Finite Group Databasebeta

← #27181530

Subgroups — #27181530

Order 12928.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15692246C211yesmaximal, characteristic, complemented
3#20265544C212yesmaximal, complemented
4#20265540C212yesmaximal, complemented
5#20265533C212yesmaximal, complemented
6#20265530C212yesmaximal, complemented
7#20265179C211yesmaximal, characteristic, complemented
8#20265178C211yesmaximal, characteristic, complemented
9#20265176C211yesmaximal, characteristic, complemented
10M16×Dic101C211yesmaximal, characteristic, complemented
11#417×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C2×#13108058C2211yescharacteristic
14#15692253C2211yescharacteristic
15#15692251C2211yescharacteristic
16#15692246—21no—
17#15692246—21no—
18#15692246C2212yes—
19#15692246C2212yes—
20#15692244C2211yescharacteristic
21#15692242C2211yescharacteristic
22#15692227—21no—
23#15692227—21no—
24C4×C22⋊D101—21no—
25C4×C22⋊D101—21no—
26C2×C4⋊Dic101—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×Dic202C2311yescharacteristic
46C2×D404C2311yescharacteristic
47C2×C22⋊D101—21no—
48C2×C22⋊D101—21no—
49C2×C22⋊D101—21no—
50C22×Dic101—21no—
51C22×Dic101D411yescharacteristic
52C22×Dic101D411yescharacteristic
53C2×C4×D101—21no—
54#13108058—22no—
55C2×#11717579C2312yes—
56C4⋊Dic101—21no—
57C4⋊Dic101—22no—
58C4⋊Dic101—22no—
59C4⋊Dic101C2312yes—
60#13108053—22no—
61#13108053—22no—
62#13108053—21no—
63#13108053—21no—
64C22⋊Dic101—21no—
65C22⋊Dic101—21no—
66#13108051—21no—
67#13108051—21no—
68C4×Dic101D411yescharacteristic
69C4×Dic101D411yescharacteristic
70C4×Dic101C2311yescharacteristic
71C4×Dic101C2311yescharacteristic
72#41×C101—22no—
73#41×C101—22no—
74C2×Q8×C101C2311yescharacteristic
75C2×D4×C101—21no—
76C2×D4×C101C2311yescharacteristic
77C22×C4×C101—21no—
78C22×C4×C101—21no—
79C22×C4×C101C2311yescharacteristic
80M16×C101—22no—
81C2×C8×C101C2312yes—
82C4⋊C4×C101—21no—
83C4⋊C4×C101C2312yes—
84C23×C101—21no—
85C23×C101C2×D411yescharacteristic
86C22×D101—21no—
87Q8×C101—21no—
88Q8×C101—21no—
89D4×C101—21no—
90D4×C101—21no—
91D4×C101—42no—
92C2×C4×C101—22no—
93C2×C4×C101—21no—
94C2×C4×C101—42no—
95C2×C4×C101—42no—
96C2×C4×C101C2×D411yescharacteristic
97C2×C4×C101#4112yes—
98C2×C4×C101#4112yes—
99C2×C4×C101C2411yescharacteristic
100C22⋊D101—42no—
101C22⋊D101—42no—
102C22⋊D101—42no—
103C2×Dic101—21no—
104C2×Dic101—22no—
105C2×Dic101—41no—
106C2×Dic101—41no—
107C2×Dic101—42no—
108C2×Dic101C2×D411yescharacteristic
109C2×Dic101C2×D411yescharacteristic
110C2×Dic101C2×D411yescharacteristic
111C2×Dic101C2×D411yescharacteristic
112D404—21no—
113D404—21no—
114C4×D101—42no—
115Dic202—21no—
116Dic202—21no—
117C8×C101—22no—
118#11717579—22no—
119C22×C101—21no—
120C22×C101—21no—
121C22×C101—41no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101#12412yes—
125C22×C101C22×D411yescharacteristic
126C2×D101—41no—
127C2×D101—41no—
128C2×D101—41no—
129C4×C101—41no—
130C4×C101—42no—
131C4×C101#12412yes—
132C4×C101C2×#4111yescharacteristic
133C4×C101C2×#4111yescharacteristic
134Dic101—21no—
135Dic101—21no—
136Dic101—21no—
137Dic101—21no—
138Dic101—41no—
139Dic101—42no—
140C2×C101—22no—
141C2×C101—81no—
142C2×C101C2×#12411yescharacteristic
143C2×C101#57511yescharacteristic
144C2×C101#57511yescharacteristic
145D101—81no—
146#3142—1011nomaximal, Sylow
147C101#314211yescharacteristic, complemented, Sylow
