Finite Group Databasebeta

← #124844

Subgroups — #124844

Order 512.

724 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 90 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#14130C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7122C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C2×#389—21no—
7C2×#385—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1749—21no—
11#1749—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1444C412yes—
15#1442C2212yes—
16C4⋊#95—21no—
17C22⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4—21no—
21C24×C4C2×C411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—44no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—41no—
37C2×#97—41no—
38#389—44no—
39#385—44no—
40#380—24no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95C2×C412yes—
47C2×#95C2×C411yescharacteristic
48C2×#95C2311yescharacteristic
49#336—41no—
50#336—41no—
51#323—24no—
52#323—24no—
53C25C22⋊C411yescharacteristic
54C23×C4—22no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4—44no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—82no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—42no—
74C22×C8—42no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—48no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—82no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—42no—
105C22⋊C8—24no—
106C22⋊C8—24no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#97—82no—
110#97—82no—
111#95—22no—
112#95—41no—
113#95—41no—
114#95—44no—
115#95—44no—
116#95—82no—
117#95M1614yes—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—41no—
123C24—41no—
124C24—41no—
125C24—41no—
126C24#10112yes—
127C24#10412yes—
128C24#10412yes—
129C24C2×C22⋊C411yescharacteristic
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—48no—
152C22×C4—48no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—48no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—88no—
180C22×C4—88no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4#9911yescharacteristic
184C22×C4#9911yescharacteristic
185C22×C4C2×M1612yes—
186C22×C4AGL(1,ℤ/8)12yes—
187C22×C4#13712yes—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—82no—
193C2×C8—82no—
194C4⋊C4—48no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—88no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C22⋊C4—48no—
203C42—42no—
204C42—42no—
205C42—81no—
206C42—81no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23#33014yes—
234C23#40711yescharacteristic
235C23C2×#9911yescharacteristic
236C23C2×#10111yescharacteristic
237C23#41811yescharacteristic
238C23#41911yescharacteristic
239C23C2×#10411yescharacteristic
240C23C2×#10411yescharacteristic
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—816no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4AΣL(1,𝔽16)12yes—
290C2×C4#35212yes—
291C8—84no—
292C8—84no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22#146311yescharacteristic
317C22#149011yescharacteristic
318C22#149111yescharacteristic
319C22#150711yescharacteristic
320C22#150811yescharacteristic
321C22C2×#33011yescharacteristic
322C22#155611yescharacteristic
323C22C2×AΣL(1,𝔽16)11yescharacteristic
324C22C2×#35211yescharacteristic
325C22#212911yescharacteristic
326C22#213411yescharacteristic
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—81no—
334C4—84no—
335C4—81no—
336C2—21no—
337C2—21no—
338C2—21no—
339C2—21no—
340C2—44no—
341C2#762711yescharacteristic
342C2#1151511yescharacteristic
343C2#1151711yescharacteristic
344C2#1177311yescharacteristic
345C2#1177911yescharacteristic
346C2#1178511yescharacteristic
347C2#1179011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#141302
#81522
#71222