Finite Group Databasebeta

← #128363

Subgroups — #128363

Order 512.

804 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C4×#381C211yesmaximal, characteristic
2C2×#1317C212yesmaximal
3#8152C212yesmaximal
4#7187C212yesmaximal
5C23×C42C411yescharacteristic
6C2×#381C411yescharacteristic
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1749—22no—
11C2×#323C2214yes—
12#1434—22no—
13C22⋊#95C411yescharacteristic
14C22⋊#95C2211yescharacteristic
15C4×#95—22no—
16#1317C418yescomplemented
17#1280—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C22×C22⋊C4C2×C411yescharacteristic
22C22×C22⋊C4C2×C411yescharacteristic
23C22×C22⋊C4C2×C411yescharacteristic
24C22×C22⋊C4C2311yescharacteristic
25C22×C42—21no—
26C22×C42—21no—
27C22×C42—22no—
28C22×C42—22no—
29C22×C42—21no—
30C22×C42—42no—
31C22×C42—42no—
32C22×C42D412yes—
33C2×C22⋊C8—24no—
34C2×#97—42no—
35#381—22no—
36#381—22no—
37#381—22no—
38#381C2×C414yes—
39C4×C22⋊C4—42no—
40C4×C22⋊C4—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—21no—
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—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—44no—
64C23×C4—44no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C4212yes—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4M1612yes—
76C23×C4M1612yes—
77C23×C4M1612yes—
78C23×C4C2×D412yes—
79C22×C8—44no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4C4212yes—
93C2×C22⋊C4M1612yes—
94C2×C22⋊C4C22×C412yes—
95C2×C22⋊C4#4112yes—
96C2×C42—24no—
97C2×C42—24no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—42no—
101C2×C42—48no—
102C2×C42—48no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—44no—
107C2×C42—42no—
108C2×C42—44no—
109C22⋊C8—28no—
110C22⋊C8—44no—
111#97—84no—
112#95—28no—
113#95—28no—
114#95—42no—
115#95—44no—
116#95—44no—
117#95—42no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—42no—
123C24—42no—
124C24C2×C4211yescharacteristic
125C24C2×C22⋊C411yescharacteristic
126C24#11711yescharacteristic
127C24C2×M1611yescharacteristic
128C24C2×M1611yescharacteristic
129C24C2×M1611yescharacteristic
130C24C2×M1611yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
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—48no—
168C22×C4—44no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4#9714yescomplemented
198C22×C4#9912yes—
199C22×C4#10012yes—
200C22×C4C4×D414yes—
201C2×C8—48no—
202C2×C8—84no—
203C2×C8—84no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—48no—
213C42—48no—
214C42—44no—
215C42—82no—
216C42—84no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—44no—
242C23—44no—
243C23—42no—
244C23—42no—
245C23C4×C22⋊C411yescharacteristic
246C23C2×#9711yescharacteristic
247C23C4×M1611yescharacteristic
248C23#40711yescharacteristic
249C23#40711yescharacteristic
250C23C2×#9911yescharacteristic
251C23#41011yescharacteristic
252C23C2×#10011yescharacteristic
253C23#41311yescharacteristic
254C23#43211yescharacteristic
255C23#43211yescharacteristic
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—48no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4#35214yes—
306C8—88no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22#146211yescharacteristic
338C22#146411yescharacteristic
339C22#146511yescharacteristic
340C22#146611yescharacteristic
341C22#174911yescharacteristic
342C22C4×#9911yescharacteristic
343C22C4×#10011yescharacteristic
344C22C2×#35212yes—
345C22#212012yes—
346C4—24no—
347C4—44no—
348C4—44no—
349C4—42no—
350C4—44no—
351C4—82no—
352C4—82no—
353C4—82no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2—41no—
359C2—41no—
360C2—41no—
361C2—41no—
362C2#815211yescharacteristic
363C2#1150011yescharacteristic
364C2#1150111yescharacteristic
365C2#1152612yes—
366C2C4×#35212yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ 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
197. C22×C4#97semidirect

Maximal subgroups

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

subgroupindex
C4×#3812
C2×#13172
#81522
#71872