Finite Group Databasebeta

← #22551

Subgroups — #22551

Order 256.

529 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3480C211yesmaximal, characteristic
2Q8×M16C211yesmaximal, characteristic, complemented
3#2295C211yesmaximal, characteristic, complemented
4#1980C211yesmaximal, characteristic, complemented
5#1980C212yesmaximal, complemented
6#1963C212yesmaximal, complemented
7#1962C211yesmaximal, characteristic, complemented
8#1806C211yesmaximal, characteristic, complemented
9#1806C212yesmaximal, complemented
10#1759C212yesmaximal, complemented
11C4×#104C211yesmaximal, characteristic, complemented
12#559C412yescomplemented
13#559C411yescharacteristic, complemented
14#557C411yescharacteristic, complemented
15#556C411yescharacteristic, complemented
16#556C412yescomplemented
17#553—22no—
18#553—21no—
19#550C411yescharacteristic, complemented
20#548—21no—
21#532C2211yescharacteristic
22#532C2212yes—
23#529—21no—
24#529—22no—
25C4×#41—21no—
26C4×#41C2211yescharacteristic
27C4×#41C2212yes—
28C2×C4×Q8C2211yescharacteristic
29#446—21no—
30#446—22no—
31C8×Q8—21no—
32#423C2211yescharacteristic
33#423C2212yes—
34C2×#104C2212yes—
35C2×#104C2211yescharacteristic
36C2×#104C2211yescharacteristic
37C2×#104C2212yes—
38C2×#104C2212yes—
39C2×#104C2212yes—
40C2×#104C2212yes—
41C4×M16C2211yescharacteristic
42C4×M16C2212yes—
43#391—21no—
44#389—21no—
45#387—22no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2212yes—
48#376C2212yes—
49C43C2211yescharacteristic
50#339C2211yescharacteristic
51#339C2212yes—
52#339C2211yescharacteristic
53C2×#41—21no—
54C2×#41C2311yescharacteristic
55C2×#41C2312yes—
56C22×Q8D411yescharacteristic
57C2×M16C2311yescharacteristic
58C2×M16C2312yes—
59C2×M16C2311yescharacteristic
60C4⋊Q8C2×C411yescharacteristic
61C4⋊Q8C2×C412yes—
62#127C2×C411yescharacteristic
63#127C2×C412yes—
64#126—22no—
65#126—22no—
66#126—22no—
67#125—22no—
68#125—21no—
69#124—21no—
70#124—22no—
71#124C2×C412yes—
72#124C2×C412yes—
73#124C2×C412yes—
74#123—22no—
75#123—22no—
76#123—22no—
77C22⋊Q8—21no—
78C22⋊Q8—22no—
79#121—21no—
80#121—22no—
81#121—21no—
82#121—22no—
83#121—22no—
84#121—21no—
85C4×Q8—21no—
86C4×Q8C2×C411yescharacteristic
87C4×Q8C2×C412yes—
88C4×Q8C2×C411yescharacteristic
89C4×Q8C2×C412yes—
90C4×Q8C2×C411yescharacteristic
91C4×Q8C2×C411yescharacteristic
92C4×Q8C2×C412yes—
93C4×Q8D411yescharacteristic
94C4×Q8D411yescharacteristic
95C4×Q8D412yes—
96C4×D4—21no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—21no—
106C4×D4C2×C411yescharacteristic
107C4×D4C2×C412yes—
108C4×D4C2×C411yescharacteristic
109C4×D4C2×C412yes—
110#117—22no—
111#117—21no—
112#117—22no—
113#117—21no—
114#117C2311yescharacteristic
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4D412yes—
118C2×C4⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42—22no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124C2×C42C2312yes—
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C2×C42C2312yes—
128C4⋊C8—21no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132#104—22no—
133#104—24no—
134#104—24no—
135#104—22no—
136#104—42no—
137#104—42no—
138#104—42no—
139#97—22no—
140#97—21no—
141C4×C8—21no—
142C4×C8—22no—
143#95—22no—
144#95—22no—
145#95—21no—
146#95—22no—
147#41—21no—
148#41—22no—
