Finite Group Databasebeta

← #25202

Subgroups — #25202

Order 256.

471 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic
2C2×#465C211yesmaximal, characteristic
3#2502C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic
5#2084C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1927C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#481—24no—
25#465—24no—
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#398—21no—
39#398—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#386—21no—
46#385—21no—
47#385—21no—
48#385—21no—
49#384C2211yescharacteristic
50#384C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—41no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80#121—22no—
81#121D414yes—
82C4×D4—22no—
83C4×D4—41no—
84C4×D4—44no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
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—41no—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#106—22no—
114#106—22no—
115#106—22no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118#103—22no—
119#103—22no—
120#103—41no—
121#102—22no—
122#102—22no—
123#102—22no—
124C22⋊C8—22no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
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×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—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D412yes—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4#4112yes—
177C22×C4C2411yescharacteristic
178SD16—44no—
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—
190C2×C8—44no—
191C2×C8#4112yes—
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—42no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4C2×D412yes—
211C4⋊C4#4112yes—
212C22⋊C4—22no—
213C22⋊C4—24no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C22⋊C4SD1614yescomplemented
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23#12112yes—
236C23C2×SD1612yes—
237C23#13712yes—
238C23C22×D411yescharacteristic
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243D4—22no—
244D4—22no—
245D4—48no—
246D4—41no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
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—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4#12112yes—
289C2×C4C2×SD1612yes—
290C2×C4C2×SD1612yes—
291C2×C4C22×D411yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—22no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22#46114yes—
315C22C2×#12111yescharacteristic
316C22#54611yescharacteristic
317C22#54711yescharacteristic
318C22C22×SD1611yescharacteristic
319C22C2×#13711yescharacteristic
320C22#57611yescharacteristic
321C22#57711yescharacteristic
322C4—22no—
323C4—22no—
324C4—22no—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#52911yescharacteristic
335C4#53911yescharacteristic
336C4#54611yescharacteristic
337C4#54711yescharacteristic
338C2—24no—
339C2—82no—
340C2#242511yescharacteristic
341C2C2×#46111yescharacteristic
342C2#304511yescharacteristic
343C2#329511yescharacteristic
344C2#331611yescharacteristic
345C2#333511yescharacteristic
346C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
3. #2502C2semidirect
5. #2084C2semidirect
6. #2071C2semidirect
7. #2066C2semidirect
8. #2055C2semidirect
9. #1948C2semidirect
10. #1927C2semidirect
11. #1877C2semidirect
12. #1872C2semidirect
13. #1850C2semidirect
14. #1802C2semidirect
15. C2×#340C2semidirect
218. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4652
#25022
#23612
#20842
#20712
#20662
#20552
#19482
#19272
#18772
#18722
#18502
#18022
C2×#3402