Finite Group Databasebeta

← #188053

Subgroups — #188053

Order 512.

3410 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 3410 conjugacy classes of subgroups — too many to draw. Only its 334 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
1C22×#536C211yesmaximal, characteristic
2#21825C214yesmaximal, complemented
3#20584C212yesmaximal
4#8544C214yesmaximal, complemented
5#8543C214yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C24×Q8C411yescharacteristic, complemented
8C2×#536—24no—
9C2×#536—24no—
10C2×#536—216no—
11C22×C4⋊Q8C412yescomplemented
12C22×#124C2212yes—
13C22×#123C412yescomplemented
14C22×#123C2212yes—
15C22×C22⋊Q8C412yescomplemented
16C22×C22⋊Q8C2212yes—
17C22×#117C411yescharacteristic, complemented
18C2×#410—24no—
19#2280C2214yes—
20#1996—216no—
21#1819C22116yes—
22#1782—24no—
23#1781C2214yes—
24#1781C2214yes—
25C22×#41—28no—
26C22×#41—48no—
27C22×#41D414yes—
28C22×#41D414yes—
29C23×Q8—28no—
30C23×Q8—22no—
31C23×Q8—22no—
32C23×Q8—48no—
33C23×Q8C2×C412yes—
34C23×Q8C2×C412yes—
35C23×Q8C2×C411yescharacteristic
36C23×D4C2×C411yescharacteristic
37C23×D4C2311yescharacteristic
38C23×D4C2311yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43#536—232no—
44#536—232no—
45#536—264no—
46C2×C4⋊Q8—28no—
47C2×C4⋊Q8—216no—
48C2×#124—28no—
49C2×#124—216no—
50C2×#123—28no—
51C2×#123—28no—
52C2×#123—216no—
53C2×#123—216no—
54C2×C22⋊Q8—28no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—28no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8D4116yes—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—28no—
65C2×#117—24no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C414yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C414yes—
71C22×C22⋊C4C2314yes—
72C22×C42C2×C412yes—
73#410—416no—
74C2×#99—28no—
75C2×#99—28no—
76#381—24no—
77#380—28no—
78#379C2318yes—
79C4×C22⋊C4—28no—
80#342—232no—
81C25C22×C411yescharacteristic
82C25C22×C411yescharacteristic
83C25C2411yescharacteristic
84C2×#41—216no—
85C2×#41—28no—
86C2×#41—28no—
87C2×#41—216no—
88C2×#41—48no—
89C2×#41—48no—
90C2×#41—48no—
91C2×#41—464no—
92C22×Q8—216no—
93C22×Q8—216no—
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—22no—
97C22×Q8—28no—
98C22×Q8—24no—
99C22×Q8—24no—
100C22×Q8—44no—
101C22×Q8—48no—
102C22×Q8—48no—
103C22×Q8—464no—
104C22×Q8—48no—
105C22×Q8—48no—
106C22×Q8—48no—
107C22×Q8—48no—
108C22×Q8C22⋊C414yes—
109C22×Q8C22⋊C418yes—
110C22×D4—24no—
111C22×D4—24no—
112C22×D4—24no—
113C22×D4—44no—
114C22×D4—44no—
115C22×D4—44no—
116C22×D4C22⋊C414yes—
117C22×D4C2×D414yes—
118C22×D4C2×D414yes—
119C23×C4—28no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—28no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—48no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C414yes—
134C23×C4C22×C414yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D418yes—
140C4⋊Q8—264no—
141#124—264no—
142#123—264no—
143#123—264no—
144C22⋊Q8—232no—
145C22⋊Q8—232no—
146C22⋊Q8—216no—
147C22⋊Q8—216no—
148C22⋊Q8—232no—
149#117—216no—
150#117—216no—
151#117—416no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4C22⋊C418yes—
162C2×C4⋊C4C22⋊C4116yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—216no—
172C2×C22⋊C4C2×D4116yes—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42—44no—
176#99—416no—
177#99—416no—
178#95—48no—
179C24—22no—
180C24—24no—
181C24—22no—
182C24—42no—
183C24—42no—
184C24—44no—
185C24—44no—
186C24—44no—
187C24C2×C22⋊C412yes—
188C24C2×C22⋊C412yes—
189C24C23×C411yescharacteristic
190C24C22×D412yes—
191C24C22×D414yes—
192#41—432no—
193#41—432no—
194#41—464no—
195C2×Q8—216no—
196C2×Q8—28no—
197C2×Q8—28no—
198C2×Q8—28no—
199C2×Q8—28no—
200C2×Q8—44no—
201C2×Q8—416no—
202C2×Q8—432no—
203C2×Q8—464no—
204C2×Q8—48no—
205C2×Q8—48no—
206C2×Q8—416no—
207C2×Q8—464no—
208C2×Q8—416no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×D4—28no—
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—28no—
215C2×D4—28no—
216C2×D4—28no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—416no—
221C2×D4—416no—
222C2×D4—416no—
223C22×C4—232no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—216no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—416no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—416no—
248C22×C4—432no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4#9918yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C414yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C418yes—
261C22×C4C2×C22⋊C418yes—
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4#12014yes—
265C22×C4#12014yes—
266C22×C4#12018yes—
267C4⋊C4—232no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—232no—
271C4⋊C4—88no—
272C22⋊C4—216no—
273C22⋊C4—232no—
274C22⋊C4—232no—
275C22⋊C4—416no—
276C22⋊C4—432no—
277C42—216no—
278C42—88no—
279C23—22no—
280C23—24no—
281C23—28no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—28no—
286C23—22no—
287C23—24no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—416no—
293C23—44no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23—88no—
300C23—88no—
301C23C2×#9914yes—
302C23C22×C22⋊C411yescharacteristic
303C23C22×C22⋊C412yes—
304C23C2×#12014yes—
305Q8—416no—
306Q8—416no—
307Q8—432no—
308Q8—416no—
309D4—416no—
310D4—416no—
311D4—416no—
312C2×C4—28no—
313C2×C4—216no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—216no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—22no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—432no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—88no—
339C2×C4—816no—
340C2×C4#37914yes—
341C2×C4#37912yes—
342C2×C4#37912yes—
343C2×C4C2×#9914yes—
344C2×C4C2×#9914yes—
345C22—28no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22—24no—
351C22—28no—
352C22—44no—
353C22—44no—
354C22—416no—
355C22—44no—
356C22—42no—
357C22—44no—
358C22—84no—
359C22—88no—
360C22—84no—
361C22—88no—
362C22C2×#37911yescharacteristic
363C22C22×#9912yes—
364C22#288414yes—
365C4—22no—
366C4—24no—
367C4—24no—
368C4—42no—
369C4—48no—
370C4—48no—
371C4—168no—
372C4#178114yes—
373C2—22no—
374C2—24no—
375C2—44no—
376C2—44no—
377C2—82no—
378C2C2×#178111yescharacteristic
379C2#2058712yes—
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21825C2semidirect
4. #8544C2semidirect
5. #8543C2semidirect
7. C24×Q8C4semidirect
11. C22×C4⋊Q8C4semidirect
13. C22×#123C4semidirect
15. C22×C22⋊Q8C4semidirect
17. C22×#117C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5362
#218252
#205842
#85442
#85432