Finite Group Databasebeta

← #27183851

Subgroups — #27183851

Order 12928.

372 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#20265899C211yesmaximal, characteristic, complemented
2D4×Dic202C211yesmaximal, characteristic, complemented
3#20265538C211yesmaximal, characteristic, complemented
4#20265537C211yesmaximal, characteristic, complemented
5#20265526C211yesmaximal, characteristic, complemented
6#20265472C211yesmaximal, characteristic, complemented
7#20265471C211yesmaximal, characteristic, complemented
8#20265456C211yesmaximal, characteristic, complemented
9#20265394C211yesmaximal, characteristic
10#20265392C211yesmaximal, characteristic
11#20265380C211yesmaximal, characteristic, complemented
12#20265305C211yesmaximal, characteristic, complemented
13#20265274C211yesmaximal, characteristic, complemented
14#20265257C211yesmaximal, characteristic
15#440×C101C211yesmaximal, characteristic, complemented
16C22×Dic202—21no—
17C2×#13108062C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19#15692255C2211yescharacteristic
20#15692254C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692253C2211yescharacteristic
23#15692250—21no—
24#15692249—21no—
25#15692244C2211yescharacteristic
26#15692243C2211yescharacteristic
27#15692234—21no—
28#15692233—21no—
29#15692232—21no—
30#15692232—21no—
31C4×Dic202C2211yescharacteristic
32#15692220C2211yescharacteristic
33D4×Dic101C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35#15692217C2211yescharacteristic
36#15692212—21no—
37#15692188C2211yescharacteristic
38#15692188C2211yescharacteristic
39C4⋊#11717579C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692177C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×Dic202—44no—
63C2×Dic202D411yescharacteristic
64C2×Dic202C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C22×Dic101—21no—
67C22×Dic101—21no—
68C2×C4×D101—21no—
69#13108062—22no—
70#13108059—22no—
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C4⋊Dic101—21no—
74C4⋊Dic101D411yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053D411yescharacteristic
83#13108053D411yescharacteristic
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91C4×Dic101—21no—
92C4×Dic101C2311yescharacteristic
93C4×Dic101C2311yescharacteristic
94C2×Q8×C101C2311yescharacteristic
95C2×D4×C101C2311yescharacteristic
96C22×C4×C101—21no—
97SD16×C101—22no—
98C2×C8×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C4⋊C4×C101—21no—
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C42×C101—21no—
105C42×C101C2311yescharacteristic
106C23×C101—21no—
107C22×D101—21no—
108Q8×C101—21no—
109Q8×C101—21no—
110D4×C101—21no—
111D4×C101#4112yes—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—42no—
116C2×C4×C101C2×D411yescharacteristic
117C2×C4×C101C2×D411yescharacteristic
118C2×C4×C101C2411yescharacteristic
119C2×Dic101—21no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—42no—
123C2×Dic101—42no—
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128D404—21no—
129D404—21no—
130C4×D101—42no—
131Dic202—22no—
132Dic202—21no—
133Dic202—21no—
134Dic202—42no—
135Dic202—42no—
136Dic202—42no—
137C8×C101—21no—
138C8×C101—21no—
139#11717579—21no—
140#11717579—21no—
141C22×C101—22no—
142C22×C101—41no—
143C22×C101—41no—
144C22×C101C22×D411yescharacteristic
145C2×D101—41no—
146C2×D101—41no—
147C2×D101—41no—
148C4×C101—21no—
149C4×C101—21no—
150C4×C101—41no—
151C4×C101—41no—
152C4×C101—41no—
153C4×C101C2×#4111yescharacteristic
154C4×C1012−1+411yescharacteristic
155Dic101—21no—
156Dic101—21no—
157Dic101—21no—
158Dic101—41no—
159Dic101—41no—
160Dic101—41no—
161Dic101AGL(1,ℤ/8)12yes—
162C2×C101—42no—
163C2×C101#54811yescharacteristic
164C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C101#57811yescharacteristic
166D101—81no—
167#3367—1011nomaximal, Sylow
