Finite Group Databasebeta

← #129799

Subgroups — #129799

Order 512.

1094 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C22×#400C211yesmaximal, characteristic
2C2×#1353C214yesmaximal, complemented
3#10727C211yesmaximal, characteristic
4#8299C211yesmaximal, characteristic
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C2×#400—28no—
9C2×#400—21no—
10C2×#400—21no—
11C2×#400C412yes—
12C2×#400C2212yes—
13C22×#95—21no—
14C22×#95—21no—
15C22×#95C411yescharacteristic
16#1779—21no—
17C2×#343—21no—
18C2×#343C2212yes—
19C2×#324—21no—
20C2×#324C2212yes—
21#1353—28no—
22C24×C4—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4—42no—
32C22×C4⋊C4—42no—
33C22×C4⋊C4—42no—
34C22×C4⋊C4—42no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40#407—24no—
41#407—42no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—42no—
44C2×C22⋊C8—42no—
45#400—28no—
46#400—28no—
47#400—24no—
48#400—24no—
49#400—24no—
50#400D418yes—
51C2×#95—28no—
52C2×#95—28no—
53C2×#95—21no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—24no—
59C2×#95C2×C412yes—
60#343—24no—
61#343—42no—
62#324—24no—
63#324—42no—
64C25C22⋊C411yescharacteristic
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—44no—
82C23×C4—44no—
83C23×C4—42no—
84C23×C4—44no—
85C23×C4—44no—
86C23×C4—48no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4SD1612yes—
94C23×C4Q1612yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—24no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—42no—
104C22×C8—42no—
105C22×C8—42no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—416no—
122C2×C4⋊C4—416no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C22⋊C414yes—
126C2×C4⋊C4C2×D414yes—
127C22⋊C8—48no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—84no—
131#95—28no—
132#95—24no—
133#95—28no—
134#95—22no—
135#95—28no—
136#95—28no—
137#95C22⋊C414yes—
138C24—24no—
139C24—24no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24#9912yes—
145C24#10312yes—
146C24#10412yes—
147C24C2×C22⋊C411yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—28no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—416no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4—84no—
205C22×C4—88no—
206C22×C4—88no—
207C22×C4—88no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4#9912yes—
211C22×C4#10312yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4#12012yes—
214C22×C4#12012yes—
215C22×C4C2×SD1611yescharacteristic
216C22×C4C2×Q1611yescharacteristic
217C22×C4#13712yes—
218M16—44no—
219M16—44no—
220M16—82no—
221M16—82no—
222C2×C8—44no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—84no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—28no—
239C23—28no—
240C23—22no—
241C23—22no—
242C23—28no—
243C23—28no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23#32814yes—
253C23#35214yes—
254C23#35514yes—
255C23#37911yescharacteristic
256C23C2×#9911yescharacteristic
257C23C2×#9911yescharacteristic
258C23C2×#10311yescharacteristic
259C23#41911yescharacteristic
260C23C2×#10411yescharacteristic
261C23#42111yescharacteristic
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—48no—
277C2×C4—48no—
278C2×C4—42no—
279C2×C4—416no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—81no—
300C2×C4—88no—
301C2×C4—816no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—816no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—81no—
308C2×C4#44812yes—
309C2×C4#45112yes—
310C8—84no—
311C8—162no—
312C22—28no—
313C22—28no—
314C22—28no—
315C22—28no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—22no—
320C22—24no—
321C22—21no—
322C22—24no—
323C22—21no—
324C22—21no—
325C22—24no—
326C22—21no—
327C22—22no—
328C22—24no—
329C22—22no—
330C22#135318yescomplemented
331C22C2×#32811yescharacteristic
332C22#150811yescharacteristic
333C22#151111yescharacteristic
334C22#152011yescharacteristic
335C22#178111yescharacteristic
336C22#178711yescharacteristic
337C22#178911yescharacteristic
338C22C2×#35212yes—
339C22C2×#35512yes—
340C4—22no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—84no—
347C4—81no—
348C4—84no—
349C2—21no—
350C2—28no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2#832911yescharacteristic
355C2C2×#135312yes—
356C2#1181412yes—
357C2#1268811yescharacteristic
358C2#1269911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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×#1353C2semidirect
330. C22#1353semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C2×#13532
#107272
#82992