Finite Group Databasebeta

← #26182

Subgroups — #26182

Order 256.

449 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#461C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic
3#2446C211yesmaximal, characteristic, complemented
4D4×C4⋊C4C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2072C211yesmaximal, characteristic, complemented
8#2043C211yesmaximal, characteristic, complemented
9#1935C211yesmaximal, characteristic, complemented
10#1917C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic
13#1877C211yesmaximal, characteristic
14#1863C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4—21no—
24#461—24no—
25#440—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#400—21no—
39#398—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#387—21no—
47#387—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380—21no—
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16D414yes—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—41no—
79C4×Q8—22no—
80C4×Q8—41no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—41no—
86C4×D4—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107—41no—
112C4⋊C8—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—41no—
119#102—22no—
120#102—22no—
121#97—22no—
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#95D411yescharacteristic
132#95D411yescharacteristic
133C24—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—21no—
144C2×Q8—21no—
145C2×Q8C2×D412yes—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C2×D412yes—
150C2×D4#4112yes—
151C2×D4#4112yes—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2411yescharacteristic
175SD16—28no—
176SD16—44no—
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—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8C2×D412yes—
189C4⋊C4—22no—
190C4⋊C4—22no—
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—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—81no—
208C4⋊C4#4112yes—
209C22⋊C4—24no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—42no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23C22×D411yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4#12314yes—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—48no—
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—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—84no—
278C2×C4—84no—
279C2×C4AGL(1,ℤ/8)12yes—
280C2×C4#13712yes—
281C2×C4C22×D411yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C2×C42−1+411yescharacteristic
287C8—22no—
288C8—41no—
289C8—41no—
290C8—41no—
291C22—24no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22#52511yescharacteristic
299C22D4211yescharacteristic
300C22#54811yescharacteristic
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22C2×#13711yescharacteristic
303C22#57711yescharacteristic
304C22#57811yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4C2×#12311yescharacteristic
319C4#53911yescharacteristic
320C4#54711yescharacteristic
321C4#55311yescharacteristic
322C2—44no—
323C2#244311yescharacteristic
324C2#311011yescharacteristic
325C2#312011yescharacteristic
326C2#326511yescharacteristic
327C2#327611yescharacteristic
328C2#336711yescharacteristic
329C2#337211yescharacteristic
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. #2446C2semidirect
5. #2095C2semidirect
6. #2086C2semidirect
7. #2072C2semidirect
8. #2043C2semidirect
9. #1935C2semidirect
10. #1917C2semidirect
14. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×#4402
#24462
D4×C4⋊C42
#20952
#20862
#20722
#20432
#19352
#19172
#18772
#18632
#18622