148#532—1011no—
149C2×#125—1011no—
150#491—1012no—
151#490—1012no—
152#488—1012no—
153#488—1012no—
154#417—1011no—
155#417—1011no—
156#417—1011no—
157#417D10111yescharacteristic, complemented
158C4×M16—1011no—
159C2×#41—1011no—
160C2×#41C2×D10111yescharacteristic
161C2×M16—1011no—
162C2×M16C2×D10111yescharacteristic
163C4⋊Q8—1011no—
164#127—1011no—
165#125—1012no—
166#125—1012no—
167#125—2021no—
168#125—2021no—
169#124—1011no—
170#124—1011no—
171#121—2021no—
172#121—2021no—
173C4×D4—2021no—
174C4×D4—2021no—
175C2×C4⋊C4—1011no—
176C2×C4⋊C4—2021no—
177C2×C4⋊C4—2021no—
178C2×C4⋊C4C2×D10111yescharacteristic
179C2×C42—1011no—
180#103—1012no—
181#103—1012no—
182#103—1012no—
183#103C2×D10112yes—
184#102—1012no—
185#102—1012no—
186#102—1012no—
187#102C2×D10112yes—
188#97—1012no—
189C4×C8—1012no—
190#41—22no—
191#41—22no—
192#41—2022no—
193#41—2022no—
194C2×Q8—1011no—
195C2×Q8C22×D10111yescharacteristic
196C2×D4—21no—
197C2×D4—1011no—
198C2×D4—2021no—
199C2×D4—2021no—
200C2×D4—2021no—
201C2×D4C22×D10111yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1011no—
205C22×C4—1011no—
206C22×C4—2021no—
207C22×C4—2021no—
208C22×C4C22×D10111yescharacteristic
209M16—22no—
210M16—2022no—
211C2×C8—1012no—
212C2×C8C22×D10112yes—
213C4⋊C4—21no—
214C4⋊C4—1012no—
215C4⋊C4—2022no—
216C4⋊C4—2022no—
217C4⋊C4—2021no—
218C4⋊C4—2021no—
219C4⋊C4—2022no—
220C4⋊C4—2022no—
221C4⋊C4—2021no—
222C4⋊C4C22×D10112yes—
223C22⋊C4—2021no—
224C22⋊C4—2021no—
225C22⋊C4—2021no—
226C22⋊C4—2021no—
227C42—1011no—
228C42—1011no—
229C42—1011no—
230C42—1011no—
231C23—21no—
232C23—2021no—
233C23D4×D10111yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—2021no—
237Q8—2021no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—4042no—
242D4—4042no—
243D4—4042no—
244D4—2021no—
245D4—2021no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—1011no—
251C2×C4—1011no—
252C2×C4—1011no—
253C2×C4—1011no—
254C2×C4—4042no—
255C2×C4—4042no—
256C2×C4—4041no—
257C2×C4—4041no—
258C2×C4—2021no—
259C2×C4—2022no—
260C2×C4D4×D10111yescharacteristic
261C2×C4#1310807312yes—
262C2×C4#1310807412yes—
263C2×C4C23×D10111yescharacteristic
264C8—22no—
265C8—2022no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—4041no—
272C22—4041no—
273C22—4041no—
274C22#1569224112yes—
275C22C2×D4×D10111yescharacteristic
276C4—41no—
277C4—42no—
278C4—4041no—
279C4—4042no—
280C4—2021no—
281C4—2021no—
282C4—2021no—
283C4—2021no—
284C4#1569224112yes—
285C4C2×#1310807311yescharacteristic
286C4C2×#1310807411yescharacteristic
287C2—22no—
288C2—81no—
289C2—8081no—
290C2C2×#1569224111yescharacteristic
291C2#2026597311yescharacteristic
292C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
prime radical#20265723C2
hypercenter#417D101
hypocenterC101#3142
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×#15692246C2semidirect
3. #20265544C2semidirect
4. #20265540C2semidirect
5. #20265533C2semidirect
6. #20265530C2semidirect
7. #20265179C2semidirect
8. #20265178C2semidirect
9. #20265176C2semidirect
10. M16×Dic101C2semidirect
11. #417×C101C2semidirect
147. C101#3142semidirect
157. #417D101semidirect

Maximal subgroups

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

subgroupindex
#202657232
C2×#156922462
#202655442
#202655402
#202655332
#202655302
#202651792
#202651782
#202651762
M16×Dic1012
#417×C1012
#3142101