Finite Group Databasebeta

← #129251

Subgroups — #129251

Order 512.

1048 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2317C211yesmaximal, characteristic
2C2×#1325C212yesmaximal, complemented
3C2×#1324C212yesmaximal, complemented
4#8299C212yesmaximal
5C22×#117C411yescharacteristic
6C22×#117C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8#2317—24no—
9#2317—24no—
10C2×#389—22no—
11C2×#385—22no—
12C2×#384C411yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#382C411yescharacteristic
15C2×#382C2211yescharacteristic
16#1779—22no—
17C2×#324—22no—
18C2×#324C2214yes—
19#1325—24no—
20#1324—24no—
21C24×C4C2×C411yescharacteristic
22C24×C4D412yes—
23C2×#117—22no—
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—42no—
31C2×#117—42no—
32C2×#117C814yescomplemented
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—42no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C42—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×C22⋊C8—24no—
51C2×C22⋊C8—44no—
52#389—216no—
53#385—216no—
54#384—22no—
55#384—22no—
56#384—22no—
57#384D414yes—
58#382—22no—
59#382—22no—
60#382—22no—
61#382D414yes—
62C2×#95—22no—
63C2×#95—22no—
64#324—24no—
65#324—24no—
66#324—44no—
67C25C22⋊C411yescharacteristic
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—42no—
83C23×C4—84no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C412yes—
86C23×C4C2×C812yes—
87C23×C4M1612yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D412yes—
90C22×C8—44no—
91#117—28no—
92#117—28no—
93#117—28no—
94#117—28no—
95#117—48no—
96#117—48no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—84no—
120C2×C4⋊C4—84no—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—24no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42C22⋊C412yes—
141C2×C42C22⋊C412yes—
142C2×C42C2×C812yes—
143C2×C42C2×C812yes—
144C2×C42M1612yes—
145C2×C42M1612yes—
146C22⋊C8—44no—
147C22⋊C8—44no—
148C22⋊C8—88no—
149#95—28no—
150#95—28no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—42no—
156C24—42no—
157C24C22⋊C812yes—
158C24#9912yes—
159C24#10112yes—
160C24C2×C22⋊C411yescharacteristic
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—48no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—84no—
213C22×C4—88no—
214C22×C4—88no—
215C22×C4C22⋊C812yes—
216C22×C4#9912yes—
217C22×C4#10112yes—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4#12014yes—
221C22×C4C22×C811yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4C2×M1611yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C2×C8—84no—
226C2×C8—88no—
227C4⋊C4—216no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—48no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—88no—
238C4⋊C4—88no—
239C22⋊C4—216no—
240C22⋊C4—48no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—48no—
248C42C22⋊C814yescomplemented
249C42C22⋊C814yescomplemented
250C23—22no—
251C23—22no—
252C23—24no—
253C23—24no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—44no—
261C23—44no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—44no—
267C23—44no—
268C23#32414yes—
269C23#37911yescharacteristic
270C23C2×C22⋊C811yescharacteristic
271C23#40711yescharacteristic
272C23C2×#9911yescharacteristic
273C23C2×#9911yescharacteristic
274C23C2×#10111yescharacteristic
275C23C2×#10111yescharacteristic
276C2×C4—28no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—816no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#32414yes—
326C2×C4C2×C22⋊C811yescharacteristic
327C2×C4C2×C22⋊C811yescharacteristic
328C2×C4#40711yescharacteristic
329C2×C4#40711yescharacteristic
330C8—164no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—24no—
340C22—28no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22C2×#32411yescharacteristic
349C22C2×#32411yescharacteristic
350C22#146311yescharacteristic
351C22#146311yescharacteristic
352C22#177911yescharacteristic
353C22#178111yescharacteristic
354C22#178411yescharacteristic
355C22#212712yes—
356C22#213312yes—
357C4—24no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—44no—
363C4—44no—
364C4—44no—
365C4—82no—
366C4—82no—
367C4—82no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—21no—
372C2—44no—
373C2#829911yescharacteristic
374C2#1144812yes—
375C2#1151412yes—
376C2#1269611yescharacteristic
377C2#1270211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
3. C2×#1324C2semidirect
32. C2×#117C8semidirect
248. C42C22⋊C8semidirect
249. C42C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#13252
C2×#13242
#82992