Finite Group Databasebeta

← #32117

Subgroups — #32117

Order 256.

387 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2491C211yesmaximal, characteristic
2#2489C211yesmaximal, characteristic, complemented
3#2075C211yesmaximal, characteristic, complemented
4#2055C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic
9#1989C211yesmaximal, characteristic, complemented
10#1987C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic
14#1892C211yesmaximal, characteristic, complemented
15#1850C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×#124—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#400—21no—
34#397—21no—
35#394—21no—
36#393—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#388—21no—
42#388—21no—
43#382—21no—
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C2×SD16—41no—
65C2×SD16—41no—
66C2×SD16—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76#125—22no—
77#125—41no—
78#124—44no—
79C22⋊Q8—22no—
80C22⋊Q8—41no—
81#121—22no—
82#121—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—22no—
111#103—41no—
112#103—41no—
113#102—22no—
114#102—41no—
115#102—41no—
116C22⋊C8—41no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24C2×D411yescharacteristic
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—81no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—81no—
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—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166SD16—44no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
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×C8—21no—
179C2×C8—44no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4#4112yes—
195C4⋊C4#4112yes—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—81no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—42no—
211C42—42no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23AGL(1,ℤ/8)12yes—
226C23C22×D411yescharacteristic
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—84no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
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#13512yes—
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C4H(5,𝔽2)11yescharacteristic
281C2×C4H(5,𝔽2)11yescharacteristic
282C2×C42−1+411yescharacteristic
283C8—41no—
284C8—41no—
285C8—41no—
286C8—41no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22#53711yescharacteristic
302C22#54611yescharacteristic
303C22#54611yescharacteristic
304C22C2×#13511yescharacteristic
305C22C2×AGL(1,ℤ/8)11yescharacteristic
306C22#57711yescharacteristic
307C22#57811yescharacteristic
308C4—22no—
309C4—41no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4#54211yescharacteristic
321C4#54211yescharacteristic
322C4#55911yescharacteristic
323C4H(3,𝔽4)11yescharacteristic
324C2—42no—
325C2—161no—
326C2#271211yescharacteristic
327C2#324711yescharacteristic
328C2#325811yescharacteristic
329C2#330011yescharacteristic
330C2#330311yescharacteristic
331C2#330811yescharacteristic
332C2#331011yescharacteristic
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
2. #2489C2semidirect
3. #2075C2semidirect
4. #2055C2semidirect
5. #2055C2semidirect
6. #2025C2semidirect
7. #2024C2semidirect
9. #1989C2semidirect
10. #1987C2semidirect
11. #1928C2semidirect
12. #1925C2semidirect
14. #1892C2semidirect
15. #1850C2semidirect

Maximal subgroups

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

subgroupindex
#24912
#24892
#20752
#20552
#20252
#20242
#20002
#19892
#19872
#19282
#19252
#18932
#18922
#18502