Finite Group Databasebeta

← #24933

Subgroups — #24933

Order 256.

427 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2486C211yesmaximal, characteristic
2D4×C4⋊C4C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2083C211yesmaximal, characteristic, complemented
5#2075C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1869C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic, complemented
11#1852C211yesmaximal, characteristic, complemented
12#1823C211yesmaximal, characteristic, complemented
13#1823C211yesmaximal, characteristic, complemented
14#1808C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#123—21no—
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#398—21no—
31#397—21no—
32#389—21no—
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#385—21no—
36#385—21no—
37#384C2211yescharacteristic
38#381—21no—
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4D411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#123—44no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#107—41no—
110#107—41no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—41no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122C24C2×D411yescharacteristic
123C24C2×D411yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4D814yes—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4#4112yes—
159C22×C4#4112yes—
160C22×C4C2411yescharacteristic
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—44no—
173C2×C8#4112yes—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4#4112yes—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—42no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23#12312yes—
219C23C2×D812yes—
220C23C2×D812yes—
221C23#13712yes—
222C23C22×D411yescharacteristic
223D4—28no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
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—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#12312yes—
270C2×C4C2×D812yes—
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C42−1+411yescharacteristic
277C8—22no—
278C8—41no—
279C8—41no—
280C8—41no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22#48014yes—
295C22C2×#12311yescharacteristic
296C22#54611yescharacteristic
297C22#54711yescharacteristic
298C22C22×D811yescharacteristic
299C22C2×#13711yescharacteristic
300C22#57711yescharacteristic
301C22#57811yescharacteristic
302C4—22no—
303C4—22no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4#52911yescharacteristic
315C4#53911yescharacteristic
316C4#55111yescharacteristic
317C4#55211yescharacteristic
318C2—24no—
319C2—42no—
320C2#241911yescharacteristic
321C2C2×#48011yescharacteristic
322C2#309711yescharacteristic
323C2#329411yescharacteristic
324C2#331211yescharacteristic
325C2#333411yescharacteristic
326C2#334411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2099C2semidirect
4. #2083C2semidirect
5. #2075C2semidirect
6. #2075C2semidirect
7. #1918C2semidirect
8. #1918C2semidirect
10. #1852C2semidirect
11. #1852C2semidirect
12. #1823C2semidirect
13. #1823C2semidirect
14. #1808C2semidirect
15. C2×#340C2semidirect

Maximal subgroups

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

subgroupindex
#24862
D4×C4⋊C42
#20992
#20832
#20752
#19182
#18692
#18522
#18232
#18082
C2×#3402