Finite Group Databasebeta

← #121133

Subgroups — #121133

Order 512.

503 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 106 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#8104C211yesmaximal, characteristic
2#8102C211yesmaximal, characteristic
3#8100C211yesmaximal, characteristic
4#8049C211yesmaximal, characteristic
5#7946C211yesmaximal, characteristic
6#7946C211yesmaximal, characteristic
7#7946C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10#2280C2211yescharacteristic
11#2280C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14#2280C2211yescharacteristic
15#1305—22no—
16#1304—22no—
17#1304—22no—
18#1297—22no—
19#1284—22no—
20#1284—22no—
21#1284C414yescomplemented
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C2×#117—21no—
26C2×#117—21no—
27C2×#117—21no—
28C2×#117—21no—
29C2×#117—21no—
30#407—21no—
31#407—21no—
32#407—21no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407C2×C412yes—
37#407D412yes—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42#381—21no—
43#381—21no—
44#381—21no—
45#381—21no—
46#381C2×C412yes—
47#380—21no—
48#380—21no—
49#380—21no—
50#380—21no—
51#380—21no—
52#380—21no—
53#380—21no—
54#380—21no—
55#380—21no—
56#380—21no—
57#379—21no—
58#379—21no—
59#379—21no—
60#379—21no—
61#379D412yes—
62C4×C22⋊C4—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C412yes—
70C4×C22⋊C4D412yes—
71C4×C22⋊C4Q812yes—
72C25C2×D411yescharacteristic
73C23×C4—82no—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×Q811yescharacteristic
78C23×C4#4111yescharacteristic
79C23×C4#4111yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—82no—
93C2×M16—82no—
94C22×C8—41no—
95C22×C8—41no—
96#117—82no—
97#117—82no—
98#117—82no—
99#117—82no—
100#117—82no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—41no—
125C2×C42—41no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134#95—22no—
135#95—22no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—41no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24C4×D411yescharacteristic
148C24#12111yescharacteristic
149C24#12111yescharacteristic
150C24C22⋊Q811yescharacteristic
151C24#12311yescharacteristic
152C24#12311yescharacteristic
153C24#12411yescharacteristic
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4#11711yescharacteristic
174C22×C4#11711yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×Q811yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4C22⋊Q811yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4#12311yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12511yescharacteristic
191C22×C4#12611yescharacteristic
192C22×C4#12611yescharacteristic
193C22×C4#12611yescharacteristic
194C22×C4#12611yescharacteristic
195C22×C4#12611yescharacteristic
196C22×C4#12611yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4#12611yescharacteristic
199C22×C4#12611yescharacteristic
200C22×C4#12611yescharacteristic
201M16—82no—
202M16—82no—
203M16—82no—
204M16—82no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—82no—
208C2×C8—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—82no—
230C42—82no—
231C42—82no—
232C42—82no—
233C42—82no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—88no—
256C23#38711yescharacteristic
257C23#38811yescharacteristic
258C23#38811yescharacteristic
259C23#39611yescharacteristic
260C23#39711yescharacteristic
261C23#39711yescharacteristic
262C23#40011yescharacteristic
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
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—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—82no—
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—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#38311yescharacteristic
307C2×C4#38711yescharacteristic
308C2×C4#38711yescharacteristic
309C2×C4#38811yescharacteristic
310C2×C4#39711yescharacteristic
311C2×C4#40011yescharacteristic
312C2×C4#40011yescharacteristic
313C2×C4#40111yescharacteristic
314C8—82no—
315C8—82no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22#144311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—162no—
353C4—162no—
354C4—162no—
355C4—162no—
356C4—162no—
357C4#210412yes—
358C2—41no—
359C2—41no—
360C2—41no—
361C2—41no—
362C2—41no—
363C2—41no—
364C2—41no—
365C2#1130711yescharacteristic
366C2#1130911yescharacteristic
367C2#1131811yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1284C4semidirect

Maximal subgroups

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

subgroupindex
#81042
#81022
#81002
#80492
#79462