Finite Group Databasebeta

← #135306

Subgroups — #135306

Order 512.

726 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 100 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#14339C211yesmaximal, characteristic
2#10727C211yesmaximal, characteristic
3#8299C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C23×C4⋊C4C2211yescharacteristic
6C2×#407—21no—
7C2×#381—21no—
8C2×#380C411yescharacteristic
9C2×#380C411yescharacteristic
10#1779—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13C2×#324—21no—
14C2×#324C2212yes—
15#1441—21no—
16#1441—21no—
17#1441C412yes—
18#1441C2212yes—
19C22⋊#95—21no—
20C24×C4C2×C411yescharacteristic
21C24×C4D411yescharacteristic
22C24×C4D411yescharacteristic
23C22×M16—22no—
24C22×C4⋊C4—24no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—42no—
27C22×C4⋊C4—42no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4C2311yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35#407—24no—
36#407—42no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—42no—
39C2×C22⋊C8—42no—
40#381—42no—
41#381—42no—
42#380—42no—
43#380—42no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C412yes—
50#343—24no—
51#343—42no—
52#324—24no—
53#324—42no—
54C25C22⋊C411yescharacteristic
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—82no—
70C23×C4—82no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4D812yes—
75C23×C4SD1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C2×M16—24no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—42no—
85C22×C8—42no—
86C22×C8—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—416no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4#4114yes—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C22⋊C8—48no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—84no—
114#95—42no—
115#95—42no—
116#95—44no—
117#95—42no—
118#95—42no—
119#95—82no—
120#95—82no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—42no—
126C24—42no—
127C24#9912yes—
128C24#10212yes—
129C24#10412yes—
130C24C2×C22⋊C411yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—28no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
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—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—84no—
185C22×C4—84no—
186C22×C4—84no—
187C22×C4—84no—
188C22×C4#9912yes—
189C22×C4#10212yes—
190C22×C4#11712yes—
191C22×C4#12312yes—
192C22×C4#12312yes—
193C22×C4C2×D811yescharacteristic
194C22×C4C2×SD1611yescharacteristic
195C22×C4#13712yes—
196M16—44no—
197M16—44no—
198M16—82no—
199M16—82no—
200C2×C8—44no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8—84no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C22⋊C4—84no—
212C22⋊C4—84no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—44no—
226C23—44no—
227C23—42no—
228C23—42no—
229C23—44no—
230C23—44no—
231C23—42no—
232C23—42no—
233C23#32714yes—
234C23#38111yescharacteristic
235C23C2×#9911yescharacteristic
236C23C2×#9911yescharacteristic
237C23C2×#10211yescharacteristic
238C23#41911yescharacteristic
239C23C2×#10411yescharacteristic
240C23#42111yescharacteristic
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—88no—
274C2×C4—88no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#48012yes—
291C2×C4#48312yes—
292C8—84no—
293C8—162no—
294C22—22no—
295C22—22no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—24no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#32711yescharacteristic
314C22#150811yescharacteristic
315C22#151111yescharacteristic
316C22#151911yescharacteristic
317C22#181911yescharacteristic
318C22#182311yescharacteristic
319C22#182611yescharacteristic
320C22#212112yes—
321C22#213312yes—
322C4—22no—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—162no—
330C4—162no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—44no—
336C2#871211yescharacteristic
337C2#1169812yes—
338C2#1181712yes—
339C2#1285411yescharacteristic
340C2#1286211yescharacteristic
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
4. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#143392
#107272
#82992
#72512