Finite Group Databasebeta

← #97043

Subgroups — #97043

Order 512.

836 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2628C211yesmaximal, characteristic
2#8300C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7593C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×#117C2211yescharacteristic
7#2628—28no—
8C2×#400—21no—
9C2×#399—21no—
10C2×#395—21no—
11C2×#393—21no—
12C2×#389C2212yes—
13C2×#382C411yescharacteristic
14C2×#380C412yes—
15#1817C2212yes—
16#1817C2212yes—
17C2×#324—21no—
18C2×#324—21no—
19C2×#323—21no—
20C2×#323—21no—
21#1271—28no—
22C23×Q8—21no—
23C24×C4C2×C411yescharacteristic
24C2×C4⋊Q8—24no—
25C2×C4⋊Q8—42no—
26C2×C4⋊Q8—42no—
27C2×#117—24no—
28C2×#117—42no—
29C2×#117—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C22⋊C4—22no—
37C22×C22⋊C4D411yescharacteristic
38C22×C22⋊C4D411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43#400—28no—
44#399—28no—
45#395—28no—
46#393—28no—
47#389—24no—
48#389D418yes—
49#382—24no—
50#380—24no—
51#380—24no—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—22no—
56#336—24no—
57#336—24no—
58#324—42no—
59#324—42no—
60#323—42no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C22×Q8—24no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—84no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—42no—
83C22×C8—42no—
84C4⋊Q8—416no—
85#117—416no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—84no—
103C2×C4⋊C4C22⋊C414yes—
104C2×C4⋊C4C22⋊C414yes—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—84no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42C2×D414yes—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—28no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—42no—
134C24—42no—
135C24#10112yes—
136C24C2×C22⋊C411yescharacteristic
137C2×Q8—48no—
138C2×Q8—88no—
139C2×Q8—88no—
140C2×Q8—88no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—28no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4#9912yes—
181C22×C4#9912yes—
182C22×C4#10112yes—
183C22×C4C2×C22⋊C412yes—
184C22×C4#12012yes—
185C22×C4#12012yes—
186C22×C4#13112yes—
187C22×C4#13112yes—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—84no—
191C2×C8—84no—
192C4⋊C4—28no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C4⋊C4—44no—
199C4⋊C4—88no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C22⋊C4—88no—
203C42—28no—
204C42—44no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—44no—
211C23—44no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23—82no—
223C23—82no—
224C23#37911yescharacteristic
225C23#40811yescharacteristic
226C23#40811yescharacteristic
227C23C2×#9911yescharacteristic
228C23C2×#9911yescharacteristic
229C23C2×#10111yescharacteristic
230C23C2×#10111yescharacteristic
231Q8—816no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—416no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4#40814yes—
273C8—162no—
274C8—162no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—44no—
284C22—42no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—81no—
290C22—81no—
291C22—82no—
292C22#147211yescharacteristic
293C22#147211yescharacteristic
294C22#147211yescharacteristic
295C22#147211yescharacteristic
296C22#178011yescharacteristic
297C22#178111yescharacteristic
298C22#178411yescharacteristic
299C22#179512yes—
300C22#179612yes—
301C4—24no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—84no—
308C4—84no—
309C2—21no—
310C2—21no—
311C2—21no—
312C2—21no—
313C2—82no—
314C2#830811yescharacteristic
315C2#832512yes—
316C2#851611yescharacteristic
317C2#852011yescharacteristic
318C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×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
4. #7593C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#83002
#75932