Finite Group Databasebeta

← #23711

Subgroups — #23711

Order 256.

479 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
1C2×#440C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3C4×C22⋊Q8C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic
5#2086C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#2055C211yesmaximal, characteristic, complemented
10#2036C211yesmaximal, characteristic, complemented
11#2034C211yesmaximal, characteristic
12#2000C211yesmaximal, characteristic
13#1776C211yesmaximal, characteristic, complemented
14#1763C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#440—24no—
24#432—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#387—21no—
46#386—21no—
47#385—21no—
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#121—22no—
78#121—22no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—44no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
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×C42—22no—
102C2×C42—22no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42D411yescharacteristic
107C2×C42D411yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107—22no—
112#106—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123#97—22no—
124#97D414yescomplemented
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8#4112yes—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4#4112yes—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2411yescharacteristic
178SD16—28no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8C2×D412yes—
190C2×C8C2×D412yes—
191C4⋊C4—24no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4#4112yes—
206C4⋊C4#4112yes—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C42—22no—
215C42—22no—
216C42—21no—
217C42—21no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42C2×D412yes—
222C42#4112yes—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23#13712yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—41no—
241Q8—41no—
242D4—22no—
243D4—22no—
244D4—48no—
245D4—41no—
246D4—41no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
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—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—44no—
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—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—84no—
287C2×C4#12112yes—
288C2×C4#12112yes—
289C2×C4AGL(1,ℤ/8)12yes—
290C2×C4C22×D411yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C8—22no—
297C8—22no—
298C8—41no—
299C8—41no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22C2×#12111yescharacteristic
315C22#52511yescharacteristic
316C22#52511yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22C2×#13711yescharacteristic
319C22#57511yescharacteristic
320C22#57511yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4#52511yescharacteristic
334C4#52911yescharacteristic
335C4#52911yescharacteristic
336C4#53211yescharacteristic
337C2—42no—
338C2—82no—
339C2#239211yescharacteristic
340C2#305511yescharacteristic
341C2#305611yescharacteristic
342C2#311011yescharacteristic
343C2#311111yescharacteristic
344C2#311511yescharacteristic
345C2#311711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#440C2semidirect
2. C2×#432C2semidirect
3. C4×C22⋊Q8C2semidirect
5. #2086C2semidirect
6. #2076C2semidirect
7. #2067C2semidirect
8. #2066C2semidirect
9. #2055C2semidirect
10. #2036C2semidirect
13. #1776C2semidirect
14. #1763C2semidirect
15. #1762C2semidirect
124. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
C2×#4322
C4×C22⋊Q82
C4×#1212
#20862
#20762
#20672
#20662
#20552
#20362
#20342
#20002
#17762
#17632
#17622