Finite Group Databasebeta

← #27181523

Subgroups — #27181523

Order 12928.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#20265723C211yesmaximal, characteristic
2C2×#15692254C211yesmaximal, characteristic, complemented
3#20265538C212yesmaximal, complemented
4#20265535C212yesmaximal, complemented
5#20265526C212yesmaximal, complemented
6#20265522C212yesmaximal, complemented
7#20265179C211yesmaximal, characteristic, complemented
8#20265178C211yesmaximal, characteristic, complemented
9#20265176C211yesmaximal, characteristic, complemented
10M16×Dic101C211yesmaximal, characteristic, complemented
11#417×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×Dic202—21no—
14C2×#13108058C2211yescharacteristic
15#15692254—21no—
16#15692254—21no—
17#15692254C2212yes—
18#15692254C2212yes—
19#15692253C2211yescharacteristic
20#15692251C2211yescharacteristic
21#15692244C2211yescharacteristic
22#15692242C2211yescharacteristic
23#15692227—21no—
24#15692227—21no—
25C4×C22⋊D101—21no—
26C4×C22⋊D101—21no—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×#13108053—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692180C2212yes—
31#15692179C2212yes—
32#15692178C2212yes—
33#15692177C2212yes—
34#15692176C2212yes—
35#15692175C2212yes—
36#15692165C2212yes—
37C8×Dic101C2212yes—
38C2×#41×C101C2211yescharacteristic
39C2×M16×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#103×C101C2212yes—
42#102×C101C2212yes—
43#13108074—22no—
44#13108074—22no—
45C2×Dic202—22no—
46C2×Dic202—22no—
47C2×Dic202—44no—
48C2×Dic202C2311yescharacteristic
49C2×D404C2311yescharacteristic
50C2×C22⋊D101—21no—
51C2×C22⋊D101—21no—
52C2×C22⋊D101—21no—
53C22×Dic101—21no—
54C22×Dic101D411yescharacteristic
55C22×Dic101D411yescharacteristic
56C2×C4×D101—21no—
57#13108058—22no—
58C2×#11717579C2312yes—
59C4⋊Dic101—21no—
60C4⋊Dic101C2312yes—
61#13108053—22no—
62#13108053—22no—
63#13108053—21no—
64#13108053—21no—
65C22⋊Dic101—21no—
66C22⋊Dic101—21no—
67#13108051—21no—
68#13108051—21no—
69C4×Dic101D411yescharacteristic
70C4×Dic101D411yescharacteristic
71C4×Dic101C2311yescharacteristic
72C4×Dic101C2311yescharacteristic
73#41×C101—22no—
74#41×C101—22no—
75C2×Q8×C101C2311yescharacteristic
76C2×D4×C101—21no—
77C2×D4×C101C2311yescharacteristic
78C22×C4×C101—21no—
79C22×C4×C101—21no—
80C22×C4×C101C2311yescharacteristic
81M16×C101—22no—
82C2×C8×C101C2312yes—
83C4⋊C4×C101—21no—
84C4⋊C4×C101C2312yes—
85C23×C101—21no—
86C23×C101C2×D411yescharacteristic
87C22×D101—21no—
88Q8×C101—21no—
89Q8×C101—21no—
90D4×C101—21no—
91D4×C101—21no—
92D4×C101—42no—
93C2×C4×C101—22no—
94C2×C4×C101—21no—
95C2×C4×C101—42no—
96C2×C4×C101—42no—
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101#4112yes—
99C2×C4×C101#4112yes—
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—42no—
102C22⋊D101—42no—
103C22⋊D101—42no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101—21no—
107C2×Dic101—22no—
108C2×Dic101—42no—
109C2×Dic101C2×D411yescharacteristic
110C2×Dic101C2×D411yescharacteristic
111C2×Dic101C2×D411yescharacteristic
112C2×Dic101C2×D411yescharacteristic
113D404—21no—
114D404—21no—
115C4×D101—42no—
116Dic202—21no—
117Dic202—21no—
118Dic202—44no—
119Dic202—44no—
120C8×C101—22no—
121#11717579—22no—
122C22×C101—21no—
123C22×C101—21no—
124C22×C101—41no—
125C22×C101—41no—
126C22×C101—41no—
127C22×C101#12412yes—
128C22×C101C22×D411yescharacteristic
129C2×D101—41no—
130C2×D101—41no—
131C2×D101—41no—
132C4×C101—41no—
133C4×C101—42no—
134C4×C101#12412yes—
135C4×C101C2×#4111yescharacteristic
136C4×C101C2×#4111yescharacteristic
137Dic101—21no—
138Dic101—21no—
139Dic101—41no—
140Dic101—42no—
141Dic101AGL(1,ℤ/8)12yes—
