Finite Group Databasebeta

← #121495

Subgroups — #121495

Order 512.

495 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21973C211yesmaximal, characteristic
2#17807C211yesmaximal, characteristic
3#17806C211yesmaximal, characteristic
4#8052C211yesmaximal, characteristic
5#7958C211yesmaximal, characteristic
6#7891C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1939—22no—
16#1850—22no—
17#1843—21no—
18#1832—21no—
19#1574—21no—
20#1574—21no—
21#1573—22no—
22#1573—21no—
23#1573—21no—
24#1571—21no—
25#1570—21no—
26#1565—22no—
27#1564—22no—
28#1297—22no—
29#1286C414yescomplemented
30#1279—22no—
31#1276—22no—
32C23×C8—21no—
33C2×#117—21no—
34C22×C42C2311yescharacteristic
35#432—21no—
36#432—21no—
37#432—21no—
38#432Q812yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431C2×C412yes—
49#431D412yes—
50#431Q812yes—
51#423C2×C412yes—
52#423D411yescharacteristic
53#423D411yescharacteristic
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8D412yes—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64#407—21no—
65#407—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C2×C4×C8—21no—
71C2×C4×C8D412yes—
72#383Q812yes—
73#382C2×C412yes—
74#381—21no—
75#380—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4Q812yes—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4D412yes—
82#376—21no—
83#376—21no—
84C43—21no—
85#336—24no—
86#335—22no—
87#335—22no—
88#335—24no—
89#334—22no—
90#334—22no—
91#334—24no—
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—22no—
96C2×M16—21no—
97C2×M16—21no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—82no—
105#117—24no—
106#117—42no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—41no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×Q811yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42C2×Q811yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—22no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—84no—
143C4⋊C8C4⋊C414yes—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152#97—22no—
153#97—41no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8C4⋊C414yes—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—41no—
165#95C4⋊C414yes—
166C24#12311yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4C2×C4⋊C411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4C22⋊Q811yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12711yescharacteristic
191C22×C4C4⋊Q811yescharacteristic
192M16—42no—
193M16—42no—
194M16—81no—
195M16—81no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—44no—
199C2×C8—42no—
200C2×C8—44no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—82no—
208C2×C8—82no—
209C4⋊C4—24no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C42—42no—
216C42—42no—
217C42—44no—
218C42—44no—
219C42—81no—
220C42—81no—
221C42C2×C4⋊C411yescharacteristic
222C42C2×C4⋊C411yescharacteristic
223C42C4×D411yescharacteristic
224C42C4×Q811yescharacteristic
225C42#12011yescharacteristic
226C42#12011yescharacteristic
227C42#12111yescharacteristic
228C42#12111yescharacteristic
229C42C22⋊Q811yescharacteristic
230C42C22⋊Q811yescharacteristic
231C42#12511yescharacteristic
232C42C4⋊Q811yescharacteristic
233C42C4⋊Q811yescharacteristic
234C42C4⋊Q811yescharacteristic
235C42C4⋊Q811yescharacteristic
236C42C4⋊Q811yescharacteristic
237C23—22no—
238C23—22no—
239C23—41no—
240C23#38511yescharacteristic
241C23#39811yescharacteristic
242C23#39911yescharacteristic
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#38411yescharacteristic
272C2×C4#38511yescharacteristic
273C2×C4#38511yescharacteristic
274C2×C4#38911yescharacteristic
275C2×C4#38911yescharacteristic
276C2×C4#38911yescharacteristic
277C2×C4#39211yescharacteristic
278C2×C4#39511yescharacteristic
279C2×C4#39511yescharacteristic
280C2×C4#39511yescharacteristic
281C2×C4#39811yescharacteristic
282C2×C4#39811yescharacteristic
283C2×C4#44312yes—
284C2×C4#44412yes—
285C8—44no—
286C8—44no—
287C8—81no—
288C8—82no—
289C8—81no—
290C8—81no—
291C8—81no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#144411yescharacteristic
301C22#186612yes—
302C22#209812yes—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—84no—
313C4—84no—
314C4#186612yes—
315C4#194511yescharacteristic
316C4#194611yescharacteristic
317C4#200811yescharacteristic
318C4#200911yescharacteristic
319C2—22no—
320C2—22no—
321C2—41no—
322C2#1132911yescharacteristic
323C2#1133311yescharacteristic
324C2#1133611yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1444
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1286C4semidirect

Maximal subgroups

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

subgroupindex
#219732
#178072
#178062
#80522
#79582
#78912
#78702