Finite Group Databasebeta

← #27182584

Subgroups — #27182584

Order 12928.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265788C211yesmaximal, characteristic
2#20265785C211yesmaximal, characteristic
3#20265486C211yesmaximal, characteristic, complemented
4#20265485C211yesmaximal, characteristic, complemented
5#20265484C211yesmaximal, characteristic, complemented
6#20265482C211yesmaximal, characteristic, complemented
7#20265434C211yesmaximal, characteristic
8#20265432C211yesmaximal, characteristic
9#20265417C211yesmaximal, characteristic
10#20265415C211yesmaximal, characteristic
11#20265385C211yesmaximal, characteristic, complemented
12#20265385C211yesmaximal, characteristic, complemented
13#20265384C211yesmaximal, characteristic, complemented
14#20265202C211yesmaximal, characteristic
15#462×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108057C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20#15692253C2211yescharacteristic
21#15692250—21no—
22#15692250—21no—
23#15692250—21no—
24#15692249—21no—
25#15692247—21no—
26#15692247C2211yescharacteristic
27#15692246C2211yescharacteristic
28#15692245—21no—
29#15692245—21no—
30#15692245—21no—
31#15692235C2211yescharacteristic
32#15692231C2211yescharacteristic
33#15692231C2211yescharacteristic
34#15692231C2211yescharacteristic
35Q8×Dic101C2211yescharacteristic
36Q8×Dic101C2211yescharacteristic
37C4×Dic202C2211yescharacteristic
38#15692212C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692187C2211yescharacteristic
42#15692186C2211yescharacteristic
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692179C2211yescharacteristic
50#15692179C2211yescharacteristic
51#15692179C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C2×Q16×C101C2211yescharacteristic
54C4⋊Q8×C101C2211yescharacteristic
55C4×Q8×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60C2×C4×D101C2311yescharacteristic
61#13108064—41no—
62#13108064—41no—
63#13108057—41no—
64C2×#11717579C2311yescharacteristic
65C2×#11717579C2311yescharacteristic
66C8×D101—41no—
67C4⋊Dic101—21no—
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101C2311yescharacteristic
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74#13108053—21no—
75#13108053—21no—
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#13108051—21no—
89#13108051D411yescharacteristic
90#13108051D411yescharacteristic
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98Q16×C101—41no—
99Q16×C101—41no—
100C2×C8×C101C2311yescharacteristic
101C2×C8×C101C2311yescharacteristic
102C4⋊C4×C101—21no—
103C4⋊C4×C101—21no—
104C4⋊C4×C101—21no—
105C4⋊C4×C101C2311yescharacteristic
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C22×D101C2×D411yescharacteristic
109Q8×C101—21no—
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101—21no—
114Q8×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2411yescharacteristic
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129C4×D101—21no—
130C4×D101—21no—
131Dic202—21no—
132Dic202—21no—
133C8×C101—21no—
134C8×C101—21no—
135#11717579—21no—
136#11717579—21no—
137C22×C101C22×D411yescharacteristic
138C2×D101—21no—
139C2×D101—21no—
140C2×D101—21no—
141C4×C101—21no—
142C4×C101—21no—
143C4×C101—41no—
144C4×C101—41no—
145C4×C101—41no—
146C4×C101—41no—
147C4×C101H(5,𝔽2)11yescharacteristic
148C4×C1012−1+411yescharacteristic
149Dic101—21no—
150Dic101—21no—
151Dic101—21no—
152Dic101—41no—
153Dic101—41no—
154Dic101—41no—
155Dic101—41no—
156C2×C101#53711yescharacteristic
157C2×C101#57811yescharacteristic
158C2×C101#57811yescharacteristic
