Finite Group Databasebeta

← #120369

Subgroups — #120369

Order 512.

454 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21362C211yesmaximal, characteristic
2#17810C211yesmaximal, characteristic
3#8109C211yesmaximal, characteristic
4#8096C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#7984C211yesmaximal, characteristic
7#7949C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18C4×#107—21no—
19C4×#106—21no—
20#1573—21no—
21#1573—21no—
22#1572—22no—
23#1565C412yescomplemented
24#1565C412yescomplemented
25#1305—22no—
26#1303—22no—
27#1297—22no—
28#1289—22no—
29#1285—21no—
30#1285—21no—
31C2×#117—21no—
32C2×#117—21no—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432Q812yes—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431C2×C412yes—
49#431D412yes—
50C2×#107—22no—
51C2×#106—22no—
52#423—21no—
53#423Q812yes—
54C2×C4⋊C8C2×C411yescharacteristic
55C2×C4⋊C8C2×C411yescharacteristic
56C2×C4⋊C8D411yescharacteristic
57C2×C4⋊C8D411yescharacteristic
58#407—21no—
59#407D412yes—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C4×M16—21no—
63C4×M16C2×C412yes—
64C2×#97—21no—
65C2×C4×C8—21no—
66#384—21no—
67#383Q812yes—
68#382—21no—
69#382—21no—
70#381—21no—
71#380—21no—
72#380—21no—
73#380D412yes—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4D411yescharacteristic
79C4×C4⋊C4D411yescharacteristic
80C4×C22⋊C4—21no—
81C4×C22⋊C4—21no—
82C4×C22⋊C4—21no—
83#376—21no—
84#376—21no—
85C43—21no—
86#340—22no—
87#340—22no—
88#335—22no—
89#335—22no—
90#334—22no—
91#334—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—41no—
96C2×M16C4⋊C414yes—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101#117—42no—
102#117—42no—
103#117—82no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42C2×Q811yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131#107—44no—
132#106—44no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—41no—
143C4⋊C8C22⋊C412yes—
144C4⋊C8C22⋊C412yes—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153#97—22no—
154#97—22no—
155#97—41no—
156#97—41no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—41no—
161#95—21no—
162#95—21no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—41no—
168#95—41no—
169#95—41no—
170C24#12111yescharacteristic
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×Q811yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12411yescharacteristic
195C22×C4#12511yescharacteristic
196C22×C4#12611yescharacteristic
197C22×C4C4⋊Q811yescharacteristic
198M16—44no—
199M16—82no—
200C2×C8—24no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—81no—
209C2×C8—81no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—82no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C22⋊C4—44no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C22⋊C4—82no—
224C42—44no—
225C42—81no—
226C42—81no—
227C42—82no—
228C42—82no—
229C42—82no—
230C42C2×C22⋊C411yescharacteristic
231C42#11711yescharacteristic
232C42C4×D411yescharacteristic
233C42C4×Q811yescharacteristic
234C42#12111yescharacteristic
235C42C22⋊Q811yescharacteristic
236C42C22⋊Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42#12311yescharacteristic
240C42#12511yescharacteristic
241C42#12511yescharacteristic
242C42#12511yescharacteristic
243C42#12611yescharacteristic
244C42C4⋊Q811yescharacteristic
245C42C4⋊Q811yescharacteristic
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23#38911yescharacteristic
251C23#39411yescharacteristic
252C23#39611yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#38011yescharacteristic
289C2×C4#38611yescharacteristic
290C2×C4#38711yescharacteristic
291C2×C4#38711yescharacteristic
292C2×C4#38911yescharacteristic
293C2×C4#39111yescharacteristic
294C2×C4#39511yescharacteristic
295C2×C4#39611yescharacteristic
296C2×C4#39811yescharacteristic
297C2×C4#39811yescharacteristic
298C2×C4#39811yescharacteristic
299C2×C4#40011yescharacteristic
300C8—44no—
301C8—42no—
302C8—42no—
303C8—82no—
304C8—81no—
305C8—81no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22#144111yescharacteristic
317C22#204811yescharacteristic
318C22#204911yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—82no—
331C4—162no—
332C4—162no—
333C4#181212yes—
334C4#190611yescharacteristic
335C4#190611yescharacteristic
336C2—21no—
337C2—21no—
338C2—41no—
339C2—41no—
340C2#1121511yescharacteristic
341C2#1122911yescharacteristic
342C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1565C4semidirect
24. #1565C4semidirect

Maximal subgroups

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

subgroupindex
#213622
#178102
#81092
#80962
#80522
#79842
#79492