Finite Group Databasebeta

← #98548

Subgroups — #98548

Order 512.

824 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2499C211yesmaximal, characteristic
2#8723C212yesmaximal, complemented
3#8559C212yesmaximal
4#7643C212yesmaximal, complemented
5C22×C22⋊Q8C411yescharacteristic
6C22×#117C411yescharacteristic
7#2499—24no—
8#2499—24no—
9C2×#397—22no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#387—22no—
13C2×#384C2211yescharacteristic
14C2×#382C411yescharacteristic
15C2×C4×C22⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#323—22no—
18#1296—24no—
19#1280—22no—
20#1272—24no—
21C23×Q8C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C2×C22⋊Q8—22no—
24C2×C22⋊Q8—22no—
25C2×C22⋊Q8—42no—
26C2×C22⋊Q8—42no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—42no—
30C2×#117—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D412yes—
39C22×C42—21no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C4⋊C8—24no—
43C2×C22⋊C8—24no—
44#397—216no—
45#391—22no—
46#391—22no—
47#391—22no—
48#391D414yes—
49#387—216no—
50#384—22no—
51#384D414yes—
52#382—22no—
53#382—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95D412yes—
60#336—24no—
61#323—44no—
62C25C22⋊C411yescharacteristic
63C22×Q8—21no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C22⋊C412yes—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4C22⋊C412yes—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D412yes—
83C22×C8—44no—
84C22⋊Q8—48no—
85C22⋊Q8—48no—
86#117—48no—
87#117—48no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C22⋊C412yes—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C22⋊C412yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C42—24no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C4⋊C8—48no—
127C22⋊C8—44no—
128C22⋊C8—44no—
129#95—28no—
130#95—28no—
131#95—28no—
132#95—24no—
133C24—21no—
134C24—21no—
135C24—22no—
136C24—22no—
137C24—41no—
138C24—41no—
139C24—42no—
140C24#10012yes—
141C24C2×C22⋊C411yescharacteristic
142C2×Q8—24no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—84no—
149C2×Q8—84no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4#10012yes—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4#12014yes—
196C22×C4#13514yes—
197C22×C4AGL(1,ℤ/8)12yes—
198C22×C4#13712yes—
199C2×C8—84no—
200C2×C8—84no—
201C2×C8—84no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—48no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—48no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C42—24no—
216C42—42no—
217C42—42no—
218C42—48no—
219C42—42no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—82no—
236C23—82no—
237C23—82no—
238C23—82no—
239C23#37911yescharacteristic
240C23C2×#10011yescharacteristic
241C23C2×#10011yescharacteristic
242C23#41612yes—
243C23#41712yes—
244Q8—84no—
245Q8—84no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#41012yes—
289C2×C4#41312yes—
290C2×C4#44914yes—
291C2×C4#45214yes—
292C8—164no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—84no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22#154311yescharacteristic
310C22#155112yes—
311C22#156011yescharacteristic
312C22#178211yescharacteristic
313C22#178311yescharacteristic
314C22#178511yescharacteristic
315C22#178812yes—
316C22#179612yes—
317C4—22no—
318C4—22no—
319C4—24no—
320C4—24no—
321C4—42no—
322C4—82no—
323C4—82no—
324C4—82no—
325C4—82no—
326C4—82no—
327C2—21no—
328C2—22no—
329C2—21no—
330C2—82no—
331C2#834411yescharacteristic
332C2#838212yes—
333C2#838912yes—
334C2#852411yescharacteristic
335C2#855311yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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. #8723C2semidirect
4. #7643C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
#87232
#85592
#76432