Finite Group Databasebeta

← #27800

Subgroups — #27800

Order 256.

412 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic, complemented
4#2490C211yesmaximal, characteristic
5#2036C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic
7#2026C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1973C211yesmaximal, characteristic, complemented
11#1927C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1808C211yesmaximal, characteristic, complemented
14#1776C211yesmaximal, characteristic, complemented
15#1752C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22#501—24no—
23#465—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#400—21no—
39#397—21no—
40#394—21no—
41#393—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#390—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#388—21no—
49#388—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78#125—22no—
79#125—41no—
80C22⋊Q8—22no—
81C22⋊Q8—41no—
82#121—22no—
83#121—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#107—22no—
111#106—22no—
112#106D414yes—
113#103—22no—
114#103—22no—
115#103—41no—
116#102—22no—
117#102—22no—
118#102—41no—
119C22⋊C8—22no—
120#97—22no—
121#97—22no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—81no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—81no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—44no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—44no—
181C2×C8C2×D412yes—
182C2×C8#4112yes—
183C2×C8#4112yes—
184C2×C8#4112yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4C2×D412yes—
200C4⋊C4C2×D412yes—
201C22⋊C4—22no—
202C22⋊C4—41no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—81no—
216C42—81no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23#13712yes—
231C23C22×D411yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236Q8—84no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—84no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
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×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4C22×D411yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C42−1+411yescharacteristic
288C8—22no—
289C8—41no—
290C8—41no—
291C8#12414yes—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#53211yescharacteristic
307C22D4211yescharacteristic
308C22#54711yescharacteristic
309C22C2×#13711yescharacteristic
310C22#57511yescharacteristic
311C22#57611yescharacteristic
312C22#57711yescharacteristic
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—42no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4C2×#12411yescharacteristic
326C4#54211yescharacteristic
327C4#54811yescharacteristic
328C4#55511yescharacteristic
329C2—42no—
330C2—161no—
331C2#249111yescharacteristic
332C2#317911yescharacteristic
333C2#318111yescharacteristic
334C2#326611yescharacteristic
335C2#326811yescharacteristic
336C2#334611yescharacteristic
337C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
3. #2492C2semidirect
5. #2036C2semidirect
7. #2026C2semidirect
8. #2023C2semidirect
9. #1973C2semidirect
10. #1973C2semidirect
11. #1927C2semidirect
12. #1925C2semidirect
13. #1808C2semidirect
14. #1776C2semidirect
15. #1752C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4652
#24922
#24902
#20362
#20342
#20262
#20232
#19732
#19272
#19252
#18082
#17762
#17522