Finite Group Databasebeta

← #136605

Subgroups — #136605

Order 512.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#467C211yesmaximal, characteristic
2C22×#335C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4C2×#1332C212yesmaximal, complemented
5#7171C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×C4×Q16C2211yescharacteristic
8C22×C4⋊C8—21no—
9C22×C4×C8C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×C22⋊C16—21no—
12C2×C4×C16C2212yes—
13#1969C411yescharacteristic
14#1942C411yescharacteristic
15#1933—21no—
16#1892—21no—
17#1808C411yescharacteristic
18C4×#107C2211yescharacteristic
19C4×#103—21no—
20C2×#335—21no—
21C2×#335—24no—
22C2×#335C2212yes—
23#1374—21no—
24#1332—24no—
25C22×Q16C2×C411yescharacteristic
26C23×C8—41no—
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C22⋊Q8—21no—
30C2×C4×Q8—21no—
31C22×C42D411yescharacteristic
32C22×C16—22no—
33#467—22no—
34#467C814yescomplemented
35C4×Q16—22no—
36C4×Q16—22no—
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×C4⋊C8—24no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—42no—
43C2×C4⋊C8—41no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C4×C8—22no—
48C2×C4×C8—22no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—21no—
54#389—21no—
55#387—41no—
56#385—41no—
57#380—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4—41no—
60C4×C22⋊C4—41no—
61C22⋊C16—24no—
62C4×C16—22no—
63C4×C16—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#335—24no—
68#335—22no—
69#335D414yes—
70C22×Q8—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4D811yescharacteristic
73C23×C4SD1611yescharacteristic
74C2×Q16—22no—
75C2×Q16—21no—
76C2×Q16—22no—
77C2×Q16C2×C812yescomplemented
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—44no—
84C22×C8—42no—
85C22×C8—82no—
86C22×C8—82no—
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C2×C812yes—
90C22×C8M1612yes—
91C22×C8C22×C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22⋊Q8—24no—
94C22⋊Q8—42no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C22⋊C411yescharacteristic
114C2×C42D811yescharacteristic
115C2×C42SD1611yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C16—24no—
118C2×C16—42no—
119C2×C16—42no—
120#107—21no—
121#107—22no—
122#107—22no—
123#107C2×C812yescomplemented
124C4⋊C8—24no—
125C4⋊C8—44no—
126C4⋊C8—42no—
127#103—24no—
128#103—42no—
129#103—42no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8C2×D412yes—
135C4×C8#4112yes—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141C24#10211yescharacteristic
142C2×Q8—22no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C22⋊C812yes—
165C22×C4#10211yescharacteristic
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10412yes—
169C22×C4SD3212yes—
170C22×C4Q3212yes—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×D811yescharacteristic
173C22×C4C2×SD1611yescharacteristic
174Q16—24no—
175Q16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—24no—
181C2×C8—24no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—44no—
186C2×C8—44no—
187C2×C8—88no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8C22⋊C812yes—
191C2×C8C4×D412yes—
192C2×C8#12112yes—
193C2×C8C22×C811yescharacteristic
194C2×C8C2×M1611yescharacteristic
195C2×C8#13112yes—
196C4⋊C4—22no—
197C4⋊C4—44no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—82no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C22⋊C4—44no—
207C22⋊C4—82no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—81no—
215C42—81no—
216C42—81no—
217C42—81no—
218C42#12011yescharacteristic
219C42#12311yescharacteristic
220C42AGL(1,ℤ/8)11yescharacteristic
221C42#13711yescharacteristic
222C16—44no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23D4⋊C812yes—
228C23#35812yes—
229C23#35912yes—
230C23C2×#10211yescharacteristic
231Q8—42no—
232Q8—42no—
233Q8—81no—
234Q8—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
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—81no—
265C2×C4—81no—
266C2×C4D4⋊C812yes—
267C2×C4#35812yes—
268C2×C4#38611yescharacteristic
269C2×C4C2×C22⋊C811yescharacteristic
270C2×C4#40811yescharacteristic
271C2×C4#41711yescharacteristic
272C2×C4C2×#10411yescharacteristic
273C2×C4#42111yescharacteristic
274C2×C4#44711yescharacteristic
275C2×C4#44811yescharacteristic
276C2×C4#48011yescharacteristic
277C2×C4#48111yescharacteristic
278C2×C4C2×SD3211yescharacteristic
279C2×C4C2×Q3211yescharacteristic
280C2×C4#50911yescharacteristic
281C2×C4#51011yescharacteristic
282C8—22no—
283C8—22no—
284C8—41no—
285C8—84no—
286C8C8×D412yes—
287C8#43512yes—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22#133214yescomplemented
295C22C2×D4⋊C811yescharacteristic
296C22#148011yescharacteristic
297C22#187411yescharacteristic
298C22C2×#35811yescharacteristic
299C22#213911yescharacteristic
300C22C2×#35911yescharacteristic
301C22#214311yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4—161no—
310C4—161no—
311C4#147711yescharacteristic
312C4#148311yescharacteristic
313C4#186911yescharacteristic
314C4#187111yescharacteristic
315C4#218711yescharacteristic
316C4#218811yescharacteristic
317C4#223511yescharacteristic
318C4#223611yescharacteristic
319C2—24no—
320C2#893911yescharacteristic
321C2C2×#133211yescharacteristic
322C2#1159211yescharacteristic
323C2#1164611yescharacteristic
324C2#1164811yescharacteristic
325C2#1299011yescharacteristic
326C2#1299311yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×#102
prime radicalC24#102
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
4. C2×#1332C2semidirect
5. #7171C2semidirect
34. #467C8semidirect
77. C2×Q16C2×C8semidirect
123. #107C2×C8semidirect
294. C22#1332semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
C22×#3352
C4×C22⋊C162
C2×#13322
#71712