Finite Group Databasebeta

← #27180368

Subgroups — #27180368

Order 12928.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×Q8×D101C211yesmaximal, characteristic, complemented
2C2×#15692288C211yesmaximal, characteristic, complemented
3C2×#15692286C211yesmaximal, characteristic, complemented
4#20265689C211yesmaximal, characteristic
5#20265334C214yesmaximal, complemented
6#20265317C214yesmaximal, complemented
7#20265185C211yesmaximal, characteristic, complemented
8#20265140C211yesmaximal, characteristic, complemented
9#419×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic
11C22×Dic202—21no—
12C22×Dic202C2211yescharacteristic
13C2×Q8×D101—24no—
14C2×Q8×D101—22no—
15C2×Q8×D101—24no—
16C2×Q8×D101C2214yes—
17C22×C4×D101—21no—
18C22×C4×D101C2211yescharacteristic
19#15692288—24no—
20#15692286—24no—
21C2×Dic404C2212yescomplemented
22C2×#13108064C2212yescomplemented
23C2×#13108063C2212yescomplemented
24C2×#13108061C2212yescomplemented
25C2×#13108058C2211yescharacteristic
26#15692239—21no—
27#15692235C2212yes—
28#15692230C2212yes—
29#15692225—21no—
30C4×D404—22no—
31#15692178C2214yescomplemented
32#15692167C2214yes—
33C22×Q8×C101C2211yescharacteristic
34C2×M16×C101C2211yescharacteristic
35#117×C101C2211yescharacteristic
36#103×C101C2214yescomplemented
37C23×D101D411yescharacteristic
38#13108074—22no—
39#13108074—22no—
40C2×Dic202—28no—
41C2×Dic202—21no—
42C2×Dic202—22no—
43C2×Dic202—24no—
44C2×Dic202D414yescomplemented
45C2×Dic202C2311yescharacteristic
46C2×Dic202C2312yes—
47Q8×D101—216no—
48Q8×D101—28no—
49Q8×D101—28no—
50C2×D404—22no—
51C2×D404C2311yescharacteristic
52C2×C22⋊D101—21no—
53C22×Dic101—21no—
54C22×Dic101D411yescharacteristic
55C2×C4×D101—28no—
56C2×C4×D101—24no—
57C2×C4×D101—22no—
58C2×C4×D101—22no—
59C2×C4×D101—21no—
60C2×C4×D101D412yes—
61C2×C4×D101C2312yes—
62Dic404—24no—
63#13108064—24no—
64#13108063—24no—
65#13108061—24no—
66#13108058—22no—
67C2×#11717579C2312yes—
68C4⋊Dic101—22no—
69#13108051—22no—
70#13108051—22no—
71C2×Q8×C101—22no—
72C2×Q8×C101—21no—
73C2×Q8×C101D414yescomplemented
74C2×Q8×C101C2312yes—
75C22×C4×C101—21no—
76C22×C4×C101C2311yescharacteristic
77M16×C101—22no—
78C2×C8×C101C2312yes—
79C4⋊C4×C101C2312yes—
80C22⋊C4×C101—21no—
81C42×C101—21no—
82C23×C101C2×D411yescharacteristic
83C22×D101—22no—
84C22×D101—24no—
85C22×D101—21no—
86C22×D101C2×D412yes—
87Q8×C101—24no—
88Q8×C101—22no—
89Q8×C101C2×D414yes—
90C2×C4×C101—24no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D412yes—
95C2×C4×C101C2×D412yes—
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—42no—
98C2×Dic101—24no—
99C2×Dic101—22no—
100C2×Dic101—22no—
101C2×Dic101—21no—
102C2×Dic101C2×D412yes—
103D404—21no—
104D404—21no—
105D404—44no—
106C4×D101—216no—
107C4×D101—24no—
108C4×D101—22no—
109C4×D101—22no—
110C4×D101—42no—
111Dic202—216no—
112Dic202—24no—
113Dic202—22no—
114Dic202—21no—
115Dic202—21no—
116Dic202C2×D414yes—
117C8×C101—22no—
118#11717579—22no—
119C22×C101—22no—
120C22×C101#12012yes—
121C22×C101C22×D411yescharacteristic
122C2×D101—28no—
123C2×D101—22no—
124C2×D101—22no—
125C2×D101—41no—
126C2×D101—42no—
127C2×D101#13714yescomplemented
128C4×C101—24no—
129C4×C101—42no—
130C4×C101#12012yes—
131C4×C101C22×D411yescharacteristic
132C4×C101C22×D411yescharacteristic
133Dic101—22no—
134Dic101—24no—
135Dic101—41no—
136C2×C101—22no—
137C2×C101C2×#12011yescharacteristic
138C2×C101C2×#13712yes—
139D101—24no—
140D101—81no—
141#3004—1011nomaximal, Sylow
142C101#300411yescharacteristic, complemented, Sylow
