Finite Group Databasebeta

← #136808

Subgroups — #136808

Order 512.

632 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 102 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#33487C211yesmaximal, characteristic, complemented
2C8×#120C211yesmaximal, characteristic
3#12679C212yesmaximal, complemented
4#11465C211yesmaximal, characteristic, complemented
5#7199C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C2×C8×D4—21no—
8C2×C8×D4C2211yescharacteristic
9C4×#120C411yescharacteristic
10#2168—24no—
11#2163—21no—
12#2112—21no—
13C2×C22⋊C16—24no—
14C2×C22⋊C16C2212yes—
15C2×C22⋊C16C2212yes—
16#1869—21no—
17#1869C411yescharacteristic
18#1850—21no—
19#1850C411yescharacteristic
20#1779C411yescharacteristic
21C8×C22⋊C4—21no—
22C8×C22⋊C4C2211yescharacteristic
23#1322—21no—
24#1314—21no—
25C24×C4C2×C411yescharacteristic
26C23×C8—22no—
27C23×C8—21no—
28C23×C8—21no—
29C23×C8—22no—
30C23×C8—44no—
31C23×C8C2×C411yescharacteristic
32C23×C8D411yescharacteristic
33C23×C8C2311yescharacteristic
34C2×#120C811yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×M32—22no—
38C22×C16—22no—
39C22×C16D414yescomplemented
40C8×D4—22no—
41C8×D4—22no—
42C8×D4—44no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×C4×C8—21no—
52C2×C4×C8D411yescharacteristic
53#386—21no—
54#386C811yescharacteristic
55#385—21no—
56#385C811yescharacteristic
57#379C811yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C411yescharacteristic
60C4⋊C16—22no—
61C22⋊C16—22no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64C22⋊C16—22no—
65C22⋊C16—22no—
66C22⋊C16—48no—
67#346—22no—
68#336—21no—
69#336—21no—
70#336—21no—
71C25C2×C811yescharacteristic
72C22×D4—21no—
73C22×D4C2×C811yescharacteristic
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C2×C811yescharacteristic
82C23×C4C22×C411yescharacteristic
83C22×C8—28no—
84C22×C8—24no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—44no—
94C22×C8—42no—
95C22×C8—44no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—44no—
99C22×C8—41no—
100C22×C8—42no—
101C22×C8—41no—
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C2×D412yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8#4112yes—
108#120—22no—
109#120—22no—
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—42no—
113C4×D4—42no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4M1611yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4C2×C811yescharacteristic
121C2×C22⋊C4C2×C811yescharacteristic
122C2×C22⋊C4C2×C811yescharacteristic
123C2×C22⋊C4M1611yescharacteristic
124C2×C42—21no—
125C2×C42C22⋊C411yescharacteristic
126M32—82no—
127C2×C16—24no—
128C2×C16—44no—
129C2×C16—44no—
130C2×C16—42no—
131C4⋊C8—22no—
132C4⋊C8—42no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—42no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—22no—
147C24—21no—
148C24—22no—
149C24—42no—
150C24—42no—
151C24C22⋊C811yescharacteristic
152C24M3212yescomplemented
153C24C22×C811yescharacteristic
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4C22⋊C811yescharacteristic
188C22×C4C22⋊C811yescharacteristic
189C22×C4C22⋊C811yescharacteristic
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C4×D412yes—
192C22×C4C2×M1611yescharacteristic
193C22×C4#13112yes—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—48no—
203C2×C8—44no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—44no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—81no—
215C2×C8#12011yescharacteristic
216C2×C8#12112yes—
217C2×C8#12311yescharacteristic
218C4⋊C4—22no—
219C4⋊C4—42no—
220C22⋊C4—24no—
221C22⋊C4—22no—
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4M3212yescomplemented
229C42—21no—
230C42—21no—
231C42—41no—
232C42—41no—
233C16—44no—
234C16—82no—
235C23—22no—
236C23—22no—
237C23—24no—
238C23—24no—
239C23—22no—
240C23—24no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—44no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—41no—
250C23—42no—
251C23—44no—
252C23—48no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23C2×C22⋊C811yescharacteristic
259C23C8×D412yes—
260C23C2×M3211yescharacteristic
261C23C2×M3211yescharacteristic
262C23#50412yes—
263D4—48no—
264D4—42no—
265D4—42no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4#38611yescharacteristic
300C2×C4#40811yescharacteristic
301C2×C4#43512yes—
302C8—22no—
303C8—21no—
304C8—42no—
305C8—41no—
306C8—41no—
307C8—81no—
308C8B(4,𝔽2)12yes—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—24no—
314C22—28no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22#186911yescharacteristic
333C22#211211yescharacteristic
334C22#211311yescharacteristic
335C22#216812yes—
336C22#216912yes—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4#188911yescharacteristic
346C4#189111yescharacteristic
347C2—21no—
348C2—21no—
349C2—22no—
350C2—22no—
351C2—44no—
352C2—81no—
353C2#896511yescharacteristic
354C2#1297711yescharacteristic
355C2#1298111yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC2×#120C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33487C2semidirect
3. #12679C2semidirect
4. #11465C2semidirect
5. #7199C2semidirect
39. C22×C16D4semidirect
152. C24M32semidirect
228. C22⋊C4M32semidirect

Maximal subgroups

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

subgroupindex
#334872
C8×#1202
#126792
#114652
#71992