Finite Group Databasebeta

← #17622

Subgroups — #17622

Order 256.

537 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic
2#2973C211yesmaximal, characteristic, complemented
3#2349C211yesmaximal, characteristic, complemented
4#1520C211yesmaximal, characteristic, complemented
5#1519C211yesmaximal, characteristic, complemented
6#1513C212yesmaximal, complemented
7#1512C212yesmaximal, complemented
8#1509C212yesmaximal, complemented
9#1504C212yesmaximal, 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×D4—21no—
27C2×C4×D4C2211yescharacteristic
28#435—22no—
29#435—22no—
30#432C2212yescomplemented
31#407C2212yescomplemented
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34#389—21no—
35#385—21no—
36#385—21no—
37#382C2211yescharacteristic, complemented
38#380C2211yescharacteristic
39#380C2211yescharacteristic, complemented
40C4×C4⋊C4C2211yescharacteristic, complemented
41C2×#95C2211yescharacteristic
42C2×#95C2211yescharacteristic, complemented
43#328C2212yescomplemented
44#328C2212yescomplemented
45#328C2212yescomplemented
46#328C2212yescomplemented
47#327C2212yes—
48#327C2212yes—
49#327C2212yes—
50#327C2212yes—
51C2×#41C2×C412yescomplemented
52C22×Q8C2×C411yescharacteristic, complemented
53C22×D4—21no—
54C22×D4C2×C411yescharacteristic, complemented
55C22×D4D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C2×M16—22no—
61C22×C8—21no—
62C22×C8—21no—
63#126—22no—
64#126—22no—
65#126—22no—
66#125—22no—
67#125C2×C412yescomplemented
68#124—22no—
69#124C2×C412yescomplemented
70#123—22no—
71#123—22no—
72#123—22no—
73#123—22no—
74#123—22no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8C2×C412yescomplemented
79C22⋊Q8C2×C412yescomplemented
80C22⋊Q8C2312yes—
81#121—22no—
82#121—22no—
83#121—21no—
84#121—21no—
85#121—22no—
86#121C2×C412yescomplemented
87#121C2×C412yescomplemented
88#121C2312yes—
89#120—22no—
90#120—22no—
91C4×Q8C2×C412yescomplemented
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—44no—
98C4×D4C2×C412yes—
99C4×D4C2×C412yes—
100C4×D4C2×C412yescomplemented
101C4×D4D412yes—
102C4×D4D412yes—
103#117—22no—
104#117—22no—
105#117C2×C412yescomplemented
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4C2×C411yescharacteristic, complemented
112C2×C4⋊C4C2×C411yescharacteristic, complemented
113C2×C4⋊C4D411yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42—21no—
122C2×C42C2311yescharacteristic
123C4⋊C8—22no—
124C22⋊C8—21no—
125C22⋊C8—21no—
126C22⋊C8—22no—
127C22⋊C8C2312yes—
128C22⋊C8C2312yes—
129#97—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95C2311yescharacteristic
134#95C2311yescharacteristic
135#95C2311yescharacteristic
136#95C2311yescharacteristic
137C24—21no—
138C24C2×D411yescharacteristic
139C24C2×D411yescharacteristic
140#41—82no—
141C2×Q8—21no—
142C2×Q8—81no—
143C2×Q8C22×C412yes—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—81no—
154C2×D4C22⋊C412yes—
155C2×D4C22⋊C412yes—
156C2×D4C22×C412yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D412yes—
184C22×C4C2411yescharacteristic
185M16—82no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—81no—
189C2×C8—81no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4C22⋊C412yes—
203C4⋊C4C22⋊C412yes—
204C4⋊C4C22×C412yes—
205C4⋊C4C22×C412yes—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4C22⋊C412yes—
219C22⋊C4C22⋊C412yes—
220C22⋊C4C22⋊C412yes—
221C22⋊C4C22⋊C412yes—
222C42—22no—
223C42—22no—
224C42—42no—
225C42C22×C412yes—
226C42C2×D412yes—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—81no—
236C23—81no—
237C23C2×C22⋊C412yes—
238C23C2×C22⋊C411yescharacteristic
239C23C2×C22⋊C411yescharacteristic
240C23AGL(1,ℤ/8)12yes—
241C23#13712yes—
242C23C22×D411yescharacteristic
243Q8—42no—
244Q8—81no—
245D4—24no—
246D4—24no—
247D4—42no—
248D4—42no—
249D4—42no—
250D4—44no—
251D4—81no—
252D4—82no—
253D4—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4C2×C22⋊C412yes—
289C2×C4C2×C22⋊C412yes—
290C2×C4C2×C22⋊C412yes—
291C2×C4C2×C22⋊C411yescharacteristic
292C2×C4C2×C22⋊C411yescharacteristic
293C2×C4C23×C411yescharacteristic
294C2×C4C22×D411yescharacteristic
295C8—42no—
296C8—42no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—162no—
314C22#41712yes—
315C22#41712yes—
316C22C22×C22⋊C411yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22C2×#13711yescharacteristic
319C4—21no—
320C4—21no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—22no—
336C2—22no—
337C2—42no—
338C2—82no—
339C2#288311yescharacteristic
340C2C2×#41711yescharacteristic
341C2#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. #2973C2semidirect
3. #2349C2semidirect
4. #1520C2semidirect
5. #1519C2semidirect
6. #1513C2semidirect
7. #1512C2semidirect
8. #1509C2semidirect
9. #1504C2semidirect
10. C2×#328C2semidirect
11. C2×#327C2semidirect
13. #553C4semidirect
14. #552C4semidirect
16. #548C4semidirect
17. #546C4semidirect
30. #432C22semidirect
31. #407C22semidirect
37. #382C22semidirect
39. #380C22semidirect
40. C4×C4⋊C4C22semidirect
42. C2×#95C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
46. #328C22semidirect
51. C2×#41C2×C4semidirect
52. C22×Q8C2×C4semidirect
54. C22×D4C2×C4semidirect
67. #125C2×C4semidirect
69. #124C2×C4semidirect
78. C22⋊Q8C2×C4semidirect
79. C22⋊Q8C2×C4semidirect
86. #121C2×C4semidirect
87. #121C2×C4semidirect
91. C4×Q8C2×C4semidirect
100. C4×D4C2×C4semidirect
105. #117C2×C4semidirect
111. C2×C4⋊C4C2×C4semidirect
112. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35052
#29732
#23492
#15202
#15192
#15132
#15122
#15092
#15042
C2×#3282
C2×#3272