Finite Group Databasebeta

← #27183840

Subgroups — #27183840

Order 12928.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265886C211yesmaximal, characteristic, complemented
2#20265848C211yesmaximal, characteristic, complemented
3#20265542C211yesmaximal, characteristic, complemented
4#20265540C211yesmaximal, characteristic, complemented
5#20265522C211yesmaximal, characteristic, complemented
6#20265477C211yesmaximal, characteristic, complemented
7#20265476C211yesmaximal, characteristic, complemented
8#20265466C211yesmaximal, characteristic, complemented
9#20265372C211yesmaximal, characteristic, complemented
10#20265370C211yesmaximal, characteristic, complemented
11#20265370C211yesmaximal, characteristic, complemented
12#20265305C211yesmaximal, characteristic, complemented
13#20265286C211yesmaximal, characteristic, complemented
14#20265257C211yesmaximal, characteristic, complemented
15#442×C101C211yesmaximal, characteristic, complemented
16C2×#13108074—21no—
17C2×#13108059C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19#15692254C2211yescharacteristic
20#15692252C2211yescharacteristic
21#15692251C2211yescharacteristic
22#15692251C2211yescharacteristic
23#15692247C2211yescharacteristic
24#15692246C2211yescharacteristic
25#15692234—21no—
26#15692232—21no—
27#15692232—21no—
28#15692229—21no—
29#15692228—21no—
30#15692228—21no—
31C4×Dic202C2211yescharacteristic
32D4×Dic101C2211yescharacteristic
33D4×Dic101C2211yescharacteristic
34#15692217C2211yescharacteristic
35#15692217C2211yescharacteristic
36C2×#13108053—21no—
37#15692188C2211yescharacteristic
38#15692186C2211yescharacteristic
39C4⋊#11717579C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692177C2211yescharacteristic
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×D8×C101C2211yescharacteristic
53C4×D4×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic
59#13108074—44no—
60C2×Dic202—21no—
61C2×Dic202D411yescharacteristic
62C2×D404—21no—
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C22×Dic101—21no—
71C22×Dic101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74#13108059—22no—
75#13108059—22no—
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C4⋊Dic101—21no—
79C4⋊Dic101D411yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87#13108053D411yescharacteristic
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90#13108051—21no—
91#13108051—21no—
92C4×Dic101C2311yescharacteristic
93C4×Dic101C2311yescharacteristic
94C2×D4×C101C2311yescharacteristic
95C2×D4×C101C2311yescharacteristic
96C22×C4×C101—21no—
97C22×C4×C101—21no—
98D8×C101—22no—
99C2×C8×C101C2311yescharacteristic
100C2×C8×C101C2311yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C22⋊C4×C101—21no—
105C42×C101C2311yescharacteristic
106C23×C101—21no—
107C23×C101—21no—
108C22×D101—21no—
109C22×D101—21no—
110D4×C101—21no—
111D4×C101—21no—
112D4×C101—21no—
113D4×C101#4112yescomplemented
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—42no—
117C2×C4×C101—81no—
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2411yescharacteristic
121C22⋊D101—42no—
122C22⋊D101—42no—
123C22⋊D101—81no—
124C22⋊D101—81no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—42no—
128C2×Dic101—81no—
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133D404—21no—
134D404—21no—
135D404—21no—
136D404—21no—
137D404—42no—
138D404—81no—
139C4×D101—42no—
140C4×D101—42no—
141Dic202—22no—
142Dic202—81no—
143C8×C101—21no—
144C8×C101—21no—
145#11717579—21no—
146#11717579—21no—
147C22×C101—22no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101C22×D411yescharacteristic
154C2×D101—41no—
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—41no—
158C2×D101—41no—
159C2×D101—41no—
160C4×C101—21no—
161C4×C101—21no—
162C4×C101—41no—
163C4×C101—41no—
164C4×C101C2×#4111yescharacteristic
165C4×C1012−1+411yescharacteristic
166Dic101—21no—
167Dic101—21no—
168Dic101—21no—
169Dic101—41no—
170Dic101—41no—
171Dic101AGL(1,ℤ/8)12yes—
172C2×C101—42no—
173C2×C101—81no—
