Finite Group Databasebeta

← #110406

Subgroups — #110406

Order 512.

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

normal subgroups only

This group has 936 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#13904C211yesmaximal, characteristic
4C2×#1305C212yesmaximal
5#7911C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9C2×#381C414yes—
10C2×#380C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12#1821—24no—
13#1821C418yescomplemented
14#1753—24no—
15#1749—21no—
16C4×C22⋊C8—21no—
17C2×#343—24no—
18#1445—22no—
19C4⋊#95—21no—
20C4⋊#95—21no—
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#381—48no—
55#380—22no—
56#380—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59C2×#95—24no—
60C2×#95—24no—
61C2×#95—21no—
62C2×#95D412yes—
63#343—216no—
64#336—22no—
65#336—22no—
66C25C22×C411yescharacteristic
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—24no—
70C23×C4—24no—
71C23×C4—22no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4C4214yes—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C412yes—
79C23×C4C22×C412yes—
80C23×C4C2×D412yes—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D412yes—
83C23×C4C2×D411yescharacteristic
84C23×C4#4112yes—
85C23×C4#4112yes—
86C23×C4#4112yes—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—42no—
90C2×M16—44no—
91C2×M16—44no—
92C2×M16—44no—
93C2×M16—44no—
94C2×M16C22⋊C418yes—
95C22×C8—24no—
96C22×C8—42no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—44no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—42no—
126C2×C42—42no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—24no—
132C22⋊C8—24no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135#97—82no—
136C4×C8—82no—
137#95—24no—
138#95—48no—
139#95—48no—
140#95—41no—
141#95—41no—
142C24—22no—
143C24—22no—
144C24—41no—
145C24—41no—
146C24—42no—
147C24C2×C4211yescharacteristic
148C24C2×C22⋊C411yescharacteristic
149C24#11711yescharacteristic
150C24C4×D412yes—
151C24C4×D412yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—28no—
162C22×C4—22no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—48no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—88no—
188C22×C4—84no—
189C22×C4—84no—
190C22×C4#10014yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4#11712yes—
193C22×C4C4×D412yes—
194C22×C4C4×D412yes—
195C22×C4C4×D412yes—
196C22×C4C4×D412yes—
197C22×C4#12011yescharacteristic
198C22×C4#12112yes—
199C22×C4#12112yes—
200C22×C4#12112yes—
201C22×C4#12312yes—
202C22×C4#12311yescharacteristic
203C22×C4#12411yescharacteristic
204C22×C4#12412yes—
205C22×C4#12612yes—
206C22×C4#12711yescharacteristic
207M16—48no—
208M16—48no—
209M16—84no—
210M16—84no—
211C2×C8—28no—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—84no—
222C22⋊C4—48no—
223C22⋊C4—84no—
224C42—42no—
225C42—42no—
226C42—81no—
227C42—81no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—24no—
234C23—24no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23—44no—
248C23—44no—
249C23C4×C22⋊C412yes—
250C23C4×C22⋊C411yescharacteristic
251C23#38611yescharacteristic
252C23#38812yes—
253C23#39011yescharacteristic
254C23C2×#10012yes—
255C23#41312yes—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—48no—
283C2×C4—82no—
284C2×C4—88no—
285C2×C4—84no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—88no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#38612yes—
296C2×C4#38812yes—
297C2×C4#38812yes—
298C2×C4#39012yes—
299C2×C4#45812yes—
300C2×C4#45812yes—
301C8—48no—
302C8—84no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22#143711yescharacteristic
324C22C4×#10012yes—
325C22#189011yescharacteristic
326C22#189011yescharacteristic
327C22#190012yes—
328C22#191512yes—
329C22#196511yescharacteristic
330C22#196711yescharacteristic
331C4—22no—
332C4—22no—
333C4—22no—
334C4—44no—
335C4—82no—
336C4—84no—
337C4—162no—
338C4#189014yes—
339C2—21no—
340C2—21no—
341C2—22no—
342C2—42no—
343C2—42no—
344C2#950712yes—
345C2#950812yes—
346C2#1093611yescharacteristic
347C2#1094011yescharacteristic
348C2#1096211yescharacteristic
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
13. #1821C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18212
#205242
#139042
C2×#13052
#79112