Finite Group Databasebeta

← #52167

Subgroups — #52167

Order 256.

1355 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#532C212yesmaximal, complemented
2#2734C211yesmaximal, characteristic, complemented
3#2733C211yesmaximal, characteristic, complemented
4#2731C212yesmaximal, complemented
5#2699C212yesmaximal, complemented
6#2686C214yesmaximal, complemented
7#2682C212yesmaximal, complemented
8#2673C214yesmaximal, complemented
9#2392C212yesmaximal, complemented
10#2388C214yesmaximal, complemented
11C4×#123C214yesmaximal, complemented
12C4×#121C212yesmaximal, complemented
13#2297C211yesmaximal, characteristic, complemented
14C22×#41C2212yes—
15C22×#41C2211yescharacteristic
16#532—216no—
17C2×C4⋊Q8C2212yes—
18C2×#127C2212yes—
19C2×#124C2212yes—
20C2×#124C2212yes—
21C2×#123C2214yes—
22C2×#121C2212yes—
23C2×#121C2214yes—
24C2×#121C2214yes—
25C2×C4×D4C2214yes—
26C2×C4×D4C2214yes—
27C2×C4×D4C2212yes—
28C2×C4×D4C2212yes—
29C2×#117C2211yescharacteristic
30C22×C42C2212yes—
31C22×C42C2211yescharacteristic
32#398C2212yes—
33#397C2212yes—
34#396C2214yes—
35#395C2214yes—
36#394C2214yes—
37#394C2214yes—
38#394C2214yes—
39#394C2214yes—
40#394C2212yes—
41#393C2214yes—
42#392C2214yes—
43#392C2214yes—
44#392C2214yes—
45#392C2212yes—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2212yes—
50#390C2211yescharacteristic
51#389C2212yes—
52#388C2214yes—
53#388C2214yes—
54#388C2214yes—
55#387C2214yes—
56#387C2214yes—
57#386C2214yes—
58#386C2214yes—
59#386C2214yes—
60#385C2214yes—
61#381C2214yes—
62#380C2212yes—
63C4×C4⋊C4C2212yes—
64C4×C4⋊C4C2212yes—
65C4×C22⋊C4C2214yes—
66C4×C22⋊C4C2214yes—
67C4×C22⋊C4C2214yes—
68C2×#41—28no—
69C2×#41—42no—
70C2×#41—42no—
71C2×#41—44no—
72C2×#41—44no—
73C2×#41D418yescomplemented
74C22×Q8C2312yes—
75C22×Q8C2311yescharacteristic
76C22×D4C2314yes—
77C22×D4C2314yes—
78C22×D4C2312yes—
79C22×D4C2314yes—
80C22×D4C2312yes—
81C22×D4C2312yes—
82C22×D4C2311yescharacteristic
83C22×D4C2312yes—
84C23×C4C2314yes—
85C23×C4C2314yes—
86C23×C4C2311yescharacteristic
87C23×C4C2312yes—
88C23×C4C2312yes—
89C4⋊Q8—28no—
90#127—28no—
91#124—28no—
92#124—28no—
93#123—216no—
94#121—216no—
95#121—216no—
96#121—28no—
97C4×D4—216no—
98C4×D4—216no—
99C4×D4—44no—
100C4×D4—44no—
101#117—42no—
102C2×C4⋊C4C2314yes—
103C2×C4⋊C4C2314yes—
104C2×C4⋊C4C2314yes—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2312yes—
108C2×C4⋊C4C2312yes—
109C2×C4⋊C4C2312yes—
110C2×C4⋊C4C2312yes—
111C2×C22⋊C4C2314yes—
112C2×C22⋊C4C2314yes—
113C2×C22⋊C4C2314yes—
114C2×C22⋊C4C2314yes—
115C2×C22⋊C4C2314yes—
116C2×C22⋊C4C2314yes—
117C2×C22⋊C4C2314yes—
118C2×C22⋊C4C2314yes—
119C2×C22⋊C4C2312yes—
120C2×C22⋊C4C2314yes—
121C2×C22⋊C4C2314yes—
122C2×C22⋊C4C2314yes—
123C2×C22⋊C4C2314yes—
124C2×C42—28no—
125C2×C42—42no—
126C2×C42C2312yes—
127C2×C42C2312yes—
128C2×C42C2312yes—
129C2×C42C2314yes—
130C2×C42C2312yes—
131C2×C42C2311yescharacteristic
132C2×C42C2311yescharacteristic
133#95C2314yes—
