Finite Group Databasebeta

← #17625

Subgroups — #17625

Order 256.

501 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic
2#2973C211yesmaximal, characteristic, complemented
3#2357C211yesmaximal, characteristic, complemented
4#1516C212yesmaximal, complemented
5#1512C212yesmaximal, complemented
6#1511C211yesmaximal, characteristic, complemented
7#1510C211yesmaximal, characteristic, complemented
8#1509C212yesmaximal, complemented
9#1508C211yesmaximal, characteristic, complemented
10#1507C211yesmaximal, characteristic, complemented
11#1501C212yesmaximal, 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
27#435—22no—
28#435—22no—
29#432C2212yescomplemented
30#407C2211yescharacteristic
31#407C2211yescharacteristic
32C2×C22⋊C8C2212yescomplemented
33#390—21no—
34#388—21no—
35#388—21no—
36#382C2211yescharacteristic, complemented
37#381C2211yescharacteristic
38#381C2211yescharacteristic, complemented
39C4×C4⋊C4—21no—
40C4×C4⋊C4C2211yescharacteristic, complemented
41C4×C22⋊C4C2211yescharacteristic
42C4×C22⋊C4C2211yescharacteristic, complemented
43#328C2212yescomplemented
44#328C2212yescomplemented
45#328C2212yescomplemented
46#328C2212yescomplemented
47#327C2212yes—
48#327C2212yes—
49#327C2212yes—
50#327C2212yes—
51C2×#41C2×C412yes—
52C22×Q8C2×C411yescharacteristic
53C22×D4—21no—
54C22×D4C2×C411yescharacteristic
55C22×D4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C2×M16—21no—
59C2×M16—21no—
60C22×C8—22no—
61#126—22no—
62#126—22no—
63#126—22no—
64#125—22no—
65#125C2×C412yes—
66#124—22no—
67#124C2×C412yes—
68#123—22no—
69#123—22no—
70#123—22no—
71#123—22no—
72#123—22no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75C22⋊Q8—22no—
76C22⋊Q8C2×C412yes—
77C22⋊Q8C2×C412yes—
78C22⋊Q8C2312yes—
79#121—22no—
80#121—22no—
81#121—21no—
82#121—21no—
83#121—22no—
84#121C2×C412yes—
85#121C2×C412yes—
86#121C2312yes—
87#120—22no—
88#120—22no—
89C4×Q8C2×C412yes—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4C2×C412yes—
96C4×D4C2×C412yes—
97C4×D4C2×C412yes—
98C4×D4D412yes—
99C4×D4D412yes—
100#117—22no—
101#117—22no—
102#117C2×C412yes—
103C2×C4⋊C4—21no—
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⋊C4—21no—
116C2×C22⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C4⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—21no—
125C22⋊C8—21no—
126C22⋊C8C2312yes—
127C22⋊C8C2312yes—
128#97—22no—
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—21no—
148C2×D4—41no—
149C2×D4—41no—
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—81no—
166C22×C4—81no—
167C22×C4C22×C411yescharacteristic
168C22×C4C22×C411yescharacteristic
169C22×C4C22×C412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D412yes—
177C22×C4C2411yescharacteristic
178M16—81no—
179M16—81no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—82no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4C22⋊C412yes—
194C4⋊C4C22⋊C412yes—
195C4⋊C4C22×C412yes—
196C4⋊C4C22×C412yes—
197C22⋊C4—22no—
198C22⋊C4—22no—
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—42no—
208C22⋊C4—42no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4C22⋊C412yes—
212C22⋊C4C22⋊C412yes—
213C22⋊C4C22⋊C412yes—
214C22⋊C4C22⋊C412yes—
215C42—22no—
216C42—22no—
217C42—42no—
218C42—81no—
219C42—81no—
220C42C22×C412yes—
221C42C2×D412yes—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—81no—
233C23—81no—
234C23C2×C22⋊C411yescharacteristic
235C23C2×C22⋊C412yes—
236C23C2×C22⋊C411yescharacteristic
237C23C22×D411yescharacteristic
238Q8—42no—
239Q8—81no—
240D4—42no—
241D4—42no—
242D4—42no—
243D4—44no—
244D4—42no—
245D4—42no—
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—42no—
267C2×C4—41no—
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×C4C2×C22⋊C412yes—
282C2×C4C2×C22⋊C412yes—
283C2×C4C2×C22⋊C412yes—
284C2×C4C2×C22⋊C411yescharacteristic
285C2×C4C2×C22⋊C411yescharacteristic
286C2×C4C23×C411yescharacteristic
287C2×C4C22×D411yescharacteristic
288C8—42no—
289C8—42no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—162no—
309C22C22×C22⋊C411yescharacteristic
310C22#57511yescharacteristic
311C22#57511yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—81no—
326C2—21no—
327C2—21no—
328C2—41no—
329C2—42no—
330C2—41no—
331C2—82no—
332C2#288311yescharacteristic
333C2#289611yescharacteristic
334C2#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. #2357C2semidirect
4. #1516C2semidirect
5. #1512C2semidirect
6. #1511C2semidirect
7. #1510C2semidirect
8. #1509C2semidirect
9. #1508C2semidirect
10. #1507C2semidirect
11. #1501C2semidirect
13. #553C4semidirect
14. #552C4semidirect
16. #548C4semidirect
17. #546C4semidirect
29. #432C22semidirect
32. C2×C22⋊C8C22semidirect
36. #382C22semidirect
38. #381C22semidirect
40. C4×C4⋊C4C22semidirect
42. C4×C22⋊C4C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
46. #328C22semidirect

Maximal subgroups

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

subgroupindex
#35052
#29732
#23572
#15162
#15122
#15112
#15102
#15092
#15082
#15072
#15012