Finite Group Databasebeta

← #27180997

Subgroups — #27180997

Order 12928.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265807C211yesmaximal, characteristic, complemented
2#20265659C211yesmaximal, characteristic
3#20265482C211yesmaximal, characteristic, complemented
4#20265480C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265468C211yesmaximal, characteristic, complemented
7#20265466C211yesmaximal, characteristic, complemented
8#20265462C211yesmaximal, characteristic, complemented
9#20265460C211yesmaximal, characteristic, complemented
10#20265218C211yesmaximal, characteristic, complemented
11#106×D101C211yesmaximal, characteristic, complemented
12#20265202C211yesmaximal, characteristic, complemented
13#20265177C211yesmaximal, characteristic
14#20265172C211yesmaximal, characteristic
15#479×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108057C2211yescharacteristic
18C2×C8×D101C2211yescharacteristic
19#15692255C2211yescharacteristic
20#15692254—21no—
21#15692254C2211yescharacteristic
22#15692246C2211yescharacteristic
23#15692245—21no—
24#15692235—21no—
25#15692233—21no—
26Q8×Dic101C2211yescharacteristic
27C4×D404—21no—
28C4×D404C2211yescharacteristic
29#15692217—21no—
30#15692217C2211yescharacteristic
31#15692213C2211yescharacteristic
32#15692212—21no—
33#15692212C2211yescharacteristic
34C4⋊C4×D101—21no—
35C4⋊C4×D101C2211yescharacteristic
36C4⋊C4×D101C2211yescharacteristic
37#15692188C2211yescharacteristic
38#15692187C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692186C2211yescharacteristic
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692177C2211yescharacteristic
49#15692177C2211yescharacteristic
50#15692175C2211yescharacteristic
51#15692175C2211yescharacteristic
52#125×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic
59#13108075—22no—
60#13108075Q814yescomplemented
61C2×Dic202—21no—
62C2×Dic202—21no—
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67C2×C4×D101—21no—
68C2×C4×D101—21no—
69C2×C4×D101—21no—
70C2×C4×D101C2311yescharacteristic
71#13108057—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C8×D101—22no—
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101—21no—
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86#13108053D411yescharacteristic
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051D411yescharacteristic
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×C8×C101C2311yescharacteristic
97C2×C8×C101C2311yescharacteristic
98C4⋊C4×C101—21no—
99C4⋊C4×C101—21no—
100C4⋊C4×C101—21no—
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C42×C101—21no—
105C42×C101C2311yescharacteristic
106C22×D101—21no—
107C22×D101C2×D411yescharacteristic
108Q8×C101—21no—
109Q8×C101C2×Q812yes—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101C2×D411yescharacteristic
115C2×C4×C101C2×D411yescharacteristic
116C2×C4×C101C2411yescharacteristic
117C2×Dic101—21no—
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101C2×D411yescharacteristic
121C2×Dic101C2×D411yescharacteristic
122C2×Dic101C2×D411yescharacteristic
123D404—24no—
124D404—21no—
125D404C2×Q812yes—
126C4×D101—24no—
127C4×D101—81no—
128C4×D101—81no—
129C4×D101—81no—
130C4×D101—81no—
131C4×D101C2×Q812yes—
132C4×D101#4112yes—
133Dic202—81no—
134Dic202—81no—
135C8×C101—21no—
136C8×C101—21no—
137#11717579—21no—
138#11717579—21no—
139C22×C101C22×D411yescharacteristic
140C2×D101—21no—
141C2×D101—21no—
142C2×D101—22no—
143C2×D101—41no—
144C2×D101—41no—
145C2×D101C22⋊Q812yes—
146C4×C101—22no—
147C4×C101—21no—
148C4×C101—21no—
149C4×C101—41no—
150C4×C101—41no—
151C4×C101—41no—
152C4×C101C22×Q811yescharacteristic
153C4×C101C2×#4111yescharacteristic
154Dic101—21no—
155Dic101—21no—
156Dic101—41no—
157Dic101—41no—
158Dic101—41no—
159Dic101C22⋊Q812yes—
160C2×C101C2×C22⋊Q811yescharacteristic
161C2×C101#57711yescharacteristic
162C2×C101#57711yescharacteristic
163D101—22no—
164D101—42no—
