Finite Group Databasebeta

← #121825

Subgroups — #121825

Order 512.

756 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 134 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#13953C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3C2×#1297C212yesmaximal
4C2×#1284C212yesmaximal
5#7911C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×#342—22no—
12#1439—22no—
13#1436—22no—
14C4⋊#95C412yescomplemented
15#1434C412yescomplemented
16#1297—24no—
17#1284—24no—
18#1280—22no—
19C24×C4C2311yescharacteristic
20C22×M16—22no—
21C22×C4⋊C4—22no—
22C22×C22⋊C4—22no—
23C22×C22⋊C4—21no—
24C22×C22⋊C4—21no—
25C22×C22⋊C4D412yes—
26C22×C42—22no—
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30#407—22no—
31#407—22no—
32#407—22no—
33#407D414yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36#381—22no—
37#381—22no—
38C4×C22⋊C4—22no—
39C4×C22⋊C4—22no—
40C4×C22⋊C4—22no—
41C4×C22⋊C4—22no—
42C4×C22⋊C4—22no—
43C4×C22⋊C4—22no—
44C4×C22⋊C4—22no—
45C4×C22⋊C4Q814yes—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—21no—
50C2×#95—22no—
51C2×#95C2×C412yes—
52C2×#95C2×C412yes—
53C2×#95D412yes—
54#342—28no—
55C25C2×D411yescharacteristic
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—42no—
59C23×C4—81no—
60C23×C4—81no—
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C412yes—
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D412yes—
66C23×C4C2×Q812yes—
67C23×C4#4112yes—
68C23×C4#4112yes—
69C23×C4#4112yes—
70C23×C4#4112yes—
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—82no—
74C2×M16—82no—
75C22×C8—42no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—82no—
79C2×C22⋊C4—28no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—81no—
92C2×C22⋊C4—81no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—24no—
98C2×C42—42no—
99C2×C42—42no—
100C2×C42—81no—
101C2×C42—81no—
102C2×C42—82no—
103C2×C42—82no—
104C2×C42—81no—
105C2×C42—81no—
106C22⋊C8—24no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#95—24no—
111#95—24no—
112#95—24no—
113#95—24no—
114#95—44no—
115#95—44no—
116#95—42no—
117#95—42no—
118#95—82no—
119C24—22no—
120C24—22no—
121C24—42no—
122C24—41no—
123C24—41no—
124C24C2×C22⋊C411yescharacteristic
125C24C22⋊Q812yes—
126C24#12312yes—
127C24#12411yescharacteristic
128C24#12711yescharacteristic
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—48no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—84no—
169C22×C4#10414yes—
170C22×C4#11711yescharacteristic
171C22×C4#11711yescharacteristic
172C22×C4C4×D412yes—
173C22×C4C4×Q812yes—
174C22×C4#12112yes—
175C22×C4C22⋊Q812yes—
176C22×C4C22⋊Q812yes—
177C22×C4#12312yes—
178C22×C4#12312yes—
179C22×C4#12312yes—
180C22×C4#12411yescharacteristic
181C22×C4#12512yes—
182C22×C4#12512yes—
183C22×C4#12612yes—
184C22×C4#12612yes—
185C22×C4C4⋊Q811yescharacteristic
186M16—84no—
187M16—84no—
188C2×C8—44no—
189C2×C8—82no—
190C2×C8—82no—
191C4⋊C4—88no—
192C22⋊C4—28no—
193C22⋊C4—48no—
194C22⋊C4—48no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—82no—
199C22⋊C4—82no—
200C42—81no—
201C42—81no—
202C42—81no—
203C42—81no—
204C42—84no—
205C42—84no—
206C42—84no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—24no—
211C23—24no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—42no—
223C23—42no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23#38111yescharacteristic
229C23#39011yescharacteristic
230C23#39111yescharacteristic
231C23#39912yes—
232C23#40012yes—
233C23C2×#10412yes—
234C23#42112yes—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—84no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—88no—
272C2×C4—88no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#38712yes—
284C2×C4#38712yes—
285C2×C4#39712yes—
286C2×C4#39812yes—
287C8—84no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—44no—
303C22—44no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#144511yescharacteristic
309C22#182612yes—
310C22#182712yes—
311C22#182811yescharacteristic
312C22#182811yescharacteristic
313C22#198012yes—
314C22#199411yescharacteristic
315C22#199511yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—44no—
321C4—84no—
322C4—88no—
323C4—82no—
324C4—162no—
325C2—22no—
326C2—21no—
327C2—21no—
328C2—42no—
329C2—42no—
330C2#877712yes—
331C2#877812yes—
332C2#1134811yescharacteristic
333C2#1135111yescharacteristic
334C2#1135911yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ 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
14. C4⋊#95C4semidirect
15. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#139532
C4×#3422
C2×#12972
C2×#12842
#79112