Finite Group Databasebeta

← #136807

Subgroups — #136807

Order 512.

645 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 114 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#33486C211yesmaximal, 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
10D4×C16—24no—
11#2162—21no—
12C2×C22⋊C16—24no—
13C2×C22⋊C16—21no—
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
37C22×C16—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—
67C4×C16—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#120C1614yescomplemented
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
126C2×C16—24no—
127C2×C16—44no—
128C2×C16—44no—
129C2×C16—42no—
130C2×C16—82no—
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
152C24C2×C1612yescomplemented
153C24C22×C811yescharacteristic
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4C2×C1612yes—
160C2×D4M3212yes—
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—22no—
221C22⋊C4—21no—
222C22⋊C4—24no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4C2×C1612yescomplemented
229C42—21no—
230C42—21no—
231C42—41no—
232C42—41no—
233C16—44no—
234C16—82no—
235C23—22no—
236C23—22no—
237C23—24no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—24no—
242C23—21no—
243C23—41no—
244C23—44no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23—41no—
249C23—42no—
250C23—44no—
251C23—48no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23C22⋊C1612yescomplemented
258C23C2×C22⋊C811yescharacteristic
259C23C8×D412yes—
260C23C22×C1611yescharacteristic
261C23C2×M3211yescharacteristic
262C23#50412yes—
263D4—24no—
264D4—24no—
265D4—48no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4C22⋊C1612yescomplemented
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—22no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—42no—
327C22—41no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—81no—
332C22#186911yescharacteristic
333C22C2×C22⋊C1611yescharacteristic
334C22#211311yescharacteristic
335C22D4×C1612yes—
336C22#216812yes—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—42no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4#188911yescharacteristic
346C4#189111yescharacteristic
347C2—21no—
348C2—21no—
349C2—22no—
350C2—22no—
351C2—44no—
352C2—42no—
353C2#896511yescharacteristic
354C2#1297611yescharacteristic
355C2#1298111yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC2×#120C8
LCSG ▷ C23 ▷ 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. #33486C2semidirect
3. #12679C2semidirect
4. #11465C2semidirect
5. #7199C2semidirect
39. C22×C16D4semidirect
109. #120C16semidirect
152. C24C2×C16semidirect
228. C22⋊C4C2×C16semidirect
257. C23C22⋊C16semidirect
298. C2×C4C22⋊C16semidirect

Maximal subgroups

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

subgroupindex
#334862
C8×#1202
#126792
#114652
#71992