Finite Group Databasebeta

← #128306

Subgroups — #128306

Order 512.

752 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 150 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#13776C211yesmaximal, characteristic
2C2×#1317C212yesmaximal
3C4×#323C212yesmaximal
4#7187C212yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C2×#381C411yescharacteristic
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—22no—
11C2×#323C2214yes—
12#1436—22no—
13#1436—22no—
14C22⋊#95C411yescharacteristic
15C22⋊#95C2211yescharacteristic
16#1317C418yescomplemented
17#1280—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—42no—
25C22×C4⋊C4—42no—
26C22×C4⋊C4C814yescomplemented
27C22×C22⋊C4C2×C411yescharacteristic
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42D412yes—
33C2×C22⋊C8—24no—
34C2×C4×C8—42no—
35#381—22no—
36#381—22no—
37#381C814yescomplemented
38#381C2×C414yes—
39C4×C22⋊C4—42no—
40C4×C22⋊C4—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95D412yes—
47#336—42no—
48#323—24no—
49#323C2×C418yes—
50C25C22×C411yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4—82no—
68C23×C4—82no—
69C23×C4C4212yes—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C2×C812yes—
74C23×C4C2×C812yes—
75C23×C4C2×C812yes—
76C23×C4C2×C812yes—
77C23×C4M1612yes—
78C23×C4C2×D412yes—
79C22×C8—44no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—48no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—48no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4C4212yes—
99C2×C22⋊C4C2×C812yes—
100C2×C22⋊C4C2×C812yes—
101C2×C22⋊C4M1612yes—
102C2×C22⋊C4C22×C412yes—
103C2×C22⋊C4#4112yes—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—42no—
107C2×C42—82no—
108C2×C42—82no—
109C22⋊C8—28no—
110C22⋊C8—44no—
111C4×C8—84no—
112#95—28no—
113#95—28no—
114#95—42no—
115#95—42no—
116#95—82no—
117#95—82no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—42no—
123C24—42no—
124C24C2×C4211yescharacteristic
125C24C2×C22⋊C411yescharacteristic
126C24#11711yescharacteristic
127C24C22×C811yescharacteristic
128C24C22×C811yescharacteristic
129C24C2×M1611yescharacteristic
130C24C2×M1611yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—84no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4C4×C814yescomplemented
194C22×C4C22⋊C814yes—
195C22×C4C22⋊C812yes—
196C22×C4C22⋊C812yes—
197C22×C4#9912yes—
198C22×C4#10012yes—
199C22×C4C4×D414yes—
200C2×C8—48no—
201C2×C8—84no—
202C2×C8—84no—
203C4⋊C4—48no—
204C4⋊C4—48no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C42—44no—
214C42—82no—
215C42—84no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—42no—
241C23—44no—
242C23—42no—
243C23—42no—
244C23C4×C22⋊C411yescharacteristic
245C23C2×C4×C811yescharacteristic
246C23C4×M1611yescharacteristic
247C23C2×C22⋊C811yescharacteristic
248C23C2×C22⋊C811yescharacteristic
249C23C2×#9911yescharacteristic
250C23#41011yescharacteristic
251C23C2×#10011yescharacteristic
252C23#41311yescharacteristic
253C23#43111yescharacteristic
254C23#43111yescharacteristic
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4#32314yes—
304C8—88no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22C2×#32311yescharacteristic
336C22#145811yescharacteristic
337C22#145911yescharacteristic
338C22#146011yescharacteristic
339C22C4×C22⋊C811yescharacteristic
340C22C4×#9911yescharacteristic
341C22C4×#10011yescharacteristic
342C22#212112yes—
343C22#212312yes—
344C4—24no—
345C4—44no—
346C4—42no—
347C4—44no—
348C4—44no—
349C4—82no—
350C4—82no—
351C4—82no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—41no—
357C2—41no—
358C2—41no—
359C2—41no—
360C2C4×#32311yescharacteristic
361C2#1142611yescharacteristic
362C2#1142811yescharacteristic
363C2#1147412yes—
364C2#1262112yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23C2×C4×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1317C4semidirect
26. C22×C4⋊C4C8semidirect
37. #381C8semidirect
193. C22×C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
#137762
C2×#13172
C4×#3232
#71872