Finite Group Databasebeta

← #17547

Subgroups — #17547

Order 256.

583 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#2275C211yesmaximal, characteristic, complemented
4#1509C211yesmaximal, characteristic, complemented
5#1507C212yesmaximal, complemented
6#1505C211yesmaximal, characteristic, complemented
7#1504C212yesmaximal, complemented
8#1502C211yesmaximal, characteristic, complemented
9#1501C212yesmaximal, complemented
10#1500C211yesmaximal, characteristic, complemented
11C2×#327C212yesmaximal, complemented
12#555C411yescharacteristic, complemented
13#554C411yescharacteristic, complemented
14#553—21no—
15#553C412yescomplemented
16#550C411yescharacteristic, complemented
17#546—22no—
18#546C412yescomplemented
19D42—21no—
20D42C411yescharacteristic, complemented
21#540—21no—
22#538—22no—
23#532C2211yescharacteristic
24#529C2211yescharacteristic
25#525C2212yes—
26C2×#121C2212yes—
27#518—21no—
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30#436—21no—
31#435—22no—
32C8×D4—21no—
33#432C2211yescharacteristic, complemented
34#431C2211yescharacteristic, complemented
35#407C2212yescomplemented
36C2×C22⋊C8C2212yescomplemented
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic, complemented
39C4×C22⋊C4—22no—
40C4×C22⋊C4C2212yescomplemented
41#376C2211yescharacteristic, complemented
42C2×#95C2212yescomplemented
43#328C2211yescharacteristic, complemented
44#328C2211yescharacteristic, complemented
45#328C2212yescomplemented
46#327C2211yescharacteristic
47#327C2212yes—
48#327C2211yescharacteristic
49#327C2212yescomplemented
50#327C2212yescomplemented
51#327C2212yescomplemented
52#327C2212yescomplemented
53C2×#41—21no—
54C2×#41C2×C412yescomplemented
55C22×D4—22no—
56C22×D4—21no—
57C22×D4C2×C412yescomplemented
58C22×D4D411yescharacteristic
59C23×C4—22no—
60C23×C4C2312yes—
61C2×M16—22no—
62C22×C8—22no—
63C4⋊Q8C2×C411yescharacteristic, complemented
64#127—21no—
65#127C2×C411yescharacteristic, complemented
66#126—22no—
67#126—21no—
68#125—21no—
69#125C2×C411yescharacteristic, complemented
70#124—22no—
71#124—21no—
72#124C2×C411yescharacteristic, complemented
73#123—22no—
74#123—21no—
75#123—22no—
76#123—22no—
77C22⋊Q8—22no—
78C22⋊Q8C2×C412yescomplemented
79C22⋊Q8C2311yescharacteristic
80#121—22no—
81#121—21no—
82#121—22no—
83#121—22no—
84#121—21no—
85#121—21no—
86#121—22no—
87#121—22no—
88#121C2×C412yescomplemented
89#121C2×C412yescomplemented
90#121C2×C412yescomplemented
91#121C2311yescharacteristic
92#121C2312yes—
93#120—22no—
94#120—22no—
95#120—22no—
96C4×Q8C2×C411yescharacteristic, complemented
97C4×D4—28no—
98C4×D4—21no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—22no—
104C4×D4—21no—
105C4×D4C2×C411yescharacteristic
106C4×D4C2×C412yes—
107C4×D4C2×C411yescharacteristic
108C4×D4C2×C411yescharacteristic, complemented
109C4×D4C2×C412yescomplemented
110C4×D4D411yescharacteristic
111C4×D4D412yes—
112C4×D4D411yescharacteristic
113#117—21no—
114#117—21no—
115#117C2×C412yescomplemented
116C2×C4⋊C4—21no—
117C2×C4⋊C4C2×C412yescomplemented
118C2×C4⋊C4D411yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4D412yes—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42C2311yescharacteristic
127C4⋊C8—21no—
128C4⋊C8—21no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8C2311yescharacteristic
132C22⋊C8C2312yes—
133C22⋊C8C2311yescharacteristic
134#97—21no—
135C4×C8—21no—
136#95—22no—
137#95—21no—
138#95C2311yescharacteristic
139#95C2312yes—
140#95C2311yescharacteristic
141C24—22no—
142C24C2×D412yes—
143#41—81no—
144#41—81no—
145#41—82no—
146C2×Q8—21no—
147C2×Q8C22×C411yescharacteristic
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4—44no—
159C2×D4—82no—
160C2×D4—82no—
161C2×D4—82no—
162C2×D4C22⋊C412yes—
