Finite Group Databasebeta

← #98509

Subgroups — #98509

Order 512.

962 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2563C211yesmaximal, characteristic
2#8718C212yesmaximal, complemented
3#8153C212yesmaximal, complemented
4#7626C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×D4C411yescharacteristic
7#2563—24no—
8#2563—24no—
9C2×#393—22no—
10C2×#390C411yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#382C411yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#380—22no—
15C2×C4×C4⋊C4C2211yescharacteristic
16#1749C2214yes—
17C2×#324—22no—
18#1295—24no—
19#1280—22no—
20#1270—24no—
21C23×Q8—21no—
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×C4×D4—22no—
29C2×C4×D4—22no—
30C2×C4×D4—42no—
31C2×C4×D4—42no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4D412yes—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43C2×C22⋊C8—24no—
44C2×#97—24no—
45#393—216no—
46#390—22no—
47#390—22no—
48#390—22no—
49#390D414yes—
50#382—22no—
51#382—22no—
52#382—22no—
53#382D414yes—
54#380—216no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95D412yes—
60#336—24no—
61#324—28no—
62C25—21no—
63C22×Q8—24no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—84no—
67C22×D4—24no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—42no—
72C22×D4C22⋊C412yes—
73C23×C4—24no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—42no—
78C23×C4—44no—
79C23×C4—42no—
80C23×C4—44no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C412yes—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D412yes—
86C22×C8—44no—
87C4⋊Q8—48no—
88C4⋊Q8—48no—
89C4×D4—48no—
90C4×D4—48no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C22⋊C412yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—84no—
115C2×C22⋊C4—84no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C22⋊C8—28no—
127C22⋊C8—44no—
128#97—48no—
129#95—28no—
130#95—28no—
131#95—24no—
132C24—24no—
133C24—22no—
134C24—44no—
135C24—42no—
136C24—41no—
137C24—41no—
138C24—42no—
139C24—42no—
140C24C2×C22⋊C411yescharacteristic
141C2×Q8—48no—
142C2×Q8—88no—
143C2×Q8—88no—
144C2×Q8—88no—
145C2×D4—24no—
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—48no—
151C2×D4—48no—
152C2×D4—42no—
153C2×D4#10114yes—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—48no—
174C22×C4—416no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—88no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4#10012yes—
192C22×C4#10012yes—
193C22×C4#10414yes—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4#12014yes—
197C2×C8—48no—
198C2×C8—84no—
199C2×C8—84no—
200C4⋊C4—48no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—48no—
208C22⋊C4—48no—
209C22⋊C4—88no—
210C22⋊C4—88no—
211C42—24no—
212C42—24no—
213C42—24no—
214C42—42no—
215C42—42no—
216C42—42no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—24no—
221C23—44no—
222C23—44no—
223C23—48no—
224C23—48no—
225C23—44no—
226C23—42no—
227C23—42no—
228C23—44no—
229C23—82no—
230C23—82no—
231C23—84no—
232C23—84no—
233C23—81no—
234C23—82no—
235C23—81no—
236C23#37911yescharacteristic
237C23C2×#10011yescharacteristic
238C23C2×#10011yescharacteristic
239C23C2×#10412yes—
240C23#42112yes—
241Q8—816no—
242D4—44no—
243D4—48no—
244D4—44no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—816no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#32914yes—
284C2×C4#41012yes—
285C2×C4C2×#10112yes—
286C8—88no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—44no—
296C22—48no—
297C22—84no—
298C22—88no—
299C22—84no—
300C22—82no—
301C22—82no—
302C22—84no—
303C22C2×#32911yescharacteristic
304C22#152712yes—
305C22#153311yescharacteristic
306C22#178211yescharacteristic
307C22#178311yescharacteristic
308C22#178411yescharacteristic
309C22#178912yes—
310C22#179512yes—
311C4—22no—
312C4—22no—
313C4—24no—
314C4—42no—
315C4—44no—
316C4—82no—
317C4—82no—
318C4—82no—
319C4—82no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—84no—
324C2#833411yescharacteristic
325C2#836012yes—
326C2#837212yes—
327C2#852211yescharacteristic
328C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
3. #8153C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#87182
#81532
#76262