Finite Group Databasebeta

← #194568

Subgroups — #194568

Order 512.

1302 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 258 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×#3140C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic
3#21866C212yesmaximal, complemented
4C2×#1533C211yesmaximal, characteristic
5#8999C218yesmaximal, complemented
6#8033C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C22×#124C2211yescharacteristic
9#3140—24no—
10#3140C418yescomplemented
11C2×#489C2214yes—
12C2×#488—22no—
13C2×#432C2212yes—
14C2×#419—21no—
15C2×#418—21no—
16C2×C4×M16C2211yescharacteristic
17#2379—24no—
18#2379—22no—
19#2379C2214yes—
20C2×#391—21no—
21C2×#388—22no—
22C2×#386—21no—
23C2×#385—22no—
24C2×#382C2212yes—
25#1993C2214yescomplemented
26#1960C2214yescomplemented
27#1872—22no—
28#1872—22no—
29#1805—24no—
30#1533—24no—
31C2×#329C2214yes—
32#1295C2218yescomplemented
33C23×Q8D411yescharacteristic
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C22×M16—21no—
37#533—22no—
38#533—24no—
39#533C2×C414yescomplemented
40C2×C4⋊Q8C2312yes—
41C2×#125—21no—
42C2×#124—22no—
43C2×#124—24no—
44C2×#124C2×C414yes—
45C2×#124D412yes—
46C2×#124D414yes—
47C2×#124C2312yes—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#489—28no—
60#489C2×C4116yescomplemented
61#488—216no—
62#432—24no—
63#432—24no—
64C2×C4⋊C8—22no—
65#419—28no—
66#418—28no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#102—24no—
71C2×C22⋊C8—22no—
72C4×M16—22no—
73C4×M16C2×C414yescomplemented
74C2×#97C2314yes—
75C2×#97C2312yes—
76C2×C4×C8—21no—
77#391—24no—
78#391—24no—
79#388—28no—
80#388—28no—
81#386—24no—
82#386—24no—
83#385—28no—
84#385—28no—
85#382—22no—
86#382—22no—
87#382D414yes—
88#382C2314yes—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92#340—24no—
93#340—24no—
94#329—28no—
95C25—21no—
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8C22⋊C414yes—
102C22×Q8C2×D412yes—
103C22×D4—21no—
104C22×D4—22no—
105C22×D4—42no—
106C22×D4—42no—
107C22×D4C22⋊C414yes—
108C22×D4C2×D412yes—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—84no—
114C23×C4—82no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C2×M16—24no—
118C2×M16—42no—
119C2×M16—42no—
120C22×C8—22no—
121C22×C8—24no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8C22×C414yescomplemented
126#125—24no—
127#125—42no—
128#124—28no—
129#124—28no—
130#124—24no—
131#124—24no—
132#124C22⋊C418yes—
133#124C22×C414yes—
134#124C2×D414yes—
135C22⋊Q8—28no—
136C22⋊Q8—44no—
137C4×Q8—28no—
138C4×Q8—44no—
139#117—24no—
140#117—42no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×D414yes—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—84no—
160C2×C22⋊C4—82no—
161C2×C42—24no—
162C2×C42—41no—
163C2×C42—82no—
164C2×C42C22×C412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42#4114yes—
169C2×C42C2411yescharacteristic
170C4⋊C8—44no—
171#103—216no—
172#103—84no—
173#103—84no—
174#102—216no—
175C22⋊C8—44no—
176#97—22no—
177#97—24no—
178#97—24no—
179#97C22×C414yescomplemented
180C4×C8—24no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186C24—22no—
187C24—24no—
188C24—44no—
189C24—44no—
190C24—42no—
191C24—42no—
192C24#12011yescharacteristic
193C2×Q8—22no—
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—48no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—42no—
202C2×Q8—84no—
203C2×Q8—81no—
204C2×Q8—81no—
205C2×Q8C2×C22⋊C414yes—
206C2×D4—22no—
207C2×D4—24no—
208C2×D4—48no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—84no—
212C2×D4—81no—
213C2×D4—81no—
214C2×D4C2×C22⋊C414yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—84no—
233C22×C4—88no—
234C22×C4—816no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C4×D414yes—
237C22×C4#12011yescharacteristic
238C22×C4#12012yes—
239C22×C4#12114yes—
240C22×C4#12312yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D412yes—
243M16—48no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—48no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—84no—
260C22⋊C4—28no—
261C22⋊C4—28no—
262C22⋊C4—216no—
263C22⋊C4—44no—
264C22⋊C4—84no—
265C22⋊C4—88no—
266C42—44no—
267C42—84no—
268C42C2×C22⋊C412yes—
269C42C4×D414yes—
270C42#12114yes—
271C42#12312yes—
272C42C23×C411yescharacteristic
273C42C22×D411yescharacteristic
274C42C2×#4112yes—
275C23—22no—
276C23—22no—
277C23—24no—
278C23—416no—
279C23—48no—
280C23—42no—
281C23—42no—
282C23—44no—
283C23—44no—
284C23—84no—
285C23—88no—
286C23—84no—
287C23#38612yes—
288C23#45512yes—
289C23#45512yes—
290C23C2×#12011yescharacteristic
291Q8—44no—
292Q8—48no—
293Q8—82no—
294Q8—84no—
295Q8—84no—
296D4—44no—
297D4—82no—
298D4—84no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—88no—
323C2×C4—88no—
324C2×C4—816no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#38612yes—
328C2×C4#38614yes—
329C2×C4C22×C22⋊C411yescharacteristic
330C2×C4C2×C4×D412yes—
331C2×C4C2×#12112yes—
332C2×C4C2×#12311yescharacteristic
333C8—44no—
334C8—84no—
335C22—22no—
336C22—24no—
337C22—22no—
338C22—24no—
339C22—48no—
340C22—44no—
341C22—816no—
342C22—82no—
343C22—84no—
344C22—82no—
345C22#188514yes—
346C22C2×#38611yescharacteristic
347C22C2×#45511yescharacteristic
348C22C2×#45511yescharacteristic
349C22#301912yes—
350C22#302012yes—
351C4—22no—
352C4—22no—
353C4—24no—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—44no—
358C4—84no—
359C4—84no—
360C4—84no—
361C2—24no—
362C2—84no—
363C2#1899711yescharacteristic
364C2#1899812yes—
365C2#1899911yescharacteristic
366C2#2094312yes—
367C2C2×#188511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21866C2semidirect
5. #8999C2semidirect
6. #8033C2semidirect
10. #3140C4semidirect
25. #1993C22semidirect
26. #1960C22semidirect
32. #1295C22semidirect
39. #533C2×C4semidirect
60. #489C2×C4semidirect
73. C4×M16C2×C4semidirect
125. C4⋊Q8C22×C4semidirect
179. #97C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31402
C2×#23792
#218662
C2×#15332
#89992
#80332