Finite Group Databasebeta

← #6594340

Subgroups — #6594340

Order 512.

1842 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
1#36180C212yesmaximal
2#33310C211yesmaximal, characteristic, complemented
3#33207C212yesmaximal, complemented
4#29206C212yesmaximal, complemented
5#28829C214yesmaximal, complemented
6#27948C214yesmaximal, complemented
7#26060C218yesmaximal, complemented
8D4×#106C214yesmaximal
9C4×#498C211yesmaximal, characteristic, complemented
10C2×#1844C212yesmaximal, complemented
11C4×C8×D4C211yesmaximal, characteristic
12C2×#554—22no—
13C2×#498C2211yescharacteristic
14C2×#498C2214yes—
15C2×C8×D4C2212yes—
16C22×#106C2218yes—
17C22×C4×C8C2212yes—
18#2868C2212yes—
19#2857C2214yes—
20#2567—22no—
21#2552—22no—
22#2552C2214yes—
23#2502—24no—
24#2440—24no—
25#2440C2218yes—
26D4×C4⋊C4—22no—
27D4×C4⋊C4C2214yes—
28C4×C4⋊Q8C2212yes—
29C2×#384C2214yes—
30C42×D4C2211yescharacteristic
31#2084C22116yes—
32#2058C2218yes—
33#1990C2214yes—
34#1990C2212yes—
35#1948C2218yes—
36#1941C2218yes—
37#1941C2214yes—
38#1853C22116yes—
39#1844C2214yes—
40#1844C2212yes—
41#1844C2212yes—
42#1844C2214yes—
43#1844C2211yescharacteristic
44#1844C2218yes—
45#1807C2218yes—
46C4×#106C2214yes—
47C8×C4⋊C4C2212yes—
48C4×C4⋊C8C2211yescharacteristic
49C8×C22⋊C4C2214yes—
50C4×C22⋊C8C2212yes—
51C42×C8C2211yescharacteristic
52C23×C8C2314yes—
53#554—416no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2312yes—
57C2×C4⋊Q8C2314yes—
58C2×#125—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—28no—
61C2×C4×Q8—22no—
62C2×C4×D4—24no—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4C2318yes—
68C22×C42C2312yes—
69#498—24no—
70#498—48no—
71C8×D4Q8116yescomplemented
72C2×#106—416no—
73C2×#106C2314yes—
74C2×#106C2318yes—
75C2×#106C2314yes—
76C2×#106C23116yes—
77C2×#106C2318yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8C2314yes—
80C2×C4×C8—28no—
81C2×C4×C8C2314yes—
82C2×C4×C8C2312yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2312yes—
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87#398—28no—
88#398—28no—
89#395—28no—
90#391—22no—
91#391—24no—
92#389—24no—
93#389—24no—
94#389—22no—
95#389—24no—
96#389C2314yes—
97#389C2318yes—
98#385—28no—
99#385—28no—
100#384—24no—
101#384—22no—
102#384—22no—
103#384—24no—
104#384C2312yes—
105#384C2318yes—
106#384C2314yes—
107#384C2314yes—
108#380—24no—
109#380C2318yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2314yes—
113C4×C4⋊C4C2311yescharacteristic
114C4×C22⋊C4D414yes—
115C4×C22⋊C4C2312yes—
116C43C2311yescharacteristic
117#340C23116yes—
118#340C2318yes—
119#336C2312yes—
120#336C2314yes—
121C22×Q8—22no—
122C22×Q8—24no—
123C22×D4C2×D411yescharacteristic
124C23×C4—28no—
125C23×C4C2×D414yes—
126C23×C4C2412yes—
127C22×C8—28no—
128C22×C8—28no—
129C22×C8C2×Q8116yes—
130C22×C8C2414yes—
131C22×C8C2412yes—
132C22×C8C2412yes—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—48no—
136C4⋊Q8—48no—
137C4⋊Q8—84no—
138#125—416no—
139C22⋊Q8—416no—
140C22⋊Q8—432no—
141C4×Q8—48no—
142C4×D4—416no—
143C4×D4SD16116yes—
144C4×D4C2×Q818yes—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—416no—
159C2×C4⋊C4—416no—
160C2×C4⋊C4—88no—
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2418yes—
165C2×C4⋊C4C2414yes—
166C2×C22⋊C4—28no—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D414yes—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42C2×D412yes—
