Finite Group Databasebeta

← #36432

Subgroups — #36432

Order 256.

1247 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×C4×#41C211yesmaximal, characteristic, complemented
3#2855C213yesmaximal, complemented
4#2499C213yesmaximal, complemented
5#2485C213yesmaximal, complemented
6#2447C213yesmaximal, complemented
7#2447C213yesmaximal, complemented
8#2422C213yesmaximal, complemented
9#2420C213yesmaximal, complemented
10#2351C211yesmaximal, characteristic, complemented
11Q8×C4⋊C4C211yesmaximal, characteristic, complemented
12#2341C211yesmaximal, characteristic, complemented
13Q8×C22⋊C4C211yesmaximal, characteristic, complemented
14Q8×C22⋊C4C211yesmaximal, characteristic, complemented
15C4×#123C213yesmaximal, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18#548—28no—
19C2×#124C2213yes—
20C2×#123C2213yes—
21C2×C22⋊Q8C2213yes—
22C2×#121C2213yes—
23C4×#41—28no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4C2213yes—
30C2×C4×D4C2213yes—
31C2×C4×D4C2213yes—
32C2×#117C2213yes—
33C22×C42C2213yes—
34#397C2216yes—
35#397C2216yes—
36#397C2216yes—
37#394C2216yes—
38#391C2213yes—
39#391C2213yes—
40#391C2213yes—
41#391C2213yes—
42#390C2213yes—
43#390C2213yes—
44#389C2213yes—
45#389C2213yes—
46#389C2213yes—
47#389C2213yes—
48#388C2213yes—
49#388C2211yescharacteristic
50#387C2213yes—
51#387C2216yes—
52#387C2213yes—
53#386C2213yes—
54#385C2216yes—
55#385C2213yes—
56#384C2213yes—
57#382C2213yes—
58#381C2216yes—
59#381C2213yes—
60#380C2213yes—
61#380C2213yes—
62#380C2213yes—
63C4×C4⋊C4C2213yes—
64C4×C4⋊C4C2213yes—
65C4×C22⋊C4C2213yes—
66C4×C22⋊C4C2213yes—
67C4×C22⋊C4C2213yes—
68C2×#41—24no—
69C2×#41—24no—
70C2×#41D418yes—
71C22×Q8—24no—
72C22×Q8—24no—
73C22×Q8—24no—
74C22×Q8C2313yes—
75C22×Q8C2311yescharacteristic
76C22×D4C2313yes—
77C22×D4C2313yes—
78C23×C4C2313yes—
79C23×C4C2313yes—
80C23×C4C2313yes—
81C23×C4C2313yes—
82#124—212no—
83#123—212no—
84C22⋊Q8—212no—
85#121—212no—
86C4×Q8—24no—
87C4×Q8—24no—
88C4×Q8—24no—
89C4×Q8—24no—
90C4×Q8—24no—
91C4×D4—212no—
92C4×D4—212no—
93C4×D4—212no—
94#117—212no—
95C2×C4⋊C4C2313yes—
96C2×C4⋊C4C2313yes—
97C2×C4⋊C4C2313yes—
98C2×C4⋊C4C2313yes—
99C2×C4⋊C4C2313yes—
100C2×C4⋊C4C2313yes—
101C2×C4⋊C4C2313yes—
102C2×C4⋊C4C2313yes—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2313yes—
105C2×C4⋊C4C2316yes—
106C2×C22⋊C4C2316yes—
107C2×C22⋊C4C2311yescharacteristic
108C2×C22⋊C4C2313yes—
109C2×C22⋊C4C2316yes—
110C2×C22⋊C4C2313yes—
111C2×C22⋊C4C2313yes—
112C2×C22⋊C4C2311yescharacteristic
113C2×C22⋊C4C2313yes—
114C2×C22⋊C4C2313yes—
115C2×C22⋊C4C2316yes—
116C2×C22⋊C4C2311yescharacteristic
117C2×C42—212no—
118C2×C42C2313yes—
119C2×C42C2313yes—
120C2×C42C2313yes—
121C2×C42C2313yes—
122C2×C42C2313yes—
123C2×C42C2313yes—
124C2×C42C2311yescharacteristic
125#95C2311yescharacteristic
126#95C2313yes—
127#95C2313yes—
128#95C2313yes—
129#95C2313yes—
130#95C2313yes—
131#95C2311yescharacteristic
132#95C2313yes—
133#95C2313yes—
134#95C2313yes—
135#95C2316yes—
136#95C2316yes—
137#95C2316yes—
138C24C2413yes—
139C24C2411yescharacteristic
140#41—232no—
141C2×Q8—26no—
142C2×Q8—26no—
143C2×Q8—26no—
144C2×Q8—416no—
145C2×Q8C2×D414yes—
146C2×Q8#4114yes—
147C2×Q8#4114yes—
148C2×D4—26no—
149C2×D4—26no—
150C2×D4—26no—
151C2×D4—26no—
152C2×D4—43no—
153C2×D4C2×D4112yes—
