Finite Group Databasebeta

← #121271

Subgroups — #121271

Order 512.

382 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1307C211yesmaximal, characteristic
2#8044C211yesmaximal, characteristic
3#8042C211yesmaximal, characteristic
4C2×#1290C211yesmaximal, characteristic
5C2×#1287C211yesmaximal, characteristic
6#7900C211yesmaximal, characteristic
7#7877C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2285C2211yescharacteristic
15#1307C414yescomplemented
16#1296—21no—
17#1296—21no—
18#1296—21no—
19#1296—21no—
20#1290—22no—
21#1287—22no—
22#1279—22no—
23#1277—22no—
24C22×M16—21no—
25C23×C8—21no—
26C2×#117—21no—
27C22×C42C2311yescharacteristic
28#432—21no—
29#432—21no—
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432C2×C412yes—
38#432C2×C412yes—
39#432D412yes—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#423—21no—
45#423—21no—
46#423—21no—
47#423C2×C412yes—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8Q812yes—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×#97—21no—
67C2×C4×C8D412yes—
68#384D411yescharacteristic
69#384D411yescharacteristic
70#382—21no—
71#380—21no—
72#380—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C23×C4#4111yescharacteristic
76C23×C4#4111yescharacteristic
77C23×C4#4111yescharacteristic
78C2×M16—22no—
79C2×M16—41no—
80C2×M16—81no—
81C2×M16—81no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—81no—
91C22×C8—81no—
92#117—82no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C22⋊C4—41no—
100C2×C42—41no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×Q811yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#97—22no—
138#97—22no—
139#97—22no—
140#97—41no—
141#97—41no—
142#97—41no—
143C4×C8—22no—
144C4×C8—22no—
145#95—41no—
146#95—41no—
147C24#12611yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4#11711yescharacteristic
159C22×C4#11711yescharacteristic
160C22×C4#11711yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4#12311yescharacteristic
164C22×C4#12311yescharacteristic
165C22×C4#12311yescharacteristic
166C22×C4#12311yescharacteristic
167C22×C4#12411yescharacteristic
168C22×C4#12411yescharacteristic
169C22×C4#12511yescharacteristic
170C22×C4#12511yescharacteristic
171C22×C4#12611yescharacteristic
172C22×C4#12611yescharacteristic
173C22×C4#12611yescharacteristic
174C22×C4#12611yescharacteristic
175C22×C4#12611yescharacteristic
176M16—82no—
177M16—82no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8—81no—
192C2×C8—82no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—82no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C22⋊C4—82no—
207C42—82no—
208C42C4×D411yescharacteristic
209C42C4×D411yescharacteristic
210C42C4×D411yescharacteristic
211C42C4×Q811yescharacteristic
212C42#12111yescharacteristic
213C42#12111yescharacteristic
214C42C22⋊Q811yescharacteristic
215C42C22⋊Q811yescharacteristic
216C42#12311yescharacteristic
217C42#12311yescharacteristic
218C42#12411yescharacteristic
219C42#12511yescharacteristic
220C42#12611yescharacteristic
221C42#12611yescharacteristic
222C42#12611yescharacteristic
223C42#12611yescharacteristic
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23#38311yescharacteristic
231C23#39711yescharacteristic
232C23#40011yescharacteristic
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#38711yescharacteristic
266C2×C4#38711yescharacteristic
267C2×C4#38711yescharacteristic
268C2×C4#38811yescharacteristic
269C2×C4#38811yescharacteristic
270C2×C4#38811yescharacteristic
271C2×C4#39611yescharacteristic
272C2×C4#39711yescharacteristic
273C2×C4#39711yescharacteristic
274C2×C4#40011yescharacteristic
275C2×C4#40011yescharacteristic
276C2×C4#40111yescharacteristic
277C2×C4#47111yescharacteristic
278C2×C4#47111yescharacteristic
279C2×C4#47211yescharacteristic
280C2×C4#47311yescharacteristic
281C8—82no—
282C8—82no—
283C8—82no—
284C8—82no—
285C8—82no—
286C8—82no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22#144311yescharacteristic
298C22#207811yescharacteristic
299C22#207911yescharacteristic
300C22#208011yescharacteristic
301C22#208011yescharacteristic
302C22#208111yescharacteristic
303C22#208211yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—161no—
315C4—161no—
316C4#193011yescharacteristic
317C4#193111yescharacteristic
318C4#206311yescharacteristic
319C4#206411yescharacteristic
320C2—41no—
321C2—41no—
322C2#1064511yescharacteristic
323C2#1064611yescharacteristic
324C2#1064711yescharacteristic
325C2#1064811yescharacteristic
326C2#1130111yescharacteristic
327C2#1130311yescharacteristic
328C2#1132011yescharacteristic
subgroupG/H
center, socleC23#397
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1307C4semidirect

Maximal subgroups

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

subgroupindex
C2×#13072
#80442
#80422
C2×#12902
C2×#12872
#79002
#78772