Finite Group Databasebeta

← #135303

Subgroups — #135303

Order 512.

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

normal subgroups only

This group has 696 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#14340C211yesmaximal, characteristic
2#10726C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×#117C2211yescharacteristic
6C2×#384C411yescharacteristic, complemented
7C2×#382C411yescharacteristic, complemented
8C2×#381—21no—
9C2×#380—21no—
10#1817—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#324—21no—
14C2×#323C2212yes—
15#1441—21no—
16#1441C412yescomplemented
17#1439—21no—
18#1439C2212yes—
19C4⋊#95—21no—
20C24×C4D411yescharacteristic
21C2×#117—24no—
22C2×#117—42no—
23C2×#117—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C22⋊C4—22no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4D411yescharacteristic
33C22×C22⋊C4C2311yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C22⋊C8—22no—
37#384—42no—
38#382—42no—
39#381—24no—
40#381—42no—
41#380—42no—
42#380—42no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—42no—
48C2×#95C2×C412yes—
49#342—24no—
50#342—42no—
51#336—42no—
52#324—42no—
53#323—24no—
54C25C2×D411yescharacteristic
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4—42no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4D812yes—
69C23×C4SD1612yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73#117—416no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—82no—
89C2×C4⋊C4—82no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—82no—
104C2×C22⋊C4—82no—
105C2×C22⋊C4#4114yes—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—41no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#95—42no—
115#95—48no—
116#95—42no—
117#95—44no—
118#95—82no—
119#95—82no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24—41no—
128C24C2×C22⋊C411yescharacteristic
129C24#12312yes—
130C24C2×D811yescharacteristic
131C24C2×SD1611yescharacteristic
132C24#13712yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
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—24no—
149C22×C4—24no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—44no—
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—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4#9912yes—
192C22×C4#9912yes—
193C22×C4#10212yes—
194C22×C4#10212yes—
195C22×C4#10412yes—
196C22×C4#11712yes—
197C22×C4#12312yes—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—82no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C22⋊C4—82no—
215C42—28no—
216C42—44no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23#38111yescharacteristic
248C23C2×#9911yescharacteristic
249C23C2×#9911yescharacteristic
250C23C2×#10211yescharacteristic
251C23#41911yescharacteristic
252C23C2×#10411yescharacteristic
253C23#42111yescharacteristic
254C23#48012yes—
255C23#48312yes—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—162no—
309C2×C4—162no—
310C2×C4—162no—
311C2×C4—162no—
312C2×C4#32714yes—
313C8—162no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—82no—
341C22—82no—
342C22C2×#32711yescharacteristic
343C22#150811yescharacteristic
344C22#151111yescharacteristic
345C22#151911yescharacteristic
346C22#181911yescharacteristic
347C22#182311yescharacteristic
348C22#182611yescharacteristic
349C22#212211yescharacteristic
350C22#212311yescharacteristic
351C22#212811yescharacteristic
352C22#212911yescharacteristic
353C4—24no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—81no—
359C4—162no—
360C4—164no—
361C4—162no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—42no—
368C2#871211yescharacteristic
369C2#1169311yescharacteristic
370C2#1169411yescharacteristic
371C2#1181211yescharacteristic
372C2#1181311yescharacteristic
373C2#1285511yescharacteristic
374C2#1285911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#384C4semidirect
7. C2×#382C4semidirect
16. #1441C4semidirect

Maximal subgroups

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

subgroupindex
#143402
#107262
#83002
#72502