Finite Group Databasebeta

← #231714

Subgroups — #231714

Order 512.

686 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 198 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#23215C212yesmaximal
2#23162C212yesmaximal, complemented
3C2×#1722C211yesmaximal, characteristic
4#9747C212yesmaximal
5#9730C212yesmaximal, complemented
6#9066C212yesmaximal, complemented
7#9064C211yesmaximal, characteristic
8#9063C211yesmaximal, characteristic
9#8289C212yesmaximal, complemented
10C2×#500C2212yes—
11C2×#494C2211yescharacteristic
12C2×#492C2211yescharacteristic
13C2×#486C2212yes—
14#2369C2211yescharacteristic
15#2369C2212yes—
16#2366C2211yescharacteristic
17#1987C2212yes—
18#1987C2212yes—
19#1987C2212yes—
20#1982C2212yes—
21#1943—22no—
22#1934—22no—
23#1926—22no—
24#1918—22no—
25#1918—22no—
26#1864—22no—
27#1776—22no—
28C4×#106C2212yes—
29#1770C2212yes—
30C8×C4⋊C4C2212yes—
31C4×#102—22no—
32#1722C418yescomplemented
33C2×#335C2212yes—
34C2×#333C2211yescharacteristic
35C2×#333C2212yes—
36C2×#331C2211yescharacteristic
37C2×#322C2211yescharacteristic
38#1279C2212yes—
39#1279C2212yes—
40C22×Q16—41no—
41C22×SD16—41no—
42C2×C4⋊Q8C2312yes—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#127C2311yescharacteristic
45C2×C4×D4—22no—
46#500—24no—
47#500C2×C418yescomplemented
48#494—41no—
49#494C2×C414yescomplemented
50#492—41no—
51#492C2×C414yes—
52#486—24no—
53#486C2×C418yes—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106D412yes—
57C2×#106D412yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#102—22no—
61C2×#102—22no—
62C2×#102—22no—
63C2×#102—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2312yes—
68#391—42no—
69#391—41no—
70#390—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#386—41no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2312yes—
79C4×C4⋊C4C2312yes—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340D412yes—
87#340D412yes—
88#336—22no—
89#336—22no—
90#335—24no—
91#333—42no—
92#333—41no—
93#331—41no—
94#322C2×C414yescomplemented
95C22×Q8—21no—
96C22×Q8—22no—
97C22×D4—22no—
98C22×D4—21no—
99C23×C4—41no—
100C2×Q16—44no—
101C2×Q16—82no—
102C2×Q16—82no—
103C2×SD16—44no—
104C2×SD16—82no—
105C2×SD16—82no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—22no—
116C4⋊Q8C22×C414yes—
117C4⋊Q8C22×C412yes—
118#127—22no—
119#127—21no—
120#127C22×C412yes—
121C4×D4—44no—
122C4×D4—82no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—42no—
137C2×C42—41no—
138C2×C42C2411yescharacteristic
139#107—28no—
140#107—28no—
141#106—24no—
142#106—42no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147#102—28no—
148#102—28no—
149#102—44no—
150#102—82no—
151#97—44no—
152C4×C8—21no—
153C4×C8—82no—
154C4×C8C22×C412yes—
155C4×C8C22×C414yes—
156C4×C8#4114yes—
157#95—22no—
158#95—22no—
159#95—42no—
160#95—42no—
161#95—42no—
162C24—41no—
163C2×Q8—22no—
164C2×Q8—24no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×D4—24no—
172C2×D4—22no—
173C2×D4—42no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—81no—
180C2×D4—81no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—84no—
188C22×C4—161no—
189C22×C4C22×D412yes—
190C22×C4H(5,𝔽2)11yescharacteristic
191Q16—88no—
192SD16—88no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C4×D414yes—
207C2×C8C4×D414yes—
208C4⋊C4—22no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—161no—
222C4⋊C4—162no—
223C4⋊C4#13714yes—
224C22⋊C4—84no—
225C22⋊C4—161no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42—42no—
230C42—162no—
231C42—161no—
232C42C23×C411yescharacteristic
233C42C2×#4112yes—
234C42H(5,𝔽2)11yescharacteristic
235C23—42no—
236C23—82no—
237C23—82no—
238C23—81no—
239C23—81no—
240C23D4211yescharacteristic
241Q8—82no—
242Q8—84no—
243Q8—161no—
244Q8—162no—
245D4—44no—
246D4—44no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—161no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—164no—
272C2×C4C2×C4×D412yes—
273C2×C4#51811yescharacteristic
274C2×C4#55012yes—
275C2×C4#55911yescharacteristic
276C2×C4C2×#13712yes—
277C2×C4#57512yes—
278C2×C4#57712yes—
279C2×C4#57812yes—
280C8—24no—
281C8—24no—
282C8—24no—
283C8—82no—
284C8—82no—
285C22—84no—
286C22—82no—
287C22—161no—
288C22—162no—
289C22—161no—
290C22#236611yescharacteristic
291C22#326612yes—
292C22#326912yes—
293C22#329211yescharacteristic
294C22#329211yescharacteristic
295C4—21no—
296C4—21no—
297C4—44no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—84no—
302C4—82no—
303C4C4×#13712yes—
304C4#294712yes—
305C4#339812yes—
306C4#340112yes—
307C2—162no—
308C2#2004011yescharacteristic
309C2#2005012yes—
310C2#2015711yescharacteristic
311C2#2319212yes—
312C2#2320111yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #23162C2semidirect
5. #9730C2semidirect
6. #9066C2semidirect
9. #8289C2semidirect
32. #1722C4semidirect
47. #500C2×C4semidirect
49. #494C2×C4semidirect
94. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232152
#231622
C2×#17222
#97472
#97302
#90662
#90642
#90632
#82892