Finite Group Databasebeta

← #27181429

Subgroups — #27181429

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#20265785C211yesmaximal, characteristic
2#20265653C211yesmaximal, characteristic
3#20265520C211yesmaximal, characteristic
4#20265517C211yesmaximal, characteristic
5#20265486C211yesmaximal, characteristic, complemented
6#20265483C211yesmaximal, characteristic, complemented
7#20265386C211yesmaximal, characteristic, complemented
8#20265385C211yesmaximal, characteristic, complemented
9#20265334C211yesmaximal, characteristic
10#20265331C211yesmaximal, characteristic
11#20265282C211yesmaximal, characteristic, complemented
12#20265282C211yesmaximal, characteristic, complemented
13C4×Dic404C211yesmaximal, characteristic, complemented
14#20265236C211yesmaximal, characteristic
15C4×Q16×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×Dic404C2211yescharacteristic
18C2×#13108064C2211yescharacteristic
19C2×#13108064C2211yescharacteristic
20#15692253C2211yescharacteristic
21#15692250—21no—
22#15692249—21no—
23#15692247—21no—
24#15692246C2211yescharacteristic
25#15692245—21no—
26#15692235C2211yescharacteristic
27#15692231C2211yescharacteristic
28Q8×Dic101C2211yescharacteristic
29C4×Dic202—21no—
30C4×Dic202C2211yescharacteristic
31C4×Dic202C2211yescharacteristic
32#15692220—21no—
33#15692220C2211yescharacteristic
34#15692213—21no—
35#15692213C2211yescharacteristic
36#15692212—21no—
37#15692212C2211yescharacteristic
38#15692189C2211yescharacteristic
39#15692186C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692178C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692166C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C4×C8×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×Dic202C2311yescharacteristic
62Q8×D101—22no—
63Q8×D101—22no—
64C2×C4×D101—21no—
65C2×C4×D101C2311yescharacteristic
66Dic404—22no—
67#13108064—22no—
68#13108064—22no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84#13108053D411yescharacteristic
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051D411yescharacteristic
89#13108051D411yescharacteristic
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98Q16×C101—22no—
99C2×C8×C101C2311yescharacteristic
100C2×C8×C101C2311yescharacteristic
101C4⋊C4×C101—21no—
102C4⋊C4×C101—21no—
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C42×C101—21no—
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C22×D101C2×D411yescharacteristic
109Q8×C101—21no—
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101#4112yes—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101C2×D411yescharacteristic
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2411yescharacteristic
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127C4×D101—24no—
128C4×D101—21no—
129C4×D101—21no—
130Dic202—24no—
131Dic202—21no—
132Dic202—21no—
133Dic202—21no—
134Dic202#4112yes—
135C8×C101—21no—
136C8×C101—21no—
137#11717579—21no—
138#11717579—21no—
139C22×C101C22×D411yescharacteristic
140C2×D101—21no—
141C2×D101—21no—
142C2×D101—21no—
143C4×C101—22no—
144C4×C101—21no—
145C4×C101—21no—
146C4×C101—41no—
147C4×C101—41no—
148C4×C101—41no—
149C4×C101C2×#4111yescharacteristic
150C4×C101C2×#4111yescharacteristic
151Dic101—21no—
152Dic101—22no—
153Dic101—21no—
154Dic101—21no—
155Dic101—41no—
156Dic101—41no—
157Dic101—41no—
158C2×C101#52511yescharacteristic
159C2×C101#57811yescharacteristic
160C2×C101#57811yescharacteristic
161D101—41no—
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#451—1011no—
174#441—1011no—
175#441—1011no—
176C4×Q16—1011no—
177C4×Q16D10111yescharacteristic, complemented
178#433—1011no—
179C22×Q8—1011no—
180C2×Q16—1011no—
181C2×Q16—1011no—
182C2×Q16—1011no—
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⋊Q8—1011no—
192C4×Q8—1011no—
193C4×Q8—1011no—
194C4×Q8—1011no—
195C4×Q8—1011no—
196C4×Q8—2021no—
197C4×Q8—2021no—
198C4×Q8C2×D10111yescharacteristic
199C4×Q8C2×D10111yescharacteristic
200#117—1011no—
201#117—1011no—
202#117—2021no—
203#117—2021no—
204#107—1011no—
205#107C2×D10111yescharacteristic
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⋊C8—1011no—
220#97—1011no—
221C4×C8C2×D10111yescharacteristic
222C2×Q8—1011no—
223C2×Q8—1011no—
224C2×Q8—2021no—
225C2×Q8—2022no—
226C2×Q8—2022no—
227C2×Q8C22×D10111yescharacteristic
228C2×Q8C22×D10111yescharacteristic
229C22×C4—1011no—
230C22×C4—2021no—
231Q16—22no—
232Q16—2022no—
233Q16—2022no—
234Q16—2022no—
235C2×C8—1011no—
236C2×C8—1011no—
237C2×C8C22×D10111yescharacteristic
238C2×C8C22×D10111yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
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⋊C4C22×D10111yescharacteristic
256C4⋊C4C22×D10111yescharacteristic
257C22⋊C4—1011no—
258C22⋊C4—1011no—
259C22⋊C4—2021no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C42—21no—
263C42—21no—
264C42—1011no—
265C42—2021no—
266C42—2021no—
267C42—2021no—
268C42—2021no—
269C42—2021no—
270C42C22×D10111yescharacteristic
271C23—1011no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—1012no—
276Q8—2024no—
277Q8—2021no—
278Q8—2021no—
279Q8—2021no—
280Q8#1310807412yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—1011no—
286C2×C4—1011no—
287C2×C4—1011no—
288C2×C4—2024no—
289C2×C4—2021no—
290C2×C4—2021no—
291C2×C4—2021no—
292C2×C4—2021no—
293C2×C4—2021no—
294C2×C4—2021no—
295C2×C4D4×D10111yescharacteristic
296C2×C4D4×D10111yescharacteristic
297C2×C4C23×D10111yescharacteristic
298C8—21no—
299C8—21no—
300C8—2021no—
301C8—2021no—
302C22—2021no—
303C22—2021no—
304C22—2021no—
305C22C2×D4×D10111yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—4041no—
313C4—4041no—
314C4—4041no—
315C4—2021no—
316C4—2021no—
317C4—2021no—
318C4—2022no—
319C4#41×D10111yescharacteristic
320C4C2×#1310807411yescharacteristic
321C2—4041no—
322C2#2026568611yescharacteristic
323C2#2026599011yescharacteristic
324C2#2026599111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×Q16×C101C2
socleC22×C101C22×D4
prime radicalC22×D101C2×D4
hypercenterC4×Q16D101
hypocenterC101#3127
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
5. #20265486C2semidirect
6. #20265483C2semidirect
7. #20265386C2semidirect
8. #20265385C2semidirect
11. #20265282C2semidirect
12. #20265282C2semidirect
13. C4×Dic404C2semidirect
15. C4×Q16×C101C2semidirect
163. C101#3127semidirect
177. C4×Q16D101semidirect

Maximal subgroups

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

subgroupindex
#202657852
#202656532
#202655202
#202655172
#202654862
#202654832
#202653862
#202653852
#202653342
#202653312
#202652822
C4×Dic4042
#202652362
C4×Q16×C1012
#3127101