Finite Group Databasebeta

← #28903

Subgroups — #28903

Order 256.

365 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2513C211yesmaximal, characteristic, complemented
2#2448C211yesmaximal, characteristic, complemented
3#2085C211yesmaximal, characteristic, complemented
4#2066C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic
6#2012C211yesmaximal, characteristic
7#1973C211yesmaximal, characteristic, complemented
8#1972C211yesmaximal, characteristic
9#1958C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1771C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#399—21no—
35#397—21no—
36#397—21no—
37#397—21no—
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#388—21no—
48#388—21no—
49#388—21no—
50#382C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C2×SD16—22no—
63C2×SD16—22no—
64C2×SD16—41no—
65C2×D8—22no—
66C2×D8—41no—
67C2×D8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#124—22no—
73#124—41no—
74C4×Q8—22no—
75C4×Q8—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—41no—
104#106—41no—
105C4⋊C8—22no—
106#103—22no—
107#103—22no—
108#103—41no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—41no—
113#102—41no—
114#97—22no—
115#97—41no—
116C4×C8—41no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8#4112yes—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159SD16—44no—
160D8—44no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8#4112yes—
171C2×C8#4112yes—
172C2×C8#4112yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—81no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—41no—
226Q8—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C8—22no—
278C8—22no—
279C8—22no—
280C8—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22#53211yescharacteristic
296C22#54611yescharacteristic
297C22#55011yescharacteristic
298C22#57511yescharacteristic
299C22#57511yescharacteristic
300C22#57611yescharacteristic
301C22#57711yescharacteristic
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4#52911yescharacteristic
315C4#53811yescharacteristic
316C4#55111yescharacteristic
317C4#55311yescharacteristic
318C2—161no—
319C2—161no—
320C2#255411yescharacteristic
321C2#317911yescharacteristic
322C2#318311yescharacteristic
323C2#330911yescharacteristic
324C2#332111yescharacteristic
325C2#340111yescharacteristic
326C2#340311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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. #2513C2semidirect
2. #2448C2semidirect
3. #2085C2semidirect
4. #2066C2semidirect
7. #1973C2semidirect
9. #1958C2semidirect
10. #1936C2semidirect
11. #1934C2semidirect
12. #1925C2semidirect
13. #1880C2semidirect
14. #1771C2semidirect
15. #1763C2semidirect

Maximal subgroups

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

subgroupindex
#25132
#24482
#20852
#20662
#20542
#20122
#19732
#19722
#19582
#19362
#19342
#19252
#18802
#17712
#17632