Finite Group Databasebeta

← #7622564

Subgroups — #7622564

Order 512.

1481 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#568C211yesmaximal, characteristic
2#33428C216yesmaximal, complemented
3#33427C216yesmaximal, complemented
4#33426C216yesmaximal, complemented
5C2×#2165C216yesmaximal, complemented
6#33401C212yesmaximal, complemented
7#33337C212yesmaximal
8#33329C212yesmaximal, complemented
9C2×#568C411yescharacteristic
10C2×#568C2211yescharacteristic
11C2×C4×#41C411yescharacteristic
12#3407C2214yes—
13#2875C413yes—
14#2875C411yescharacteristic
15#2875C2213yes—
16#2875C2211yescharacteristic
17C4×#131C413yes—
18C4×#131C411yescharacteristic
19C4×#131C2213yes—
20C4×#131C2211yescharacteristic
21#2873C413yes—
22#2873C2213yes—
23C2×C4×M16C413yes—
24C2×C4×M16C2213yes—
25#2166C22124yescomplemented
26#2165C2214yes—
27#2165C22112yes—
28#2165C22112yescomplemented
29#2165C22112yescomplemented
30#2153C22112yescomplemented
31C2×#364C22112yescomplemented
32#2115C22112yes—
33C2×#349C22112yes—
34#2110C22112yescomplemented
35C2×#347C22112yescomplemented
36C22×#41C2×C411yescharacteristic
37#568—28no—
38#568—28no—
39C2×#131C2×C413yes—
40C2×#131C2×C411yescharacteristic
41C2×#131C2×C413yes—
42C2×#131C2×C411yescharacteristic
43C2×#131C2×C413yes—
44C2×#131C2×C411yescharacteristic
45C2×#131C2313yes—
46C2×#131C2311yescharacteristic
47C22×M16C2×C413yes—
48C22×M16C2×C413yes—
49C22×M16C2×C413yes—
50C22×M16C2313yes—
51C4×#41—21no—
52C4×#41—23no—
53C4×#41C2×C413yes—
54C4×#41C2×C411yescharacteristic
55C4×#41C2×C413yes—
56C4×#41C2×C411yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×D4—23no—
59C2×#117C2×C413yes—
60C22×C42C2×C413yes—
61#504—216no—
62C2×M32C23124yes—
63#405C2×C416yes—
64#405C2×C413yes—
65#405C2×C413yes—
66#405C2×C413yes—
67#405C2×C416yes—
68#405C2×C413yes—
69#405C2×C413yes—
70#405C2×C416yes—
71#405C2×C413yes—
72#405C2313yes—
73#405C2316yes—
74#405C2313yes—
75C4×M16C2×C413yes—
76C4×M16C2×C416yes—
77C4×M16C2×C413yes—
78C4×M16C2×C413yes—
79C4×M16C2×C413yes—
80C4×M16C2×C413yes—
81C4×M16C2×C413yes—
82C4×M16C2×C413yes—
83C4×M16C2×C413yes—
84C4×M16C2311yescharacteristic
85C4×M16C2313yes—
86C4×M16C2313yes—
87C4×M16C2313yes—
88C2×#97C2×C416yes—
89C2×#97C2×C413yes—
90C2×#97C2313yes—
91C2×C4×C8C2×C416yes—
92C2×C4×C8C2×C413yes—
93C2×C4×C8C2313yes—
94#364C2318yes—
95#364C23124yes—
96#349C2318yes—
97#349C23124yes—
98#347C2318yes—
99#347C23124yes—
100C2×#41—24no—
101C2×#41—24no—
102C2×#41M1611yescharacteristic
103C2×#41M1613yes—
104C2×#41C22×C413yes—
105C2×#41C22×C411yescharacteristic
106C22×Q8M1611yescharacteristic
107C22×D4M1613yes—
108C23×C4—23no—
109C23×C4C22×C413yes—
110#131—24no—
111#131—24no—
112#131—212no—
113#131—212no—
114#131—212no—
115#131—212no—
116#131M1618yes—
117#131#4118yes—
118C2×M16—212no—
119C2×M16—212no—
120C2×M16—212no—
121C2×M16—212no—
122C2×M16C22×C413yes—
123C2×M16C22×C411yescharacteristic
124C2×M16C22×C413yes—
125C2×M16C22×C413yes—
126C2×M16C22×C413yes—
127C2×M16C22×C413yes—
128C2×M16C22×C413yes—
129C2×M16C22×C413yes—
130C2×M16C22×C413yes—
131C2×M16C22×C413yes—
132C2×M16C2411yescharacteristic
133C2×M16C2413yes—
134C22×C8C22×C416yes—
135C22×C8C22×C413yes—
136C22×C8C22×C416yes—
137C22×C8C22×C413yes—
138C22×C8C22×C413yes—
139C22×C8C2413yes—
140C4×Q8—23no—
141C4×Q8—23no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×D4—26no—
145C4×D4—23no—
146C4×D4—23no—
147C4×D4—26no—
148C4×D4—23no—
149C4×D4—23no—
150#117—23no—
151#117—23no—
152#117C22×C413yes—
153#117C22×C416yes—
154#117C22×C413yes—
155C2×C4⋊C4—23no—
156C2×C22⋊C4—23no—
157C2×C42—23no—
158C2×C42—23no—
159C2×C42—23no—
160C2×C42C22×C413yes—
161C2×C42C22×C413yes—