134#95C2314yes—
135#95C2314yes—
136#95C2314yes—
137#95C2314yes—
138#95C2314yes—
139#95C2312yes—
140#95C2314yes—
141#95C2312yes—
142C24C2414yes—
143C24C2412yes—
144C24C2411yescharacteristic
145#41—416no—
146#41—432no—
147C2×Q8—24no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8C2×D414yes—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—28no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—28no—
159C2×D4—28no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4C2×D414yes—
175C2×D4C2×D418yes—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—28no—
185C22×C4—28no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4C2×D418yes—
190C22×C4C2×D414yes—
191C22×C4C2412yes—
192C22×C4C2412yes—
193C22×C4C2414yes—
194C22×C4C2412yes—
195C22×C4C2414yes—
196C22×C4C2412yes—
197C22×C4C2411yescharacteristic
198C22×C4C2412yes—
199C22×C4C2414yes—
200C22×C4C2411yescharacteristic
201C4⋊C4—28no—
202C4⋊C4—28no—
203C4⋊C4—28no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4#4118yescomplemented
210C22⋊C4—28no—
211C22⋊C4—28no—
212C22⋊C4—28no—
213C22⋊C4—28no—
214C22⋊C4—28no—
215C22⋊C4—28no—
216C22⋊C4—28no—
217C22⋊C4—28no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C42—28no—
224C42—24no—
225C42—24no—
226C42—24no—
227C42—24no—
228C42—41no—
229C42—41no—
230C23—24no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23C22×D412yes—
246C23C22×D414yes—
247C23C2511yescharacteristic
248Q8—44no—
249Q8—48no—
250D4—48no—
251D4—44no—
252D4—416no—
253D4—416no—
254D4—48no—
255D4—48no—
256D4—416no—
257D4—48no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—216no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—416no—
282C2×C4—48no—
283C2×C4—416no—
284C2×C4C22×D414yes—
285C2×C4C22×D412yes—
286C2×C4C22×D412yes—
287C2×C4C2×#4114yes—
288C2×C4C2×#4118yes—
289C2×C4H(5,𝔽2)12yes—
290C2×C4H(5,𝔽2)12yes—
291C22—24no—
292C22—22no—
293C22—28no—
294C22—24no—
295C22—42no—
296C22—44no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22C23×D411yescharacteristic
309C22C22×#4112yes—
310C22C2×H(5,𝔽2)11yescharacteristic
311C22C2×H(5,𝔽2)11yescharacteristic
312C22#58512yes—
313C4—24no—
314C4—22no—
315C4—24no—
316C4—28no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—42no—
321C4—42no—
322C4#53414yes—
323C2—44no—
324C2—42no—
325C2—84no—
326C2C2×#53411yescharacteristic
327C2#346012yes—
328C2D4×#4112yes—
329C2#348311yescharacteristic
330C2#352911yescharacteristic
subgroupG/H
centerC22×C4C24
commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#532C2semidirect
2. #2734C2semidirect
3. #2733C2semidirect
4. #2731C2semidirect
5. #2699C2semidirect
6. #2686C2semidirect
7. #2682C2semidirect
8. #2673C2semidirect
9. #2392C2semidirect
10. #2388C2semidirect
11. C4×#123C2semidirect
12. C4×#121C2semidirect
13. #2297C2semidirect
73. C2×#41D4semidirect
209. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
#27342
#27332
#27312
#26992
#26862
#26822
#26732
#23922
#23882
C4×#1232
C4×#1212
#22972