Finite Group Databasebeta

← #98525

Subgroups — #98525

Order 512.

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

normal subgroups only

This group has 930 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×#2566C211yesmaximal, characteristic
2#8718C212yesmaximal, complemented
3#8153C212yesmaximal, complemented
4#7626C212yesmaximal
5C22×#125C411yescharacteristic
6C22×C4×D4C411yescharacteristic
7#2566—24no—
8#2566—24no—
9C2×#396—22no—
10C2×#390C411yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#385—22no—
13C2×#384C411yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1749C2214yes—
17C2×#324—22no—
18#1295—24no—
19#1280—22no—
20#1270—24no—
21C23×D4C2×C411yescharacteristic
22C24×C4—21no—
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—42no—
26C2×#125—42no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—42no—
30C2×C4×D4—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4D412yes—
38C22×C22⋊C4—21no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—22no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2311yescharacteristic
44C2×C22⋊C8—24no—
45C2×#97—24no—
46#396—216no—
47#390—22no—
48#390—22no—
49#390—22no—
50#390D414yes—
51#385—216no—
52#384—22no—
53#384—22no—
54#382—22no—
55#382D414yes—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C2×#95—22no—
59C2×#95D412yes—
60#336—24no—
61#324—28no—
62C25—21no—
63C22×D4—24no—
64C22×D4—21no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—42no—
68C22×D4C22⋊C412yes—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—24no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—84no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D412yes—
81C22×C8—44no—
82#125—48no—
83#125—48no—
84C4×D4—48no—
85C4×D4—48no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—84no—
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—44no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—84no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42C2×D412yes—
126C2×C42C2×D412yes—
127C22⋊C8—28no—
128C22⋊C8—44no—
129#97—48no—
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×D4—24no—
142C2×D4—22no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—42no—
149C2×D4#10114yes—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—48no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—88no—
184C22×C4—88no—
185C22×C4—88no—
186C22×C4#10012yes—
187C22×C4#10012yes—
188C22×C4#10414yes—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4#12014yes—
192C2×C8—48no—
193C2×C8—84no—
194C2×C8—84no—
195C4⋊C4—48no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—48no—
203C4⋊C4—48no—
204C4⋊C4—88no—
205C22⋊C4—416no—
206C22⋊C4—48no—
207C22⋊C4—88no—
208C42—24no—
209C42—24no—
210C42—24no—
211C42—42no—
212C42—42no—
213C42—42no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—24no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—48no—
222C23—44no—
223C23—42no—
224C23—42no—
225C23—44no—
226C23—82no—
227C23—82no—
228C23—84no—
229C23—84no—
230C23—81no—
231C23—81no—
232C23—82no—
233C23#37911yescharacteristic
234C23C2×#10011yescharacteristic
235C23C2×#10011yescharacteristic
236C23C2×#10412yes—
237C23#42112yes—
238D4—44no—
239D4—48no—
240D4—44no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—816no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4#41012yes—
281C2×C4C2×#10112yes—
282C8—88no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—44no—
292C22—48no—
293C22—84no—
294C22—84no—
295C22—88no—
296C22—82no—
297C22—82no—
298C22—84no—
299C22#152511yescharacteristic
300C22#153012yes—
301C22#153611yescharacteristic
302C22#178211yescharacteristic
303C22#178311yescharacteristic
304C22#178411yescharacteristic
305C22#178912yes—
306C22#179512yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—44no—
312C4—44no—
313C4—82no—
314C4—82no—
315C4—82no—
316C2—21no—
317C2—21no—
318C2—22no—
319C2—84no—
320C2#833711yescharacteristic
321C2#836212yes—
322C2#837212yes—
323C2#852211yescharacteristic
324C2#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×#25662
#87182
#81532
#76262