Finite Group Databasebeta

← #98506

Subgroups — #98506

Order 512.

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

normal subgroups only

This group has 886 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×#2503C211yesmaximal, characteristic
2#8718C212yesmaximal, complemented
3#8152C212yesmaximal
4#7629C212yesmaximal, complemented
5C22×C22⋊Q8C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7#2503—24no—
8#2503—24no—
9C2×#400—22no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#382C411yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#381—22no—
15C2×C4×C22⋊C4C2211yescharacteristic
16#1749C2214yes—
17C2×#323—22no—
18#1295—24no—
19#1280—22no—
20#1272—24no—
21C23×Q8C2×C411yescharacteristic
22C24×C4—21no—
23C24×C4C2×C411yescharacteristic
24C2×C22⋊Q8—22no—
25C2×C22⋊Q8—22no—
26C2×C22⋊Q8—42no—
27C2×C22⋊Q8—42no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—42no—
33C22×C4⋊C4—42no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4D412yes—
39C22×C42C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—24no—
42C2×#97—24no—
43#400—216no—
44#391—22no—
45#391—22no—
46#391—22no—
47#391D414yes—
48#382—22no—
49#382—22no—
50#382—22no—
51#382D414yes—
52#381—216no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95D412yes—
60#336—24no—
61#323—28no—
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—24no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—44no—
82C23×C4—42no—
83C23×C4—44no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D412yes—
88C22×C8—44no—
89C22⋊Q8—48no—
90C22⋊Q8—48no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—48no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4C22⋊C412yes—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42C2×D412yes—
122C2×C42C2×D412yes—
123C22⋊C8—28no—
124C22⋊C8—44no—
125#97—48no—
126#95—28no—
127#95—28no—
128#95—28no—
129#95—28no—
130#95—24no—
131C24—21no—
132C24—21no—
133C24—22no—
134C24—41no—
135C24—41no—
136C24—42no—
137C24#10012yes—
138C24#10414yes—
139C24C2×C22⋊C411yescharacteristic
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—84no—
147C2×Q8—84no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
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—48no—
176C22×C4—44no—
177C22×C4—416no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—88no—
195C22×C4—84no—
196C22×C4—84no—
197C22×C4#10012yes—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4#12014yes—
201C2×C8—48no—
202C2×C8—84no—
203C2×C8—84no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—48no—
210C4⋊C4—42no—
211C4⋊C4—48no—
212C22⋊C4—48no—
213C22⋊C4—48no—
214C42—24no—
215C42—42no—
216C42—42no—
217C42—42no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23—24no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—44no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—44no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23#32914yes—
237C23#37911yescharacteristic
238C23C2×#10011yescharacteristic
239C23C2×#10011yescharacteristic
240C23C2×#10412yes—
241C23#42112yes—
242Q8—84no—
243Q8—84no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—816no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4#41012yes—
288C2×C4#41312yes—
289C8—88no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—24no—
299C22—44no—
300C22—48no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—44no—
305C22—44no—
306C22C2×#32911yescharacteristic
307C22#152712yes—
308C22#153311yescharacteristic
309C22#178211yescharacteristic
310C22#178311yescharacteristic
311C22#178511yescharacteristic
312C22#178912yes—
313C22#179612yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—44no—
318C4—82no—
319C4—82no—
320C4—82no—
321C4—82no—
322C4—82no—
323C2—21no—
324C2—22no—
325C2—21no—
326C2—44no—
327C2#833411yescharacteristic
328C2#836312yes—
329C2#837112yes—
330C2#852411yescharacteristic
331C2#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. #8718C2semidirect
4. #7629C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#87182
#81522
#76292