154C22×C4—26no—
155C22×C4—26no—
156C22×C4—26no—
157C22×C4—26no—
158C22×C4—26no—
159C22×C4—26no—
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—26no—
163C22×C4—43no—
164C22×C4—43no—
165C22×C4C2×D4112yes—
166C22×C4C2411yescharacteristic
167C22×C4C2413yes—
168C22×C4C2413yes—
169C22×C4C2413yes—
170C22×C4C2413yes—
171C22×C4C2411yescharacteristic
172C22×C4C2413yes—
173C22×C4C2413yes—
174C22×C4C2411yescharacteristic
175C22×C4C2411yescharacteristic
176C22×C4C2411yescharacteristic
177C22×C4C2411yescharacteristic
178C22×C4C2413yes—
179C4⋊C4—26no—
180C4⋊C4—26no—
181C4⋊C4—26no—
182C4⋊C4—26no—
183C4⋊C4—26no—
184C4⋊C4—26no—
185C4⋊C4—26no—
186C4⋊C4—26no—
187C4⋊C4—22no—
188C4⋊C4—43no—
189C4⋊C4—46no—
190C22⋊C4—212no—
191C22⋊C4—26no—
192C22⋊C4—212no—
193C22⋊C4—26no—
194C22⋊C4—22no—
195C22⋊C4—26no—
196C22⋊C4—22no—
197C22⋊C4—46no—
198C22⋊C4—43no—
199C22⋊C4—43no—
200C22⋊C4#4114yescomplemented
201C42—22no—
202C42—26no—
203C42—26no—
204C42—26no—
205C42—26no—
206C42—26no—
207C42—26no—
208C23—21no—
209C23—23no—
210C23—21no—
211C23—21no—
212C23—23no—
213C23—23no—
214C23—21no—
215C23—23no—
216C23—21no—
217C23—23no—
218C23—23no—
219C23—21no—
220C23C22×D416yes—
221C23C2×#4112yes—
222C23C2511yescharacteristic
223Q8—412no—
224Q8#123116yescomplemented
225D4—224no—
226D4—412no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—224no—
236C2×C4—21no—
237C2×C4—23no—
238C2×C4—23no—
239C2×C4—23no—
240C2×C4—23no—
241C2×C4—21no—
242C2×C4—23no—
243C2×C4—23no—
244C2×C4—23no—
245C2×C4—23no—
246C2×C4—23no—
247C2×C4—23no—
248C2×C4—23no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—23no—
252C2×C4—21no—
253C2×C4—23no—
254C2×C4—23no—
255C2×C4—23no—
256C2×C4—23no—
257C2×C4—21no—
258C2×C4—23no—
259C2×C4—23no—
260C2×C4—412no—
261C2×C4—412no—
262C2×C4—412no—
263C2×C4C22×D412yes—
264C2×C4C22×D416yes—
265C2×C4C2×#4112yes—
266C2×C4C2×#4112yes—
267C2×C4C2×#4116yes—
268C2×C4C2×#4116yes—
269C2×C42−1+412yes—
270C22—26no—
271C22—26no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—26no—
276C22—41no—
277C22—43no—
278C22—43no—
279C22—41no—
280C22—41no—
281C22—43no—
282C22—41no—
283C22—43no—
284C22C23×D411yescharacteristic
285C22C22×#4111yescharacteristic
286C22C22×#4111yescharacteristic
287C22C22×#4111yescharacteristic
288C22C2×2−1+411yescharacteristic
289C22#58511yescharacteristic
290C22#58511yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—43no—
296C4—43no—
297C4—43no—
298C4—43no—
299C4—41no—
300C4—43no—
301C4—41no—
302C4—43no—
303C4C2×#123112yes—
304C2—46no—
305C2—42no—
306C2C22×#12311yescharacteristic
307C2D4×#4111yescharacteristic
308C2#348011yescharacteristic
309C2#348211yescharacteristic
310C2#348911yescharacteristic
311C2#351811yescharacteristic
312C2#352411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#41C2semidirect
3. #2855C2semidirect
4. #2499C2semidirect
5. #2485C2semidirect
6. #2447C2semidirect
7. #2447C2semidirect
8. #2422C2semidirect
9. #2420C2semidirect
10. #2351C2semidirect
11. Q8×C4⋊C4C2semidirect
12. #2341C2semidirect
13. Q8×C22⋊C4C2semidirect
14. Q8×C22⋊C4C2semidirect
15. C4×#123C2semidirect
200. C22⋊C4#41semidirect
224. Q8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×C4×#412
#28552
#24992
#24852
#24472
#24222
#24202
#23512
Q8×C4⋊C42
#23412
Q8×C22⋊C42
C4×#1232