Finite Group Databasebeta

← #24764

Subgroups — #24764

Order 256.

452 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic
2C2×#467C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#424C211yesmaximal, characteristic
5#2621C211yesmaximal, characteristic
6#2323C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#2056C211yesmaximal, characteristic, complemented
10#2056C211yesmaximal, characteristic, complemented
11#1981C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1894C211yesmaximal, characteristic
14#1892C211yesmaximal, characteristic
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#470—24no—
27#467—24no—
28#462—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31#424—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44#398—21no—
45#398—21no—
46#397—21no—
47#397—21no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic
53#388—21no—
54#388—21no—
55#387—21no—
56#382C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8D414yes—
91C4×Q8—22no—
92C4×Q8—22no—
93#117—22no—
94#117—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#106—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8D414yes—
125#103—22no—
126#103—22no—
127#103—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#103—41no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24C2×D411yescharacteristic
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—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8C2×D412yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
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×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4#4112yes—
182C22×C4C2411yescharacteristic
183Q16—44no—
184Q16—44no—
185M16—44no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—44no—
197C2×C8C2×D412yes—
198C2×C8C2×D412yes—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—22no—
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⋊C4C2×D412yes—
218C22⋊C4—24no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—81no—
228C42C2×D412yes—
229C42#4112yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#12112yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
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—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#12112yes—
291C2×C4#12112yes—
292C2×C4#12112yes—
293C2×C4C22×D411yescharacteristic
294C2×C4C22×D411yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C2×C42−1+411yescharacteristic
299C8—22no—
300C8—22no—
301C8—41no—
302C8—41no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22C2×#12111yescharacteristic
311C22C2×#12111yescharacteristic
312C22#53711yescharacteristic
313C22#57711yescharacteristic
314C22#57811yescharacteristic
315C22#57811yescharacteristic
316C22#57811yescharacteristic
317C4—22no—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4D4211yescharacteristic
331C4#54711yescharacteristic
332C4#54811yescharacteristic
333C4#55311yescharacteristic
334C2—42no—
335C2#241611yescharacteristic
336C2#304411yescharacteristic
337C2#304611yescharacteristic
338C2#305911yescharacteristic
339C2#306311yescharacteristic
340C2#325811yescharacteristic
341C2#326111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#462C2semidirect
7. #2068C2semidirect
8. #2066C2semidirect
9. #2056C2semidirect
10. #2056C2semidirect
11. #1981C2semidirect
12. #1949C2semidirect
15. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4672
C2×#4622
C2×#4242
#26212
#23232
#20682
#20662
#20562
#19812
#19492
#18942
#18922
#18052