Finite Group Databasebeta

← #10574336

Subgroups — #10574336

Order 512.

6196 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#41C211yesmaximal, characteristic
2#60253C2124yesmaximal, complemented
3#60250C2118yesmaximal, complemented
4#60248C213yesmaximal, complemented
5C2×#2982C213yesmaximal, complemented
6Q8×#131C218yesmaximal, complemented
7#60127C216yesmaximal
8Q8×#41C4112yes—
9Q8×#41C414yes—
10Q8×#41C2214yes—
11Q8×#41C22112yes—
12C2×Q82C411yescharacteristic
13C2×#557C413yes—
14C2×#554C419yes—
15C2×D4×Q8C413yes—
16C2×#533C2219yes—
17C2×C4×#41C2213yes—
18C22×C4×Q8C2213yes—
19#2986C22172yescomplemented
20#2985C22172yescomplemented
21#2984C22124yes—
22#2984C22124yescomplemented
23#2983C22118yes—
24#2982C2211yescharacteristic
25#2982C2213yes—
26#2982C22118yes—
27#2982C22124yescomplemented
28Q8×M16C22124yescomplemented
29#2962C22172yescomplemented
30#2960C22172yescomplemented
31C2×C8×Q8C22124yescomplemented
32#2925C22136yes—
33#2924C22136yes—
34#2922C22136yes—
35#2920C22118yes—
36C2×#431C22118yes—
37#2906C22112yes—
38#2880C2214yes—
39C4×#131C22112yes—
40C22×#41C2311yescharacteristic
41C23×Q8C2×C413yes—
42C2×#131C2318yes—
43Q82C2×C414yes—
44Q82C2×C4112yes—
45#557C2×C4112yes—
46#557C2×C4136yes—
47#554C2×C4136yes—
48#554C2×C4172yes—
49#554C2×C4136yes—
50D4×Q8C2×C4112yes—
51D4×Q8C2×C4112yes—
52D4×Q8C2×C4124yes—
53#533C2×C4136yes—
54#533C2×C4136yes—
55#533C23136yes—
56#533C23136yes—
57C2×C4⋊Q8C2×C419yes—
58C2×C4⋊Q8C2×C419yes—
59C2×#125C2×C4118yes—
60C2×C22⋊Q8C2×C4118yes—
61C2×C22⋊Q8C2×C4118yes—
62C4×#41C2×C416yes—
63C4×#41C2×C4118yes—
64C4×#41C2316yes—
65C4×#41C23118yes—
66C2×C4×Q8C2×C413yes—
67C2×C4×Q8C2×C419yes—
68C2×C4×Q8C2×C4112yes—
69C2×C4×Q8C2×C4112yes—
70C2×C4×Q8C2×C413yes—
71C2×C4×Q8C2×C413yes—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×Q8C2319yes—
74C2×C4×Q8C23112yes—
75C2×C4×Q8C23112yes—
76C2×C4×Q8C2316yes—
77C2×C4×D4C2×C419yes—
78C2×C4×D4C2×C419yes—
79C2×#117C2319yes—
80C22×C4⋊C4C2×C419yes—
81C22×C42C2319yes—
82#446C231144yes—
83#446C231144yes—
84C8×Q8C2318yes—
85C8×Q8C23172yes—
86C8×Q8C23172yes—
87C8×Q8C23124yes—
88C8×Q8C23148yes—
89#433C231144yes—
90#432C23136yes—
91#432C23136yes—
92#431C2316yes—
93#431C23118yes—
94#431C23136yes—
95#431C23136yes—
96#431C23136yes—
97#431C23136yes—
98#424C23172yes—
99#423C23136yes—
100C2×C4⋊C8C23136yes—
101#408C23124yes—
102#407C23112yes—
103C2×C22⋊C8C23112yes—
