Finite Group Databasebeta

← #27181426

Subgroups — #27181426

Order 12928.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265787C211yesmaximal, characteristic
2#20265655C211yesmaximal, characteristic
3#20265519C211yesmaximal, characteristic
4#20265515C211yesmaximal, characteristic
5#20265486C211yesmaximal, characteristic, complemented
6#20265482C211yesmaximal, characteristic, complemented
7#20265385C211yesmaximal, characteristic, complemented
8#20265385C211yesmaximal, characteristic, complemented
9#20265333C211yesmaximal, characteristic
10#20265284C211yesmaximal, characteristic, complemented
11#20265281C211yesmaximal, characteristic, complemented
12Q16×Dic101C211yesmaximal, characteristic, complemented
13#20265269C211yesmaximal, characteristic, complemented
14#20265235C211yesmaximal, characteristic
15#451×C101C211yesmaximal, characteristic
16C2×#13108064C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18#15692253C2211yescharacteristic
19#15692248—21no—
20#15692248—21no—
21#15692246—21no—
22#15692246C2211yescharacteristic
23#15692245—21no—
24#15692234C2211yescharacteristic
25Q8×Dic101—21no—
26Q8×Dic101—21no—
27Q8×Dic101C2211yescharacteristic
28Q8×Dic101C2211yescharacteristic
29Q8×Dic101C2211yescharacteristic
30Q8×Dic101C2211yescharacteristic
31Q8×Dic101C2211yescharacteristic
32#15692220C2211yescharacteristic
33#15692214—21no—
34#15692214—21no—
35#15692214C2211yescharacteristic
36#15692214C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692179C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic
53C2×Q16×C101C2211yescharacteristic
54C2×Q16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60#13108064—22no—
61#13108064—22no—
62C2×#11717579C2311yescharacteristic
63C2×#11717579C2311yescharacteristic
64C4⋊Dic101—21no—
65C4⋊Dic101—21no—
66C4⋊Dic101—21no—
67C4⋊Dic101—21no—
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101D411yescharacteristic
71C4⋊Dic101D411yescharacteristic
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80C22⋊Dic101—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101D411yescharacteristic
83C22⋊Dic101D411yescharacteristic
84C4×Dic101—21no—
85C4×Dic101—21no—
86C4×Dic101—21no—
87C4×Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93C2×Q8×C101—22no—
94C2×Q8×C101—22no—
95C2×Q8×C101—21no—
96C2×Q8×C101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98C2×Q8×C101C2311yescharacteristic
99C22×C4×C101—21no—
100C22×C4×C101C2311yescharacteristic
101Q16×C101—22no—
102Q16×C101—22no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C23×C101C2×D411yescharacteristic
109Q8×C101—24no—
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101—21no—
114Q8×C101#4112yes—
115Q8×C101#4112yes—
116C2×C4×C101—24no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2411yescharacteristic
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133Dic202—21no—
134Dic202—21no—
135C8×C101—21no—
136C8×C101—21no—
137#11717579—21no—
138#11717579—21no—
139C22×C101—21no—
140C22×C101—21no—
141C22×C101—21no—
142C22×C101C22×D411yescharacteristic
143C4×C101—22no—
144C4×C101—22no—
145C4×C101—21no—
146C4×C101—41no—
147C4×C101—41no—
148C4×C101C2×#4111yescharacteristic
149C4×C101C2×#4111yescharacteristic
150Dic101—21no—
151Dic101—21no—
152Dic101—21no—
153Dic101—21no—
154Dic101—41no—
155Dic101—41no—
156Dic101—41no—
157Dic101—41no—
158C2×C101—41no—
159C2×C101#52511yescharacteristic
160C2×C101#57811yescharacteristic
161C2×C101#57811yescharacteristic
162#3127—1011nomaximal, Sylow
163C101#312711yescharacteristic, complemented, Sylow
164#541—1011no—
165#519—1011no—
166#485—1011no—
