Finite Group Databasebeta

← #6376675

Subgroups — #6376675

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2402C211yesmaximal, characteristic
2#22306C218yesmaximal, complemented
3#22304C212yesmaximal, complemented
4#22302C214yesmaximal, complemented
5#22300C212yesmaximal, complemented
6#21146C212yesmaximal
7#21146C214yesmaximal
8#21023C214yesmaximal
9#20792C211yesmaximal, characteristic
10C4⋊C4×M16C212yesmaximal
11#20365C211yesmaximal, characteristic
12C2×#423C2212yes—
13C2×#423C2214yes—
14C2×#423C2214yes—
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2214yes—
17C2×C4×M16C2211yescharacteristic
18#2402C414yes—
19#2402C414yes—
20#2402C2211yescharacteristic
21#2402C2212yes—
22#2402C2211yescharacteristic
23#2402C2214yes—
24C2×#400C414yes—
25C2×#398C414yes—
26C2×C4×C4⋊C4C2214yes—
27C2×C4×C4⋊C4C2212yes—
28C2×#376C2211yescharacteristic
29#1940C2214yes—
30#1940C2214yes—
31#1940C2218yes—
32#1940C2218yes—
33#1939C2218yes—
34#1939C2218yes—
35#1939C2218yes—
36#1917C2218yes—
37#1917C2218yes—
38#1916C2218yes—
39#1916C2218yes—
40#1843C2212yes—
41#1843C2212yes—
42#1843C2214yes—
43#1771C2218yes—
44#1770C2214yes—
45C8×C4⋊C4C2214yes—
46#1746C2212yes—
47#1746C2212yes—
48#1746C2214yes—
49#1728C2212yes—
50#1728C2214yes—
51#1728C2214yes—
52#1728C2212yes—
53C22×M16C2311yescharacteristic
54C22×M16C2314yes—
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C22×C4⋊C4C2×C414yes—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4C2×C414yes—
60C22×C42C2314yes—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63#423—28no—
64#423—48no—
65#423—48no—
66C2×C4⋊C8C2314yes—
67C2×C4⋊C8C2314yes—
68C2×C4⋊C8C2318yes—
69C2×C4⋊C8C2318yes—
70C2×C4⋊C8C2318yes—
71C2×C4⋊C8C2318yes—
72C4×M16—24no—
73C4×M16—48no—
74C4×M16Q818yes—
75C2×#97C2312yes—
76C2×#97C2318yes—
77C2×#97C2312yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2318yes—
80C2×C4×C8C2312yes—
81#400C2×C414yes—
82#400C2×C414yes—
83#400C2×C414yes—
84#400C2×C414yes—
85#400C2×C414yes—
86#400C2×C414yes—
87#400C2×C414yes—
88#400C2×C414yes—
89#398C2×C414yes—
90#398C2×C414yes—
91#398C2×C414yes—
92#398C2×C414yes—
93#398C2×C414yes—
94#398C2×C414yes—
95#398C2×C414yes—
96#398C2×C414yes—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C414yes—
101C4×C4⋊C4C2×C414yes—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2314yes—
105C4×C4⋊C4C2314yes—
106C4×C4⋊C4C2314yes—
107C4×C4⋊C4C2314yes—
108C4×C4⋊C4C2314yes—
109C4×C4⋊C4C2312yes—
110C4×C4⋊C4C2312yes—
111#376C2×C414yes—
112#376C2311yescharacteristic
113#376C2311yescharacteristic
114#376C2312yes—
115C2×#95C2×C412yes—
116C2×#95C2×C414yes—
117C2×#95C2×C414yes—
118C2×#95C2×C414yes—
119C2×#95C2×C412yes—
120#336C2318yes—
121#336C2314yes—
122#336C2314yes—
123#336C2318yes—
124#336C2318yes—
125#336C2318yes—
126#336C2314yes—
127#336C2314yes—
128C23×C4C22×C414yes—
129C23×C4C22×C414yes—
130C23×C4C22×C414yes—
131C23×C4C22×C412yes—
132C23×C4C2411yescharacteristic
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—28no—
138C2×M16—28no—
139C2×M16—24no—
140C2×M16—28no—
141C2×M16—24no—
142C2×M16C2×Q818yes—
143C22×C8C2412yes—
144C22×C8C2412yes—
145C22×C8C2418yes—