165#3084—1011nomaximal, Sylow
166C101#308411yescharacteristic, complemented, Sylow
167#544—1011no—
168#520—1011no—
169#479—1011no—
170#479D10111yescharacteristic, complemented
171#478—1011no—
172#478—1011no—
173#476—1011no—
174#476—1011no—
175#475—1011no—
176#475—1011no—
177#428—1011no—
178C2×#106—1011no—
179#424—1011no—
180#417—1011no—
181#416—1011no—
182C2×#41—1011no—
183C2×M16—1011no—
184C22×C8—1011no—
185C4⋊Q8—1011no—
186C4⋊Q8—1011no—
187C4⋊Q8—2021no—
188#125—1011no—
189#125—2021no—
190#125C2×D10111yescharacteristic
191C22⋊Q8—2021no—
192C22⋊Q8—2021no—
193C4×Q8—1011no—
194C4×Q8C2×D10111yescharacteristic
195C4×D4—1011no—
196C4×D4—1011no—
197C4×D4—2021no—
198C4×D4—2021no—
199#117—1011no—
200#117—1011no—
201#117—2021no—
202C2×C4⋊C4—1011no—
203C2×C4⋊C4—1011no—
204C2×C4⋊C4—2021no—
205#107—1011no—
206#107C2×D10111yescharacteristic
207#106—1011no—
208#106—1011no—
209#106—1011no—
210#106—1011no—
211#106—1011no—
212#106C2×D10111yescharacteristic
213C4⋊C8—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216C4⋊C8C2×D10111yescharacteristic
217#103—1011no—
218#103—1011no—
219#103C2×D10111yescharacteristic
220#103C2×D10111yescharacteristic
221#102—1011no—
222#102—1011no—
223#102—1011no—
224#102—1011no—
225#41—1014no—
226#41—2022no—
227C2×Q8—2021no—
228C2×Q8—2021no—
229C2×Q8C22×D10111yescharacteristic
230C2×D4—1011no—
231C2×D4—2021no—
232C22×C4—1011no—
233C22×C4—2021no—
234C22×C4—2021no—
235C22×C4—2021no—
236C22×C4—2021no—
237C22×C4—2021no—
238M16—2022no—
239C2×C8—1011no—
240C2×C8—1011no—
241C2×C8—2022no—
242C2×C8C22×D10111yescharacteristic
243C2×C8C22×D10111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—1011no—
248C4⋊C4—1011no—
249C4⋊C4—1011no—
250C4⋊C4—1011no—
251C4⋊C4—1011no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4C22×D10111yescharacteristic
260C4⋊C4C22×D10111yescharacteristic
261C4⋊C4C22×D10111yescharacteristic
262C22⋊C4—1011no—
263C22⋊C4—1011no—
264C22⋊C4—2021no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C42—21no—
268C42—1011no—
269C42—2021no—
270C42—2021no—
271C42C22×D10111yescharacteristic
272C23—1011no—
273C23—2021no—
274Q8—21no—
275Q8—8081no—
276Q8—8081no—
277Q8Q8×D10112yes—
278D4—1012no—
279D4—2024no—
280D4—2021no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—8081no—
286C2×C4—8081no—
287C2×C4—8081no—
288C2×C4—8081no—
289C2×C4—1011no—
290C2×C4—1012no—
291C2×C4—1012no—
292C2×C4—1011no—
293C2×C4—1011no—
294C2×C4—2024no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4—2021no—
298C2×C4D4×D10111yescharacteristic
299C2×C4D4×D10111yescharacteristic
300C2×C4C23×D10111yescharacteristic
301C8—21no—
302C8—21no—
303C8—2021no—
304C8—2021no—
305C22—1012no—
306C22—4041no—
307C22—4041no—
308C22—2022no—
309C22—2021no—
310C22—2021no—
311C22C2×D4×D10111yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—1012no—
319C4—4041no—
320C4—4041no—
321C4—4041no—
322C4—2021no—
323C4—2021no—
324C4C2×Q8×D10111yescharacteristic
325C4#41×D10111yescharacteristic
326C2—4042no—
327C2—2022no—
328C2C22⋊Q8×D10111yescharacteristic
329C2#2026598411yescharacteristic
330C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#479×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108075C22
hypercenter#479D101
hypocenterC101#3084
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #20265807C2semidirect
3. #20265482C2semidirect
4. #20265480C2semidirect
5. #20265474C2semidirect
6. #20265468C2semidirect
7. #20265466C2semidirect
8. #20265462C2semidirect
9. #20265460C2semidirect
10. #20265218C2semidirect
11. #106×D101C2semidirect
12. #20265202C2semidirect
15. #479×C101C2semidirect
60. #13108075Q8semidirect
166. C101#3084semidirect
170. #479D101semidirect

Maximal subgroups

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

subgroupindex
#202658072
#202656592
#202654822
#202654802
#202654742
#202654682
#202654662
#202654622
#202654602
#202652182
#106×D1012
#202652022
#202651772
#202651722
#479×C1012
#3084101