Finite Group Databasebeta

← #27180752

Subgroups — #27180752

Order 12928.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692283C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265689C211yesmaximal, characteristic, complemented
4C2×#15692230C211yesmaximal, characteristic, complemented
5#20265431C212yesmaximal, complemented
6#20265424C212yesmaximal, complemented
7#20265414C212yesmaximal, complemented
8#20265407C212yesmaximal, complemented
9#20265181C211yesmaximal, characteristic, complemented
10#20265176C211yesmaximal, characteristic, complemented
11#428×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404—21no—
14C22×D404C2211yescharacteristic
15C22×C4×D101C2211yescharacteristic
16#15692283—24no—
17C2×#13108063C2212yescomplemented
18C2×D808C2212yescomplemented
19C2×#13108058C2211yescharacteristic
20C2×#13108057C2212yescomplemented
21M16×D101—24no—
22C2×C8×D101C2212yescomplemented
23#15692239—21no—
24#15692235C2211yescharacteristic, complemented
25#15692235C2211yescharacteristic, complemented
26#15692230—21no—
27#15692230—21no—
28#15692230C2212yes—
29#15692230C2212yes—
30#15692230C2211yescharacteristic, complemented
31#15692230C2211yescharacteristic, complemented
32#15692225—21no—
33C4×D404—21no—
34C4×D404—21no—
35C2×#13108051—21no—
36#15692178C2212yescomplemented
37#15692175C2212yescomplemented
38#15692175C2212yescomplemented
39#15692175C2212yescomplemented
40C2×M16×C101C2211yescharacteristic
41#117×C101C2211yescharacteristic
42C2×C4⋊C4×C101C2211yescharacteristic
43#107×C101C2212yescomplemented
44#106×C101C2212yescomplemented
45C23×D101—21no—
46C23×D101D411yescharacteristic
47#13108074—22no—
48#13108074—22no—
49C2×Dic202C2311yescharacteristic
50C2×D404—22no—
51C2×D404—22no—
52C2×D404—22no—
53C2×D404—22no—
54C2×D404—21no—
55C2×D404—21no—
56C2×D404—21no—
57C2×D404—44no—
58C2×D404C2312yes—
59C2×D404C2311yescharacteristic
60C2×C22⋊D101—21no—
61C22×Dic101D411yescharacteristic
62C2×C4×D101—22no—
63C2×C4×D101—22no—
64C2×C4×D101—21no—
65C2×C4×D101D411yescharacteristic
66C2×C4×D101D411yescharacteristic
67C2×C4×D101C2311yescharacteristic
68C2×C4×D101C2311yescharacteristic
69#13108063—24no—
70D808—24no—
71#13108058—22no—
72#13108057—24no—
73C2×#11717579C2312yes—
74C8×D101—24no—
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77#13108051—21no—
78#13108051—21no—
79#13108051—22no—
80#13108051—22no—
81#13108051—21no—
82#13108051—21no—
83C22×C4×C101—21no—
84C22×C4×C101C2311yescharacteristic
85M16×C101D414yescomplemented
86C2×C8×C101C2312yes—
87C4⋊C4×C101—21no—
88C4⋊C4×C101C2311yescharacteristic
89C4⋊C4×C101C2312yes—
90C4⋊C4×C101C2311yescharacteristic
91C22⋊C4×C101—21no—
92C42×C101—21no—
93C23×C101C2×D411yescharacteristic
94C22×D101—22no—
95C22×D101—22no—
96C22×D101—21no—
97C22×D101—22no—
98C22×D101—41no—
99C22×D101—42no—
100C22×D101—42no—
101C22×D101—42no—
102C22×D101C2×D411yescharacteristic
103C22×D101C2×D411yescharacteristic
104C2×C4×C101—21no—
105C2×C4×C101—21no—
106C2×C4×C101—22no—
107C2×C4×C101—42no—
108C2×C4×C101C2×D411yescharacteristic
109C2×C4×C101C2×D412yes—
110C2×C4×C101#4112yes—
111C2×C4×C101C2411yescharacteristic
112C22⋊D101—42no—
113C2×Dic101—22no—
114C2×Dic101—21no—
115C2×Dic101C2×D411yescharacteristic
116C2×Dic101C2×D411yescharacteristic
117D404—21no—
118D404—22no—
119D404—22no—
120D404—21no—
121D404—42no—
122D404—44no—
123D404—42no—
124D404—44no—
125D404—42no—
126C4×D101—22no—
127C4×D101—22no—
128C4×D101—21no—
129C4×D101—21no—
130C4×D101—21no—
131C4×D101—21no—
132C4×D101—42no—
133Dic202—21no—
134Dic202—21no—
135C8×C101C2×D414yescomplemented
136#11717579—22no—
137C22×C101—21no—
138C22×C101—21no—
139C22×C101#12112yes—
140C22×C101C22×D411yescharacteristic
141C2×D101—21no—
142C2×D101—21no—
143C2×D101—21no—
144C2×D101—21no—
145C2×D101—21no—
146C2×D101—44no—
147C2×D101—41no—
148C2×D101—42no—
149C2×D101—42no—
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—42no—
153C2×D101—84no—
154C2×D101AGL(1,ℤ/8)12yescomplemented
155C4×C101—42no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101#12112yes—
159C4×C101C22×D411yescharacteristic
160C4×C101C2×#4111yescharacteristic
161Dic101—21no—
162Dic101—21no—
163Dic101—41no—
164C2×C101—22no—
165C2×C101C2×#12111yescharacteristic
166C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C101#57511yescharacteristic
168D101—22no—
169D101—41no—
170D101—81no—
171D101—82no—
172#3055—1011nomaximal, Sylow