104#405C23172yes—
105C4×M16C23136yes—
106C2×C4×C8C23136yes—
107C2×#41—28no—
108C2×#41C22×C411yescharacteristic
109C2×#41C22×C413yes—
110C2×#41C2413yes—
111C2×#41C2411yescharacteristic
112C22×Q8—224no—
113C22×Q8C22×C411yescharacteristic
114C22×Q8C22×C419yes—
115C22×Q8C22×C4112yes—
116C22×Q8C22×C4112yes—
117C22×Q8C22×C416yes—
118C22×Q8C22×C411yescharacteristic
119C22×D4C22×C413yes—
120C23×C4C22×C419yes—
121C23×C4C22×C419yes—
122C23×C4C2413yes—
123#131—232no—
124C2×M16C24124yes—
125C22×C8C24124yes—
126C4⋊Q8C22×C4136yes—
127C4⋊Q8C22×C4136yes—
128C4⋊Q8C22×C4136yes—
129C4⋊Q8C22×C4136yes—
130#125C22×C4172yes—
131#125C22×C4172yes—
132C22⋊Q8C22×C4172yes—
133C22⋊Q8C22×C4172yes—
134C22⋊Q8C22×C4172yes—
135C22⋊Q8C22×C4172yes—
136C4×Q8C22×C414yes—
137C4×Q8C22×C4136yes—
138C4×Q8C22×C4112yes—
139C4×Q8C22×C4112yes—
140C4×Q8C22×C4136yes—
141C4×Q8C22×C4136yes—
142C4×Q8C22×C4118yes—
143C4×Q8C22×C416yes—
144C4×Q8C22×C4118yes—
145C4×Q8C22×C416yes—
146C4×Q8C22×C4124yes—
147C4×Q8C2414yes—
148C4×Q8C24136yes—
149C4×Q8C24124yes—
150C4×D4C22×C4136yes—
151C4×D4C22×C4118yes—
152C4×D4C22×C4118yes—
153C4×D4C22×C4136yes—
154C4×D4C22×C4118yes—
155C4×D4C22×C4118yes—
156#117C22×C4118yes—
157#117C22×C4118yes—
158#117C24118yes—
159#117C24118yes—
160C2×C4⋊C4C22×C413yes—
161C2×C4⋊C4C22×C4118yes—
162C2×C4⋊C4C22×C419yes—
163C2×C4⋊C4C22×C4136yes—
164C2×C4⋊C4C22×C4136yes—
165C2×C4⋊C4C22×C419yes—
166C2×C4⋊C4C22×C419yes—
167C2×C4⋊C4C22×C4118yes—
168C2×C22⋊C4C22×C4118yes—
169C2×C42C22×C419yes—
170C2×C42C22×C419yes—
171C2×C42C22×C4118yes—
172C2×C42C22×C4118yes—
173C2×C42C2419yes—
174C2×C42C2413yes—
175C2×C42C24118yes—
176C2×C42C24118yes—
177C4⋊C8C24112yes—
178C4⋊C8C24136yes—
179C4⋊C8C24172yes—
180C4⋊C8C24172yes—
181C22⋊C8C2414yes—
182C22⋊C8C24112yes—
183C22⋊C8C24124yes—
184C22⋊C8C24124yes—
185#97C24172yes—
186C4×C8C24112yes—
187C4×C8C24136yes—
188C4×C8C24172yes—
189C24C23×C413yes—
190#41—24no—
191#41—212no—
192#41—212no—
193#412−1+418yes—
194C2×Q8—24no—
195C2×Q8—248no—
196C2×Q8—248no—
197C2×Q8—236no—
198C2×Q8—86no—
199C2×Q8#13118yescomplemented
200C2×Q8C23×C413yes—
201C2×Q8C23×C4136yes—
202C2×Q8C23×C4124yes—
203C2×Q8C23×C411yescharacteristic
204C2×Q8C23×C414yes—
205C2×D4—212no—
206C2×D4C23×C416yes—
207C2×D4C23×C413yes—
208C2×D4C23×C413yes—
209C22×C4—236no—
210C22×C4—212no—
211C22×C4—89no—
212C22×C4C23×C413yes—
213C22×C4C23×C413yes—
214C22×C4C23×C413yes—
