Finite Group Databasebeta

← #184986

Subgroups — #184986

Order 512.

2586 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 2586 conjugacy classes of subgroups — too many to draw. Only its 318 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×#540C211yesmaximal, characteristic
2#17448C214yesmaximal, complemented
3#17357C212yesmaximal, complemented
4#8326C218yesmaximal, complemented
5C2×#540—28no—
6C2×#540—28no—
7C2×#540—24no—
8C2×#540C414yes—
9C2×#540C2214yes—
10C22×#127C411yescharacteristic
11C22×#125C411yescharacteristic
12C22×#123—22no—
13C22×#121—22no—
14C22×#121C412yes—
15C22×C4×D4C2212yes—
16C22×#117C2211yescharacteristic
17#2920C2214yes—
18#1490—28no—
19#1486—28no—
20#1478C22116yescomplemented
21#1464—24no—
22#1463—22no—
23#1462—22no—
24#1458C2218yescomplemented
25C23×D4—22no—
26C23×D4—22no—
27C23×D4C2×C412yes—
28C24×C4—22no—
29C24×C4C2×C411yescharacteristic
30#540—232no—
31#540—232no—
32#540—232no—
33#540—232no—
34C2×#127—24no—
35C2×#127—24no—
36C2×#127—22no—
37C2×#127C2×C414yes—
38C2×#125—24no—
39C2×#125—24no—
40C2×#125—22no—
41C2×#125C2×C414yes—
42C2×#123—216no—
43C2×#123—216no—
44C2×#123—28no—
45C2×#123—28no—
46C2×#121—216no—
47C2×#121—216no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121—28no—
54C2×#121—28no—
55C2×#121C2×C418yes—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C414yes—
61C2×C4×D4D418yes—
62C2×C4×D4D418yes—
63C2×C4×D4C2314yes—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117C2×C412yes—
67C2×#117D414yes—
68C2×#117C2312yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4D412yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42C2×C411yescharacteristic
76C22×C42C2311yescharacteristic
77#432—24no—
78#432—24no—
79#431—24no—
80#431C2318yes—
81#423—24no—
82#407—24no—
83C4×M16—24no—
84D4⋊C8—232no—
85#324—28no—
86#323—28no—
87C25—22no—
88C25C22⋊C411yescharacteristic
89C22×D4—28no—
90C22×D4—28no—
91C22×D4—28no—
92C22×D4—28no—
93C22×D4—28no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—28no—
98C22×D4—48no—
99C22×D4—88no—
100C22×D4—88no—
101C22×D4C22⋊C414yes—
102C22×D4C22⋊C414yes—
103C22×D4C22×C414yes—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—28no—
109C23×C4—28no—
110C23×C4—24no—
111C23×C4—44no—
112C23×C4—88no—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D412yes—
119C2×M16—44no—
120#127—28no—
121#127—28no—
122#127—216no—
123#125—28no—
124#125—28no—
125#125—216no—
126#123—264no—
127#123—264no—
128#121—216no—
129#121—216no—
130#121—264no—
131#121—264no—
132#121—216no—
133#121—216no—
134C4×D4—28no—
135C4×D4—28no—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4C22⋊C4116yes—
141C4×D4C2×D4116yes—
142#117—28no—
143#117—28no—
144#117—28no—
145#117—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C22×C414yes—
158C2×C4⋊C4C2×D414yes—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4C22⋊C414yes—
169C2×C22⋊C4C2×D414yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C22⋊C412yes—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D414yes—
176C2×C42C2×D414yes—
177C2×C42C2×D412yes—
178C2×C42C2411yescharacteristic
179C4⋊C8—28no—
180C4⋊C8—44no—
181C22⋊C8—28no—
182C22⋊C8—44no—
183#97—44no—
184C4×C8—28no—
185C24—24no—
186C24—22no—
187C24—24no—
188C24—24no—
189C24—42no—
190C24—48no—
191C24—48no—
192C24—48no—
193C24—44no—
194C24C2×C22⋊C411yescharacteristic
195C24C2×C22⋊C412yes—
196C2×D4—232no—
197C2×D4—232no—
198C2×D4—24no—
199C2×D4—216no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—48no—
203C2×D4—48no—
204C2×D4—48no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D4—816no—
208C2×D4—816no—
209C2×D4—816no—
210C2×D4—816no—
211C2×D4—816no—
212C2×D4—816no—
213C2×D4C2×C22⋊C418yes—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—28no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—216no—
226C22×C4—216no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—816no—
235C22×C4—88no—
236C22×C4—816no—
237C22×C4—88no—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4#12014yes—
245C22×C4#12014yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—88no—
249C2×C8—48no—
250C4⋊C4—232no—
251C4⋊C4—24no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4C2×C22⋊C418yes—
257C22⋊C4—232no—
258C22⋊C4—216no—
259C22⋊C4—28no—
260C22⋊C4—216no—
261C22⋊C4—28no—
262C42—24no—
263C42—22no—
264C42—22no—
265C42C2×C22⋊C414yes—
266C42#12014yes—
267C42#12014yes—
268C42C22×D414yes—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—28no—
273C23—24no—
274C23—44no—
275C23—48no—
276C23—48no—
277C23—48no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—48no—
282C23—44no—
283C23—48no—
284C23—44no—
285C23—48no—
286C23—48no—
287C23—88no—
288C23—832no—
289C23#37912yes—
290C23C22×C22⋊C411yescharacteristic
291D4—216no—
292D4—48no—
293D4—48no—
294D4—48no—
295D4—832no—
296D4—832no—
297C2×C4—28no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—28no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—48no—
317C2×C4—44no—
318C2×C4—816no—
319C2×C4—816no—
320C2×C4#37911yescharacteristic
321C2×C4#37911yescharacteristic
322C2×C4#37914yes—
323C2×C4C22×C22⋊C412yes—
324C2×C4C2×#12014yes—
325C2×C4#56814yes—
326C8—88no—
327C22—28no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—42no—
332C22—48no—
333C22—48no—
334C22—44no—
335C22—48no—
336C22—84no—
337C22—88no—
338C22—816no—
339C22—816no—
340C22—816no—
341C22#179514yes—
342C22C2×#37911yescharacteristic
343C22#287912yes—
344C4—21no—
345C4—21no—
346C4—24no—
347C4—24no—
348C4—42no—
349C4—48no—
350C4—44no—
351C4#287914yes—
352C2—24no—
353C2—22no—
354C2—82no—
355C2—88no—
356C2#2058011yescharacteristic
357C2C2×#179512yes—
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17448C2semidirect
3. #17357C2semidirect
4. #8326C2semidirect
20. #1478C22semidirect
24. #1458C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
#174482
#173572
#83262