146C22×C8C2414yes—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C414yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C22×C414yes—
162C2×C42—28no—
163C2×C42—24no—
164C2×C42—28no—
165C2×C42—24no—
166C2×C42C22×C414yes—
167C2×C42C22×C414yes—
168C2×C42C22×C414yes—
169C2×C42C22×C412yes—
170C2×C42C2×Q814yes—
171C2×C42#4114yes—
172C2×C42C2412yes—
173C2×C42C2412yes—
174C2×C42C2414yes—
175C2×C42C2414yes—
176C2×C42C2411yescharacteristic
177C2×C42C2411yescharacteristic
178C4⋊C8—28no—
179C4⋊C8—28no—
180C4⋊C8—48no—
181C4⋊C8—48no—
182C4⋊C8—48no—
183C4⋊C8—48no—
184#97—24no—
185#97—48no—
186#97C2×Q818yes—
187C4×C8—24no—
188C4×C8—48no—
189C4×C8C2×Q818yes—
190#95C22×C414yes—
191#95C22×C414yes—
192#95C22×C414yes—
193#95C22×C414yes—
194#95C22×C412yes—
195#95C22×C414yes—
196#95C22×C412yes—
197#95C22×C414yes—
198#95C22×C414yes—
199#95C22×C414yes—
200#95C22×C414yes—
201#95C22×C414yes—
202#95C22×C414yes—
203#95C22×C414yes—
204#95C22×C414yes—
205#95C22×C414yes—
206#95C22×C412yes—
207#95C22×C412yes—
208C24C23×C411yescharacteristic
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4C4×Q818yes—
220C22×C4C23×C414yes—
221C22×C4C23×C414yes—
222C22×C4C23×C414yes—
223C22×C4C23×C414yes—
224C22×C4C23×C414yes—
225C22×C4C23×C412yes—
226C22×C4C23×C412yes—
227C22×C4C23×C414yes—
228C22×C4C23×C411yescharacteristic
229C22×C4C22×Q812yes—
230C22×C4C2×#4112yes—
231C22×C4H(5,𝔽2)14yes—
232C22×C42−1+414yes—
233C22×C4C2511yescharacteristic
234M16—216no—
235M16—44no—
236M16—416no—
237M16—44no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—28no—
243C2×C8—28no—
244C2×C8—24no—
245C2×C8—28no—
246C2×C8—24no—
247C2×C8C22×Q818yes—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C42—28no—
261C42—28no—
262C42—28no—
263C42—24no—
264C42—24no—
265C42—22no—
266C42—22no—
267C42—24no—
268C42—22no—
269C42—22no—
270C42—24no—
271C42C4×Q818yes—
272C42C22×Q812yes—
273C42C22×Q812yes—
274C42C2×#4112yes—
275C42C2×#4112yes—
276C23—24no—
277C23C2×C4×Q812yes—
278C23#52014yes—
279C23C24×C411yescharacteristic
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—48no—
300C2×C4—416no—
301C2×C4—88no—
302C2×C4—88no—
303C2×C4C2×C4×Q814yes—
304C2×C4C2×C4×Q814yes—
305C2×C4C2×C4×Q812yes—
306C2×C4#52014yes—
307C2×C4#54014yes—
308C2×C4#54114yes—
309C2×C4#54418yes—
310C2×C4#56818yes—
311C2×C4C23×Q811yescharacteristic
312C2×C4C22×#4111yescharacteristic
313C2×C4C2×H(5,𝔽2)12yes—
314C2×C4C2×2−1+412yes—
315C8—28no—
316C8—42no—
317C8—48no—
318C8—42no—
319C22—24no—
320C22—28no—
321C22#233014yes—
322C22C22×C4×Q811yescharacteristic
323C22C2×#52012yes—
324C22C2×#56814yes—
325C4—24no—
326C4—24no—
327C4—28no—
328C4—44no—
329C4—44no—
330C4—44no—
331C4—44no—
332C4—44no—
333C4#233014yes—
334C4C2×#54011yescharacteristic
335C4C2×#54111yescharacteristic
336C4C2×#54412yes—
337C2—24no—
338C2C2×#233011yescharacteristic
339C2#6019212yes—
340C2#6026414yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22306C2semidirect
3. #22304C2semidirect
4. #22302C2semidirect
5. #22300C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24022
#223062
#223042
#223022
#223002
#211462
#210232
#207922
C4⋊C4×M162
#203652