159D101—41no—
160#3261—1011nomaximal, Sylow
161C101#326111yescharacteristic, complemented, Sylow
162#541—1011no—
163#541—1011no—
164#479—1011no—
165#479—1011no—
166#479—1011no—
167#479—1011no—
168#470—1011no—
169#470—1011no—
170#467—1011no—
171#467—1011no—
172#462—1011no—
173#462—1011no—
174#462—1011no—
175#462D10111yescharacteristic, complemented
176#424—1011no—
177C2×Q16—1011no—
178C2×Q16—1011no—
179C2×Q16C2×D10111yescharacteristic
180C2×Q16C2×D10111yescharacteristic
181C2×M16—1011no—
182C22×C8—1011no—
183C4⋊Q8—1011no—
184C4⋊Q8C2×D10111yescharacteristic
185#126—2021no—
186#126—2021no—
187#126—2021no—
188#126—2021no—
189#125—1011no—
190#125—1011no—
191#125—2021no—
192#125—2021no—
193#125—2021no—
194#125—2021no—
195C22⋊Q8—1011no—
196C22⋊Q8—1011no—
197C22⋊Q8—1011no—
198C22⋊Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—1011no—
201C4×Q8—1011no—
202C4×Q8C2×D10111yescharacteristic
203#117—1011no—
204#107—1011no—
205#107—1011no—
206#106—1011no—
207#106—1011no—
208C4⋊C8—1011no—
209C4⋊C8—1011no—
210C4⋊C8—1011no—
211C4⋊C8C2×D10111yescharacteristic
212#103—1011no—
213#103—1011no—
214#103—1011no—
215#103—1011no—
216#103—1011no—
217#103—1011no—
218#103C2×D10111yescharacteristic
219#103C2×D10111yescharacteristic
220C2×Q8—1011no—
221C2×Q8C22×D10111yescharacteristic
222C2×Q8C22×D10111yescharacteristic
223C2×Q8C22×D10111yescharacteristic
224C22×C4—1011no—
225Q16—41no—
226Q16—41no—
227Q16—4041no—
228Q16—4041no—
229M16—4041no—
230C2×C8—1011no—
231C2×C8—1011no—
232C2×C8—4041no—
233C2×C8C22×D10111yescharacteristic
234C2×C8C22×D10111yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—1011no—
239C4⋊C4—1011no—
240C4⋊C4—1011no—
241C4⋊C4—1011no—
242C4⋊C4—1011no—
243C4⋊C4—2021no—
244C4⋊C4—2021no—
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—1011no—
258C22⋊C4—1011no—
259C22⋊C4—2021no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C42—21no—
264C42—1011no—
265C42—2021no—
266C42—2021no—
267C42—2021no—
268C42C22×D10111yescharacteristic
269C23—1011no—
270Q8—21no—
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—2021no—
277Q8—2021no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—1011no—
283C2×C4—1011no—
284C2×C4—1011no—
285C2×C4—2021no—
286C2×C4—2021no—
287C2×C4—2021no—
288C2×C4—2021no—
289C2×C4—2021no—
290C2×C4—2021no—
291C2×C4D4×D10111yescharacteristic
292C2×C4D4×D10111yescharacteristic
293C2×C4C23×D10111yescharacteristic
294C8—21no—
295C8—21no—
296C8—2021no—
297C8—2021no—
298C22—2021no—
299C22—2021no—
300C22—2021no—
301C22C2×D4×D10111yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—4041no—
309C4—4041no—
310C4—4041no—
311C4—4041no—
312C4—2021no—
313C4—2021no—
314C4—2021no—
315C4#1569230011yescharacteristic
316C4#1569230211yescharacteristic
317C2—4041no—
318C2#2026575611yescharacteristic
319C2#2026599011yescharacteristic
320C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#462×C101C2
socleC22×C101C22×D4
prime radicalC22×D101C2×D4
hypercenter#462D101
hypocenterC101#3261
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #20265486C2semidirect
4. #20265485C2semidirect
5. #20265484C2semidirect
6. #20265482C2semidirect
11. #20265385C2semidirect
12. #20265385C2semidirect
13. #20265384C2semidirect
15. #462×C101C2semidirect
161. C101#3261semidirect
175. #462D101semidirect

Maximal subgroups

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

subgroupindex
#202657882
#202657852
#202654862
#202654852
#202654842
#202654822
#202654342
#202654322
#202654172
#202654152
#202653852
#202653842
#202652022
#462×C1012
#3261101