Finite Group Databasebeta

← #29762

Subgroups — #29762

Order 256.

419 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic
3#2850C211yesmaximal, characteristic
4#2296C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2085C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2067C211yesmaximal, characteristic, complemented
9#2066C211yesmaximal, characteristic, complemented
10#2065C211yesmaximal, characteristic, complemented
11#2042C211yesmaximal, characteristic, complemented
12#2042C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic
14#1873C211yesmaximal, characteristic, complemented
15#1797C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#485—24no—
23#482—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#388—21no—
48#386—21no—
49#386—21no—
50#385—21no—
51#385—21no—
52#382C2211yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#124—41no—
72#124—41no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75#121—22no—
76#121—22no—
77#117—22no—
78#117—41no—
79#117—41no—
80#117D414yes—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
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⋊C4—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#106—22no—
111#106—22no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114#103—22no—
115#103—22no—
116#103—41no—
117#103—41no—
118#102—22no—
119#102—22no—
120#102—41no—
121#102—41no—
122C22⋊C8—22no—
123C22⋊C8—22no—
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×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D411yescharacteristic
172C22×C4#4112yes—
173C22×C4C2411yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—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—21no—
195C4⋊C4—21no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4C2×D412yes—
202C4⋊C4C2×D412yes—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42—24no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—41no—
218C42—42no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23#12112yes—
233C23C22×D411yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
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—44no—
263C2×C4—44no—
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—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4#12112yes—
284C2×C4#13512yes—
285C2×C4C22×D411yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4H(5,𝔽2)11yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C42−1+411yescharacteristic
290C2×C42−1+411yescharacteristic
291C8—41no—
292C8—41no—
293C8—41no—
294C8—41no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22C2×#12111yescharacteristic
310C22#53411yescharacteristic
311C22#53611yescharacteristic
312C22C2×#13511yescharacteristic
313C22#57511yescharacteristic
314C22#57511yescharacteristic
315C22#57511yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4#53711yescharacteristic
329C4#53711yescharacteristic
330C4#53911yescharacteristic
331C4#53911yescharacteristic
332C2—42no—
333C2—161no—
334C2#261811yescharacteristic
335C2#303511yescharacteristic
336C2#304011yescharacteristic
337C2#319111yescharacteristic
338C2#319711yescharacteristic
339C2#323311yescharacteristic
340C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#485C2semidirect
4. #2296C2semidirect
5. #2085C2semidirect
6. #2085C2semidirect
7. #2068C2semidirect
8. #2067C2semidirect
9. #2066C2semidirect
10. #2065C2semidirect
11. #2042C2semidirect
12. #2042C2semidirect
14. #1873C2semidirect
15. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4822
#28502
#22962
#20852
#20682
#20672
#20662
#20652
#20422
#18752
#18732
#17972