Finite Group Databasebeta

← #137607

Subgroups — #137607

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C16C211yesmaximal, characteristic, complemented
2C4×#466C211yesmaximal, characteristic
3C2×#1331C212yesmaximal, complemented
4#8596C211yesmaximal, characteristic, complemented
5#7171C212yesmaximal, complemented
6C23×C16—21no—
7C2×#466C411yescharacteristic
8C2×C4×D8C2211yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C4×#121—21no—
11C2×C4×C16—21no—
12C2×C4×C16—24no—
13C2×C4×C16C2212yes—
14#1970C411yescharacteristic
15#1942C411yescharacteristic
16#1934—21no—
17#1893—21no—
18#1808C411yescharacteristic
19#1797—21no—
20C4×#107C2211yescharacteristic
21C4×#102—21no—
22C2×#335C2212yes—
23#1331—24no—
24#1277—21no—
25C22×D8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121—21no—
29C2×C4×D4—21no—
30C2×C4×D4—41no—
31C22×C42D411yescharacteristic
32C22×C16—24no—
33C22×C16—24no—
34C22×C16—24no—
35C22×C16—22no—
36#466—22no—
37#466C814yescomplemented
38C4×D8—22no—
39C4×D8—22no—
40C2×#107C2×C411yescharacteristic
41C2×#107C2×C411yescharacteristic
42C2×#107C2×C411yescharacteristic
43C2×C4⋊C8—22no—
44C2×#102—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×C22⋊C8—41no—
48C2×C4×C8—22no—
49C2×C4×C8—22no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#390—21no—
55#389—21no—
56#388—41no—
57#386—41no—
58#380—21no—
59C4×C4⋊C4—21no—
60C4×C22⋊C4—41no—
61C4×C16—24no—
62C4×C16—22no—
63C4×C16D414yescomplemented
64#340—21no—
65#340—21no—
66#340—21no—
67#336—41no—
68#335—22no—
69#335—22no—
70C22×D4—21no—
71C22×D4—41no—
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4SD1611yescharacteristic
76C2×D8—22no—
77C2×D8—21no—
78C2×D8—22no—
79C2×D8C2×C812yes—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—42no—
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C2×C812yes—
90C22×C8M1612yes—
91C22×C8C22×C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93#121—24no—
94#121—42no—
95C4×D4—42no—
96C4×D4—42no—
97C4×D4—82no—
98C4×D4—82no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—41no—
111C2×C42C22⋊C411yescharacteristic
112C2×C42D811yescharacteristic
113C2×C42SD1611yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C16—216no—
116C2×C16—24no—
117C2×C16—24no—
118C2×C16—24no—
119#107—21no—
120#107—22no—
121#107—22no—
122#107C2×C812yes—
123C4⋊C8—42no—
124C4⋊C8—42no—
125#102—24no—
126#102—42no—
127#102—41no—
128#102—41no—
129C22⋊C8—82no—
130C4×C8—22no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—22no—
134C4×C8C2×D412yes—
135C4×C8#4112yes—
136#95—41no—
137#95—41no—
138#95—41no—
139C24—41no—
140C24#10211yescharacteristic
141C2×D4—22no—
142C2×D4—44no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—82no—
149C2×D4—82no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—82no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—82no—
166C22×C4C22⋊C812yes—
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4#10211yescharacteristic
170C22×C4#10412yes—
171C22×C4D1612yes—
172C22×C4SD3212yes—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C2×SD1611yescharacteristic
176D8—24no—
177D8—42no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8C22⋊C812yes—
192C2×C8#10412yes—
193C2×C8C4×D412yes—
194C2×C8#12011yescharacteristic
195C2×C8#12311yescharacteristic
196C2×C8C22×C811yescharacteristic
197C2×C8C2×M1611yescharacteristic
198C2×C8#13112yes—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C22⋊C4—44no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—81no—
216C42—81no—
217C42#12112yes—
218C42#13512yes—
219C16—28no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—42no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23D4⋊C812yes—
232C23#35712yes—
233C23#35912yes—
234C23C2×#10211yescharacteristic
235D4—42no—
236D4—42no—
237D4—88no—
238D4—81no—
239D4—81no—
240C2×C4—28no—
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—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—162no—
265C2×C4—162no—
266C2×C4D4⋊C812yes—
267C2×C4#35712yes—
268C2×C4#35912yes—
269C2×C4#38611yescharacteristic
270C2×C4C2×C22⋊C811yescharacteristic
271C2×C4#40811yescharacteristic
272C2×C4#41611yescharacteristic
273C2×C4C2×#10411yescharacteristic
274C2×C4C2×#10411yescharacteristic
275C2×C4#46512yes—
276C2×C4#46612yes—
277C2×C4C2×D1611yescharacteristic
278C2×C4C2×SD3211yescharacteristic
279C2×C4#50812yes—
280C8—22no—
281C8—21no—
282C8—22no—
283C8—21no—
284C8—162no—
285C8C8×D412yes—
286C8#43512yes—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—161no—
297C22—161no—
298C22—161no—
299C22—161no—
300C22#133114yescomplemented
301C22C2×D4⋊C811yescharacteristic
302C22#149211yescharacteristic
303C22#189311yescharacteristic
304C22C2×#35711yescharacteristic
305C22#213811yescharacteristic
306C22C2×#35911yescharacteristic
307C22C2×#35911yescharacteristic
308C4—22no—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—82no—
313C4—161no—
314C4—161no—
315C4#147611yescharacteristic
316C4#149411yescharacteristic
317C4#186911yescharacteristic
318C4#189611yescharacteristic
319C4#221512yes—
320C4#221712yes—
321C2—24no—
322C2—162no—
323C2#907211yescharacteristic
324C2C2×#133111yescharacteristic
325C2#1159111yescharacteristic
326C2#1167111yescharacteristic
327C2#1167311yescharacteristic
328C2#1305911yescharacteristic
329C2#1306311yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×#102
prime radicalC2×#466C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×C16C2semidirect
3. C2×#1331C2semidirect
4. #8596C2semidirect
5. #7171C2semidirect
37. #466C8semidirect
63. C4×C16D4semidirect
300. C22#1331semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C162
C4×#4662
C2×#13312
#85962
#71712