Finite Group Databasebeta

← #111111

Subgroups — #111111

Order 512.

988 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C2×#1824C212yesmaximal
2#20526C211yesmaximal, characteristic
3#13924C211yesmaximal, characteristic
4#10846C211yesmaximal, characteristic
5C2×#1289C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2212yes—
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1824—24no—
12#1824C418yescomplemented
13#1752—24no—
14#1749—22no—
15C2×#342—22no—
16C2×#340—24no—
17#1445C414yescomplemented
18C4⋊#95—22no—
19C22⋊#95—22no—
20#1289—24no—
21C24×C4—21no—
22C24×C4C2311yescharacteristic
23C23×C8D414yes—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C414yes—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—22no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4D412yes—
37C22×C42D412yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8C2×C414yes—
43C2×C22⋊C8D414yes—
44C2×#97—24no—
45#380—22no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380—22no—
50#380C2×C414yes—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C2×#95—22no—
54C2×#95—28no—
55C2×#95C2×C412yes—
56C2×#95C2×C412yes—
57#342—44no—
58#340—216no—
59#336—24no—
60C25C2×D411yescharacteristic
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—22no—
64C23×C4—44no—
65C23×C4—42no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4—42no—
69C23×C4—84no—
70C23×C4C22⋊C414yes—
71C23×C4C22×C412yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D412yes—
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4112yes—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—48no—
82C22×C8C22⋊C418yes—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—48no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C4214yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—44no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123#97—84no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—42no—
128#95—88no—
129C24—42no—
130C24—42no—
131C24C2×C22⋊C411yescharacteristic
132C24C4×D412yes—
133C24#12112yes—
134C24#12411yescharacteristic
135C24#12711yescharacteristic
136C24AGL(1,ℤ/8)12yes—
137C24#13712yes—
138C24#13714yes—
139C22×C4—24no—
140C22×C4—28no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—88no—
164C22×C4—88no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4—84no—
170C22×C4C2×C4211yescharacteristic
171C22×C4C2×C22⋊C412yes—
172C22×C4#11712yes—
173C22×C4#11711yescharacteristic
174C22×C4C4×D412yes—
175C22×C4C4×D412yes—
176C22×C4C4×D412yes—
177C22×C4C4×D412yes—
178C22×C4C4×D412yes—
179C22×C4#12011yescharacteristic
180C22×C4#12112yes—
181C22×C4#12112yes—
182C22×C4#12312yes—
183C22×C4#12311yescharacteristic
184C22×C4#12412yes—
185C22×C4#12612yes—
186C2×C8—28no—
187C2×C8—48no—
188C2×C8—48no—
189C2×C8—84no—
190C4⋊C4—28no—
191C4⋊C4—48no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—48no—
200C22⋊C4—82no—
201C22⋊C4—81no—
202C22⋊C4—82no—
203C22⋊C4—81no—
204C22⋊C4—82no—
205C42—44no—
206C42—82no—
207C23—22no—
208C23—24no—
209C23—22no—
210C23—24no—
211C23—42no—
212C23—41no—
213C23—41no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23—44no—
218C23—48no—
219C23—44no—
220C23C4×C22⋊C411yescharacteristic
221C23#38611yescharacteristic
222C23#38812yes—
223C23#39011yescharacteristic
224C23#39012yes—
225C23#41712yes—
226C23#41912yes—
227C23#44014yes—
228C23#44114yes—
229C23#46814yes—
230C23#47014yes—
231C23#48912yes—
232C23#49112yes—
233C23#49611yescharacteristic
234C23#49711yescharacteristic
235C23#49712yes—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—48no—
252C2×C4—48no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—84no—
258C2×C4—88no—
259C2×C4—88no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4—88no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—168no—
270C2×C4C4×C22⋊C412yes—
271C2×C4#38612yes—
272C2×C4#38812yes—
273C2×C4#38812yes—
274C2×C4#45814yes—
275C8—48no—
276C22—24no—
277C22—24no—
278C22—28no—
279C22—28no—
280C22—21no—
281C22—22no—
282C22—21no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—44no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—42no—
295C22#143711yescharacteristic
296C22#176312yes—
297C22#189012yes—
298C22#189412yes—
299C22#191012yes—
300C22#193514yes—
301C22#195911yescharacteristic
302C22#196011yescharacteristic
303C22#197112yes—
304C22#197312yes—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—84no—
309C4—84no—
310C4—162no—
311C4—164no—
312C4#189014yes—
313C2—21no—
314C2—22no—
315C2—24no—
316C2—21no—
317C2—22no—
318C2—22no—
319C2#947314yes—
320C2#1092811yescharacteristic
321C2#1093611yescharacteristic
322C2#1094811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1824C4semidirect
17. #1445C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18242
#205262
#139242
#108462
C2×#12892