142Dic101#13712yes—
143C2×C101—22no—
144C2×C101—81no—
145C2×C101C2×#12411yescharacteristic
146C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C101C2×#13711yescharacteristic
148D101—81no—
149#3141—1011nomaximal, Sylow
150C101#314111yescharacteristic, complemented, Sylow
151#532—1011no—
152C2×C4⋊Q8—1011no—
153#489—1012no—
154#489—1012no—
155#487—1012no—
156#486—1012no—
157#417—1011no—
158#417—1011no—
159#417—1011no—
160#417D10111yescharacteristic, complemented
161C4×M16—1011no—
162C2×#41—1011no—
163C2×#41C2×D10111yescharacteristic
164C22×Q8—2021no—
165C2×M16—1011no—
166C2×M16C2×D10111yescharacteristic
167C4⋊Q8—1012no—
168C4⋊Q8—1012no—
169C4⋊Q8—1011no—
170C4⋊Q8—2021no—
171C4⋊Q8—2021no—
172#127—1011no—
173#124—1011no—
174#124—1011no—
175#121—2021no—
176#121—2021no—
177C4×D4—2021no—
178C4×D4—2021no—
179C2×C4⋊C4—1011no—
180C2×C4⋊C4—2021no—
181C2×C4⋊C4C2×D10111yescharacteristic
182C2×C42—1011no—
183#103—1012no—
184#103—1012no—
185#103—1012no—
186#103C2×D10112yes—
187#102—1012no—
188#102—1012no—
189#102—1012no—
190#102C2×D10112yes—
191#97—1012no—
192C4×C8—1012no—
193#41—22no—
194#41—22no—
195#41—2022no—
196#41—2022no—
197C2×Q8—1011no—
198C2×Q8—4044no—
199C2×Q8—2022no—
200C2×Q8—2022no—
201C2×Q8C22×D10111yescharacteristic
202C2×D4—21no—
203C2×D4—1011no—
204C2×D4—2021no—
205C2×D4—2021no—
206C2×D4—2021no—
207C2×D4C22×D10111yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—1011no—
211C22×C4—1011no—
212C22×C4—2021no—
213C22×C4—2021no—
214C22×C4C22×D10111yescharacteristic
215M16—22no—
216M16—2022no—
217C2×C8—1012no—
218C2×C8C22×D10112yes—
219C4⋊C4—21no—
220C4⋊C4—1012no—
221C4⋊C4—2022no—
222C4⋊C4—2022no—
223C4⋊C4—2021no—
224C4⋊C4—2021no—
225C4⋊C4—2021no—
226C4⋊C4C22×D10112yes—
227C22⋊C4—2021no—
228C22⋊C4—2021no—
229C22⋊C4—2021no—
230C22⋊C4—2021no—
231C42—1011no—
232C42—1011no—
233C42—1011no—
234C42—1011no—
235C23—21no—
236C23—2021no—
237C23D4×D10111yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—4044no—
241Q8—4044no—
242Q8—2021no—
243Q8—2021no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—4042no—
248D4—4042no—
249D4—4042no—
250D4—2021no—
251D4—2021no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—1011no—
257C2×C4—1011no—
258C2×C4—1011no—
259C2×C4—1011no—
260C2×C4—4042no—
261C2×C4—4042no—
262C2×C4—2022no—
263C2×C4—2022no—
264C2×C4—2021no—
265C2×C4—2022no—
266C2×C4D4×D10111yescharacteristic
267C2×C4#1310807312yes—
268C2×C4#1310807412yes—
269C2×C4C23×D10111yescharacteristic
270C8—22no—
271C8—2022no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—4041no—
278C22—4041no—
279C22—4041no—
280C22#1569224112yes—
281C22C2×D4×D10111yescharacteristic
282C4—41no—
283C4—42no—
284C4—1012no—
285C4—1012no—
286C4—4041no—
287C4—4042no—
288C4—2021no—
289C4—2021no—
290C4#1569224112yes—
291C4C2×#1310807311yescharacteristic
292C4C2×#1310807411yescharacteristic
293C2—22no—
294C2—81no—
295C2—8081no—
296C2C2×#1569224111yescharacteristic
297C2AGL(1,ℤ/8)×D10111yescharacteristic
298C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
prime radical#20265723C2
hypercenter#417D101
hypocenterC101#3141
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#15692254C2semidirect
3. #20265538C2semidirect
4. #20265535C2semidirect
5. #20265526C2semidirect
6. #20265522C2semidirect
7. #20265179C2semidirect
8. #20265178C2semidirect
9. #20265176C2semidirect
10. M16×Dic101C2semidirect
11. #417×C101C2semidirect
150. C101#3141semidirect
160. #417D101semidirect

Maximal subgroups

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

subgroupindex
#202657232
C2×#156922542
#202655382
#202655352
#202655262
#202655222
#202651792
#202651782
#202651762
M16×Dic1012
#417×C1012
#3141101