Finite Group Databasebeta

← #25019

Subgroups — #25019

Order 256.

448 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3#2535C211yesmaximal, characteristic
4#2352C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1954C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1853C211yesmaximal, characteristic, complemented
13#1846C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C2×#121C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#482—24no—
23#480—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#397—21no—
41#392—21no—
42#388—21no—
43#387—21no—
44#386—21no—
45#386—21no—
46#386—21no—
47#382C2211yescharacteristic
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4D411yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#121—22no—
74#121—22no—
75#121—41no—
76#121—41no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81#117—22no—
82#117—41no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#107—22no—
110#106—22no—
111#106—41no—
112#106—41no—
113C4⋊C8—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102D414yescomplemented
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C4×C8—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C24C2×D411yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—41no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4C2×D412yes—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
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—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2411yescharacteristic
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×C8—21no—
190C2×C8C2×D412yes—
191C2×C8#4112yes—
192C4⋊C4—22no—
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—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—42no—
207C4⋊C4C2×D412yes—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C42—42no—
219C42—81no—
220C42#4112yes—
221C42#4112yes—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23#13512yes—
243C23C22×D411yescharacteristic
244D4—22no—
245D4—22no—
246D4—44no—
247D4—41no—
248D4—41no—
249D4—84no—
250D4—84no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—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—84no—
291C2×C4—84no—
292C2×C4#12312yes—
293C2×C4#12312yes—
294C2×C4C22×D411yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C42−1+411yescharacteristic
300C8—22no—
301C8—22no—
302C8—41no—
303C8—41no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#12311yescharacteristic
326C22D4211yescharacteristic
327C22#54711yescharacteristic
328C22C2×#13511yescharacteristic
329C22#57511yescharacteristic
330C22#57611yescharacteristic
331C22#57711yescharacteristic
332C4—22no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—42no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4#53211yescharacteristic
344C4#54211yescharacteristic
345C4#54711yescharacteristic
346C4#55311yescharacteristic
347C2—42no—
348C2—81no—
349C2—82no—
350C2#242211yescharacteristic
351C2#309111yescharacteristic
352C2#309611yescharacteristic
353C2#327511yescharacteristic
354C2#328011yescharacteristic
355C2#333711yescharacteristic
356C2#335511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. C2×#480C2semidirect
4. #2352C2semidirect
5. #2076C2semidirect
6. #2075C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #1954C2semidirect
10. #1949C2semidirect
11. #1854C2semidirect
12. #1853C2semidirect
13. #1846C2semidirect
14. #1804C2semidirect
15. C4×#102C2semidirect
117. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4802
#25352
#23522
#20762
#20752
#20342
#19542
#19492
#18542
#18532
#18462
#18042
C4×#1022