Finite Group Databasebeta

← #27183734

Subgroups — #27183734

Order 12928.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265875C211yesmaximal, characteristic, complemented
2#20265871C211yesmaximal, characteristic, complemented
3#20265577C211yesmaximal, characteristic, complemented
4#20265503C211yesmaximal, characteristic, complemented
5#20265501C211yesmaximal, characteristic, complemented
6#20265492C211yesmaximal, characteristic, complemented
7#20265490C211yesmaximal, characteristic, complemented
8#20265480C211yesmaximal, characteristic, complemented
9#20265476C211yesmaximal, characteristic, complemented
10#20265431C211yesmaximal, characteristic, complemented
11#20265430C211yesmaximal, characteristic, complemented
12#20265259C211yesmaximal, characteristic, complemented
13#20265253C211yesmaximal, characteristic, complemented
14#20265218C211yesmaximal, characteristic, complemented
15#428×C101C211yesmaximal, characteristic, complemented
16C2×D808C2211yescharacteristic
17C2×#13108057C2211yescharacteristic
18#15692250—21no—
19#15692249—21no—
20#15692246C2211yescharacteristic
21#15692246C2211yescharacteristic
22#15692240—21no—
23#15692237—21no—
24#15692230C2211yescharacteristic
25#15692230C2211yescharacteristic
26#15692229—21no—
27#15692229C2211yescharacteristic
28#15692229C2211yescharacteristic
29#15692228—21no—
30C4×D404—21no—
31#15692217C2211yescharacteristic
32#15692217C2211yescharacteristic
33C4×C22⋊D101—21no—
34C4×C22⋊D101C2211yescharacteristic
35#15692213C2211yescharacteristic
36C4⋊C4×D101C2211yescharacteristic
37#15692188C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692175C2211yescharacteristic
46#15692175C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×M16×C101C2211yescharacteristic
53#117×C101C2211yescharacteristic
54C2×C4⋊C4×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#106×C101C2211yescharacteristic
58#106×C101C2211yescharacteristic
59C2×D404—21no—
60C2×D404—21no—
61C2×D404C2311yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101D411yescharacteristic
66C2×C4×D101—21no—
67C2×C4×D101—21no—
68C2×C4×D101—21no—
69C2×C4×D101C2311yescharacteristic
70D808—22no—
71#13108057—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82C22⋊Dic101D411yescharacteristic
83#13108051—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051D411yescharacteristic
91C4×Dic101—21no—
92C4×Dic101C2311yescharacteristic
93C22×C4×C101—21no—
94C22×C4×C101C2311yescharacteristic
95M16×C101—22no—
96C2×C8×C101C2311yescharacteristic
97C2×C8×C101C2311yescharacteristic
98C4⋊C4×C101—21no—
99C4⋊C4×C101C2311yescharacteristic
100C4⋊C4×C101C2311yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C42×C101—21no—
105C23×C101C2×D411yescharacteristic
106C22×D101—21no—
107C22×D101—21no—
108C22×D101C2×D411yescharacteristic
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—81no—
116C2×C4×C101C2×D411yescharacteristic
117C2×C4×C101C2411yescharacteristic
118C22⋊D101—41no—
119C22⋊D101—42no—
120C22⋊D101—81no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101C2×D411yescharacteristic
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126D404—21no—
127D404—21no—
128D404—21no—
129D404—21no—
130D404—42no—
131D404—81no—
132C4×D101—21no—
133C4×D101—21no—
134C4×D101—81no—
135C4×D101—81no—
136C4×D101—81no—
137C8×C101—21no—
138C8×C101#4112yes—
139#11717579—21no—
140#11717579—21no—
141C22×C101—21no—
142C22×C101—21no—
143C22×C101—21no—
144C22×C101C22×D411yescharacteristic
145C2×D101—21no—
146C2×D101—21no—
147C2×D101—21no—
148C2×D101—41no—
149C2×D101—41no—
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—41no—
153C2×D101—41no—
154C4×C101—21no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101—41no—
159C4×C101C2×#4111yescharacteristic
160C4×C1012−1+411yescharacteristic
161Dic101—21no—
162Dic101—21no—
163Dic101—21no—
164Dic101—41no—
165Dic101—41no—
166C2×C101—41no—
167C2×C101#54711yescharacteristic
168C2×C101#57511yescharacteristic
169C2×C101#57611yescharacteristic
170D101—41no—
