Finite Group Databasebeta

← #25334

Subgroups — #25334

Order 256.

464 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic, complemented
4#2367C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic
9#1987C211yesmaximal, characteristic, complemented
10#1940C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic
13#1803C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#489—24no—
26#463—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#399—21no—
40#399—21no—
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#388—21no—
46#388—21no—
47#385—21no—
48#384C2211yescharacteristic
49#384C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—41no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—41no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79#124—22no—
80#124D414yes—
81C22⋊Q8—44no—
82C22⋊Q8—44no—
83C4×Q8—22no—
84C4×Q8—41no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—41no—
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⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
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⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117C4⋊C8—22no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120#103—22no—
121#103—22no—
122#103—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#97—22no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C2×D412yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4C2×D412yes—
156C2×D4#4112yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—81no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2411yescharacteristic
176SD16—44no—
177D8—44no—
178C2×C8—21no—
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×C8—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
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—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4#4112yes—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—81no—
224C42C2×D412yes—
225C42#4112yes—
226C23—22no—
227C23—22no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C22×D411yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—41no—
244Q8—41no—
245Q8—84no—
246D4—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
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—84no—
288C2×C4#12112yes—
289C2×C4#12112yes—
290C2×C4AGL(1,ℤ/8)12yes—
291C2×C4AGL(1,ℤ/8)12yes—
292C2×C4C22×D411yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C2×C42−1+411yescharacteristic
298C8—41no—
299C8—41no—
300C8—41no—
301C8—41no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22C2×#12111yescharacteristic
317C22#54611yescharacteristic
318C22#54811yescharacteristic
319C22C2×AGL(1,ℤ/8)11yescharacteristic
320C22C2×AGL(1,ℤ/8)11yescharacteristic
321C22#57711yescharacteristic
322C22#57811yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—41no—
329C4—42no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4#52911yescharacteristic
336C4#54211yescharacteristic
337C4#54611yescharacteristic
338C4#54711yescharacteristic
339C2—82no—
340C2—82no—
341C2#242711yescharacteristic
342C2#303811yescharacteristic
343C2#304611yescharacteristic
344C2#330611yescharacteristic
345C2#330811yescharacteristic
346C2#336711yescharacteristic
347C2#337311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#489C2semidirect
3. #2542C2semidirect
4. #2367C2semidirect
5. #2072C2semidirect
6. #2071C2semidirect
7. #2055C2semidirect
9. #1987C2semidirect
10. #1940C2semidirect
11. #1880C2semidirect
13. #1803C2semidirect
14. #1802C2semidirect
15. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4632
#25422
#23672
#20722
#20712
#20552
#20542
#19872
#19402
#18802
#18792
#18032
#18022
#17622