Finite Group Databasebeta

← #24906

Subgroups — #24906

Order 256.

470 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#451C211yesmaximal, characteristic
2C2×C4×Q16C211yesmaximal, characteristic, complemented
3#2423C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic
7#2056C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic
9#1933C211yesmaximal, characteristic, complemented
10#1923C211yesmaximal, characteristic
11#1878C211yesmaximal, characteristic, complemented
12#1878C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#451—24no—
24C4×Q16—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#399—21no—
40#398—21no—
41#391—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#387—21no—
49#384C2211yescharacteristic
50#380—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16D414yes—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22⋊Q8—22no—
84C22⋊Q8—41no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×Q8—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—22no—
114#107—41no—
115C4⋊C8—41no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—41no—
123#103—41no—
124C22⋊C8—22no—
125C22⋊C8—41no—
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24C2×D411yescharacteristic
139C2×Q8—24no—
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—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8C2×D412yes—
160C2×Q8C2×D412yes—
161C2×Q8#4112yes—
162C2×Q8#4112yes—
163C2×Q8#4112yes—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2411yescharacteristic
187Q16—28no—
188Q16—44no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8C2×D412yes—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—81no—
222C22⋊C4Q1614yescomplemented
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—42no—
229C42—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23C2×Q1612yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—22no—
245Q8—48no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8#12314yes—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
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—48no—
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—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×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4C2×Q1612yes—
294C2×C4C2×Q1612yes—
295C2×C4C22×D411yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C8—22no—
302C8—41no—
303C8—41no—
304C8—41no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22#52511yescharacteristic
313C22D4211yescharacteristic
314C22#54611yescharacteristic
315C22C22×Q1611yescharacteristic
316C22#57511yescharacteristic
317C22#57711yescharacteristic
318C22#57811yescharacteristic
319C4—24no—
320C4—22no—
321C4—22no—
322C4—41no—
323C4—42no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4C2×#12311yescharacteristic
333C4#53811yescharacteristic
334C4#54611yescharacteristic
335C4#55111yescharacteristic
336C2—42no—
337C2#241811yescharacteristic
338C2#310411yescharacteristic
339C2#312111yescharacteristic
340C2#327111yescharacteristic
341C2D4×Q1611yescharacteristic
342C2#329911yescharacteristic
343C2#332711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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. C2×C4×Q16C2semidirect
5. #2095C2semidirect
7. #2056C2semidirect
9. #1933C2semidirect
11. #1878C2semidirect
12. #1878C2semidirect
13. #1863C2semidirect
14. #1803C2semidirect
222. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
C2×C4×Q162
#24232
Q8×C22⋊C42
#20952
#20762
#20562
#20362
#19332
#19232
#18782
#18632
#18032
#17872