174C2×C101#54811yescharacteristic
175C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C101#57611yescharacteristic
177D101—81no—
178D101—81no—
179#3366—1011nomaximal, Sylow
180C101#336611yescharacteristic, complemented, Sylow
181#554—1011no—
182#550—1011no—
183#490—1011no—
184#490—1011no—
185#486—1011no—
186#478—1011no—
187#478—1011no—
188#476—1011no—
189#459—1011no—
190#459—1011no—
191#459—1011no—
192#446—1011no—
193#442—1011no—
194#442D10111yescharacteristic, complemented
195C4×D8—1011no—
196C2×#41—2021no—
197C2×D8—1011no—
198C2×D8—1011no—
199C2×D8C2×D10111yescharacteristic
200C4⋊Q8—1011no—
201#127—1011no—
202#127—1011no—
203#127—1011no—
204#125—1011no—
205#125—1011no—
206C22⋊Q8—2021no—
207C22⋊Q8—2021no—
208C22⋊Q8—2021no—
209#121—2021no—
210#121—2021no—
211#121—2021no—
212C4×Q8—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—1011no—
217C4×D4C2×D10111yescharacteristic
218C4×D4C2×D10111yescharacteristic
219C2×C4⋊C4—2021no—
220#107—1011no—
221#106—1011no—
222#106C2×D10111yescharacteristic
223C4⋊C8—1011no—
224C4⋊C8—1011no—
225C4⋊C8—1011no—
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102C2×D10111yescharacteristic
235#102C2×D10111yescharacteristic
236#97—1011no—
237#97C2×D10111yescharacteristic
238C4×C8—1011no—
239#41—4044no—
240C2×Q8—1011no—
241C2×Q8—2021no—
242C2×D4—1011no—
243C2×D4—1011no—
244C2×D4—2021no—
245C2×D4—2021no—
246C2×D4—2021no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4—2021no—
250C2×D4C22×D10111yescharacteristic
251C2×D4C22×D10111yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4—2021no—
257C22×C4—2021no—
258D8—22no—
259D8—2022no—
260D8—2022no—
261C2×C8—1011no—
262C2×C8—1011no—
263C2×C8C22×D10111yescharacteristic
264C2×C8C22×D10111yescharacteristic
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—1011no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C42—1011no—
284C42—1011no—
285C42C22×D10111yescharacteristic
286C23—21no—
287C23—21no—
288C23—2021no—
289C23—2021no—
290Q8—8081no—
291Q8—2022no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—8081no—
296D4—8081no—
297D4—8081no—
298D4—4042no—
299D4—4042no—
300D4—4042no—
301D4—2021no—
302D4—2021no—
303D4—2021no—
304D4—2021no—
305D4#1310807412yescomplemented
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—8081no—
311C2×C4—1011no—
312C2×C4—1011no—
313C2×C4—1011no—
314C2×C4—1011no—
315C2×C4—4042no—
316C2×C4—4042no—
317C2×C4—4042no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4D4×D10111yescharacteristic
321C2×C4D4×D10111yescharacteristic
322C2×C4C23×D10111yescharacteristic
323C8—21no—
324C8—21no—
325C8—2021no—
326C8—2021no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—4041no—
334C22—4041no—
335C22—4041no—
336C22—4041no—
337C22—4041no—
338C22—4041no—
339C22C2×D4×D10111yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—1012no—
345C4—4041no—
346C4—4041no—
347C4—2021no—
348C4—2021no—
349C4—2021no—
350C4C2×#1310807411yescharacteristic
351C4#1569230211yescharacteristic
352C2—42no—
353C2—81no—
354C2—8081no—
355C2—8081no—
356C2#2026583611yescharacteristic
357C2AGL(1,ℤ/8)×D10111yescharacteristic
358C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#442×C101C2
socleC22×C101C22×D4
hypercenter#442D101
hypocenterC101#3366
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #20265886C2semidirect
2. #20265848C2semidirect
3. #20265542C2semidirect
4. #20265540C2semidirect
5. #20265522C2semidirect
6. #20265477C2semidirect
7. #20265476C2semidirect
8. #20265466C2semidirect
9. #20265372C2semidirect
10. #20265370C2semidirect
11. #20265370C2semidirect
12. #20265305C2semidirect
13. #20265286C2semidirect
14. #20265257C2semidirect
15. #442×C101C2semidirect
113. D4×C101#41semidirect
180. C101#3366semidirect
194. #442D101semidirect
305. D4#13108074semidirect

Maximal subgroups

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

subgroupindex
#202658862
#202658482
#202655422
#202655402
#202655222
#202654772
#202654762
#202654662
#202653722
#202653702
#202653052
#202652862
#202652572
#442×C1012
#3366101