Finite Group Databasebeta

← #26338

Subgroups — #26338

Order 256.

483 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476C211yesmaximal, characteristic, complemented
2C22×#106C211yesmaximal, characteristic, complemented
3#2440C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1920C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic
12#1874C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1850C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#476—24no—
25C2×#106—24no—
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#398—21no—
40#395—21no—
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#386—21no—
47#385—21no—
48#385—21no—
49#384C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77C22⋊Q8—44no—
78#121—22no—
79#121Q814yescomplemented
80C4×D4—22no—
81C4×D4—41no—
82C4×D4—44no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—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—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#106—22no—
113#106—22no—
114#106—22no—
115#106—22no—
116#106—41no—
117#106D414yescomplemented
118C4⋊C8—22no—
119C4⋊C8—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—41no—
124C22⋊C8—22no—
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—42no—
135C2×Q8—42no—
136C2×Q8—81no—
137C2×D4—24no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4C2×Q812yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
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—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×Q812yes—
172C22×C4#4112yes—
173C22×C4C2411yescharacteristic
174C2×C8—28no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8C2×D412yes—
186C2×C8#4112yes—
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—21no—
197C4⋊C4—21no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4C2×D412yes—
207C4⋊C4C2×D412yes—
208C4⋊C4C2×Q812yes—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—81no—
215C22⋊C4SD1614yes—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—42no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C22⋊Q812yes—
233C23C2×SD1612yes—
234C23AGL(1,ℤ/8)12yes—
235C23C22×D411yescharacteristic
236Q8—84no—
237D4—22no—
238D4—22no—
239D4—48no—
240D4—41no—
241D4—41no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
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—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4C22⋊Q812yes—
283C2×C4C2×SD1612yes—
284C2×C4C2×SD1612yes—
285C2×C4C22×D411yescharacteristic
286C2×C4C22×Q811yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C2×C42−1+411yescharacteristic
291C8—22no—
292C8—22no—
293C8—41no—
294C8—41no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#47614yescomplemented
309C22C2×C22⋊Q811yescharacteristic
310C22D4211yescharacteristic
311C22#54711yescharacteristic
312C22C22×SD1611yescharacteristic
313C22C2×AGL(1,ℤ/8)11yescharacteristic
314C22#57711yescharacteristic
315C22#57711yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4#52511yescharacteristic
329C4#53911yescharacteristic
330C4D4×Q811yescharacteristic
331C4#55411yescharacteristic
332C2—24no—
333C2—82no—
334C2#244911yescharacteristic
335C2C2×#47611yescharacteristic
336C2#308411yescharacteristic
337C2#326511yescharacteristic
338C2D4×SD1611yescharacteristic
339C2#333511yescharacteristic
340C2#334511yescharacteristic
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×#476C2semidirect
2. C22×#106C2semidirect
3. #2440C2semidirect
5. #2094C2semidirect
6. #2088C2semidirect
7. #2065C2semidirect
8. #2058C2semidirect
9. #1948C2semidirect
10. #1920C2semidirect
13. #1864C2semidirect
14. #1853C2semidirect
15. #1850C2semidirect
79. #121Q8semidirect
117. #106D4semidirect
308. C22#476semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C22×#1062
#24402
#23612
#20942
#20882
#20652
#20582
#19482
#19202
#18932
#18742
#18642
#18532
#18502