Finite Group Databasebeta

← #27182995

Subgroups — #27182995

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265840C211yesmaximal, characteristic, complemented
2#20265828C211yesmaximal, characteristic
3#20265526C211yesmaximal, characteristic, complemented
4#20265509C211yesmaximal, characteristic, complemented
5#20265461C211yesmaximal, characteristic, complemented
6#20265408C211yesmaximal, characteristic, complemented
7#20265407C211yesmaximal, characteristic, complemented
8#20265381C211yesmaximal, characteristic, complemented
9#20265318C211yesmaximal, characteristic, complemented
10#20265317C211yesmaximal, characteristic, complemented
11#20265264C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13C2×#15692178C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#483×C101C211yesmaximal, characteristic, complemented
16C2×#13108074—21no—
17C22×Dic202C2211yescharacteristic
18C2×Q8×D101C2211yescharacteristic
19C2×#13108063C2211yescharacteristic
20C2×#13108063C2211yescharacteristic
21C2×#13108061C2211yescharacteristic
22C22×#11717579C2211yescharacteristic
23C2×C8×D101C2211yescharacteristic
24#15692254—21no—
25#15692254C2211yescharacteristic
26#15692253C2211yescharacteristic
27#15692240—21no—
28#15692238—21no—
29#15692235C2211yescharacteristic
30#15692233—21no—
31#15692233—21no—
32#15692232—21no—
33#15692230C2211yescharacteristic
34#15692229C2211yescharacteristic
35#15692220C2211yescharacteristic
36#15692217C2211yescharacteristic
37C4×C22⋊D101C2211yescharacteristic
38#15692215—21no—
39C4⋊C4×D101C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692178C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C22⋊Q8×C101C2211yescharacteristic
53C2×C4⋊C4×C101C2211yescharacteristic
54#106×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108074—44no—
60C2×Dic202—22no—
61C2×Dic202—21no—
62C2×Dic202—21no—
63C2×Dic202—21no—
64C2×Dic202—41no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—41no—
69C2×D404—21no—
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C2×C22⋊D101D411yescharacteristic
73C22×Dic101—21no—
74C2×C4×D101—21no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101C2311yescharacteristic
78#13108063—22no—
79#13108063—41no—
80#13108061—41no—
81C2×#11717579—41no—
82C2×#11717579C2311yescharacteristic
83C2×#11717579C2311yescharacteristic
84C8×D101—41no—
85C4⋊Dic101—21no—
86C4⋊Dic101C2311yescharacteristic
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053—21no—
92#13108053D411yescharacteristic
93C22⋊Dic101D411yescharacteristic
94#13108051—21no—
95#13108051—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051D411yescharacteristic
99C4×Dic101—21no—
100C4×Dic101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101—21no—
107C4⋊C4×C101—21no—
108C4⋊C4×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101—21no—
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×C101—42no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C22⋊D101—42no—
127C22⋊D101SD1614yescomplemented
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—42no—
132C2×Dic101C2×D411yescharacteristic
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135D404—21no—
136D404—21no—
137D404—42no—
138C4×D101—21no—
139C4×D101—21no—
140C4×D101—42no—
141C4×D101—42no—
142C4×D101—42no—
143Dic202—21no—
144Dic202—21no—
145Dic202—21no—
146Dic202—42no—
147Dic202—42no—
148Dic202—42no—
149Dic202#4112yescomplemented
150C8×C101—21no—
151C8×C101—21no—
152#11717579—21no—
153#11717579—21no—
154C22×C101—21no—
155C22×C101—21no—
156C22×C101C2×SD1612yes—
157C22×C101C22×D411yescharacteristic
158C2×D101—21no—
159C2×D101—21no—
160C2×D101—41no—
161C2×D101—41no—
162C2×D101—41no—
163C2×D101C2×SD1612yes—
164C4×C101—21no—
165C4×C101—41no—
166C4×C101—41no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101C2×#4111yescharacteristic
170C4×C101H(5,𝔽2)11yescharacteristic
171Dic101—22no—
172Dic101—21no—
173Dic101—21no—
174Dic101—41no—
175Dic101—41no—
176Dic101C2×SD1612yes—
177C2×C101—22no—
178C2×C101#54611yescharacteristic
179C2×C101C22×SD1611yescharacteristic
180C2×C101#57811yescharacteristic
181D101—22no—
182D101—81no—
183#3298—1011nomaximal, Sylow
184C101#329811yescharacteristic, complemented, Sylow
