Finite Group Databasebeta

← #480728

Subgroups — #480728

Order 512.

1188 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2547C211yesmaximal, characteristic
2#17362C212yesmaximal
3#11564C218yesmaximal, complemented
4#11506C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×#124C412yes—
7C22×C4×D4C2211yescharacteristic
8C2×#407—22no—
9#2547—22no—
10#2547—22no—
11#2547C414yes—
12#2547C2214yes—
13C2×#397—22no—
14C2×#391C2212yes—
15C2×#380—22no—
16C2×C4×C4⋊C4C411yescharacteristic
17#1465—22no—
18#1465C2214yes—
19#1463C2214yes—
20#1462—22no—
21#1459—22no—
22#1459C2214yes—
23#1319C22116yescomplemented
24C23×Q8C2×C412yes—
25C23×D4C2×C411yescharacteristic
26C24×C4D412yes—
27C22×M16—42no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—41no—
32C2×C4⋊Q8C2×C414yes—
33C2×#124—22no—
34C2×#124—22no—
35C2×#124—48no—
36C2×#124C2×C418yes—
37C2×C4×D4—41no—
38C2×C4×D4—42no—
39C2×C4×D4—41no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4—24no—
49C22×C22⋊C4D412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C412yes—
52#407—24no—
53#407—24no—
54#407—42no—
55#407—42no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—42no—
58#397—24no—
59#397—24no—
60#397—24no—
61#397—24no—
62#391—24no—
63#391C2×C414yes—
64#391C2314yes—
65#380—24no—
66#380—24no—
67#380—24no—
68#380—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2×C414yes—
71C2×#95—21no—
72C2×#95—24no—
73#324—24no—
74#324C2318yes—
75#323—24no—
76#323—24no—
77C25C22⋊C412yes—
78C22×Q8—22no—
79C22×Q8—44no—
80C22×Q8—44no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8C22×C414yes—
84C22×D4—24no—
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4C22⋊C411yescharacteristic
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22×C412yes—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—48no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C2×M16—44no—
107C2×M16—44no—
108C2×M16—88no—
109C22×C8—44no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114#124—48no—
115#124—48no—
116#124—416no—
117C4×D4—42no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2411yescharacteristic
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—416no—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C22⋊C412yes—
146C2×C22⋊C4C2×D412yes—
147C2×C22⋊C4C2×D412yes—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—44no—
153C2×C42C22⋊C412yes—
154C2×C42C22×C412yes—
155C2×C42C22×C414yes—
156C22⋊C8—28no—
157C22⋊C8—44no—
158C22⋊C8—44no—
159C22⋊C8—44no—
160#95—24no—
161#95—28no—
162#95—28no—
163C24—24no—
164C24—22no—
165C24—22no—
166C24—42no—
167C24—44no—
168C24—44no—
169C24C2×C22⋊C411yescharacteristic
170C24C2×C22⋊C412yes—
171C24C2×C22⋊C412yes—
172C2×Q8—28no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—88no—
179C2×Q8—88no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—48no—
189C2×D4—48no—
190C2×D4#10014yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4#9911yescharacteristic
228C22×C4#9912yes—
229C22×C4#9911yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C22×D412yes—
236M16—88no—
237M16—88no—
238C2×C8—48no—
239C2×C8—88no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—416no—
252C22⋊C4—416no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—44no—
258C42—44no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—44no—
269C23—44no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—416no—
275C23—44no—
276C23—44no—
277C23—82no—
278C23—82no—
279C23—88no—
280C23C2×#9911yescharacteristic
281C23C2×#9911yescharacteristic
282C23C2×#10014yes—
283C23C22×C22⋊C411yescharacteristic
284Q8—88no—
285Q8—88no—
286D4—44no—
287D4—48no—
288D4—44no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—88no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4#35414yes—
326C2×C4C2×#9912yes—
327C2×C4C2×#9912yes—
328C2×C4C2×#10012yes—
329C2×C4#56812yes—
330C8—88no—
331C22—24no—
332C22—24no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—24no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—88no—
349C22—88no—
350C22—82no—
351C22C2×#35412yes—
352C22#213512yes—
353C22#287911yescharacteristic
354C22C22×#9911yescharacteristic
355C22C22×#10011yescharacteristic
356C4—22no—
357C4—22no—
358C4—22no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—84no—
363C4—84no—
364C4C2×#35414yes—
365C2—21no—
366C2—21no—
367C2—22no—
368C2—22no—
369C2—44no—
370C2—82no—
371C2#1739611yescharacteristic
372C2C22×#35411yescharacteristic
373C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11564C2semidirect
4. #11506C2semidirect
23. #1319C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#173622
#115642
#115062