Finite Group Databasebeta

← #33047

Subgroups — #33047

Order 256.

432 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic, complemented
2C2×#405C211yesmaximal, characteristic, complemented
3#2629C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic
5#1992C211yesmaximal, characteristic, complemented
6#1959C211yesmaximal, characteristic, complemented
7#1936C211yesmaximal, characteristic, complemented
8#1935C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1933C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic
12#1894C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic
14#1892C211yesmaximal, characteristic, complemented
15#1834C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#488—24no—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32#405—24no—
33C2×#97C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#398—21no—
38#398—21no—
39#396—21no—
40#396—21no—
41#396—21no—
42#396—21no—
43#394—21no—
44#394—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#384C2211yescharacteristic
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#125—22no—
76#125—22no—
77#124—22no—
78#124—22no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81#121—22no—
82#121—22no—
83#117—22no—
84#117—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#97—22no—
123#97—22no—
124C4×C8—22no—
125C4×C8D414yescomplemented
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—81no—
148C2×D4—81no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
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×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4#4112yes—
166C22×C4#4112yes—
167C22×C4C2411yescharacteristic
168M16—28no—
169C2×C8—28no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8C2×D412yes—
179C2×C8C2×D412yes—
180C2×C8#4112yes—
181C2×C8#4112yes—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
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—21no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—21no—
216C42—21no—
217C42C2×D412yes—
218C42#4112yes—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23#12412yes—
233C23C22×D411yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
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#12112yes—
281C2×C4#12112yes—
282C2×C4#12412yes—
283C2×C4C22×D411yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C8—22no—
290C8—22no—
291C8—22no—
292C8—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22C2×#12111yescharacteristic
308C22C2×#12411yescharacteristic
309C22#53211yescharacteristic
310C22#57611yescharacteristic
311C22#57711yescharacteristic
312C22#57711yescharacteristic
313C22#57811yescharacteristic
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4#52511yescharacteristic
327C4#52511yescharacteristic
328C4#52911yescharacteristic
329C4#52911yescharacteristic
330C2—42no—
331C2—161no—
332C2#285011yescharacteristic
333C2#306011yescharacteristic
334C2#306111yescharacteristic
335C2#313611yescharacteristic
336C2#313711yescharacteristic
337C2#317811yescharacteristic
338C2#317911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#488C2semidirect
2. C2×#405C2semidirect
3. #2629C2semidirect
5. #1992C2semidirect
6. #1959C2semidirect
7. #1936C2semidirect
8. #1935C2semidirect
9. #1934C2semidirect
10. #1933C2semidirect
12. #1894C2semidirect
14. #1892C2semidirect
15. #1834C2semidirect
125. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4052
#26292
#26202
#19922
#19592
#19362
#19352
#19342
#19332
#18952
#18942
#18932
#18922
#18342