185D4×Q8—1011no—
186#547—1011no—
187#487—1011no—
188#483—1011no—
189#483D10111yescharacteristic, complemented
190#475—1011no—
191#465—1011no—
192#465—1011no—
193#461—1011no—
194#448—1011no—
195#448—1011no—
196C4×SD16—1011no—
197C8×D4—1011no—
198C2×#103—1011no—
199C2×#103—1011no—
200C2×#41—2021no—
201C22×Q8—1011no—
202C22×Q8—1011no—
203C2×SD16—1011no—
204C2×SD16—1011no—
205C2×SD16—1011no—
206C22×C8—1011no—
207C22×C8—1011no—
208C4⋊Q8—1011no—
209C4⋊Q8—1011no—
210C4⋊Q8—2021no—
211#123—2021no—
212#123—2021no—
213C22⋊Q8—1011no—
214C22⋊Q8—2021no—
215C22⋊Q8—2021no—
216C22⋊Q8—2021no—
217C22⋊Q8C2×D10111yescharacteristic
218#121—1011no—
219#121—1011no—
220C4×Q8—1011no—
221C4×D4—1011no—
222C4×D4—1011no—
223C4×D4—2021no—
224C2×C4⋊C4—1011no—
225C2×C4⋊C4C2×D10111yescharacteristic
226#106—1011no—
227#106C2×D10111yescharacteristic
228#106C2×D10111yescharacteristic
229C4⋊C8—1011no—
230#103—1011no—
231#103—1011no—
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103—1011no—
236#103—1011no—
237#103C2×D10111yescharacteristic
238#103C2×D10111yescharacteristic
239#102—1011no—
240C22⋊C8—1011no—
241C22⋊C8C2×D10111yescharacteristic
242C4×C8—1011no—
243#41—4044no—
244C2×Q8—1011no—
245C2×Q8—1011no—
246C2×Q8—4041no—
247C2×Q8—4041no—
248C2×Q8—2021no—
249C2×Q8—2021no—
250C2×Q8—2022no—
251C2×Q8—2022no—
252C2×Q8—2021no—
253C2×Q8C22×D10111yescharacteristic
254C2×D4—1011no—
255C2×D4—1011no—
256C2×D4—2021no—
257C2×D4—2021no—
258C22×C4—21no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4—2021no—
263C22×C4—2021no—
264C22×C4C22×D10111yescharacteristic
265SD16—4041no—
266SD16—4041no—
267SD16—2022no—
268C2×C8—1011no—
269C2×C8—1011no—
270C2×C8—4041no—
271C2×C8—4041no—
272C2×C8C22×D10111yescharacteristic
273C2×C8C22×D10111yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1011no—
277C4⋊C4—1011no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4—2021no—
282C4⋊C4—2021no—
283C4⋊C4—2021no—
284C4⋊C4C22×D10111yescharacteristic
285C4⋊C4C22×D10111yescharacteristic
286C4⋊C4C22×D10111yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—1011no—
289C22⋊C4—1011no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C42—1011no—
295C42—2021no—
296C23—1011no—
297C23—2021no—
298C23D4×D10111yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—1012no—
302Q8—4042no—
303Q8—4042no—
304Q8—4042no—
305Q8—2021no—
306Q8—2021no—
307Q8—2021no—
308D4—1014no—
309D4—4042no—
310D4—4042no—
311D4—2021no—
312D4—2021no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—1011no—
321C2×C4—1011no—
322C2×C4—1011no—
323C2×C4—4042no—
324C2×C4—4042no—
325C2×C4—4042no—
326C2×C4—4042no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4D4×D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2021no—
337C8—2021no—
338C22—21no—
339C22—21no—
340C22—1012no—
341C22—4041no—
342C22—4041no—
343C22—4041no—
344C22—2021no—
345C22—2021no—
346C22SD16×D10112yes—
347C22C2×D4×D10111yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1012no—
354C4—4041no—
355C4—4041no—
356C4—2021no—
357C4—2021no—
358C4—2022no—
359C4#41×D10111yescharacteristic
360C4#1569230111yescharacteristic
361C2—22no—
362C2—8081no—
363C2—2022no—
364C2#2026581911yescharacteristic
365C2C2×SD16×D10111yescharacteristic
366C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#483×C101C2
socleC22×C101C22×D4
prime radical#20265828C2
hypercenter#483D101
hypocenterC101#3298
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #20265840C2semidirect
3. #20265526C2semidirect
4. #20265509C2semidirect
5. #20265461C2semidirect
6. #20265408C2semidirect
7. #20265407C2semidirect
8. #20265381C2semidirect
9. #20265318C2semidirect
10. #20265317C2semidirect
11. #20265264C2semidirect
12. #20265240C2semidirect
13. C2×#15692178C2semidirect
14. #103×D101C2semidirect
15. #483×C101C2semidirect
127. C22⋊D101SD16semidirect
149. Dic202#41semidirect
184. C101#3298semidirect
189. #483D101semidirect

Maximal subgroups

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

subgroupindex
#202658402
#202658282
#202655262
#202655092
#202654612
#202654082
#202654072
#202653812
#202653182
#202653172
#202652642
#202652402
C2×#156921782
#103×D1012
#483×C1012
#3298101