Finite Group Databasebeta

← #28412

Subgroups — #28412

Order 256.

476 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3C2×#464C211yesmaximal, characteristic, complemented
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2565C211yesmaximal, characteristic, complemented
6#2544C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2094C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic
11#1973C211yesmaximal, characteristic, complemented
12#1971C211yesmaximal, characteristic, complemented
13#1961C211yesmaximal, characteristic
14#1959C211yesmaximal, characteristic, complemented
15#1770C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#488—24no—
26#478—24no—
27#464—24no—
28C4×SD16—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#398—22no—
47#394—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#387—22no—
53#386—22no—
54#384C2211yescharacteristic
55#382C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—22no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—41no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#127—22no—
79#127—41no—
80#125—22no—
81#125—22no—
82#124—22no—
83#124—22no—
84#124—22no—
85#124—41no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×D4—22no—
89C4×D4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4D412yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#106—22no—
110#106—22no—
111C4⋊C8D414yescomplemented
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#97—41no—
123#97—41no—
124C4×C8D414yescomplemented
125#95—22no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95D412yes—
130C24—21no—
131C24—21no—
132C2×Q8—21no—
133C2×Q8—21no—
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#4112yes—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4—82no—
158C2×D4#4112yes—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—82no—
167C22×C4C2×D412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172Q16—44no—
173SD16—28no—
174SD16—44no—
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×C8C2×D412yes—
184C2×C8C2×D412yes—
185C2×C8C2×D412yes—
186C2×C8C2×D412yes—
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—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—44no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42C2×D412yes—
211C42C2×D412yes—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23C22×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—88no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
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—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4#12712yes—
264C2×C4#12712yes—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C2×C42−1+411yescharacteristic
271C8—22no—
272C8—22no—
273C8—22no—
274C8—22no—
275C22—42no—
276C22—44no—
277C22—82no—
278C22—81no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—82no—
285C22C2×#12711yescharacteristic
286C22#54611yescharacteristic
287C22#54811yescharacteristic
288C22#57512yes—
289C22#57712yes—
290C4—22no—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—42no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4#52511yescharacteristic
301C4#53711yescharacteristic
302C4#54611yescharacteristic
303C4#54811yescharacteristic
304C2—82no—
305C2—161no—
306C2#254011yescharacteristic
307C2#315011yescharacteristic
308C2#315411yescharacteristic
309C2#332312yes—
310C2#337112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#488C2semidirect
2. C2×#478C2semidirect
3. C2×#464C2semidirect
4. C2×C4×SD16C2semidirect
5. #2565C2semidirect
7. #2097C2semidirect
8. #2094C2semidirect
9. #2015C2semidirect
11. #1973C2semidirect
12. #1971C2semidirect
14. #1959C2semidirect
15. #1770C2semidirect
111. C4⋊C8D4semidirect
124. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4782
C2×#4642
C2×C4×SD162
#25652
#25442
#20972
#20942
#20152
#20142
#19732
#19712
#19612
#19592
#17702