Finite Group Databasebeta

← #111112

Subgroups — #111112

Order 512.

1016 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1016 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×#1823C212yesmaximal
2#20526C211yesmaximal, characteristic
3#13904C211yesmaximal, characteristic
4#10846C211yesmaximal, characteristic
5C2×#1289C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#381—22no—
9C2×#380C2212yes—
10C2×#380C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12#1823—24no—
13#1823C418yescomplemented
14#1752—24no—
15#1749—22no—
16C2×#342—22no—
17C2×#340—24no—
18#1445C414yescomplemented
19C4⋊#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#381—48no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380—22no—
50#380—22no—
51#380C2×C414yes—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C2×#95—22no—
55C2×#95—28no—
56C2×#95C2×C412yes—
57C2×#95C2×C412yes—
58#342—44no—
59#340—216no—
60#336—24no—
61C25C2×D411yescharacteristic
62C23×C4—22no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—44no—
66C23×C4—42no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—42no—
70C23×C4—44no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4C22⋊C414yes—
74C23×C4C22×C412yes—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4#4112yes—
80C23×C4#4112yes—
81C23×C4#4112yes—
82C22×C8—24no—
83C22×C8—24no—
84C22×C8—48no—
85C22×C8C22⋊C418yes—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—48no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C4214yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—44no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—44no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125#97—84no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—416no—
130#95—42no—
131C24—42no—
132C24—42no—
133C24C2×C22⋊C411yescharacteristic
134C24C4×D412yes—
135C24#12112yes—
136C24#12411yescharacteristic
137C24#12711yescharacteristic
138C24AGL(1,ℤ/8)12yes—
139C24AGL(1,ℤ/8)14yes—
140C24#13712yes—
141C22×C4—24no—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—84no—
169C22×C4—84no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—81no—
173C22×C4—82no—
174C22×C4—81no—
175C22×C4—84no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4C2×C4211yescharacteristic
179C22×C4C2×C22⋊C412yes—
180C22×C4#11712yes—
181C22×C4#11711yescharacteristic
182C22×C4C4×D412yes—
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4C4×D412yes—
186C22×C4C4×D412yes—
187C22×C4#12011yescharacteristic
188C22×C4#12112yes—
189C22×C4#12112yes—
190C22×C4#12312yes—
191C22×C4#12311yescharacteristic
192C22×C4#12412yes—
193C22×C4#12612yes—
194C2×C8—28no—
195C2×C8—48no—
196C2×C8—48no—
197C2×C8—84no—
198C4⋊C4—28no—
199C4⋊C4—48no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—48no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—44no—
213C42—82no—
214C23—22no—
215C23—24no—
216C23—22no—
217C23—24no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—48no—
225C23—44no—
226C23—44no—
227C23C4×C22⋊C411yescharacteristic
228C23#38611yescharacteristic
229C23#38812yes—
230C23#39011yescharacteristic
231C23#39012yes—
232C23#41712yes—
233C23#41812yes—
234C23#44014yes—
235C23#44214yes—
236C23#46814yes—
237C23#46914yes—
238C23#48912yes—
239C23#49012yes—
240C23#49611yescharacteristic
241C23#49612yes—
242C23#49711yescharacteristic
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—48no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—84no—
265C2×C4—88no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—88no—
274C2×C4—82no—
275C2×C4—84no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—168no—
279C2×C4C4×C22⋊C412yes—
280C2×C4#38612yes—
281C2×C4#38812yes—
282C2×C4#38812yes—
283C2×C4#45814yes—
284C8—48no—
285C22—24no—
286C22—24no—
287C22—28no—
288C22—28no—
289C22—21no—
290C22—22no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—22no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22#143711yescharacteristic
305C22#176212yes—
306C22#189012yes—
307C22#189512yes—
308C22#190912yes—
309C22#193614yes—
310C22#195911yescharacteristic
311C22#196111yescharacteristic
312C22#197212yes—
313C22#197312yes—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—84no—
318C4—84no—
319C4—162no—
320C4—164no—
321C4#189014yes—
322C2—21no—
323C2—22no—
324C2—24no—
325C2—21no—
326C2—22no—
327C2—22no—
328C2#947414yes—
329C2#1092711yescharacteristic
330C2#1093611yescharacteristic
331C2#1094711yescharacteristic
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
13. #1823C4semidirect
18. #1445C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18232
#205262
#139042
#108462
C2×#12892