Finite Group Databasebeta

← #30120

Subgroups — #30120

Order 256.

391 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
1#2621C211yesmaximal, characteristic, complemented
2#2324C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic
10#2000C211yesmaximal, characteristic
11#1928C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1818C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#398—21no—
37#398—21no—
38#397—21no—
39#397—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#390—21no—
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#386—21no—
47#385—21no—
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—41no—
68C2×SD16—41no—
69C2×SD16—41no—
70C2×SD16—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C22⋊Q8—41no—
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81#121—22no—
82#121—22no—
83C4×D4—41no—
84C4×D4—41no—
85#117—41no—
86#117—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—41no—
112#106—41no—
113#103—22no—
114#103—22no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119#102—41no—
120#102—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×D4—24no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—81no—
153C2×D4—81no—
154C2×D4#4112yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—42no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177SD16—44no—
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—21no—
191C4⋊C4—21no—
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—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—81no—
207C4⋊C4#4112yes—
208C22⋊C4—24no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—81no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23C22×D411yescharacteristic
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245D4—22no—
246D4—22no—
247D4—41no—
248D4—41no—
249D4—84no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
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—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—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—84no—
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C8—41no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—41no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#52511yescharacteristic
316C22#53411yescharacteristic
317C22#53511yescharacteristic
318C22#57711yescharacteristic
319C22#57711yescharacteristic
320C22#57711yescharacteristic
321C22#57811yescharacteristic
322C4—42no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#53811yescharacteristic
335C4#53811yescharacteristic
336C4#54011yescharacteristic
337C4#54011yescharacteristic
338C2—81no—
339C2—82no—
340C2#263511yescharacteristic
341C2#312011yescharacteristic
342C2#312311yescharacteristic
343C2#320011yescharacteristic
344C2#320111yescharacteristic
345C2#322111yescharacteristic
346C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2621C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2043C2semidirect
6. #2036C2semidirect
7. #2035C2semidirect
8. #2013C2semidirect
11. #1928C2semidirect
12. #1927C2semidirect
13. #1877C2semidirect
14. #1877C2semidirect
15. #1818C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#23242
#21012
#20922
#20432
#20362
#20352
#20132
#20002
#19282
#19272
#18772
#18182