Finite Group Databasebeta

← #32541

Subgroups — #32541

Order 256.

343 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2513C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic, complemented
3#2093C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic
8#2054C211yesmaximal, characteristic
9#1972C211yesmaximal, characteristic
10#1949C211yesmaximal, characteristic, complemented
11#1948C211yesmaximal, characteristic, complemented
12#1943C211yesmaximal, characteristic, complemented
13#1940C211yesmaximal, characteristic, complemented
14#1926C211yesmaximal, characteristic, complemented
15#1926C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#126—21no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#397—21no—
34#397—21no—
35#397—21no—
36#396—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#388—21no—
48#384C2211yescharacteristic
49#382C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C2×D8—22no—
62C2×D8—41no—
63C2×D8—41no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#126—44no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
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×C42—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107—41no—
100#106—41no—
101#106—41no—
102#106—41no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105C4⋊C8—41no—
106#102—22no—
107#102—22no—
108#102—41no—
109#102—41no—
110#102—41no—
111#102—41no—
112#97—41no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121C24—21no—
122C24—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2411yescharacteristic
148D8—44no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8#4112yes—
160C2×C8#4112yes—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—81no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23C22×D411yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—41no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
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—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4AGL(1,ℤ/8)12yes—
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C2×C42−1+411yescharacteristic
263C2×C42−1+411yescharacteristic
264C8—22no—
265C8—22no—
266C8—41no—
267C8—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22#53711yescharacteristic
283C22#54711yescharacteristic
284C22#55011yescharacteristic
285C22C2×AGL(1,ℤ/8)11yescharacteristic
286C22#57511yescharacteristic
287C22#57611yescharacteristic
288C22#57711yescharacteristic
289C4—22no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#53811yescharacteristic
302C4#53911yescharacteristic
303C4#55611yescharacteristic
304C4#56311yescharacteristic
305C2—161no—
306C2—161no—
307C2#272711yescharacteristic
308C2#325111yescharacteristic
309C2#325711yescharacteristic
310C2#334511yescharacteristic
311C2#334711yescharacteristic
312C2#339911yescharacteristic
313C2#340111yescharacteristic
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. #2512C2semidirect
3. #2093C2semidirect
4. #2092C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
10. #1949C2semidirect
11. #1948C2semidirect
12. #1943C2semidirect
13. #1940C2semidirect
14. #1926C2semidirect
15. #1926C2semidirect

Maximal subgroups

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

subgroupindex
#25132
#25122
#20932
#20922
#20672
#20652
#20542
#19722
#19492
#19482
#19432
#19402
#19262