Finite Group Databasebeta

← #24686

Subgroups — #24686

Order 256.

389 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic
3#2565C211yesmaximal, characteristic, complemented
4#2357C211yesmaximal, characteristic
5#2073C211yesmaximal, characteristic
6#2071C211yesmaximal, characteristic
7#2055C211yesmaximal, characteristic
8#2055C211yesmaximal, characteristic
9#1991C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1948C211yesmaximal, characteristic, complemented
12#1939C211yesmaximal, characteristic, complemented
13#1928C211yesmaximal, characteristic, complemented
14#1927C211yesmaximal, characteristic, complemented
15#1805C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#478—24no—
21C4×SD16—24no—
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#398—21no—
36#398—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#388—21no—
42#388—21no—
43#387—21no—
44#387—21no—
45#384C2211yescharacteristic
46#382C2211yescharacteristic
47#382C2211yescharacteristic
48#381—21no—
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#125—22no—
72#125—22no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×D4—22no—
76C4×D4D414yes—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#106—22no—
103#106—22no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113C4×C8—22no—
114#95—21no—
115#95—21no—
116#95—21no—
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—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8#4112yes—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C2×D412yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156SD16—28no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8#4112yes—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
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—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4C2×D412yes—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—81no—
199C42—81no—
200C42C2×D412yes—
201C42#4112yes—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23C22×D411yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—84no—
255C2×C4#12112yes—
256C2×C4#12112yes—
257C2×C4#13512yes—
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C42−1+411yescharacteristic
263C2×C42−1+411yescharacteristic
264C8—22no—
265C8—41no—
266C8—41no—
267C8—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22C2×#12111yescharacteristic
276C22#54711yescharacteristic
277C22#54811yescharacteristic
278C22C2×#13511yescharacteristic
279C22#57511yescharacteristic
280C22#57711yescharacteristic
281C22#57711yescharacteristic
282C4—22no—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4#52511yescharacteristic
296C4#53611yescharacteristic
297C4#55211yescharacteristic
298C4#55611yescharacteristic
299C2—82no—
300C2#241511yescharacteristic
301C2#303511yescharacteristic
302C2#304211yescharacteristic
303C2#333711yescharacteristic
304C2#335111yescharacteristic
305C2#336511yescharacteristic
306C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. #2565C2semidirect
9. #1991C2semidirect
10. #1949C2semidirect
11. #1948C2semidirect
12. #1939C2semidirect
13. #1928C2semidirect
14. #1927C2semidirect
15. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×C4×SD162
#25652
#23572
#20732
#20712
#20552
#19912
#19492
#19482
#19392
#19282
#19272
#18052