Finite Group Databasebeta

← #26372

Subgroups — #26372

Order 256.

432 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic
3#2451C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2085C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#1942C211yesmaximal, characteristic, complemented
10#1916C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic
14#1863C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#474—24no—
23C4×SD16—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#398—21no—
38#395—21no—
39#391—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#387—21no—
47#385—21no—
48#385—21no—
49#382C2211yescharacteristic
50#380—21no—
51C4×C4⋊C4C2211yescharacteristic
52C2×#95—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16Q814yescomplemented
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C4×Q8—22no—
77C4×Q8—41no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—41no—
83C4×D4—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—22no—
107#107—22no—
108#106—22no—
109#106—41no—
110C4⋊C8—22no—
111#103—22no—
112#103—22no—
113#103—41no—
114#102—22no—
115#102—22no—
116#102D414yes—
117C4×C8—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95D411yescharacteristic
129#95D411yescharacteristic
130C24—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8C2×Q812yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C2×D412yes—
144C2×D4C2×Q812yes—
145C2×D4#4112yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2411yescharacteristic
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×C8C2×D412yes—
181C2×C8C2×Q812yes—
182C4⋊C4—22no—
183C4⋊C4—22no—
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—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4C2×D412yes—
201C4⋊C4#4112yes—
202C22⋊C4—24no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—42no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—41no—
223Q8—41no—
224Q8—84no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4C22⋊Q814yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
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—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4#13512yes—
270C2×C4#13712yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C22×Q811yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C42−1+411yescharacteristic
277C8—22no—
278C8—22no—
279C8—41no—
280C8—41no—
281C22—24no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22#52511yescharacteristic
289C22D4211yescharacteristic
290C22D4×Q811yescharacteristic
291C22C2×#13511yescharacteristic
292C22C2×#13711yescharacteristic
293C22#57611yescharacteristic
294C22#57711yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—42no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4C2×C22⋊Q811yescharacteristic
309C4#53911yescharacteristic
310C4#54711yescharacteristic
311C4#55411yescharacteristic
312C2—44no—
313C2#244911yescharacteristic
314C2#310711yescharacteristic
315C2#311811yescharacteristic
316C2#326611yescharacteristic
317C2#328011yescharacteristic
318C2#338111yescharacteristic
319C2#338511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. #2451C2semidirect
5. #2097C2semidirect
6. #2085C2semidirect
7. #2068C2semidirect
8. #2058C2semidirect
9. #1942C2semidirect
10. #1916C2semidirect
14. #1863C2semidirect
69. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×C4×SD162
#24512
#23492
#20972
#20852
#20682
#20582
#19422
#19162
#18772
#18642
#18632
#18622