Finite Group Databasebeta

← #25967

Subgroups — #25967

Order 256.

502 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475C211yesmaximal, characteristic, complemented
2C2×#417C211yesmaximal, characteristic
3#2440C211yesmaximal, characteristic, complemented
4#2341C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1863C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#475—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30#417—24no—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#398—21no—
41#395—21no—
42#390—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#385—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
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41Q814yescomplemented
67C22×Q8—21no—
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83C22⋊Q8—44no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—41no—
87C4×Q8—41no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
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—41no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#107—41no—
117#106—22no—
118#106—22no—
119#106—41no—
120C4⋊C8—22no—
121C4⋊C8—41no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103D414yes—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130C22⋊C8—22no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95D411yescharacteristic
136#95D411yescharacteristic
137C24—21no—
138C24C2×D411yescharacteristic
139#41—28no—
140#41—44no—
141#41—44no—
142#41—44no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—81no—
149C2×Q8C2×D412yes—
150C2×Q8C2×Q812yes—
151C2×Q8#4112yes—
152C2×D4—24no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4C2×Q812yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—81no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×Q812yes—
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186M16—44no—
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—21no—
197C2×C8—21no—
198C2×C8C2×D412yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
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—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4C2×D412yes—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—42no—
232C42—42no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C22⋊Q812yes—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—84no—
252Q8C22⋊Q814yescomplemented
253D4—22no—
254D4—22no—
255D4—48no—
256D4—41no—
257D4—41no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—48no—
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×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—84no—
297C2×C4C22⋊Q812yes—
298C2×C4AGL(1,ℤ/8)12yes—
299C2×C4#13712yes—
300C2×C4C22×D411yescharacteristic
301C2×C4C22×Q811yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C8—22no—
307C8—41no—
308C8—41no—
309C8—41no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C2×C22⋊Q811yescharacteristic
325C22D4211yescharacteristic
326C22#54611yescharacteristic
327C22C2×AGL(1,ℤ/8)11yescharacteristic
328C22C2×#13711yescharacteristic
329C22#57711yescharacteristic
330C22#57711yescharacteristic
331C4—22no—
332C4—24no—
333C4—22no—
334C4—22no—
335C4—41no—
336C4—42no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4C2×C22⋊Q811yescharacteristic
344C4#54311yescharacteristic
345C4#54611yescharacteristic
346C4#55111yescharacteristic
347C2—42no—
348C2—82no—
349C2#243711yescharacteristic
350C2#307911yescharacteristic
351C2#308411yescharacteristic
352C2#326511yescharacteristic
353C2#327811yescharacteristic
354C2#331611yescharacteristic
355C2#333211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
1. C2×#475C2semidirect
3. #2440C2semidirect
5. #2094C2semidirect
6. #2067C2semidirect
7. #2065C2semidirect
8. #2023C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect
11. #1853C2semidirect
12. #1851C2semidirect
13. #1809C2semidirect
14. #1803C2semidirect
66. C2×#41Q8semidirect
252. Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4172
#24402
#23412
#20942
#20672
#20652
#20232
#18652
#18632
#18532
#18512
#18092
#18032
#17882