Finite Group Databasebeta

← #115172

Subgroups — #115172

Order 512.

1130 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 158 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×#2502C211yesmaximal, characteristic
2#21753C211yesmaximal, characteristic
3C2×#1821C212yesmaximal
4#17397C211yesmaximal, characteristic
5#7929C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9C2×#407C2212yes—
10#2502—24no—
11#2502—24no—
12C2×#398—22no—
13C2×#391—21no—
14C2×#385C414yes—
15C2×#384—21no—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1901—24no—
18#1821—24no—
19#1821—24no—
20#1821—21no—
21#1784—21no—
22C2×#343—24no—
23#1472—24no—
24#1463—22no—
25#1282—24no—
26C23×Q8D411yescharacteristic
27C24×C4C2×C412yes—
28C24×C4C2311yescharacteristic
29C22×M16—22no—
30C22×M16D414yes—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4D414yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D412yes—
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C412yes—
50C22×C42—21no—
51C2×#101—24no—
52#407—22no—
53#407—22no—
54#407—22no—
55#407—22no—
56#407—22no—
57#407—22no—
58#407—22no—
59#407D414yes—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64#398—216no—
65#391—28no—
66#385—28no—
67#385—28no—
68#384—28no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4D414yes—
71C2×#95—24no—
72C2×#95—21no—
73#343—216no—
74#343—24no—
75#324—28no—
76C25C22×C411yescharacteristic
77C22×Q8—22no—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—21no—
81C22×Q8—44no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—44no—
88C23×C4—84no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C414yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4#4112yes—
99C23×C4#4112yes—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—42no—
104C2×M16—48no—
105C2×M16—48no—
106C22×C8—24no—
107C22×C8—42no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112C22⋊Q8—42no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C22⋊Q8—42no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—216no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—88no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4C22⋊C414yes—
149C2×C42—24no—
150C2×C42—44no—
151#101—84no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160#95—216no—
161#95—24no—
162C24—24no—
163C24—42no—
164C24—42no—
165C24C2×C22⋊C411yescharacteristic
166C24C2×C22⋊C411yescharacteristic
167C24C2×C22⋊C411yescharacteristic
168C24C4×D412yes—
169C24C4×D412yes—
170C2×Q8—48no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—44no—
174C2×Q8—82no—
175C2×Q8—84no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—48no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—88no—
205C22×C4—88no—
206C22×C4#12012yes—
207C22×C4#12012yes—
208C22×C4#12011yescharacteristic
209C22×C4#12012yes—
210C22×C4#12011yescharacteristic
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4#12112yes—
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12312yes—
220C22×C4#12412yes—
221C22×C4#12712yes—
222M16—48no—
223M16—48no—
224M16—48no—
225M16—84no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—84no—
229C2×C8—84no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—816no—
239C22⋊C4—28no—
240C22⋊C4—24no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4#10118yescomplemented
244C42—48no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—22no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—416no—
260C23#37911yescharacteristic
261C23#38612yes—
262C23#38611yescharacteristic
263C23#38611yescharacteristic
264C23#39012yes—
265C23C2×#10114yes—
266Q8—84no—
267Q8—84no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—28no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—416no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—84no—
296C2×C4—88no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—88no—
301C2×C4#39212yes—
302C2×C4#39212yes—
303C2×C4#39412yes—
304C2×C4#39412yes—
305C2×C4#45814yes—
306C8—84no—
307C8—84no—
308C22—28no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—44no—
321C22—48no—
322C22—48no—
323C22—44no—
324C22#143811yescharacteristic
325C22#178412yes—
326C22#189012yes—
327C22#191412yes—
328C22#196614yes—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—48no—
333C4—82no—
334C4—84no—
335C4—82no—
336C4#202214yes—
337C2—22no—
338C2—22no—
339C2—22no—
340C2—24no—
341C2—42no—
342C2#1097311yescharacteristic
343C2#1098311yescharacteristic
344C2#1099211yescharacteristic
subgroupG/H
center, socleC22#1438
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
243. C22⋊C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#217532
C2×#18212
#173972
#79292