143C23×Q8—1011no—
144C2×#137—1011no—
145C2×#137—1011no—
146#525—1011no—
147#451—1014no—
148#448—1014no—
149#419—1011no—
150#419D10111yescharacteristic, complemented
151#407—1011no—
152C2×#41—1011no—
153C22×Q8—1014no—
154C22×Q8—1011no—
155C22×Q8—2021no—
156C22×Q8—2022no—
157C22×Q8—2024no—
158C22×Q8—2024no—
159C22×Q8C2×D10111yescharacteristic
160C23×C4—1011no—
161C23×C4—2021no—
162#137—2024no—
163#137—2024no—
164C2×Q16—1012no—
165C2×Q16—1012no—
166C2×SD16—1012no—
167C2×SD16—1012no—
168C2×M16—1011no—
169C2×M16C2×D10111yescharacteristic
170#123—2021no—
171C22⋊Q8—1012no—
172#121—1012no—
173#120—2021no—
174C4×D4—2022no—
175#117C2×D10111yescharacteristic
176#103—1014no—
177#103C2×D10114yescomplemented
178C22⋊C8—1014no—
179C24—1011no—
180#41—2022no—
181#41—2022no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—1014no—
185C2×Q8—1012no—
186C2×Q8—1011no—
187C2×Q8—2024no—
188C2×Q8—2028no—
189C2×Q8—2028no—
190C2×Q8—20216no—
191C2×Q8—2028no—
192C2×Q8—2021no—
193C2×Q8—2022no—
194C2×Q8D40414yescomplemented
195C2×Q8C22×D10112yes—
196C2×D4—1011no—
197C2×D4—2022no—
198C2×D4—2021no—
199C22×C4—21no—
200C22×C4—1011no—
201C22×C4—1012no—
202C22×C4—1012no—
203C22×C4—2024no—
204C22×C4—2028no—
205C22×C4—2021no—
206C22×C4—2021no—
207C22×C4—2022no—
208C22×C4—2022no—
209C22×C4C22×D10111yescharacteristic
210Q16—2024no—
211Q16—2024no—
212SD16—2024no—
213SD16—2024no—
214M16—22no—
215M16—2022no—
216C2×C8—1012no—
217C2×C8C22×D10112yes—
218C4⋊C4—2022no—
219C4⋊C4C22×D10112yes—
220C22⋊C4—21no—
221C22⋊C4—2022no—
222C22⋊C4—2022no—
223C42—21no—
224C23—1012no—
225C23—2024no—
226C23—2021no—
227C23—2022no—
228C23D4×D10111yescharacteristic
229Q8—22no—
230Q8—24no—
231Q8—1014no—
232Q8—20216no—
233Q8—2021no—
234Q8—2021no—
235Q8—2022no—
236Q8—2024no—
237Q8C2×D40414yes—
238D4—4044no—
239D4—4042no—
240D4—2021no—
241D4—2021no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—1012no—
246C2×C4—4042no—
247C2×C4—20216no—
248C2×C4—2021no—
249C2×C4—2022no—
250C2×C4—2022no—
251C2×C4—2024no—
252C2×C4—2022no—
253C2×C4—2022no—
254C2×C4—2024no—
255C2×C4D4×D10112yes—
256C2×C4D4×D10111yescharacteristic
257C2×C4C2×D40412yes—
258C2×C4C23×D10111yescharacteristic
259C8—22no—
260C8—2022no—
261C22—22no—
262C22—1014no—
263C22—4041no—
264C22—4042no—
265C22—2028no—
266C22—2022no—
267C22—2022no—
268C22#1569222512yes—
269C22C2×D4×D10111yescharacteristic
270C4—24no—
271C4—42no—
272C4—4041no—
273C4—2022no—
274C4—2024no—
275C4#1569222512yes—
276C4C2×D4×D10111yescharacteristic
277C4C22×D40411yescharacteristic
278C2—22no—
279C2—8081no—
280C2—2024no—
281C2C2×#1569222511yescharacteristic
282C2#137×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#419×C101C2
socleC22×C101C22×D4
prime radical#20265689C2
hypercenter#419D101
hypocenterC101#3004
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D101C2semidirect
2. C2×#15692288C2semidirect
3. C2×#15692286C2semidirect
5. #20265334C2semidirect
6. #20265317C2semidirect
7. #20265185C2semidirect
8. #20265140C2semidirect
9. #419×C101C2semidirect
21. C2×Dic404C22semidirect
22. C2×#13108064C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108061C22semidirect
31. #15692178C22semidirect
36. #103×C101C22semidirect
44. C2×Dic202D4semidirect
73. C2×Q8×C101D4semidirect
127. C2×D101#137semidirect
142. C101#3004semidirect
150. #419D101semidirect
177. #103C2×D101semidirect
194. C2×Q8D404semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1012
C2×#156922882
C2×#156922862
#202656892
#202653342
#202653172
#202651852
#202651402
#419×C1012
#3004101