171D101—81no—
172D101—81no—
173#3355—1011nomaximal, Sylow
174C101#335511yescharacteristic, complemented, Sylow
175#553—1011no—
176#553—1011no—
177#499—1011no—
178#482—1011no—
179#482—1011no—
180#480—1011no—
181#480—1011no—
182#478—1011no—
183#478—1011no—
184#469—1011no—
185#469—1011no—
186C4×D8—1011no—
187#436—1011no—
188#428—1011no—
189#428D10111yescharacteristic, complemented
190C2×D8—1011no—
191C2×M16—1011no—
192C2×M16C2×D10111yescharacteristic
193#126—2021no—
194#126—2021no—
195#125—1011no—
196#125—1011no—
197#123—2021no—
198#123—2021no—
199#121—1011no—
200#121—1011no—
201#121—1011no—
202#121—1011no—
203#121—2021no—
204#121—2021no—
205C4×D4—1011no—
206C4×D4—1011no—
207C4×D4—1011no—
208C4×D4—2021no—
209C4×D4—2021no—
210#117—1011no—
211#117C2×D10111yescharacteristic
212C2×C4⋊C4—1011no—
213C2×C4⋊C4C2×D10111yescharacteristic
214#107—1011no—
215#107—1011no—
216#107—1011no—
217#107C2×D10111yescharacteristic
218#107C2×D10111yescharacteristic
219#106—1011no—
220#106—1011no—
221#106C2×D10111yescharacteristic
222#106C2×D10111yescharacteristic
223C4⋊C8—1011no—
224#102—1011no—
225#102—1011no—
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230C22⋊C8—1011no—
231C22⋊C8—1011no—
232C4×C8—1011no—
233C2×D4—1011no—
234C2×D4—1011no—
235C2×D4—1011no—
236C2×D4—2021no—
237C2×D4—2021no—
238C2×D4—2021no—
239C2×D4—2021no—
240C22×C4—21no—
241C22×C4—1011no—
242C22×C4—2021no—
243C22×C4—2021no—
244C22×C4—2021no—
245C22×C4C22×D10111yescharacteristic
246D8—2022no—
247M16—22no—
248M16—2022no—
249C2×C8—1011no—
250C2×C8—1011no—
251C2×C8C22×D10111yescharacteristic
252C2×C8C22×D10111yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1011no—
255C4⋊C4—1011no—
256C4⋊C4—1011no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4C22×D10111yescharacteristic
263C4⋊C4C22×D10111yescharacteristic
264C4⋊C4C22×D10111yescharacteristic
265C4⋊C4C22×D10111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—1011no—
268C22⋊C4—1011no—
269C22⋊C4—2021no—
270C22⋊C4—2021no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C42—21no—
277C42—1011no—
278C42—2021no—
279C23—1011no—
280C23—2021no—
281C23—2021no—
282C23D4×D10111yescharacteristic
283D4—8081no—
284D4—8081no—
285D4—4042no—
286D4—4042no—
287D4—4041no—
288D4—2021no—
289D4—2021no—
290D4—2021no—
291D4—2021no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—81no—
299C2×C4—8081no—
300C2×C4—8081no—
301C2×C4—8081no—
302C2×C4—1011no—
303C2×C4—1011no—
304C2×C4—1011no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4D4×D10111yescharacteristic
310C2×C4C23×D10111yescharacteristic
311C8—21no—
312C8—2021no—
313C8—2021no—
314C8#1310807412yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—4041no—
319C22—4041no—
320C22—4041no—
321C22—4041no—
322C22—4041no—
323C22—4041no—
324C22—2021no—
325C22—2021no—
326C22—2021no—
327C22C2×D4×D10111yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4041no—
334C4—4041no—
335C4—2021no—
336C4—2021no—
337C4—2021no—
338C4C2×#1310807411yescharacteristic
339C4#1569230311yescharacteristic
340C2—41no—
341C2—8081no—
342C2—8081no—
343C2—4041no—
344C2#2026582811yescharacteristic
345C2#2026597611yescharacteristic
346C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
hypercenter#428D101
hypocenterC101#3355
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20265875C2semidirect
2. #20265871C2semidirect
3. #20265577C2semidirect
4. #20265503C2semidirect
5. #20265501C2semidirect
6. #20265492C2semidirect
7. #20265490C2semidirect
8. #20265480C2semidirect
9. #20265476C2semidirect
10. #20265431C2semidirect
11. #20265430C2semidirect
12. #20265259C2semidirect
13. #20265253C2semidirect
14. #20265218C2semidirect
15. #428×C101C2semidirect
174. C101#3355semidirect
189. #428D101semidirect

Maximal subgroups

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

subgroupindex
#202658752
#202658712
#202655772
#202655032
#202655012
#202654922
#202654902
#202654802
#202654762
#202654312
#202654302
#202652592
#202652532
#202652182
#428×C1012
#3355101