149#41—22no—
150#41—21no—
151#41—22no—
152#41—21no—
153#41—21no—
154#41—22no—
155#41—44no—
156C2×Q8—42no—
157C2×Q8C22⋊C411yescharacteristic
158C2×Q8C22⋊C411yescharacteristic
159C2×Q8C22⋊C412yes—
160C2×Q8C22×C411yescharacteristic
161C2×Q8C22×C412yes—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4C22×C411yescharacteristic
169C2×D4C22×C412yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D412yes—
183C22×C4C2411yescharacteristic
184M16—21no—
185M16—22no—
186M16—21no—
187M16—22no—
188M16—21no—
189M16#4112yescomplemented
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—22no—
193C4⋊C4—22no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4C22⋊C412yes—
204C4⋊C4C22⋊C411yescharacteristic
205C4⋊C4C22⋊C412yes—
206C4⋊C4C22⋊C411yescharacteristic
207C4⋊C4C22⋊C412yes—
208C4⋊C4C22⋊C412yes—
209C4⋊C4C22⋊C412yes—
210C4⋊C4C22×C411yescharacteristic
211C4⋊C4C22×C411yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C42—21no—
224C42—22no—
225C42—22no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—22no—
231C42—21no—
232C42—22no—
233C42—42no—
234C42—42no—
235C42C22×C411yescharacteristic
236C42C22×C412yes—
237C42C22×C412yes—
238C42C22×C411yescharacteristic
239C42C2×D411yescharacteristic
240C42C2×D412yes—
241C42C2×D412yes—
242C42C2×D411yescharacteristic
243C42#4112yescomplemented
244C23—22no—
245C23—21no—
246C23C22×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—21no—
250Q8—22no—
251Q8—21no—
252Q8—22no—
253Q8—42no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—44no—
259D4—41no—
260D4—42no—
261D4—41no—
262D4—42no—
263D4—42no—
264D4—81no—
265D4—82no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4C2×C22⋊C411yescharacteristic
292C2×C4C2×C22⋊C412yes—
293C2×C4C2×C22⋊C412yes—
294C2×C4C2×C22⋊C411yescharacteristic
295C2×C4C2×C22⋊C412yes—
296C2×C4C2×C22⋊C412yes—
297C2×C4C2×C22⋊C411yescharacteristic
298C2×C4C2×C22⋊C411yescharacteristic
299C2×C4C23×C411yescharacteristic
300C2×C4C22×D411yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C42−1+411yescharacteristic
305C8—22no—
306C8—41no—
307C8—42no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—41no—
312C22—42no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22C22×C22⋊C411yescharacteristic
317C22#54811yescharacteristic
318C22#55011yescharacteristic
319C4—21no—
320C4—22no—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—82no—
333C4#42112yes—
334C4C4×#4111yescharacteristic
335C4#52011yescharacteristic
336C4#54811yescharacteristic
337C4#55011yescharacteristic
338C2—21no—
339C2—21no—
340C2—82no—
341C2—81no—
342C2#234111yescharacteristic
343C2C2×#42111yescharacteristic
344C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radical#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×M16C2semidirect
3. #2295C2semidirect
4. #1980C2semidirect
5. #1980C2semidirect
6. #1963C2semidirect
7. #1962C2semidirect
8. #1806C2semidirect
9. #1806C2semidirect
10. #1759C2semidirect
11. C4×#104C2semidirect
12. #559C4semidirect
13. #559C4semidirect
14. #557C4semidirect
15. #556C4semidirect
16. #556C4semidirect
19. #550C4semidirect
189. M16#41semidirect
243. C42#41semidirect

Maximal subgroups

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

subgroupindex
#34802
Q8×M162
#22952
#19802
#19632
#19622
#18062
#17592
C4×#1042