Finite Group Databasebeta

← #107334

Subgroups — #107334

Order 512.

936 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 186 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
1#34369C211yesmaximal, characteristic
2#21209C211yesmaximal, characteristic
3#17663C211yesmaximal, characteristic
4#8052C212yesmaximal, complemented
5#7962C212yesmaximal, complemented
6C2×#532C2211yescharacteristic
7#2920C2212yes—
8C2×#423C2211yescharacteristic
9C2×#421—22no—
10#2381C412yescomplemented
11#2380C412yescomplemented
12#2372—22no—
13#2352—24no—
14#2333—24no—
15#2311—22no—
16#2283C2212yes—
17C4×#117C2211yescharacteristic
18#1963C414yescomplemented
19#1962—22no—
20#1536—22no—
21#1533—22no—
22#1525C414yescomplemented
23#1297—24no—
24#1286—24no—
25C22×#41D412yes—
26C22×M16—21no—
27#532—22no—
28#532—22no—
29#532C2×C412yes—
30#532C2×C412yes—
31#532D412yes—
32#532Q812yes—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×#127C2×C411yescharacteristic
35C2×#124D412yes—
36C2×#121—22no—
37C2×C4×D4—22no—
38C2×#117—22no—
39C2×#117—21no—
40C22×C4⋊C4—22no—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#431—22no—
45#431—22no—
46#423—22no—
47#423—21no—
48#423—22no—
49#423C2×C412yescomplemented
50C2×C4⋊C8—22no—
51#421—44no—
52#421—44no—
53C2×#104—28no—
54#407—22no—
55C4×M16C2×C414yescomplemented
56#391—24no—
57#390—24no—
58#388—28no—
59#387—28no—
60#386—24no—
61#386—24no—
62#386—28no—
63#383C2×C414yes—
64#382D414yes—
65#381—22no—
66#380—24no—
67#380—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C22⋊C4—28no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4D414yes—
73#376—22no—
74#376C2×C412yescomplemented
75C43—21no—
76#329—28no—
77C2×#41—22no—
78C2×#41—22no—
79C2×#41—42no—
80C2×#41—42no—
81C2×#41—42no—
82C2×#41—42no—
83C2×#41C22⋊C414yes—
84C2×#41C4⋊C414yes—
85C22×Q8C22⋊C412yes—
86C22×D4—22no—
87C22×D4—22no—
88C22×D4C22⋊C412yes—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C2×M16—22no—
94C2×M16—24no—
95C2×M16—41no—
96C2×M16—42no—
97C2×M16—42no—
98C22×C8—41no—
99C4⋊Q8—22no—
100C4⋊Q8C4212yes—
101C4⋊Q8C4⋊C412yes—
102#127—22no—
103#127C4212yes—
104#127C4⋊C412yes—
105#124—24no—
106#124C22⋊C414yes—
107#124C4⋊C414yes—
108#121—24no—
109#121—24no—
110#121—24no—
111#121—24no—
112C4×D4—24no—
113C4×D4—24no—
114C4×D4—24no—
115C4×D4—24no—
116#117—24no—
117#117—42no—
118#117—82no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4—82no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—41no—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—42no—
135C2×C42C22×C411yescharacteristic
136C2×C42C22×C411yescharacteristic
137C2×C42C22×C411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×Q811yescharacteristic
142C2×C42#4112yes—
143C2×C42#4112yes—
144C4⋊C8—24no—
145C4⋊C8—42no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#104—416no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151#97—24no—
152C4×C8—24no—
153#95—28no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95C22⋊C418yes—
159C24—22no—
160C24#12011yescharacteristic
161#41—44no—
162#41—44no—
163#41—44no—
164#41—44no—
165#41—82no—
166#41—82no—
167#41—82no—
168#41—82no—
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8#9514yes—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—22no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—82no—
181C2×D4—82no—
182C2×D4—82no—
183C2×D4—82no—
184C2×D4#9514yes—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—82no—
197C22×C4—84no—
198C22×C4—88no—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C4⋊C411yescharacteristic
203C22×C4C4×D412yes—
204C22×C4C4×D412yes—
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4#12112yes—
208C22×C4C22⋊Q812yes—
209C22×C4#12311yescharacteristic
210C22×C4#12311yescharacteristic
211M16—44no—
212M16—82no—
213M16—84no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—81no—
217C2×C8—81no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—84no—
224C22⋊C4—28no—
225C22⋊C4—28no—
226C22⋊C4—44no—
227C22⋊C4—88no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C42—24no—
231C42—42no—
232C42—84no—
233C42—84no—
234C42—84no—
235C42C2×C4211yescharacteristic
236C42C2×C22⋊C411yescharacteristic
237C42C2×C4⋊C411yescharacteristic
238C42C2×C4⋊C411yescharacteristic
239C42#11712yes—
240C42C4×D412yes—
241C42C4×D412yes—
242C42#12112yes—
243C42C22⋊Q812yes—
244C42#12312yes—
245C23—22no—
246C23—24no—
247C23—42no—
248C23—44no—
249C23—44no—
250C23—42no—
251C23—42no—
252C23#37911yescharacteristic
253C23#38511yescharacteristic
254C23#38611yescharacteristic
255Q8—44no—
256Q8—82no—
257D4—44no—
258D4—82no—
259D4—84no—
260D4—84no—
261D4—84no—
262D4—84no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—84no—
288C2×C4—168no—
289C2×C4C2×#9511yescharacteristic
290C2×C4C4×C22⋊C412yes—
291C2×C4#38012yes—
292C2×C4#38111yescharacteristic
293C2×C4#38512yes—
294C2×C4#38511yescharacteristic
295C2×C4#38612yes—
296C2×C4#38611yescharacteristic
297C2×C4#45312yes—
298C2×C4#45512yes—
299C8—84no—
300C8—82no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—48no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—84no—
312C22C22⋊#9511yescharacteristic
313C22#179212yes—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—88no—
322C4—84no—
323C4#185612yes—
324C4#188511yescharacteristic
325C4#188711yescharacteristic
326C2—22no—
327C2—41no—
328C2—41no—
329C2—84no—
330C2#1074911yescharacteristic
331C2#1078312yes—
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8052C2semidirect
5. #7962C2semidirect
10. #2381C4semidirect
11. #2380C4semidirect
18. #1963C4semidirect
22. #1525C4semidirect
49. #423C2×C4semidirect
55. C4×M16C2×C4semidirect
74. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343692
#212092
#176632
#80522
#79622