Finite Group Databasebeta

← #27655

Subgroups — #27655

Order 256.

414 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic
3#2324C211yesmaximal, characteristic
4#2317C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2073C211yesmaximal, characteristic, complemented
8#2071C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1825C211yesmaximal, characteristic, complemented
12#1824C211yesmaximal, characteristic, complemented
13#1804C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1798C211yesmaximal, characteristic, complemented
16C2×#121C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#482—24no—
22#480—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#390—21no—
36#390C2211yescharacteristic
37#389—21no—
38#389—21no—
39#389C2211yescharacteristic
40#388—21no—
41#386—21no—
42#385—21no—
43#385—21no—
44#385—21no—
45#384C2211yescharacteristic
46#384C2211yescharacteristic
47#382C2211yescharacteristic
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70#121—22no—
71#121D414yes—
72C4×D4—22no—
73C4×D4—22no—
74#117—22no—
75#117—22no—
76#117—22no—
77#117—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⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#107—22no—
110#106—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117C22⋊C8—22no—
118C22⋊C8—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—81no—
135C2×D4—81no—
136C2×D4C2×D412yes—
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—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4#4112yes—
161C22×C4C2411yescharacteristic
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—21no—
173C2×C8—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
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—42no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4C2×D412yes—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—81no—
208C42#4112yes—
209C42#4112yes—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23#12112yes—
224C23C22×D411yescharacteristic
225D4—22no—
226D4—22no—
227D4—41no—
228D4—41no—
229D4—84no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—41no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4#12112yes—
270C2×C4#12312yes—
271C2×C4#12312yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C42−1+411yescharacteristic
277C2×C42−1+411yescharacteristic
278C8—41no—
279C8—41no—
280C8—41no—
281C8—41no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22C2×#12111yescharacteristic
297C22C2×#12311yescharacteristic
298C22#53611yescharacteristic
299C22#57611yescharacteristic
300C22#57611yescharacteristic
301C22#57711yescharacteristic
302C22#57811yescharacteristic
303C4—22no—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4#54711yescharacteristic
316C4#54711yescharacteristic
317C4#55211yescharacteristic
318C4#55211yescharacteristic
319C2—42no—
320C2—82no—
321C2#248611yescharacteristic
322C2#304111yescharacteristic
323C2#304611yescharacteristic
324C2#309611yescharacteristic
325C2#309811yescharacteristic
326C2#324111yescharacteristic
327C2#324311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2317C2semidirect
5. #2085C2semidirect
6. #2083C2semidirect
7. #2073C2semidirect
8. #2071C2semidirect
9. #1949C2semidirect
10. #1947C2semidirect
11. #1825C2semidirect
12. #1824C2semidirect
13. #1804C2semidirect
14. #1802C2semidirect
15. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4802
#23242
#23172
#20852
#20832
#20732
#20712
#19492
#19472
#18252
#18242
#18042
#18022
#17982