Finite Group Databasebeta

← #26210

Subgroups — #26210

Order 256.

408 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2417C211yesmaximal, characteristic
2#2352C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2075C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#1928C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic
13#1852C211yesmaximal, characteristic, complemented
14#1825C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32#398—21no—
33#396—21no—
34#391C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#387—21no—
38#386—21no—
39#386—21no—
40#382C2211yescharacteristic
41#380—21no—
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4—21no—
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—41no—
76C4×D4—22no—
77C4×D4—22no—
78#117—22no—
79#117—41no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—41no—
110#103—22no—
111#103—41no—
112#103—41no—
113#103—41no—
114C22⋊C8—22no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24C2×D411yescharacteristic
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4#4112yes—
166C22×C4#4112yes—
167C22×C4C2411yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—44no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4#4112yes—
200C4⋊C4#4112yes—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C42—21no—
210C42—21no—
211C42—42no—
212C42—81no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23#12312yes—
225C23#13512yes—
226C23#13712yes—
227C23C22×D411yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232D4—44no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
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—84no—
277C2×C4—84no—
278C2×C4#12312yes—
279C2×C4C2×#4111yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C8—41no—
286C8—41no—
287C8—41no—
288C8—41no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22#48514yes—
303C22C2×#12311yescharacteristic
304C22#54611yescharacteristic
305C22#54611yescharacteristic
306C22C2×#13511yescharacteristic
307C22C2×#13711yescharacteristic
308C22#57611yescharacteristic
309C22#57811yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4#52911yescharacteristic
323C4#53811yescharacteristic
324C4#55311yescharacteristic
325C4#55311yescharacteristic
326C2—24no—
327C2—81no—
328C2#244411yescharacteristic
329C2C2×#48511yescharacteristic
330C2#309811yescharacteristic
331C2#329711yescharacteristic
332C2#330011yescharacteristic
333C2#331211yescharacteristic
334C2#331411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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
3. #2101C2semidirect
4. #2075C2semidirect
5. #2074C2semidirect
6. #2036C2semidirect
7. #1928C2semidirect
8. #1918C2semidirect
9. #1892C2semidirect
10. #1873C2semidirect
11. #1872C2semidirect
13. #1852C2semidirect
14. #1825C2semidirect
15. C2×#340C2semidirect

Maximal subgroups

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

subgroupindex
#24172
#23522
#21012
#20752
#20742
#20362
#19282
#19182
#18922
#18732
#18722
#18692
#18522
#18252
C2×#3402