Finite Group Databasebeta

← #139351

Subgroups — #139351

Order 512.

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

normal subgroups only

This group has 400 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
2#28553C211yesmaximal, characteristic
3#11887C212yesmaximal, complemented
4#8609C211yesmaximal, characteristic
5#7258C212yesmaximal, complemented
6C2×#467C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2417—21no—
9#2168—24no—
10#2163—21no—
11#2112—21no—
12C2×C22⋊C16C2212yes—
13#2087C411yescharacteristic, complemented
14#2076C411yescharacteristic, complemented
15#2056C411yescharacteristic, complemented
16#2001C411yescharacteristic, complemented
17#1933—21no—
18#1892—21no—
19#1872—21no—
20#1808C2212yes—
21#1799—21no—
22C2×#340C2211yescharacteristic
23#1348—24no—
24#1289—21no—
25C22×Q16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8—21no—
30C2×C4×D4D411yescharacteristic
31C22×C4⋊C4—21no—
32C2×M32—22no—
33C22×C16—22no—
34#467—22no—
35#467—22no—
36C8×D4—22no—
37C8×D4—22no—
38C2×#107—22no—
39C2×#107C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—21no—
44C2×#99—41no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#398—41no—
49#396—41no—
50#391—21no—
51#389—21no—
52#386—41no—
53#380—21no—
54#380—22no—
55#380—41no—
56C4⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16D414yes—
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×Q8—21no—
69C22×D4D811yescharacteristic
70C23×C4—41no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×Q16—22no—
74C2×Q16—81no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—81no—
78C22×C8—81no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D412yes—
83C22×C8C2×D411yescharacteristic
84C22×C8#4112yes—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—81no—
89C4×D4—22no—
90C4×D4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4SD1611yescharacteristic
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4D811yescharacteristic
107C2×C22⋊C4SD1611yescharacteristic
108C2×C42C22⋊C411yescharacteristic
109M32—82no—
110C2×C16—24no—
111C2×C16—82no—
112#107—22no—
113#107—82no—
114C4⋊C8—22no—
115#103—42no—
116#103—41no—
117#103—41no—
118#103—81no—
119#99—82no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C4×C8—21no—
123C4×C8—21no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—42no—
129C24#10211yescharacteristic
130C24C2×D811yescharacteristic
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—82no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—82no—
155C22×C4—161no—
156C22×C4#10211yescharacteristic
157C22×C4#10211yescharacteristic
158C22×C4#10211yescharacteristic
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4#12112yes—
161C22×C4C2×SD1611yescharacteristic
162C22×C4#13512yes—
163Q16—82no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—81no—
174C2×C8C4×D412yes—
175C2×C8#12011yescharacteristic
176C2×C8#12311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—161no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—161no—
197C42—21no—
198C42—21no—
199C16—44no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23C2×#10211yescharacteristic
208C23#46612yes—
209C23#50911yescharacteristic
210C23#51011yescharacteristic
211C23#51012yes—
212Q8—81no—
213Q8—81no—
214Q8—81no—
215Q8—81no—
216D4—42no—
217D4—42no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—162no—
247C2×C4—162no—
248C2×C4—162no—
249C2×C4—162no—
250C2×C4—162no—
251C2×C4#38611yescharacteristic
252C2×C4#41611yescharacteristic
253C2×C4#46512yes—
254C8—21no—
255C8—21no—
256C8—22no—
257C8—41no—
258C8—41no—
259C22—21no—
260C22—22no—
261C22—21no—
262C22—22no—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22#189311yescharacteristic
272C22#213911yescharacteristic
273C22#214111yescharacteristic
274C22#221912yes—
275C22#222012yes—
276C4—21no—
277C4—22no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—82no—
282C4—161no—
283C4—161no—
284C4—161no—
285C4—322no—
286C4#188411yescharacteristic
287C4#191311yescharacteristic
288C2—21no—
289C2—22no—
290C2—21no—
291C2—22no—
292C2—81no—
293C2#938411yescharacteristic
294C2#1306211yescharacteristic
295C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4D8
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
3. #11887C2semidirect
5. #7258C2semidirect
13. #2087C4semidirect
14. #2076C4semidirect
15. #2056C4semidirect
16. #2001C4semidirect

Maximal subgroups

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

subgroupindex
#334872
#285532
#118872
#86092
#72582