163C2×D4C22⋊C411yescharacteristic
164C2×D4C22⋊C411yescharacteristic
165C2×D4C22×C411yescharacteristic
166C2×D4C22×C412yes—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—28no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4C22×C412yes—
179C22×C4C22×C412yes—
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2×D412yes—
185C22×C4C2411yescharacteristic
186M16—82no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8—82no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4C22⋊C411yescharacteristic
200C4⋊C4C22⋊C411yescharacteristic
201C4⋊C4C22⋊C412yes—
202C4⋊C4C22×C411yescharacteristic
203C4⋊C4C22×C412yes—
204C4⋊C4C22×C411yescharacteristic
205C22⋊C4—28no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—21no—
213C22⋊C4—22no—
214C22⋊C4—21no—
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—22no—
218C22⋊C4C22⋊C412yes—
219C22⋊C4C22⋊C412yes—
220C22⋊C4C22⋊C412yes—
221C22⋊C4C22⋊C412yes—
222C42—24no—
223C42—22no—
224C42—21no—
225C42—81no—
226C42—82no—
227C42C22×C411yescharacteristic
228C42C22×C411yescharacteristic
229C42C2×D411yescharacteristic
230C42C2×D411yescharacteristic
231C23—22no—
232C23—22no—
233C23—24no—
234C23—21no—
235C23—22no—
236C23—42no—
237C23—44no—
238C23—82no—
239C23—82no—
240C23C2×C22⋊C412yes—
241C23C2×C22⋊C412yes—
242C23C22×D411yescharacteristic
243Q8—41no—
244Q8—81no—
245D4—24no—
246D4—22no—
247D4—41no—
248D4—42no—
249D4—41no—
250D4—42no—
251D4—41no—
252D4—42no—
253D4—44no—
254D4—42no—
255D4—82no—
256D4—81no—
257D4—82no—
258D4—82no—
259D4—82no—
260D4#10414yes—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—88no—
288C2×C4C2×C22⋊C412yes—
289C2×C4C2×C22⋊C411yescharacteristic
290C2×C4C2×C22⋊C411yescharacteristic
291C2×C4C2×C22⋊C411yescharacteristic
292C2×C4C2×C22⋊C412yes—
293C2×C4C2×C22⋊C411yescharacteristic
294C2×C4C23×C411yescharacteristic
295C2×C4C22×D411yescharacteristic
296C8—41no—
297C8—42no—
298C8—41no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—44no—
307C22—44no—
308C22—42no—
309C22—41no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—164no—
315C22C2×#10414yes—
316C22C22×C22⋊C411yescharacteristic
317C22#57611yescharacteristic
318C22#57711yescharacteristic
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—44no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—42no—
332C4—41no—
333C4C2×#10412yes—
334C2—21no—
335C2—21no—
336C2—24no—
337C2—42no—
338C2—82no—
339C2—81no—
340C2#288311yescharacteristic
341C2#289711yescharacteristic
342C2C22×#10411yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3519C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2974C2semidirect
3. #2275C2semidirect
4. #1509C2semidirect
5. #1507C2semidirect
6. #1505C2semidirect
7. #1504C2semidirect
8. #1502C2semidirect
9. #1501C2semidirect
10. #1500C2semidirect
11. C2×#327C2semidirect
12. #555C4semidirect
13. #554C4semidirect
15. #553C4semidirect
16. #550C4semidirect
18. #546C4semidirect
20. D42C4semidirect
33. #432C22semidirect
34. #431C22semidirect
35. #407C22semidirect
36. C2×C22⋊C8C22semidirect
38. C4×C4⋊C4C22semidirect
40. C4×C22⋊C4C22semidirect
41. #376C22semidirect
42. C2×#95C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
49. #327C22semidirect
50. #327C22semidirect
51. #327C22semidirect
52. #327C22semidirect
54. C2×#41C2×C4semidirect
57. C22×D4C2×C4semidirect
63. C4⋊Q8C2×C4semidirect
65. #127C2×C4semidirect
69. #125C2×C4semidirect
72. #124C2×C4semidirect
78. C22⋊Q8C2×C4semidirect
88. #121C2×C4semidirect
89. #121C2×C4semidirect
90. #121C2×C4semidirect
96. C4×Q8C2×C4semidirect
108. C4×D4C2×C4semidirect
109. C4×D4C2×C4semidirect
115. #117C2×C4semidirect
117. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35192
#29742
#22752
#15092
#15072
#15052
#15042
#15022
#15012
#15002
C2×#3272