Finite Group Databasebeta

← #111108

Subgroups — #111108

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1823C212yesmaximal
2#20523C211yesmaximal, 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
14C8×C22⋊C4—24no—
15C4×C22⋊C8—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×C4×C8—24no—
45#381—44no—
46#381—44no—
47#380—22no—
48#380—22no—
49#380—22no—
50#380—22no—
51#380—22no—
52#380C2×C414yes—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C2×#95—22no—
56C2×#95—28no—
57C2×#95C2×C412yes—
58C2×#95C2×C412yes—
59#342—44no—
60#340—216no—
61#336—24no—
62C25C2×D411yescharacteristic
63C23×C4—22no—
64C23×C4—24no—
65C23×C4—22no—
66C23×C4—44no—
67C23×C4—42no—
68C23×C4—41no—
69C23×C4—41no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—44no—
73C23×C4—42no—
74C23×C4C22⋊C414yes—
75C23×C4C22×C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D412yes—
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4112yes—
83C22×C8—24no—
84C22×C8—24no—
85C22×C8—48no—
86C22×C8C22⋊C418yes—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4C4214yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4D814yes—
119C2×C22⋊C4SD1614yes—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—44no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C4×C8—84no—
128#95—24no—
129#95—24no—
130#95—24no—
131#95—416no—
132#95—42no—
133C24—42no—
134C24—42no—
135C24C2×C22⋊C411yescharacteristic
136C24C4×D412yes—
137C24#12112yes—
138C24#12411yescharacteristic
139C24#12711yescharacteristic
140C24C2×D812yes—
141C24C2×SD1612yes—
142C24#13514yes—
143C22×C4—24no—
144C22×C4—28no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—84no—
170C22×C4—84no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—82no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4#10218yes—
180C22×C4C2×C4211yescharacteristic
181C22×C4C2×C22⋊C412yes—
182C22×C4#11712yes—
183C22×C4#11711yescharacteristic
184C22×C4C4×D412yes—
185C22×C4C4×D412yes—
186C22×C4C4×D412yes—
187C22×C4C4×D412yes—
188C22×C4C4×D412yes—
189C22×C4#12011yescharacteristic
190C22×C4#12112yes—
191C22×C4#12112yes—
192C22×C4#12312yes—
193C22×C4#12311yescharacteristic
194C22×C4#12412yes—
195C22×C4#12612yes—
196C2×C8—28no—
197C2×C8—48no—
198C2×C8—48no—
199C2×C8—84no—
200C4⋊C4—28no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C22⋊C4—28no—
208C22⋊C4—28no—
209C22⋊C4—48no—
210C22⋊C4—82no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C42—44no—
216C42—82no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—24no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—44no—
228C23—44no—
229C23—48no—
230C23C4×C22⋊C411yescharacteristic
231C23#38611yescharacteristic
232C23#38812yes—
233C23#39011yescharacteristic
234C23#39012yes—
235C23C2×#10212yes—
236C23#41612yes—
237C23C4×D814yes—
238C23C4×SD1614yes—
239C23#46514yes—
240C23#46614yes—
241C23#48612yes—
242C23#48812yes—
243C23#49211yescharacteristic
244C23#49311yescharacteristic
245C23#49512yes—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—28no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—48no—
265C2×C4—44no—
266C2×C4—48no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—88no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—88no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—168no—
282C2×C4C4×C22⋊C412yes—
283C2×C4#38612yes—
284C2×C4#38812yes—
285C2×C4#38812yes—
286C2×C4#45814yes—
287C8—48no—
288C22—21no—
289C22—22no—
290C22—21no—
291C22—22no—
292C22—21no—
293C22—22no—
294C22—21no—
295C22—22no—
296C22—24no—
297C22—24no—
298C22—28no—
299C22—28no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—42no—
307C22#143711yescharacteristic
308C22C4×#10212yes—
309C22#189012yes—
310C22#189312yes—
311C22#190912yes—
312C22#193414yes—
313C22#195711yescharacteristic
314C22#195811yescharacteristic
315C22#196812yes—
316C22#197012yes—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—48no—
321C4—84no—
322C4—162no—
323C4—164no—
324C4#189014yes—
325C2—21no—
326C2—22no—
327C2—24no—
328C2—21no—
329C2—22no—
330C2—22no—
331C2#947014yes—
332C2#1092511yescharacteristic
333C2#1093611yescharacteristic
334C2#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
#205232
#139042
#108462
C2×#12892