Finite Group Databasebeta

← #23681

Subgroups — #23681

Order 256.

440 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q16C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3C4×#124C211yesmaximal, characteristic
4#2295C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2066C211yesmaximal, characteristic, complemented
9#2056C211yesmaximal, characteristic, complemented
10#2055C211yesmaximal, characteristic
11#2042C211yesmaximal, characteristic
12#2015C211yesmaximal, characteristic
13C4×C4⋊C8C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C4×Q16—24no—
23C4×SD16—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#387—21no—
48#382C2211yescharacteristic
49C4×C4⋊C4—21no—
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53#376—21no—
54C43C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#124—22no—
76#124—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8D414yes—
81C4×D4—22no—
82C4×D4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
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×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42D411yescharacteristic
101C2×C42D411yescharacteristic
102C2×C42D411yescharacteristic
103C2×C42D411yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119C4×C8—22no—
120C4×C8—22no—
121#95—21no—
122#95—21no—
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—
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×Q8C2×D412yes—
144C2×Q8#4112yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4#4112yes—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168Q16—28no—
169SD16—28no—
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#4112yes—
181C2×C8#4112yes—
182C4⋊C4—24no—
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—42no—
195C4⋊C4C2×D412yes—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C42—24no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—42no—
211C42—42no—
212C42C2×D412yes—
213C42#4112yes—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23C22×D411yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—22no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230D4—22no—
231D4—22no—
232D4—41no—
233D4—41no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
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—84no—
269C2×C4#12112yes—
270C2×C4#12112yes—
271C2×C4#13512yes—
272C2×C4#13512yes—
273C2×C4#13712yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C8—22no—
281C8—22no—
282C8—41no—
283C8—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22C2×#12111yescharacteristic
292C22#52511yescharacteristic
293C22#53211yescharacteristic
294C22C2×#13511yescharacteristic
295C22C2×#13511yescharacteristic
296C22C2×#13711yescharacteristic
297C22#57511yescharacteristic
298C4—22no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4#46414yes—
311C4#52511yescharacteristic
312C4#52511yescharacteristic
313C4#52911yescharacteristic
314C4#52911yescharacteristic
315C2—82no—
316C2#239211yescharacteristic
317C2C2×#46411yescharacteristic
318C2#303511yescharacteristic
319C2#310211yescharacteristic
320C2#310711yescharacteristic
321C2#316811yescharacteristic
322C2#317211yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×Q16C2semidirect
4. #2295C2semidirect
5. #2085C2semidirect
6. #2074C2semidirect
8. #2066C2semidirect
9. #2056C2semidirect
13. C4×C4⋊C8C2semidirect
14. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q162
C2×C4×SD162
C4×#1242
#22952
#20852
#20742
#20672
#20662
#20562
#20552
#20422
#20152
C4×C4⋊C82
C4×#1032
C4×#1022