Finite Group Databasebeta

← #185097

Subgroups — #185097

Order 512.

2468 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 2468 conjugacy classes of subgroups — too many to draw. Only its 338 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×#3501C211yesmaximal, characteristic
2#21790C211yesmaximal, characteristic
3C2×#1515C212yesmaximal, complemented
4C2×#1506C212yesmaximal, complemented
5#13959C211yesmaximal, characteristic
6#8329C218yesmaximal, complemented
7#3501—28no—
8#3501—28no—
9C2×#554—24no—
10C2×#554C414yescomplemented
11C2×D4×Q8C414yescomplemented
12C2×#544—21no—
13C2×#543—22no—
14C2×#533C2212yes—
15C22×C22⋊Q8C2214yes—
16C2×#518—21no—
17C22×C4×D4C2211yescharacteristic
18C2×#432C2212yes—
19C2×#383C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1955—24no—
22#1779C2214yes—
23#1515—28no—
24#1506—28no—
25C2×#328C2218yes—
26C2×#328C2218yes—
27#1445—21no—
28#1438—21no—
29#1437—22no—
30C22⋊#95C2214yes—
31C23×Q8C2×C414yes—
32C23×D4D411yescharacteristic
33C24×C4C2312yes—
34#554—232no—
35#554—232no—
36#554—216no—
37#554—216no—
38D4×Q8—216no—
39D4×Q8—216no—
40#544—28no—
41#544—28no—
42#543—216no—
43#543—216no—
44#533—24no—
45#533—24no—
46#533D418yes—
47#533D418yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C412yes—
51C2×#125—24no—
52C2×#125—22no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—216no—
57C2×C22⋊Q8—216no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8—28no—
60C2×C22⋊Q8C2×C418yes—
61C2×C22⋊Q8C2314yes—
62#518—28no—
63#518—28no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C2×#117—22no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4C2×C414yes—
75C22×C4⋊C4D411yescharacteristic
76C22×C22⋊C4—24no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#432—24no—
82#432—24no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—28no—
85C2×C22⋊C8C2314yes—
86C2×#97—22no—
87#383—24no—
88#376—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95C2312yes—
94C2×#95C2312yes—
95#328—216no—
96#328—216no—
97C25C2×D412yes—
98C22×Q8—216no—
99C22×Q8—22no—
100C22×Q8—24no—
101C22×Q8—88no—
102C22×Q8C22×C414yes—
103C22×D4—28no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4C22⋊C414yes—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—84no—
114C23×C4C22×C414yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D414yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C4⋊Q8—232no—
122C4⋊Q8—24no—
123C4⋊Q8—24no—
124C4⋊Q8C22⋊C418yescomplemented
125C4⋊Q8C22⋊C418yescomplemented
126#125—232no—
127#125—28no—
128#125—28no—
129C22⋊Q8—216no—
130C22⋊Q8—216no—
131C22⋊Q8—232no—
132C22⋊Q8—264no—
133C22⋊Q8—264no—
134C22⋊Q8—216no—
135C22⋊Q8—28no—
136C22⋊Q8—28no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—216no—
140C22⋊Q8C2×D418yes—
141C22⋊Q8C2×D418yes—
142C4×Q8—28no—
143C4×Q8—28no—
144C4×Q8—28no—
145C4×Q8C22⋊C4116yescomplemented
146C4×D4—232no—
147C4×D4—232no—
148C4×D4—28no—
149C4×D4—28no—
150C4×D4—28no—
151C4×D4—28no—
152#117—28no—
153#117—28no—
154C2×C4⋊C4—232no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—216no—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22×C414yes—
167C2×C22⋊C4—216no—
168C2×C22⋊C4—216no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—84no—
176C2×C22⋊C4C22⋊C418yes—
177C2×C42—24no—
178C2×C42—44no—
179C2×C42—42no—
180C2×C42C22×C412yes—
181C2×C42C2×D414yes—
182C2×C42C2×D414yes—
183C2×C42C2×D412yes—
184C4⋊C8—44no—
185C22⋊C8—24no—
186C22⋊C8—24no—
187C22⋊C8—88no—
188#97—44no—
189#95—24no—
190#95—24no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—84no—
195C24—22no—
196C24—22no—
197C24—24no—
198C24—44no—
199C24—44no—
200C24C2×C22⋊C414yes—
201C24#13718yes—
202C24C22×D411yescharacteristic
203C2×Q8—216no—
204C2×Q8—24no—
205C2×Q8—28no—
206C2×Q8—48no—
207C2×Q8—84no—
208C2×Q8—88no—
209C2×Q8—832no—
210C2×Q8C2×C22⋊C418yes—
211C2×D4—216no—
212C2×D4—216no—
213C2×D4—216no—
214C2×D4—28no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—216no—
219C22×C4—216no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—28no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—48no—
228C22×C4—44no—
229C22×C4—48no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—88no—
240C22×C4—88no—
241C22×C4—88no—
242C22×C4—88no—
243C22×C4C2×C22⋊C414yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4#12012yes—
247C22×C4#12012yes—
248C22×C4#12012yes—
249C22×C4#12012yes—
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D412yes—
253C22×C4C22×D412yes—
254C2×C8—48no—
255C2×C8—44no—
256C4⋊C4—216no—
257C4⋊C4—232no—
258C4⋊C4—24no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—216no—
266C4⋊C4C2×C22⋊C418yes—
267C22⋊C4—232no—
268C22⋊C4—28no—
269C22⋊C4—28no—
270C22⋊C4—216no—
271C22⋊C4—816no—
272C42—216no—
273C42—48no—
274C42C2×C22⋊C414yes—
275C42C2×C22⋊C414yes—
276C42#12014yes—
277C42#12014yes—
278C23—22no—
279C23—22no—
280C23—28no—
281C23—24no—
282C23—28no—
283C23—24no—
284C23—28no—
285C23—28no—
286C23—48no—
287C23—48no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—84no—
293C23—84no—
294C23#41918yes—
295C23C22×C22⋊C411yescharacteristic
296C23C2×#12011yescharacteristic
297C23C2×#12011yescharacteristic
298C23C2×#13714yes—
299Q8—48no—
300Q8—48no—
301Q8—88no—
302Q8—816no—
303D4—232no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—416no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—88no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—816no—
327C2×C4—816no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4#37914yes—
337C2×C4#37914yes—
338C2×C4C22×C22⋊C412yes—
339C2×C4C2×#12012yes—
340C8—84no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—24no—
347C22—28no—
348C22—216no—
349C22—44no—
350C22—44no—
351C22—48no—
352C22—88no—
353C22—88no—
354C22C2×#37911yescharacteristic
355C22#288312yes—
356C22C2×#41912yes—
357C22#290112yes—
358C22#300412yes—
359C22#300412yes—
360C4—22no—
361C4—22no—
362C4—24no—
363C4—24no—
364C4—44no—
365C4—48no—
366C4—48no—
367C4—84no—
368C4—162no—
369C2—22no—
370C2—22no—
371C2—48no—
372C2#1762114yes—
373C2#2058611yescharacteristic
374C2#2059911yescharacteristic
375C2#2060611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1515C2semidirect
4. C2×#1506C2semidirect
6. #8329C2semidirect
10. C2×#554C4semidirect
11. C2×D4×Q8C4semidirect
124. C4⋊Q8C22⋊C4semidirect
125. C4⋊Q8C22⋊C4semidirect
145. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
#217902
C2×#15152
C2×#15062
#139592
#83292