Finite Group Databasebeta

← #110408

Subgroups — #110408

Order 512.

908 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 174 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×#1821C212yesmaximal
2#20524C211yesmaximal, characteristic
3#13924C211yesmaximal, characteristic
4C2×#1305C212yesmaximal
5#7911C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1821—24no—
12#1821C418yescomplemented
13#1753—24no—
14#1749—21no—
15C4×C22⋊C8—21no—
16C2×#343—24no—
17#1445—22no—
18C4⋊#95—21no—
19C4⋊#95—21no—
20C22⋊#95C414yes—
21#1305—24no—
22#1280—22no—
23C24×C4C2×C412yes—
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26C22×M16D414yes—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C414yes—
36C22×C22⋊C4C2×C412yes—
37C22×C22⋊C4C2×C412yes—
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407C2×C414yes—
46#407C2×C414yes—
47#407D414yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×#97—22no—
53C2×C4×C8—22no—
54#380—22no—
55#380—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C2×#95—21no—
59C2×#95—24no—
60C2×#95—24no—
61C2×#95D412yes—
62#343—216no—
63#336—22no—
64#336—22no—
65C25C22×C411yescharacteristic
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—24no—
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4—84no—
74C23×C4C4214yes—
75C23×C4C22⋊C412yes—
76C23×C4C22⋊C412yes—
77C23×C4C22×C412yes—
78C23×C4C2×D412yes—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D412yes—
81C23×C4C2×D411yescharacteristic
82C23×C4#4112yes—
83C23×C4#4112yes—
84C23×C4#4112yes—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—42no—
88C2×M16—44no—
89C2×M16—44no—
90C2×M16—44no—
91C2×M16—44no—
92C2×M16C22⋊C418yes—
93C22×C8—24no—
94C22×C8—42no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—44no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—42no—
124C2×C42—42no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—82no—
134C4×C8—82no—
135#95—24no—
136#95—41no—
137#95—41no—
138#95—84no—
139#95—84no—
140C24—22no—
141C24—22no—
142C24—41no—
143C24—41no—
144C24—42no—
145C24C2×C4211yescharacteristic
146C24C2×C22⋊C411yescharacteristic
147C24#11711yescharacteristic
148C24C4×D412yes—
149C24C4×D412yes—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—28no—
160C22×C4—22no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4—88no—
186C22×C4—88no—
187C22×C4#10114yes—
188C22×C4C2×C22⋊C412yes—
189C22×C4#11712yes—
190C22×C4C4×D412yes—
191C22×C4C4×D412yes—
192C22×C4C4×D412yes—
193C22×C4C4×D412yes—
194C22×C4#12011yescharacteristic
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4#12312yes—
199C22×C4#12311yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#12412yes—
202C22×C4#12612yes—
203C22×C4#12711yescharacteristic
204M16—48no—
205M16—48no—
206M16—84no—
207M16—84no—
208C2×C8—28no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—82no—
213C2×C8—82no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—84no—
219C22⋊C4—48no—
220C22⋊C4—84no—
221C42—42no—
222C42—42no—
223C42—81no—
224C42—81no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—24no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—44no—
243C23—44no—
244C23—44no—
245C23—44no—
246C23C4×C22⋊C412yes—
247C23C4×C22⋊C411yescharacteristic
248C23#38611yescharacteristic
249C23#38812yes—
250C23#39011yescharacteristic
251C23C2×#10112yes—
252C23#41312yes—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—88no—
284C2×C4—84no—
285C2×C4—82no—
286C2×C4—88no—
287C2×C4—82no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#38612yes—
293C2×C4#38812yes—
294C2×C4#38812yes—
295C2×C4#39012yes—
296C2×C4#45812yes—
297C2×C4#45812yes—
298C8—48no—
299C8—84no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22#143711yescharacteristic
321C22C4×#10112yes—
322C22#189011yescharacteristic
323C22#189011yescharacteristic
324C22#190112yes—
325C22#191412yes—
326C22#196611yescharacteristic
327C22#196711yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—44no—
332C4—82no—
333C4—84no—
334C4—162no—
335C4#189014yes—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—42no—
340C2—42no—
341C2#950912yes—
342C2#951012yes—
343C2#1093611yescharacteristic
344C2#1093911yescharacteristic
345C2#1096311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1821C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18212
#205242
#139242
C2×#13052
#79112