173C101#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1011no—
175C22×M16—1011no—
176#525—1011no—
177C2×#121—1011no—
178#469—1012no—
179#468—1012no—
180#466—1012no—
181#465—1012no—
182#428D10111yescharacteristic, complemented
183#418—1011no—
184#417—1011no—
185C2×#41—1011no—
186C22×D4—1011no—
187C22×D4—2021no—
188C23×C4—1011no—
189AGL(1,ℤ/8)—2024no—
190C2×SD16—1012no—
191C2×D8—1012no—
192C2×M16—1011no—
193C2×M16—1012no—
194C2×M16—2024no—
195C2×M16C2×D10111yescharacteristic
196C22×C8—1012no—
197#123—2021no—
198C22⋊Q8—1011no—
199C22⋊Q8—1011no—
200#121—1012no—
201#121—1012no—
202#121—1011no—
203#121—1011no—
204#121—2021no—
205#121—2021no—
206#120—2021no—
207C4×D4—2021no—
208C4×D4—2021no—
209#117C2×D10111yescharacteristic
210C2×C4⋊C4C2×D10111yescharacteristic
211C2×C22⋊C4—2021no—
212#107C2×D10112yescomplemented
213#106C2×D10112yescomplemented
214#103—1012no—
215#102—1012no—
216#102—1012no—
217#102—1012no—
218C24—1011no—
219C24—2021no—
220#41—2022no—
221#41—2022no—
222C2×Q8—1011no—
223C2×D4—1011no—
224C2×D4—1012no—
225C2×D4—4044no—
226C2×D4—2021no—
227C2×D4—2021no—
228C2×D4—2022no—
229C2×D4—2022no—
230C2×D4—2021no—
231C2×D4—2021no—
232C2×D4—2022no—
233C2×D4—2022no—
234C22×C4—21no—
235C22×C4—1011no—
236C22×C4—1011no—
237C22×C4—1011no—
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—2021no—
241C22×C4—2022no—
242C22×C4—2022no—
243C22×C4C22×D10111yescharacteristic
244SD16—2024no—
245D8—2024no—
246M16—2022no—
247M16—2024no—
248M16D40414yescomplemented
249C2×C8—1012no—
250C2×C8—2024no—
251C2×C8C22×D10112yes—
252C4⋊C4—21no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4C22×D10111yescharacteristic
256C4⋊C4C22×D10111yescharacteristic
257C4⋊C4C22×D10112yes—
258C22⋊C4—21no—
259C22⋊C4—2022no—
260C22⋊C4—2022no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C42—21no—
266C23—1011no—
267C23—1011no—
268C23—4042no—
269C23—4042no—
270C23—4041no—
271C23—4042no—
272C23—2022no—
273C23—2021no—
274C23—2022no—
275C23—2022no—
276C23D4×D10111yescharacteristic
277Q8—2021no—
278Q8—2021no—
279D4—4044no—
280D4—4042no—
281D4—4044no—
282D4—4042no—
283D4—4042no—
284D4—4042no—
285D4—2021no—
286D4—2022no—
287D4—2022no—
288D4—2021no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—1011no—
294C2×C4—1011no—
295C2×C4—4042no—
296C2×C4—2021no—
297C2×C4—2022no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4—2021no—
302C2×C4—2022no—
303C2×C4—2022no—
304C2×C4D4×D10111yescharacteristic
305C2×C4C2×D40412yes—
306C2×C4#1310807512yes—
307C2×C4C23×D10111yescharacteristic
308C8—2022no—
309C8C2×D40414yescomplemented
310C22—21no—
311C22—21no—
312C22—8084no—
313C22—1012no—
314C22—4044no—
315C22—4041no—
316C22—4042no—
317C22—4042no—
318C22—4041no—
319C22—4041no—
320C22—4042no—
321C22—2021no—
322C22—2021no—
323C22—2021no—
324C22—2021no—
325C22—2021no—
326C22#1569223012yes—
327C22C2×D4×D10111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—4041no—
332C4—2021no—
333C4—2021no—
334C4#1569223012yes—
335C4C22×D40411yescharacteristic
336C4C2×#1310807511yescharacteristic
337C2—22no—
338C2—8082no—
339C2—8081no—
340C2—4041no—
341C2—2022no—
342C2C2×#1569223011yescharacteristic
343C2AGL(1,ℤ/8)×D10111yescharacteristic
344C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
hypercenter#428D101
hypocenterC101#3055
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. C2×#15692283C2semidirect
2. C2×M16×D101C2semidirect
3. #20265689C2semidirect
4. C2×#15692230C2semidirect
5. #20265431C2semidirect
6. #20265424C2semidirect
7. #20265414C2semidirect
8. #20265407C2semidirect
9. #20265181C2semidirect
10. #20265176C2semidirect
11. #428×C101C2semidirect
17. C2×#13108063C22semidirect
18. C2×D808C22semidirect
20. C2×#13108057C22semidirect
22. C2×C8×D101C22semidirect
24. #15692235C22semidirect
25. #15692235C22semidirect
30. #15692230C22semidirect
31. #15692230C22semidirect
36. #15692178C22semidirect
37. #15692175C22semidirect
38. #15692175C22semidirect
39. #15692175C22semidirect
43. #107×C101C22semidirect
44. #106×C101C22semidirect
85. M16×C101D4semidirect
135. C8×C101C2×D4semidirect
154. C2×D101AGL(1,ℤ/8)semidirect
173. C101#3055semidirect
182. #428D101semidirect
212. #107C2×D101semidirect
213. #106C2×D101semidirect
248. M16D404semidirect
309. C8C2×D404semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C2×M16×D1012
#202656892
C2×#156922302
#202654312
#202654242
#202654142
#202654072
#202651812
#202651762
#428×C1012
#3055101