Finite Group Databasebeta

← #25481

Subgroups — #25481

Order 256.

385 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474C211yesmaximal, characteristic
2C2×#446C211yesmaximal, characteristic, complemented
3#2580C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2065C211yesmaximal, characteristic
9#1993C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1944C211yesmaximal, characteristic, complemented
12#1942C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic
14#1934C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#474—24no—
20#446—24no—
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×#97C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#399—21no—
36#395—21no—
37#395—21no—
38#393—21no—
39#391—21no—
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#385—21no—
49#385—21no—
50#384—21no—
51#382C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8D411yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×D4—22no—
74C4×D4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#107—22no—
104#106—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8D414yes—
108#102—22no—
109#102—22no—
110#102—22no—
111#102Q814yescomplemented
112#97—22no—
113#97—22no—
114C4×C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124C24—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—41no—
128C2×Q8—42no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C2×Q812yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8C2×D412yes—
164C2×C8C2×D412yes—
165C2×C8C2×Q812yes—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4C2×Q812yes—
189C4⋊C4#4112yes—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—42no—
200C42C2×D412yes—
201C42#4112yes—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23C22×D411yescharacteristic
210Q8—44no—
211Q8—84no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—84no—
253C2×C4#12112yes—
254C2×C4#12112yes—
255C2×C4#13712yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×Q811yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C42−1+411yescharacteristic
261C2×C42−1+411yescharacteristic
262C8—22no—
263C8—22no—
264C8—22no—
265C8—41no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22C2×#12111yescharacteristic
274C22#54811yescharacteristic
275C22D4×Q811yescharacteristic
276C22C2×#13711yescharacteristic
277C22#57511yescharacteristic
278C22#57611yescharacteristic
279C22#57611yescharacteristic
280C4—22no—
281C4—22no—
282C4—22no—
283C4—42no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#53311yescharacteristic
294C4#54111yescharacteristic
295C4#54711yescharacteristic
296C4D4×Q811yescharacteristic
297C2—82no—
298C2#242911yescharacteristic
299C2#305611yescharacteristic
300C2#305811yescharacteristic
301C2#337011yescharacteristic
302C2#337511yescharacteristic
303C2#338511yescharacteristic
304C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#446C2semidirect
4. #2567C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
9. #1993C2semidirect
10. #1949C2semidirect
11. #1944C2semidirect
12. #1942C2semidirect
111. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4462
#25802
#25672
#20872
#20862
#20672
#20652
#19932
#19492
#19442
#19422
#19362
#19342
#17622