173C2×C42C2×D414yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C2×C42C2412yes—
177C2×C42C2411yescharacteristic
178C2×C42C2411yescharacteristic
179#106—28no—
180#106—28no—
181#106—216no—
182#106—416no—
183#106—48no—
184C4⋊C8C2×Q818yes—
185C22⋊C8C2×Q8116yes—
186C4×C8—28no—
187C4×C8—24no—
188C4×C8—24no—
189C4×C8—22no—
190C4×C8C2×Q818yes—
191C4×C8#4114yes—
192#95—24no—
193#95—24no—
194#95—28no—
195#95—28no—
196#95C2×D412yes—
197#95C2×D414yes—
198C24C22×D412yes—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—48no—
202C2×Q8—44no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×D4C4⋊Q814yes—
206C2×D4C2×SD1618yes—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—416no—
216C22×C4—88no—
217C22×C4—88no—
218C22×C4C2×SD16116yes—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D412yes—
221C22×C4C22×D412yes—
222C22×C4C22×D414yes—
223C22×C4C22×D411yescharacteristic
224C22×C4C22×D412yes—
225C22×C4C22×Q818yes—
226C22×C4C2511yescharacteristic
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—232no—
232C2×C8C22×Q814yes—
233C2×C8C22×Q818yes—
234C2×C8C22×Q814yes—
235C2×C8C2×#4114yes—
236C2×C8H(5,𝔽2)14yes—
237C4⋊C4—24no—
238C4⋊C4—28no—
239C4⋊C4—216no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—416no—
244C4⋊C4—48no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—416no—
248C4⋊C4—44no—
249C4⋊C4—48no—
250C4⋊C4—88no—
251C4⋊C4—88no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C4⋊C4C4⋊Q814yes—
255C4⋊C4C2×SD1618yes—
256C22⋊C4—416no—
257C22⋊C4C4⋊Q818yes—
258C22⋊C4C2×SD16116yes—
259C42—28no—
260C42—24no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—48no—
265C42C2×SD1618yes—
266C42C22×Q814yes—
267C42C2×#4112yes—
268C422−1+412yes—
269C23—22no—
270C23—22no—
271C23—24no—
272C23C2×C4⋊Q814yes—
273C23C22×SD1618yes—
274C23C23×D411yescharacteristic
275Q8—88no—
276Q8—168no—
277D4#498116yescomplemented
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—232no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—44no—
297C2×C4—1616no—
298C2×C4C2×C4⋊Q812yes—
299C2×C4C2×C4⋊Q814yes—
300C2×C4C2×C4⋊Q812yes—
301C2×C4#54714yes—
302C2×C4C22×SD1614yes—
303C2×C4C22×SD1618yes—
304C2×C4C22×SD1614yes—
305C2×C4C22×SD1614yes—
306C2×C4#57714yes—
307C2×C4C23×Q812yes—
308C2×C4C22×#4111yescharacteristic
309C2×C4C2×H(5,𝔽2)12yes—
310C2×C4C2×2−1+411yescharacteristic
311C8—28no—
312C8—24no—
313C8#55418yescomplemented
314C22—28no—
315C22—24no—
316C22—28no—
317C22—24no—
318C22C2×#49818yes—
319C22C22×C4⋊Q811yescharacteristic
320C22C2×#54711yescharacteristic
321C22#349811yescharacteristic
322C22C23×SD1612yes—
323C22C2×#57712yes—
324C4—28no—
325C4—24no—
326C4—24no—
327C4—22no—
328C4—84no—
329C4—84no—
330C4—88no—
331C4C2×#49814yes—
332C4#333518yes—
333C4C2×#55412yes—
334C4#350312yes—
335C2—28no—
336C2#3618011yescharacteristic
337C2C22×#49811yescharacteristic
338C2#6066511yescharacteristic
339C2C2×#333512yes—
340C2#6241312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33310C2semidirect
3. #33207C2semidirect
4. #29206C2semidirect
5. #28829C2semidirect
6. #27948C2semidirect
7. #26060C2semidirect
9. C4×#498C2semidirect
10. C2×#1844C2semidirect
71. C8×D4Q8semidirect
277. D4#498semidirect
313. C8#554semidirect

Maximal subgroups

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

subgroupindex
#361802
#333102
#332072
#292062
#288292
#279482
#260602
D4×#1062
C4×#4982
C2×#18442
C4×C8×D42