Finite Group Databasebeta

← #17600

Subgroups — #17600

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#2319C211yesmaximal, characteristic, complemented
4#1520C212yesmaximal, complemented
5#1516C212yesmaximal, complemented
6#1513C212yesmaximal, complemented
7#1511C212yesmaximal, complemented
8#1509C211yesmaximal, characteristic, complemented
9#1506C211yesmaximal, characteristic, complemented
10#1503C211yesmaximal, characteristic, complemented
11#1500C211yesmaximal, characteristic, complemented
12#556C411yescharacteristic, complemented
13#554—21no—
14#552C412yescomplemented
15#551—22no—
16D4×Q8C411yescharacteristic, complemented
17#548C412yescomplemented
18#547C411yescharacteristic, complemented
19#547C411yescharacteristic, complemented
20#546—21no—
21#542—21no—
22#539—22no—
23#533C2211yescharacteristic
24#529C2211yescharacteristic
25#525C2212yes—
26C2×C22⋊Q8C2212yes—
27#520—21no—
28C2×C4×D4C2211yescharacteristic
29#436—21no—
30#435—22no—
31C8×D4—21no—
32#432C2211yescharacteristic
33#431C2211yescharacteristic
34#407C2212yes—
35C2×C22⋊C8C2212yes—
36#390—21no—
37#389—21no—
38#388—22no—
39#381C2212yes—
40#380C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#376C2211yescharacteristic
43#328C2211yescharacteristic
44#328C2211yescharacteristic
45#328C2212yes—
46#328C2212yes—
47#328C2212yes—
48#328C2212yes—
49#328C2212yes—
50#327C2211yescharacteristic
51#327C2211yescharacteristic
52#327C2212yes—
53C2×#41—21no—
54C2×#41C2×C412yes—
55C22×Q8C2×C412yes—
56C22×D4D411yescharacteristic
57C23×C4C2312yes—
58C2×M16—22no—
59C22×C8—22no—
60C4⋊Q8—21no—
61C4⋊Q8C2×C411yescharacteristic
62C4⋊Q8C2×C411yescharacteristic
63#126—22no—
64#126—22no—
65#126—21no—
66#125—22no—
67#125—21no—
68#125C2×C411yescharacteristic
69#124—21no—
70#124—22no—
71#124C2×C411yescharacteristic
72#123—22no—
73#123—22no—
74#123—21no—
75#123—22no—
76#123—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—21no—
80C22⋊Q8—22no—
81C22⋊Q8—21no—
82C22⋊Q8—22no—
83C22⋊Q8C2×C412yes—
84C22⋊Q8C2×C412yes—
85C22⋊Q8C2×C412yes—
86C22⋊Q8C2311yescharacteristic
87C22⋊Q8C2312yes—
88#121—22no—
89#121—21no—
90#121C2×C412yes—
91#121C2311yescharacteristic
92#120—22no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8C2×C411yescharacteristic
96C4×Q8C2×C412yes—
97C4×D4—21no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—21no—
101C4×D4—22no—
102C4×D4C2×C411yescharacteristic
103C4×D4C2×C411yescharacteristic
104C4×D4C2×C411yescharacteristic
105C4×D4C2×C412yes—
106C4×D4D411yescharacteristic
107C4×D4D412yes—
108C4×D4D411yescharacteristic
109#117—21no—
110#117—22no—
111#117—21no—
112#117C2×C412yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4C2×C412yes—
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×C42C2311yescharacteristic
126C4⋊C8—21no—
127C4⋊C8—21no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8C2311yescharacteristic
131C22⋊C8C2312yes—
132C22⋊C8C2311yescharacteristic
133#97—21no—
134C4×C8—21no—
135#95—21no—
136#95—22no—
137#95C2311yescharacteristic
138#95C2311yescharacteristic
139#95C2312yes—
140C24C2×D412yes—
141#41—44no—
142#41—82no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8—82no—
146C2×Q8C22×C411yescharacteristic
147C2×Q8C22×C412yes—
148C2×D4—22no—
149C2×D4—21no—
150C2×D4—22no—
151C2×D4—42no—
152C2×D4C22⋊C411yescharacteristic
153C2×D4C22⋊C411yescharacteristic
154C2×D4C22⋊C412yes—
155C2×D4C22×C411yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—82no—
164C22×C4C22×C412yes—
165C22×C4C22×C412yes—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4C2411yescharacteristic
172M16—82no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—22no—
176C2×C8—82no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4C22⋊C411yescharacteristic
193C4⋊C4C22⋊C412yes—
194C4⋊C4C22⋊C411yescharacteristic
195C4⋊C4C22×C411yescharacteristic
196C4⋊C4C22×C412yes—
197C4⋊C4C22×C411yescharacteristic
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—22no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4C22⋊C412yes—
212C22⋊C4C22⋊C412yes—
213C22⋊C4C22⋊C412yes—
214C22⋊C4C22⋊C412yes—
215C42—22no—
216C42—22no—
217C42—21no—
218C42—42no—
219C42—81no—
220C42C22×C411yescharacteristic
221C42C22×C411yescharacteristic
222C42C2×D411yescharacteristic
223C42C2×D411yescharacteristic
224C23—22no—
225C23—22no—
226C23—21no—
227C23—42no—
228C23—42no—
229C23C2×C22⋊C412yes—
230C23C2×C22⋊C412yes—
231C23C22×D411yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—42no—
235Q8—82no—
236D4—22no—
237D4—22no—
238D4—42no—
239D4—41no—
240D4—44no—
241D4—41no—
242D4—41no—
243D4—82no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4C2×C22⋊C411yescharacteristic
272C2×C4C2×C22⋊C412yes—
273C2×C4C2×C22⋊C411yescharacteristic
274C2×C4C2×C22⋊C411yescharacteristic
275C2×C4C2×C22⋊C412yes—
276C2×C4C2×C22⋊C411yescharacteristic
277C2×C4C23×C411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C8—41no—
280C8—42no—
281C8—41no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—82no—
294C22—82no—
295C22C22×C22⋊C411yescharacteristic
296C22#57711yescharacteristic
297C22#57811yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—81no—
311C4—82no—
312C4—81no—
313C4#42112yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2—81no—
319C2#288311yescharacteristic
320C2#289811yescharacteristic
321C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3520C2
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. #2319C2semidirect
4. #1520C2semidirect
5. #1516C2semidirect
6. #1513C2semidirect
7. #1511C2semidirect
8. #1509C2semidirect
9. #1506C2semidirect
10. #1503C2semidirect
11. #1500C2semidirect
12. #556C4semidirect
14. #552C4semidirect
16. D4×Q8C4semidirect
17. #548C4semidirect
18. #547C4semidirect
19. #547C4semidirect

Maximal subgroups

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

subgroupindex
#35202
#29742
#23192
#15202
#15162
#15132
#15112
#15092
#15062
#15032
#15002