Finite Group Databasebeta

← #32192

Subgroups — #32192

Order 256.

392 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2545C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic, complemented
3#2072C211yesmaximal, characteristic, complemented
4#2054C211yesmaximal, characteristic
5#2042C211yesmaximal, characteristic, complemented
6#2013C211yesmaximal, characteristic, complemented
7#2012C211yesmaximal, characteristic
8#2012C211yesmaximal, characteristic
9#1981C211yesmaximal, characteristic, complemented
10#1973C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1880C211yesmaximal, characteristic, complemented
15#1863C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#126—21no—
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#397—21no—
38#396—21no—
39#396—21no—
40#396—21no—
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#388—21no—
50#384C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C2×SD16—22no—
64C2×SD16—22no—
65C2×SD16—41no—
66C2×SD16—41no—
67C2×SD16—41no—
68C2×SD16—41no—
69C2×D8—41no—
70C2×D8—41no—
71C2×D8—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78#126—44no—
79#124—22no—
80#124—41no—
81#124—41no—
82#124—41no—
83C4×Q8—41no—
84C4×Q8—41no—
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⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
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×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—41no—
115C4⋊C8—41no—
116#103—22no—
117#103—41no—
118#103—41no—
119#103—41no—
120#103—41no—
121#103—41no—
122#102—22no—
123#102—41no—
124#102—41no—
125#102—41no—
126#97—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8#4112yes—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—44no—
173SD16—44no—
174D8—44no—
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×C8—21no—
186C2×C8#4112yes—
187C4⋊C4—22no—
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—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—42no—
222C42—42no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
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—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4AGL(1,ℤ/8)12yes—
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C8—22no—
297C8—41no—
298C8—41no—
299C8—41no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22#53511yescharacteristic
315C22#54611yescharacteristic
316C22#55011yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22#57511yescharacteristic
319C22#57711yescharacteristic
320C22#57811yescharacteristic
321C4—22no—
322C4—41no—
323C4—41no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4#53811yescharacteristic
334C4#53811yescharacteristic
335C4#55311yescharacteristic
336C4#56011yescharacteristic
337C2—161no—
338C2—161no—
339C2#271411yescharacteristic
340C2#321411yescharacteristic
341C2#322511yescharacteristic
342C2#330811yescharacteristic
343C2#332611yescharacteristic
344C2#339711yescharacteristic
345C2#340211yescharacteristic
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. #2545C2semidirect
2. #2512C2semidirect
3. #2072C2semidirect
5. #2042C2semidirect
6. #2013C2semidirect
9. #1981C2semidirect
10. #1973C2semidirect
11. #1959C2semidirect
12. #1925C2semidirect
13. #1917C2semidirect
14. #1880C2semidirect
15. #1863C2semidirect

Maximal subgroups

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

subgroupindex
#25452
#25122
#20722
#20542
#20422
#20132
#20122
#19812
#19732
#19592
#19252
#19172
#18802
#18632