Finite Group Databasebeta

← #17607

Subgroups — #17607

Order 256.

495 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3494C211yesmaximal, characteristic
2#2924C211yesmaximal, characteristic, complemented
3#2347C211yesmaximal, characteristic, complemented
4#1521C212yesmaximal, complemented
5#1519C211yesmaximal, characteristic, complemented
6#1514C212yesmaximal, complemented
7#1513C211yesmaximal, characteristic, complemented
8#1509C212yesmaximal, complemented
9#1507C211yesmaximal, characteristic, complemented
10#1502C212yesmaximal, complemented
11#1501C211yesmaximal, characteristic, complemented
12#555—21no—
13#553—21no—
14#553—21no—
15#551—22no—
16#547—21no—
17#546—22no—
18#544C411yescharacteristic, complemented
19#542C412yescomplemented
20#540C412yescomplemented
21#535C411yescharacteristic, complemented
22#532C2212yes—
23#529C2212yes—
24#525C2211yescharacteristic
25C2×#121C2211yescharacteristic
26#520C411yescharacteristic, complemented
27#518C411yescharacteristic, complemented
28C2×#117C2211yescharacteristic
29#433—22no—
30#432C2212yescomplemented
31#431C2212yescomplemented
32#424—21no—
33#407C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#405—21no—
36#389—21no—
37#388—22no—
38#383C2212yescomplemented
39#380C2211yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2212yescomplemented
42C4×C22⋊C4C2211yescharacteristic
43#328C2211yescharacteristic, complemented
44#328C2211yescharacteristic, complemented
45#328C2212yescomplemented
46#327C2212yes—
47#327C2212yes—
48#327C2212yes—
49#327C2212yes—
50#327C2211yescharacteristic, complemented
51#327C2211yescharacteristic, complemented
52#327C2212yescomplemented
53C2×#41—22no—
54C2×#41C2×C411yescharacteristic
55C22×D4—21no—
56C22×D4C2×C411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×M16—21no—
59C22×C8—21no—
60C4⋊Q8—21no—
61C4⋊Q8C2×C412yes—
62#127C2×C412yes—
63#126—22no—
64#126—22no—
65#125—21no—
66#125C2×C412yes—
67#124—22no—
68#124—22no—
69#124C2×C412yes—
70#123—21no—
71#123—22no—
72#123—22no—
73#123—22no—
74#123—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8C2×C411yescharacteristic
78C22⋊Q8C2311yescharacteristic
79#121—21no—
80#121—21no—
81#121—22no—
82#121—21no—
83#121—21no—
84#121—22no—
85#121—22no—
86#121C2×C412yes—
87#121C2×C411yescharacteristic
88#121C2312yes—
89#121C2311yescharacteristic
90#120—21no—
91#120—22no—
92C4×Q8C2×C412yes—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—22no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4C2×C412yes—
101C4×D4C2×C412yes—
102C4×D4C2×C412yes—
103#117—21no—
104#117—22no—
105#117C2×C411yescharacteristic
106#117C2×C411yescharacteristic
107#117C2×C411yescharacteristic
108#117C2×C412yes—
109#117D412yes—
110#117D411yescharacteristic
111#117D411yescharacteristic
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4C2×C411yescharacteristic
116C2×C4⋊C4D411yescharacteristic
117C2×C4⋊C4D411yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D412yes—
123C2×C42—21no—
124C2×C42—22no—
125C2×C42C2312yes—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130C22⋊C8C2311yescharacteristic
131C22⋊C8C2311yescharacteristic
132C22⋊C8C2312yes—
133#97—22no—
134C4×C8—22no—
135#95—22no—
136#95—22no—
137#95C2311yescharacteristic
138#95C2311yescharacteristic
139#95C2312yes—
140C24—21no—
141C24C2×D411yescharacteristic
142#41—42no—
143#41—82no—
144#41—82no—
145C2×Q8—22no—
146C2×Q8C22×C411yescharacteristic
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—42no—
156C2×D4—81no—
157C2×D4—81no—
158C2×D4C22×C412yes—
159C2×D4C22×C411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4C22×C411yescharacteristic
170C22×C4C22×C411yescharacteristic
171C22×C4C2×D412yes—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D412yes—
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177M16—42no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—21no—
181C2×C8—42no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4C22⋊C411yescharacteristic
192C4⋊C4C22⋊C411yescharacteristic
193C4⋊C4C22⋊C411yescharacteristic
194C4⋊C4C22⋊C411yescharacteristic
195C4⋊C4C22⋊C412yes—
196C4⋊C4C22⋊C412yes—
197C4⋊C4C22×C412yes—
198C4⋊C4C22×C411yescharacteristic
199C4⋊C4C22×C411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—21no—
210C22⋊C4—22no—
211C22⋊C4—21no—
212C22⋊C4—42no—
213C22⋊C4—81no—
214C22⋊C4C22⋊C412yes—
215C22⋊C4C22⋊C412yes—
216C22⋊C4C22⋊C412yes—
217C22⋊C4C22⋊C412yes—
218C42—22no—
219C42—21no—
220C42—42no—
221C42—82no—
222C42C22×C412yes—
223C42C22×C412yes—
224C42C2×D412yes—
225C42C2×D412yes—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—81no—
234C23—81no—
235C23C2×C22⋊C411yescharacteristic
236C23C2×C22⋊C411yescharacteristic
237C23C22×D411yescharacteristic
238Q8—41no—
239Q8—82no—
240D4—42no—
241D4—41no—
242D4—42no—
243D4—42no—
244D4—41no—
245D4—42no—
246D4—41no—
247D4—42no—
248D4—82no—
249D4—82no—
250D4—81no—
251D4—82no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4C2×C22⋊C412yes—
280C2×C4C2×C22⋊C412yes—
281C2×C4C2×C22⋊C412yes—
282C2×C4C2×C22⋊C412yes—
283C2×C4C2×C22⋊C411yescharacteristic
284C2×C4C2×C22⋊C411yescharacteristic
285C2×C4C23×C411yescharacteristic
286C2×C4C22×D411yescharacteristic
287C8—41no—
288C8—41no—
289C8—42no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22—162no—
307C22C22×C22⋊C411yescharacteristic
308C22#57611yescharacteristic
309C22#57711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—42no—
318C4—41no—
319C4—42no—
320C4—41no—
321C4—41no—
322C4—82no—
323C4—81no—
324C4#41012yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2—81no—
330C2—82no—
331C2C2×#41011yescharacteristic
332C2#289711yescharacteristic
333C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3494C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2924C2semidirect
3. #2347C2semidirect
4. #1521C2semidirect
5. #1519C2semidirect
6. #1514C2semidirect
7. #1513C2semidirect
8. #1509C2semidirect
9. #1507C2semidirect
10. #1502C2semidirect
11. #1501C2semidirect
18. #544C4semidirect
19. #542C4semidirect
20. #540C4semidirect
21. #535C4semidirect
26. #520C4semidirect
27. #518C4semidirect
30. #432C22semidirect
31. #431C22semidirect
38. #383C22semidirect
41. C4×C4⋊C4C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
50. #327C22semidirect
51. #327C22semidirect
52. #327C22semidirect

Maximal subgroups

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

subgroupindex
#34942
#29242
#23472
#15212
#15192
#15142
#15132
#15092
#15072
#15022
#15012