Finite Group Databasebeta

← #32182

Subgroups — #32182

Order 256.

369 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2546C211yesmaximal, characteristic, complemented
2#2513C211yesmaximal, characteristic
3#2074C211yesmaximal, characteristic, complemented
4#2055C211yesmaximal, characteristic
5#2043C211yesmaximal, characteristic, complemented
6#2013C211yesmaximal, characteristic
7#2013C211yesmaximal, characteristic
8#2013C211yesmaximal, characteristic
9#1981C211yesmaximal, characteristic, complemented
10#1973C211yesmaximal, characteristic
11#1960C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1880C211yesmaximal, characteristic
15#1863C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#398—21no—
37#398—21no—
38#397—21no—
39#397—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#389—21no—
47#389C2211yescharacteristic
48#388—21no—
49#388—21no—
50#382C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C2×SD16—22no—
64C2×SD16—22no—
65C2×SD16—41no—
66C2×SD16—41no—
67C2×SD16—41no—
68C2×SD16—41no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×SD16—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4×Q8—41no—
83C4×Q8—41no—
84C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—41no—
110C4⋊C8—41no—
111#103—22no—
112#103—22no—
113#103—41no—
114#103—41no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119#103—41no—
120#102—41no—
121#97—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C2×Q8—21no—
134C2×Q8—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—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8#4112yes—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—44no—
172SD16—44no—
173SD16—44no—
174C2×C8—21no—
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#4112yes—
186C4⋊C4—22no—
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—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—81no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—42no—
217C42—81no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23C22×D411yescharacteristic
226Q8—22no—
227Q8—22no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
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—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4#13712yes—
280C2×C4#13712yes—
281C2×C4C2×#4111yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C8—22no—
288C8—41no—
289C8—41no—
290C8—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#53511yescharacteristic
299C22#54611yescharacteristic
300C22#55011yescharacteristic
301C22C2×#13711yescharacteristic
302C22C2×#13711yescharacteristic
303C22#57711yescharacteristic
304C22#57711yescharacteristic
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4#53811yescharacteristic
319C4#53811yescharacteristic
320C4#55311yescharacteristic
321C4#56011yescharacteristic
322C2—161no—
323C2#271411yescharacteristic
324C2#321811yescharacteristic
325C2#322111yescharacteristic
326C2#332011yescharacteristic
327C2#332911yescharacteristic
328C2#340011yescharacteristic
329C2#340011yescharacteristic
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
1. #2546C2semidirect
3. #2074C2semidirect
5. #2043C2semidirect
9. #1981C2semidirect
11. #1960C2semidirect
12. #1928C2semidirect
13. #1917C2semidirect
15. #1863C2semidirect

Maximal subgroups

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

subgroupindex
#25462
#25132
#20742
#20552
#20432
#20132
#19812
#19732
#19602
#19282
#19172
#18802
#18632