167#485—1011no—
168#479—1011no—
169#479—1011no—
170#462—1011no—
171#462—1011no—
172#451—1011no—
173#451D10111yescharacteristic
174#441—1011no—
175#441—1011no—
176C4×Q16—1011no—
177C4×Q16—1011no—
178#433—1011no—
179C22×Q8C2×D10111yescharacteristic
180C2×Q16—1011no—
181C2×Q16—1011no—
182C2×Q16C2×D10111yescharacteristic
183C2×Q16C2×D10111yescharacteristic
184C4⋊Q8—1011no—
185#126—2021no—
186#126—2021no—
187#125—1011no—
188#125—2021no—
189#125—2021no—
190C22⋊Q8—1011no—
191C22⋊Q8C2×D10111yescharacteristic
192C4×Q8—1011no—
193C4×Q8—1011no—
194C4×Q8—1011no—
195C4×Q8—1011no—
196C4×Q8—1011no—
197C4×Q8—1011no—
198C4×Q8—2021no—
199C4×Q8—2021no—
200#117—1011no—
201#117—1011no—
202#117—2021no—
203#117—2021no—
204#107—1011no—
205#107—1011no—
206#106—1011no—
207#106—1011no—
208C4⋊C8—1011no—
209C4⋊C8—1011no—
210#103—1011no—
211#103—1011no—
212#103—1011no—
213#103—1011no—
214#103—1011no—
215#103—1011no—
216#103C2×D10111yescharacteristic
217#103C2×D10111yescharacteristic
218C22⋊C8—1011no—
219C22⋊C8C2×D10111yescharacteristic
220#97—1011no—
221C4×C8—1011no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—1011no—
226C2×Q8C22×D10111yescharacteristic
227C2×Q8C22×D10111yescharacteristic
228C2×Q8C22×D10111yescharacteristic
229C22×C4—21no—
230C22×C4C22×D10111yescharacteristic
231Q16—22no—
232Q16—22no—
233Q16—2022no—
234Q16—2022no—
235C2×C8—1011no—
236C2×C8—1011no—
237C2×C8C22×D10111yescharacteristic
238C2×C8C22×D10111yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—1011no—
241C4⋊C4—1011no—
242C4⋊C4—1011no—
243C4⋊C4—1011no—
244C4⋊C4—1011no—
245C4⋊C4—2021no—
246C4⋊C4—2021no—
247C4⋊C4—2021no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4—2021no—
251C4⋊C4—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4C22×D10111yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—1011no—
259C22⋊C4—1011no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C42—1011no—
263C42—1011no—
264C42—2021no—
265C42—2021no—
266C42—2021no—
267C42—2021no—
268C42—2021no—
269C42—2021no—
270C42—2021no—
271C23D4×D10111yescharacteristic
272Q8—24no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—2021no—
278Q8—2021no—
279Q8#1310807312yes—
280Q8#1310807312yes—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—1011no—
289C2×C4—1011no—
290C2×C4—1011no—
291C2×C4—1011no—
292C2×C4—2021no—
293C2×C4—2021no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4D4×D10111yescharacteristic
297C2×C4C23×D10111yescharacteristic
298C8—21no—
299C8—21no—
300C8—2021no—
301C8—2021no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D10111yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—4041no—
312C4—4041no—
313C4—4041no—
314C4—4041no—
315C4—2021no—
316C4—2021no—
317C4—2021no—
318C4—2021no—
319C4C2×#1310807311yescharacteristic
320C4C2×#1310807311yescharacteristic
321C2—41no—
322C2#2026568311yescharacteristic
323C2#2026599011yescharacteristic
324C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#451×C101C2
socleC22×C101C22×D4
prime radicalC23×C101C2×D4
hypercenter#451D101
hypocenterC101#3127
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #20265486C2semidirect
6. #20265482C2semidirect
7. #20265385C2semidirect
8. #20265385C2semidirect
10. #20265284C2semidirect
11. #20265281C2semidirect
12. Q16×Dic101C2semidirect
13. #20265269C2semidirect
163. C101#3127semidirect

Maximal subgroups

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

subgroupindex
#202657872
#202656552
#202655192
#202655152
#202654862
#202654822
#202653852
#202653332
#202652842
#202652812
Q16×Dic1012
#202652692
#202652352
#451×C1012
#3127101