Finite Group Databasebeta

← #31778

Subgroups — #31778

Order 256.

369 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2485C211yesmaximal, characteristic
2#2417C211yesmaximal, characteristic
3#2087C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2066C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1954C211yesmaximal, characteristic
10#1941C211yesmaximal, characteristic, complemented
11#1933C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1823C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#398—21no—
36#397—21no—
37#396—21no—
38#394—21no—
39#391C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#387—21no—
45#386—21no—
46#380—21no—
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22⋊Q8—22no—
69C22⋊Q8—41no—
70C22⋊Q8—41no—
71C22⋊Q8—41no—
72C4×D4—41no—
73C4×D4—41no—
74#117—41no—
75#117—41no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4D411yescharacteristic
100C2×C22⋊C4D411yescharacteristic
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103#107—41no—
104#106—22no—
105#106—41no—
106#106—41no—
107C4⋊C8—41no—
108#103—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C4×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—41no—
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—21no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—81no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
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#4112yes—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
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—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4#4112yes—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—81no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C42—21no—
208C42—21no—
209C42—42no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23#13512yes—
224C23C22×D411yescharacteristic
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229D4—44no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
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—84no—
274C2×C4#13712yes—
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—22no—
282C8—41no—
283C8—41no—
284C8—41no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22#53411yescharacteristic
300C22#54611yescharacteristic
301C22#54711yescharacteristic
302C22C2×#13511yescharacteristic
303C22C2×#13711yescharacteristic
304C22#57711yescharacteristic
305C22#57711yescharacteristic
306C4—22no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—42no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4#54011yescharacteristic
319C4#54211yescharacteristic
320C4#55311yescharacteristic
321C4#56211yescharacteristic
322C2—42no—
323C2—81no—
324C2#270111yescharacteristic
325C2#319011yescharacteristic
326C2#320411yescharacteristic
327C2#329611yescharacteristic
328C2#332911yescharacteristic
329C2#333711yescharacteristic
330C2#335611yescharacteristic
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. #2087C2semidirect
4. #2077C2semidirect
5. #2066C2semidirect
6. #2036C2semidirect
7. #2035C2semidirect
8. #2035C2semidirect
10. #1941C2semidirect
11. #1933C2semidirect
12. #1873C2semidirect
13. #1854C2semidirect
14. #1853C2semidirect
15. #1823C2semidirect

Maximal subgroups

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

subgroupindex
#24852
#24172
#20872
#20772
#20662
#20362
#20352
#19542
#19412
#19332
#18732
#18542
#18532
#18232