168C101#336711yescharacteristic, complemented, Sylow
169#556—1011no—
170D4×Q8—1011no—
171#489—1011no—
172#489—1011no—
173#487—1011no—
174#477—1011no—
175#477—1011no—
176#474—1011no—
177#463—1011no—
178#463—1011no—
179#461—1011no—
180#446—1011no—
181#440—1011no—
182#440D10111yescharacteristic, complemented
183C4×D8—1011no—
184C22×Q8—2021no—
185C2×SD16—1011no—
186C2×SD16C2×D10111yescharacteristic
187C2×D8—1011no—
188C4⋊Q8—1011no—
189C4⋊Q8—1011no—
190C4⋊Q8—1011no—
191C4⋊Q8—1011no—
192#126—2021no—
193#126—2021no—
194#124—1011no—
195#124—1011no—
196C22⋊Q8—2021no—
197C22⋊Q8—2021no—
198C22⋊Q8—2021no—
199C22⋊Q8—2021no—
200C4×Q8—1011no—
201C4×Q8—1011no—
202C4×Q8C2×D10111yescharacteristic
203C4×D4—1011no—
204C4×D4—1011no—
205C4×D4—1011no—
206C4×D4C2×D10111yescharacteristic
207#117—2021no—
208#107—1011no—
209#107—1011no—
210#107C2×D10111yescharacteristic
211C4⋊C8—1011no—
212C4⋊C8—1011no—
213C4⋊C8—1011no—
214#103—1011no—
215#103—1011no—
216#103—1011no—
217#103—1011no—
218#103—1011no—
219#103C2×D10111yescharacteristic
220#102—1011no—
221#102—1011no—
222#102—1011no—
223#102C2×D10111yescharacteristic
224#97—1011no—
225#97C2×D10111yescharacteristic
226C4×C8—1011no—
227C2×Q8—1011no—
228C2×Q8—1011no—
229C2×Q8—4044no—
230C2×Q8—2021no—
231C2×Q8—2021no—
232C2×Q8—2021no—
233C2×Q8C22×D10111yescharacteristic
234C2×D4—1011no—
235C2×D4C22×D10111yescharacteristic
236C22×C4—21no—
237C22×C4—2021no—
238C22×C4—2021no—
239C22×C4—2021no—
240SD16—22no—
241SD16—2022no—
242D8—2022no—
243C2×C8—1011no—
244C2×C8—1011no—
245C2×C8C22×D10111yescharacteristic
246C2×C8C22×D10111yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—1011no—
249C4⋊C4—1011no—
250C4⋊C4—1011no—
251C4⋊C4—1011no—
252C4⋊C4—1011no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4C22×D10111yescharacteristic
260C4⋊C4C22×D10111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2021no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C42—21no—
270C42—1011no—
271C42—1011no—
272C42—2021no—
273C42C22×D10111yescharacteristic
274C23—21no—
275C23—2021no—
276Q8—21no—
277Q8—21no—
278Q8—4042no—
279Q8—4042no—
280Q8—4042no—
281Q8—2022no—
282Q8—2021no—
283Q8—2021no—
284D4—21no—
285D4—2021no—
286D4—2021no—
287D4#1310807412yes—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—1011no—
293C2×C4—1011no—
294C2×C4—1011no—
295C2×C4—1011no—
296C2×C4—4042no—
297C2×C4—4042no—
298C2×C4—4042no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4—2021no—
302C2×C4D4×D10111yescharacteristic
303C2×C4D4×D10111yescharacteristic
304C2×C4C23×D10111yescharacteristic
305C8—21no—
306C8—21no—
307C8—2021no—
308C8—2021no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—4041no—
313C22—4041no—
314C22—4041no—
315C22C2×D4×D10111yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—1012no—
322C4—4041no—
323C4—4041no—
324C4—4041no—
325C4—2021no—
326C4—2021no—
327C4—2021no—
328C4C2×#1310807411yescharacteristic
329C4#1569230211yescharacteristic
330C2—42no—
331C2—8081no—
332C2#2026583611yescharacteristic
333C2AGL(1,ℤ/8)×D10111yescharacteristic
334C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108059C22
hypercenter#440D101
hypocenterC101#3367
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20265899C2semidirect
2. D4×Dic202C2semidirect
3. #20265538C2semidirect
4. #20265537C2semidirect
5. #20265526C2semidirect
6. #20265472C2semidirect
7. #20265471C2semidirect
8. #20265456C2semidirect
11. #20265380C2semidirect
12. #20265305C2semidirect
13. #20265274C2semidirect
15. #440×C101C2semidirect
168. C101#3367semidirect
182. #440D101semidirect

Maximal subgroups

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

subgroupindex
#202658992
D4×Dic2022
#202655382
#202655372
#202655262
#202654722
#202654712
#202654562
#202653942
#202653922
#202653802
#202653052
#202652742
#202652572
#440×C1012
#3367101