
299C22#146212yes—
300C22#177911yescharacteristic
301C22#178111yescharacteristic
302C22#178311yescharacteristic
303C22#179312yes—
304C22#179412yes—
305C4—22no—
306C4—22no—
307C4—28no—
308C4—22no—
309C4—22no—
310C4—42no—
311C4—44no—
312C4—84no—
313C4—82no—
314C2—22no—
315C2—22no—
316C2—84no—
317C2#829811yescharacteristic
318C2#831912yes—
319C2#845511yescharacteristic
320C2#845912yes—
321C2#846411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC22×2−1+4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11380C2semidirect
3. #8303C2semidirect
4. #7578C2semidirect
24. C2×2−1+4C8semidirect

Maximal subgroups

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

subgroupindex
#342572
#113802
#83032
#75782