Finite Group Databasebeta

← #139348

Subgroups — #139348

Order 512.

446 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 96 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
2#28551C211yesmaximal, characteristic
3#11886C212yesmaximal, complemented
4#8609C211yesmaximal, characteristic, complemented
5#7258C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2416—21no—
9D4×C16—24no—
10#2162—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#2086C411yescharacteristic, complemented
14#2075C411yescharacteristic, complemented
15#2054C411yescharacteristic, complemented
16#1999C411yescharacteristic, complemented
17#1934—21no—
18#1893—21no—
19#1874—21no—
20#1808C2212yes—
21#1799—21no—
22C2×#340C2211yescharacteristic
23#1347—24no—
24#1289—21no—
25C22×D8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#121—21no—
29C2×#121—21no—
30C2×#121—41no—
31C2×C4×D4D411yescharacteristic
32C22×C4⋊C4—21no—
33C22×C16—22no—
34C22×C16—22no—
35#466—22no—
36#466—22no—
37C8×D4—22no—
38C8×D4—22no—
39C2×#107—22no—
40C2×#107C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×#99—41no—
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C22⋊C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#396—41no—
50#394—41no—
51#390—21no—
52#389—21no—
53#386—41no—
54#380—21no—
55#380—22no—
56C4⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16D414yescomplemented
60C4×C16—22no—
61#342—41no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340C2×C411yescharacteristic
67#340C2×C411yescharacteristic
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4D811yescharacteristic
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×D8—22no—
76C2×D8—81no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—81no—
80C22×C8—81no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8C2×D411yescharacteristic
86C22×C8#4112yes—
87#121—42no—
88#121—42no—
89#121—42no—
90#121—81no—
91#121—82no—
92#121—82no—
93C4×D4—22no—
94C4×D4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4SD1611yescharacteristic
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4D811yescharacteristic
111C2×C22⋊C4SD1611yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113C2×C16—24no—
114C2×C16—82no—
115C2×C16—82no—
116#107—22no—
117#107—82no—
118C4⋊C8—22no—
119#102—42no—
120#102—41no—
121#102—41no—
122#102—81no—
123#99—82no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C4×C8—21no—
127C4×C8—21no—
128#95—41no—
129#95—41no—
130#95—42no—
131C24—41no—
132C24#10211yescharacteristic
133C24C2×D811yescharacteristic
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—81no—
146C2×D4—81no—
147C2×D4—82no—
148C2×D4—161no—
149C2×D4D1612yes—
150C2×D4SD3212yes—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—82no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4—82no—
164C22×C4—161no—
165C22×C4#10211yescharacteristic
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4#12112yes—
170C22×C4C2×SD1611yescharacteristic
171C22×C4#13512yes—
172D8—82no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—81no—
183C2×C8C4×D412yes—
184C2×C8#12011yescharacteristic
185C2×C8#12311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—82no—
204C22⋊C4—82no—
205C22⋊C4—161no—
206C42—21no—
207C42—21no—
208C16—44no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23#35712yes—
223C23C2×#10211yescharacteristic
224C23#46612yes—
225C23C2×D1611yescharacteristic
226C23C2×SD3211yescharacteristic
227C23#50812yes—
228D4—24no—
229D4—24no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—81no—
234D4—164no—
235D4—162no—
236D4—162no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—162no—
262C2×C4—162no—
263C2×C4—162no—
264C2×C4—162no—
265C2×C4—162no—
266C2×C4#35712yes—
267C2×C4#38611yescharacteristic
268C2×C4#41611yescharacteristic
269C2×C4#46512yes—
270C8—21no—
271C8—21no—
272C8—22no—
273C8—41no—
274C8—41no—
275C22—21no—
276C22—22no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—161no—
288C22—161no—
289C22—161no—
290C22—161no—
291C22—161no—
292C22—161no—
293C22—161no—
294C22#189311yescharacteristic
295C22C2×#35711yescharacteristic
296C22#213811yescharacteristic
297C22#221512yes—
298C22#221712yes—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—161no—
306C4—161no—
307C4—322no—
308C4#188411yescharacteristic
309C4#191311yescharacteristic
310C2—21no—
311C2—22no—
312C2—21no—
313C2—22no—
314C2—42no—
315C2—321no—
316C2#938411yescharacteristic
317C2#1305911yescharacteristic
318C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radical#28551C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ 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. #11886C2semidirect
4. #8609C2semidirect
5. #7258C2semidirect
13. #2086C4semidirect
14. #2075C4semidirect
15. #2054C4semidirect
16. #1999C4semidirect
59. C22⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
#334862
#285512
#118862
#86092
#72582