Finite Group Databasebeta

← #139350

Subgroups — #139350

Order 512.

435 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 88 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
2#28551C211yesmaximal, characteristic
3#11886C212yesmaximal, complemented
4#8609C211yesmaximal, characteristic, complemented
5#7258C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2416—21no—
9#2168—24no—
10#2163—21no—
11#2112—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—
33C2×M32—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
60#346—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—84no—
92C4×D4—22no—
93C4×D4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4SD1611yescharacteristic
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4D811yescharacteristic
110C2×C22⋊C4SD1611yescharacteristic
111C2×C42C22⋊C411yescharacteristic
112M32—82no—
113C2×C16—24no—
114C2×C16—82no—
115#107—22no—
116#107—82no—
117C4⋊C8—22no—
118#102—42no—
119#102—41no—
120#102—41no—
121#102—81no—
122#99—82no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C4×C8—21no—
126C4×C8—21no—
127#95—41no—
128#95—41no—
129#95—42no—
130C24—41no—
131C24#10211yescharacteristic
132C24C2×D811yescharacteristic
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—82no—
146C2×D4—82no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—161no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—161no—
163C22×C4#10211yescharacteristic
164C22×C4#10211yescharacteristic
165C22×C4#10211yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4#12112yes—
168C22×C4C2×SD1611yescharacteristic
169C22×C4#13512yes—
170D8—82no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—81no—
181C2×C8C4×D412yes—
182C2×C8#12011yescharacteristic
183C2×C8#12311yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—161no—
204C42—21no—
205C42—21no—
206C16—44no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23C2×#10211yescharacteristic
222C23#46612yes—
223C23#50911yescharacteristic
224C23#50912yes—
225C23#51011yescharacteristic
226D4—42no—
227D4—42no—
228D4—81no—
229D4—81no—
230D4—81no—
231D4—81no—
232D4—164no—
233D4—162no—
234D4—162no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—81no—
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—164no—
261C2×C4—162no—
262C2×C4—162no—
263C2×C4—162no—
264C2×C4#38611yescharacteristic
265C2×C4#41611yescharacteristic
266C2×C4#46512yes—
267C8—21no—
268C8—21no—
269C8—22no—
270C8—41no—
271C8—41no—
272C22—21no—
273C22—22no—
274C22—21no—
275C22—22no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22—161no—
285C22—161no—
286C22—161no—
287C22—161no—
288C22—161no—
289C22—161no—
290C22—161no—
291C22#189311yescharacteristic
292C22#213911yescharacteristic
293C22#214011yescharacteristic
294C22#221812yes—
295C22#222012yes—
296C4—21no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—161no—
303C4—161no—
304C4—322no—
305C4#188411yescharacteristic
306C4#191311yescharacteristic
307C2—21no—
308C2—22no—
309C2—21no—
310C2—22no—
311C2—81no—
312C2—321no—
313C2#938411yescharacteristic
314C2#1306111yescharacteristic
315C2#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. #33487C2semidirect
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
#334872
#285512
#118862
#86092
#72582