215C22×C4C23×C419yes—
216C22×C4C23×C4118yes—
217C22×C4C23×C4118yes—
218C22×C4C23×C413yes—
219C22×C4C23×C419yes—
220C22×C4C23×C4118yes—
221C22×C4C23×C4118yes—
222C22×C4C23×C413yes—
223C22×C4C2511yescharacteristic
224C22×C4C2513yes—
225C22×C4C2513yes—
226M16—248no—
227C2×C8—248no—
228C2×C8C2518yes—
229C2×C8C25124yes—
230C4⋊C4C23×C4112yes—
231C4⋊C4C23×C4172yes—
232C4⋊C4C23×C4136yes—
233C4⋊C4C23×C4118yes—
234C4⋊C4C23×C4118yes—
235C4⋊C4C23×C4118yes—
236C4⋊C4C23×C4118yes—
237C4⋊C4C23×C4172yes—
238C22⋊C4C23×C4136yes—
239C22⋊C4C23×C4136yes—
240C42C23×C416yes—
241C42C23×C4118yes—
242C42C23×C4118yes—
243C42C23×C4118yes—
244C42C23×C4118yes—
245C42C23×C4118yes—
246C42C2516yes—
247C42C25118yes—
248C23—26no—
249C23—26no—
250C23—43no—
251C23C24×C413yes—
252C23C24×C413yes—
253C23C24×C411yescharacteristic
254Q8—26no—
255Q8—272no—
256Q8—424no—
257Q8#51914yes—
258Q8C2×#131116yes—
259D4—212no—
260D4—26no—
261D4#519112yes—
262C2×C4—26no—
263C2×C4—218no—
264C2×C4—22no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—236no—
269C2×C4—236no—
270C2×C4—43no—
271C2×C4—418no—
272C2×C4—418no—
273C2×C4—49no—
274C2×C4C2×#131112yes—
275C2×C4C24×C411yescharacteristic
276C2×C4C24×C413yes—
277C2×C4C24×C416yes—
278C2×C4C24×C4118yes—
279C2×C4C24×C4118yes—
280C2×C4C24×C413yes—
281C2×C4C24×C416yes—
282C2×C4C2×2−1+4112yes—
283C2×C4C2611yescharacteristic
284C8—28no—
285C8—224no—
286C22—23no—
287C22—21no—
288C22—23no—
289C22—21no—
290C22—23no—
291C22—23no—
292C22—23no—
293C22—412no—
294C22C2×#51916yes—
295C22C22×#13112yes—
296C22C25×C411yescharacteristic
297C4—218no—
298C4—26no—
299C4—218no—
300C4—21no—
301C4—23no—
302C4C2×#51916yes—
303C4C22×#131112yes—
304C4C22×2−1+412yes—
305C2—22no—
306C2—26no—
307C2—43no—
308C2C22×#51911yescharacteristic
309C2C23×#13111yescharacteristic
310C2#6307611yescharacteristic
subgroupG/H
center, Frattini, cosocleC2×C4C26
commutator, socleC22C25×C4
prime radicalC22×#41C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60253C2semidirect
3. #60250C2semidirect
4. #60248C2semidirect
5. C2×#2982C2semidirect
6. Q8×#131C2semidirect
19. #2986C22semidirect
20. #2985C22semidirect
22. #2984C22semidirect
27. #2982C22semidirect
28. Q8×M16C22semidirect
29. #2962C22semidirect
30. #2960C22semidirect
31. C2×C8×Q8C22semidirect
199. C2×Q8#131semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#412
#602532
#602502
#602482
C2×#29822
Q8×#1312
#601272