Finite Group Databasebeta

← #28430

Subgroups — #28430

Order 256.

532 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×#442C211yesmaximal, characteristic, complemented
5#2544C211yesmaximal, characteristic, complemented
6#2542C211yesmaximal, characteristic, complemented
7#2096C211yesmaximal, characteristic, complemented
8#2094C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic
11#1972C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1959C211yesmaximal, characteristic, complemented
14#1957C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8—22no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#489—24no—
27#476—24no—
28#463—24no—
29#442—24no—
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×#97C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#399—22no—
48#394—22no—
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#386—22no—
54#384C2211yescharacteristic
55#384C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×D4—22no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—41no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82#127—22no—
83#127—41no—
84#124—22no—
85#124—22no—
86#124—22no—
87#124—41no—
88C22⋊Q8—48no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D412yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—22no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#106—22no—
113#106—22no—
114#106—22no—
115C4⋊C8D414yescomplemented
116#103—22no—
117#103—22no—
118#103—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#97—41no—
127#97D414yescomplemented
128C4×C8—41no—
129#95—22no—
130#95D412yes—
131C24—21no—
132C24—21no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—44no—
141C2×Q8—42no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—82no—
161C2×D4#4112yes—
162C2×D4#4112yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—82no—
172C22×C4C2×D412yes—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D412yes—
176C22×C4C2411yescharacteristic
177SD16—44no—
178D8—28no—
179D8—44no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8C2×D412yes—
189C2×C8C2×D412yes—
190C2×C8C2×D412yes—
191C2×C8C2×D412yes—
192C4⋊C4—24no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—44no—
211C22⋊C4—82no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42C2×D412yes—
217C42C2×D412yes—
218C23—22no—
219C23—22no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23C22×D411yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—84no—
239D4—22no—
240D4—22no—
241D4—22no—
242D4—22no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—88no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4#12712yes—
273C2×C4#12712yes—
274C2×C4AGL(1,ℤ/8)14yes—
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—22no—
282C8—22no—
283C8—22no—
284C8—22no—
285C22—44no—
286C22—44no—
287C22—42no—
288C22—42no—
289C22—82no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—82no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22—82no—
299C22—81no—
300C22C2×#12711yescharacteristic
301C22#54611yescharacteristic
302C22#54811yescharacteristic
303C22C2×AGL(1,ℤ/8)12yes—
304C22#57712yes—
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4#52511yescharacteristic
315C4#53711yescharacteristic
316C4#54611yescharacteristic
317C4#54811yescharacteristic
318C2—82no—
319C2—82no—
320C2—161no—
321C2#254011yescharacteristic
322C2#315111yescharacteristic
323C2#315411yescharacteristic
324C2#330612yes—
325C2#337312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#489C2semidirect
2. C2×#476C2semidirect
3. C2×#463C2semidirect
4. C2×#442C2semidirect
5. #2544C2semidirect
6. #2542C2semidirect
7. #2096C2semidirect
8. #2094C2semidirect
9. #2012C2semidirect
11. #1972C2semidirect
12. #1968C2semidirect
13. #1959C2semidirect
14. #1957C2semidirect
15. #1771C2semidirect
115. C4⋊C8D4semidirect
127. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4762
C2×#4632
C2×#4422
#25442
#25422
#20962
#20942
#20122
#20112
#19722
#19682
#19592
#19572
#17712