Finite Group Databasebeta

← #17615

Subgroups — #17615

Order 256.

519 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic
2#2972C211yesmaximal, characteristic, complemented
3#2353C211yesmaximal, characteristic, complemented
4#1521C212yesmaximal, complemented
5#1520C211yesmaximal, characteristic, complemented
6#1519C211yesmaximal, characteristic, complemented
7#1516C212yesmaximal, complemented
8#1501C212yesmaximal, complemented
9#1500C212yesmaximal, complemented
10C2×#328C211yesmaximal, characteristic, complemented
11C2×#327C211yesmaximal, characteristic, complemented
12#553—22no—
13#553C412yescomplemented
14#552C412yescomplemented
15#551—22no—
16#548C412yescomplemented
17#546C412yescomplemented
18#539—21no—
19#538—21no—
20#534—21no—
21#529C2212yes—
22#525C2212yes—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25#518—21no—
26C2×C4×D4C2211yescharacteristic
27C8×D4—22no—
28C8×D4—22no—
29#431C2212yescomplemented
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2212yescomplemented
33#388—21no—
34#387—21no—
35#386—21no—
36#383C2211yescharacteristic, complemented
37#381C2211yescharacteristic, complemented
38#380C2211yescharacteristic
39C4×C22⋊C4—21no—
40C4×C22⋊C4C2211yescharacteristic, complemented
41#376C2211yescharacteristic, complemented
42C2×#95C2211yescharacteristic
43#328C2212yescomplemented
44#328C2212yescomplemented
45#328C2212yescomplemented
46#328C2212yescomplemented
47#327C2212yes—
48#327C2212yes—
49#327C2212yes—
50#327C2212yes—
51C2×#41C2×C412yescomplemented
52C22×Q8C2×C411yescharacteristic
53C22×D4—21no—
54C22×D4C2×C411yescharacteristic
55C22×D4D411yescharacteristic
56C23×C4—21no—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—22no—
62#126—22no—
63#126—22no—
64#126—22no—
65#125—22no—
66#125C2×C412yescomplemented
67#124—22no—
68#124C2×C412yescomplemented
69#123—22no—
70#123—22no—
71#123—22no—
72#123—22no—
73#123—22no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76C22⋊Q8—22no—
77C22⋊Q8C2×C412yes—
78C22⋊Q8C2×C412yescomplemented
79C22⋊Q8C2312yes—
80#121—22no—
81#121—22no—
82#121—21no—
83#121—21no—
84#121—22no—
85#121C2×C412yes—
86#121C2×C412yescomplemented
87#121C2312yes—
88#120—22no—
89#120—22no—
90C4×Q8C2×C412yescomplemented
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4C2×C412yes—
97C4×D4C2×C412yes—
98C4×D4C2×C412yescomplemented
99C4×D4D412yes—
100C4×D4D412yes—
101#117—22no—
102#117—22no—
103#117C2×C412yescomplemented
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C2×C411yescharacteristic
107C2×C4⋊C4C2×C411yescharacteristic
108C2×C4⋊C4D411yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C4⋊C8—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123C22⋊C8—22no—
124C22⋊C8C2312yes—
125C22⋊C8C2312yes—
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95C2311yescharacteristic
133#95C2311yescharacteristic
134#95C2311yescharacteristic
135#95C2311yescharacteristic
136C24—21no—
137C24C2×D411yescharacteristic
138C24C2×D411yescharacteristic
139#41—82no—
140C2×Q8—21no—
141C2×Q8—81no—
142C2×Q8C22×C412yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—21no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—81no—
153C2×D4C22⋊C412yes—
154C2×D4C22⋊C412yes—
155C2×D4C22×C412yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4C22×C411yescharacteristic
171C22×C4C22×C411yescharacteristic
172C22×C4C22×C412yes—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D412yes—
180C22×C4C2411yescharacteristic
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8—82no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—81no—
195C4⋊C4C22⋊C412yes—
196C4⋊C4C22⋊C412yes—
197C4⋊C4C22×C412yes—
198C4⋊C4C22×C412yes—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—81no—
212C22⋊C4C22⋊C412yes—
213C22⋊C4C22⋊C412yes—
214C22⋊C4C22⋊C412yes—
215C22⋊C4C22⋊C412yes—
216C42—22no—
217C42—22no—
218C42—42no—
219C42—81no—
220C42C22×C412yes—
221C42C2×D412yes—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—81no—
231C23—81no—
232C23C2×C22⋊C412yes—
233C23C2×C22⋊C411yescharacteristic
234C23C2×C22⋊C411yescharacteristic
235C23#13512yes—
236C23#13512yes—
237C23C22×D411yescharacteristic
238Q8—42no—
239Q8—81no—
240D4—24no—
241D4—24no—
242D4—42no—
243D4—42no—
244D4—42no—
245D4—44no—
246D4—81no—
247D4—82no—
248D4—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4C2×C22⋊C412yes—
283C2×C4C2×C22⋊C412yes—
284C2×C4C2×C22⋊C412yes—
285C2×C4C2×C22⋊C411yescharacteristic
286C2×C4C2×C22⋊C411yescharacteristic
287C2×C4C23×C411yescharacteristic
288C2×C4C22×D411yescharacteristic
289C8—42no—
290C8—42no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—81no—
306C22—81no—
307C22—162no—
308C22#41612yes—
309C22#41612yes—
310C22C22×C22⋊C411yescharacteristic
311C22C2×#13511yescharacteristic
312C22C2×#13511yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—81no—
325C4—81no—
326C4—81no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—22no—
331C2—42no—
332C2—82no—
333C2#288311yescharacteristic
334C2C2×#41611yescharacteristic
335C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3505C2
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. #2972C2semidirect
3. #2353C2semidirect
4. #1521C2semidirect
5. #1520C2semidirect
6. #1519C2semidirect
7. #1516C2semidirect
8. #1501C2semidirect
9. #1500C2semidirect
10. C2×#328C2semidirect
11. C2×#327C2semidirect
13. #553C4semidirect
14. #552C4semidirect
16. #548C4semidirect
17. #546C4semidirect
29. #431C22semidirect
32. C2×C22⋊C8C22semidirect
36. #383C22semidirect
37. #381C22semidirect
40. C4×C22⋊C4C22semidirect
41. #376C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
46. #328C22semidirect
51. C2×#41C2×C4semidirect
66. #125C2×C4semidirect
68. #124C2×C4semidirect
78. C22⋊Q8C2×C4semidirect
86. #121C2×C4semidirect
90. C4×Q8C2×C4semidirect
98. C4×D4C2×C4semidirect
103. #117C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35052
#29722
#23532
#15212
#15202
#15192
#15162
#15012
#15002
C2×#3282
C2×#3272