Finite Group Databasebeta

← #26676

Subgroups — #26676

Order 256.

401 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2420C211yesmaximal, characteristic
2#2360C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2000C211yesmaximal, characteristic
7#1935C211yesmaximal, characteristic, complemented
8#1924C211yesmaximal, characteristic, complemented
9#1918C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic
13#1824C211yesmaximal, characteristic, complemented
14#1805C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#397—21no—
34#394—21no—
35#391C2211yescharacteristic
36#390C2211yescharacteristic
37#389—21no—
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#387—21no—
41#386—21no—
42#385—21no—
43#382C2211yescharacteristic
44#381—21no—
45#380C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C22⋊C4—21no—
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74#121—22no—
75#121—41no—
76C4×Q8—22no—
77C4×Q8—41no—
78C4×D4—22no—
79C4×D4—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106C4⋊C8—41no—
107#103—22no—
108#103—41no—
109#103—41no—
110#102—22no—
111#102—22no—
112#102—41no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115#97—41no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8#4112yes—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—81no—
142C2×D4#4112yes—
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—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168SD16—44no—
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—21no—
181C4⋊C4—24no—
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—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4#4112yes—
197C4⋊C4#4112yes—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—81no—
204C22⋊C4—81no—
205C22⋊C4—81no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—42no—
211C42—81no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23#13712yes—
226C23C22×D411yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—41no—
230Q8—41no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
235D4—84no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
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×C4C2×#4111yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4H(5,𝔽2)11yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C8—41no—
285C8—41no—
286C8—41no—
287C8—41no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22#52511yescharacteristic
303C22#54611yescharacteristic
304C22#54611yescharacteristic
305C22C2×#13711yescharacteristic
306C22#57511yescharacteristic
307C22#57611yescharacteristic
308C22#57711yescharacteristic
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4#52911yescharacteristic
322C4#54011yescharacteristic
323C4#55111yescharacteristic
324C4#55311yescharacteristic
325C2—42no—
326C2—82no—
327C2#245711yescharacteristic
328C2#311711yescharacteristic
329C2#311911yescharacteristic
330C2#331411yescharacteristic
331C2#331911yescharacteristic
332C2#332011yescharacteristic
333C2#332311yescharacteristic
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
2. #2360C2semidirect
3. #2101C2semidirect
4. #2067C2semidirect
5. #2036C2semidirect
7. #1935C2semidirect
8. #1924C2semidirect
9. #1918C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
13. #1824C2semidirect
14. #1805C2semidirect
15. #1762C2semidirect

Maximal subgroups

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

subgroupindex
#24202
#23602
#21012
#20672
#20362
#20002
#19352
#19242
#19182
#18802
#18772
#18742
#18242
#18052
#17622