162C2×C42C22×C413yes—
163C2×C42C22×C413yes—
164C2×C42C22×C413yes—
165C2×C42C22×C411yescharacteristic
166M32—224no—
167M32C24116yes—
168C2×C16—224no—
169#97C22×C413yes—
170#97C22×C411yescharacteristic
171#97C22×C413yes—
172#97C22×C416yes—
173#97C22×C416yes—
174#97C22×C413yes—
175#97C22×C413yes—
176#97C22×C413yes—
177#97C2411yescharacteristic
178#97C2413yes—
179C4×C8C22×C413yes—
180C4×C8C22×C411yescharacteristic
181C4×C8C22×C413yes—
182C4×C8C22×C416yes—
183C4×C8C22×C416yes—
184C4×C8C22×C413yes—
185C4×C8C22×C413yes—
186C4×C8C22×C413yes—
187C4×C8C2411yescharacteristic
188C4×C8C2413yes—
189C24C2×M1613yes—
190#41—24no—
191#41—212no—
192#41—26no—
193#41—26no—
194#41#11714yes—
195#41C2×M1614yes—
196C2×Q8—24no—
197C2×Q8C2×M1613yes—
198C2×Q8C2×M1611yescharacteristic
199C2×D4—212no—
200C2×D4C2×M1616yes—
201C2×D4C2×M1613yes—
202C2×D4C2×M1613yes—
203C22×C4—26no—
204C22×C4—26no—
205C22×C4—21no—
206C22×C4—23no—
207C22×C4—23no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—86no—
212C22×C4C2×M1613yes—
213C22×C4C2×M1613yes—
214C22×C4C2×M1613yes—
215C22×C4C23×C411yescharacteristic
216C22×C4C23×C413yes—
217C22×C4C23×C413yes—
218M16—26no—
219M16—26no—
220M16—26no—
221M16—26no—
222M16—26no—
223M16—22no—
224M16—26no—
225M16—26no—
226M16—22no—
227M16—26no—
228M16#117112yes—
229M16C2×M16112yes—
230C2×C8—26no—
231C2×C8—26no—
232C2×C8—212no—
233C2×C8—212no—
234C2×C8C2×M16112yes—
235C2×C8C23×C413yes—
236C2×C8C23×C411yescharacteristic
237C2×C8C23×C413yes—
238C2×C8C23×C411yescharacteristic
239C2×C8C23×C413yes—
240C2×C8C23×C413yes—
241C2×C8C23×C411yescharacteristic
242C2×C8C2×#41112yes—
243C2×C8C2511yescharacteristic
244C4⋊C4—26no—
245C4⋊C4—23no—
246C4⋊C4—23no—
247C22⋊C4—26no—
248C22⋊C4—23no—
249C22⋊C4—23no—
250C42—21no—
251C42—23no—
252C42—23no—
253C42—23no—
254C42—23no—
255C42—23no—
256C42C23×C411yescharacteristic
257C42C23×C413yes—
258C42C23×C413yes—
259C42C23×C411yescharacteristic
260C16—216no—
261C23—26no—
262C23—83no—
263C23C22×M1613yes—
264C23C22×M1611yescharacteristic
265C23C22×M1613yes—
266Q8—26no—
267Q8#43214yes—
268D4—212no—
269D4—26no—
270D4#432112yes—
271C2×C4—26no—
272C2×C4—21no—
273C2×C4—23no—
274C2×C4—21no—
275C2×C4—26no—
276C2×C4—26no—
277C2×C4—23no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—23no—
281C2×C4—23no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—42no—
285C2×C4—83no—
286C2×C4—83no—
287C2×C4—83no—
288C2×C4C2×#11716yes—
289C2×C4C22×M1611yescharacteristic
290C2×C4C22×M1613yes—
291C2×C4C22×M1613yes—
292C2×C4C22×M1616yes—
293C2×C4C24×C411yescharacteristic
294C8—21no—
295C8—23no—
296C8—21no—
297C8—23no—
298C8C2×#11716yes—
299C8C22×M1616yes—
300C8C22×M1612yes—
301C8C22×#4112yes—
302C22—21no—
303C22—23no—
304C22—43no—
305C22—41no—
306C22—43no—
307C22—86no—
308C22C2×#43216yes—
309C22C23×M1611yescharacteristic
310C4—22no—
311C4—21no—
312C4—23no—
313C4—43no—
314C4—43no—
315C4—41no—
316C4C2×#43216yes—
317C4C22×#11711yescharacteristic
318C4C23×M1611yescharacteristic
319C2—21no—
320C2—26no—
321C2—43no—
322C2—41no—
323C2C22×#43211yescharacteristic
subgroupG/H
centerC4C22×#117
commutatorC2×C4C24×C4
Frattini, cosocleC2×C8C25
socleC2C22×#432
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ #2875 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33428C2semidirect
3. #33427C2semidirect
4. #33426C2semidirect
5. C2×#2165C2semidirect
6. #33401C2semidirect
8. #33329C2semidirect
25. #2166C22semidirect
28. #2165C22semidirect
29. #2165C22semidirect
30. #2153C22semidirect
31. C2×#364C22semidirect
34. #2110C22semidirect
35. C2×#347C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5682
#334282
#334272
#334262
C2×#21652
#334012
#333372
#333292