Finite Group Databasebeta

← #139367

Subgroups — #139367

Order 512.

418 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 82 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#33489C211yesmaximal, characteristic, complemented
2#28551C211yesmaximal, characteristic
3#11895C212yesmaximal, complemented
4#8612C211yesmaximal, characteristic, complemented
5#7264C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2416—21no—
9#2169—24no—
10#2162—21no—
11#2112—21no—
12#2112C2212yes—
13#2086C411yescharacteristic
14#2075C411yescharacteristic
15#2054C411yescharacteristic
16#1999C411yescharacteristic
17#1934—21no—
18#1893—21no—
19#1874—21no—
20#1810C2212yes—
21C2×#340C2211yescharacteristic
22#1465—21no—
23#1349—24no—
24#1293—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—
34C2×M32—22no—
35#466—22no—
36#466—22no—
37C8×D4—22no—
38C8×D4—22no—
39C2×#108—22no—
40C2×#107C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45#407—22no—
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—
55C4⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16D414yescomplemented
59C4×C16—22no—
60#341—22no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#324—41no—
67#323—41no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4SD1611yescharacteristic
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×D8—22no—
76C2×D8—81no—
77C2×M16—42no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—81no—
81C22×C8—81no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22×C411yescharacteristic
85C22×C8C2×D412yes—
86C22×C8C2×D411yescharacteristic
87C22×C8#4112yes—
88#121—42no—
89#121—42no—
90#121—42no—
91#121—81no—
92#121—84no—
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—81no—
102C2×C4⋊C4D811yescharacteristic
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4D811yescharacteristic
109C2×C22⋊C4SD1611yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111M32—82no—
112M32—82no—
113C2×C16—24no—
114#108—82no—
115#107—22no—
116C4⋊C8—22no—
117#102—42no—
118#102—41no—
119#102—41no—
120#102—81no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C4×C8—21no—
126C4×C8—21no—
127#95—41no—
128#95—41no—
129C24—41no—
130C24#10211yescharacteristic
131C24C2×SD1611yescharacteristic
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—82no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—82no—
146C2×D4—161no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—161no—
159C22×C4#10211yescharacteristic
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4#12112yes—
164C22×C4C2×D811yescharacteristic
165C22×C4#13512yes—
166D8—82no—
167M16—82no—
168M16—82no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—81no—
179C2×C8—82no—
180C2×C8C4×D412yes—
181C2×C8#12011yescharacteristic
182C2×C8#12311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C22⋊C4—161no—
199C42—21no—
200C42—21no—
201C16—44no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23C2×#10211yescharacteristic
219C23#46512yes—
220D4—44no—
221D4—81no—
222D4—81no—
223D4—81no—
224D4—81no—
225D4—164no—
226D4—164no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—164no—
252C2×C4#36112yes—
253C2×C4#38611yescharacteristic
254C2×C4#41611yescharacteristic
255C2×C4#46612yes—
256C8—21no—
257C8—21no—
258C8—22no—
259C8—41no—
260C8—41no—
261C8—162no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—161no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22—161no—
280C22—161no—
281C22#189311yescharacteristic
282C22C2×#36111yescharacteristic
283C22#214811yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—41no—
288C4—41no—
289C4—82no—
290C4—161no—
291C4—161no—
292C4#188411yescharacteristic
293C4#191311yescharacteristic
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2—321no—
299C2#938411yescharacteristic
300C2#1306711yescharacteristic
301C2#1307611yescharacteristic
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. #33489C2semidirect
3. #11895C2semidirect
4. #8612C2semidirect
5. #7264C2semidirect
58. C22⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
#334892
#285512
#118952
#86122
#72642