Finite Group Databasebeta

← #27180614

Subgroups — #27180614

Order 12928.

480 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692284C212yesmaximal
2C4⋊Q8×D101C211yesmaximal, characteristic, complemented
3C4×#13108073C211yesmaximal, characteristic
4#20265388C211yesmaximal, characteristic, complemented
5#20265386C212yesmaximal, complemented
6#20265384C211yesmaximal, characteristic, complemented
7#20265382C211yesmaximal, characteristic, complemented
8#20265380C212yesmaximal, complemented
9#20265199C211yesmaximal, characteristic, complemented
10#20265184C212yesmaximal, complemented
11#461×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2211yescharacteristic
13C2×Q8×D101C2212yes—
14#15692284—28no—
15C2×Dic404C2212yes—
16C2×#13108064C2212yes—
17C2×#13108062C2212yes—
18C2×#13108057C2212yes—
19#15692255C2211yescharacteristic
20#15692254—21no—
21#15692254—21no—
22#15692253C2212yes—
23C4×Dic202C2211yescharacteristic
24#15692220C2211yescharacteristic
25D4×Dic101C2211yescharacteristic
26C4×C22⋊D101—21no—
27#15692215—21no—
28#15692213C2211yescharacteristic
29#15692211—21no—
30C4⋊C4×D101—21no—
31C4⋊C4×D101—21no—
32C2×C4×Dic101—21no—
33C42×D101C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35#15692184C2212yes—
36#15692180C2212yes—
37#15692179C2212yes—
38#15692178C2212yes—
39C2×SD16×C101C2212yes—
40C4⋊Q8×C101C2211yescharacteristic
41C4×D4×C101C2211yescharacteristic
42C4⋊C8×C101C2211yescharacteristic
43#103×C101C2212yes—
44#13108073—22no—
45#13108073D414yes—
46C2×Dic202—22no—
47C2×Dic202C2312yes—
48C2×Dic202C2311yescharacteristic
49Q8×D101—24no—
50Q8×D101—24no—
51C2×C22⋊D101—21no—
52C22×Dic101—21no—
53C22×Dic101—21no—
54C2×C4×D101—22no—
55C2×C4×D101D411yescharacteristic
56C2×C4×D101D411yescharacteristic
57C2×C4×D101C2311yescharacteristic
58Dic404—24no—
59#13108064—24no—
60#13108062—24no—
61#13108057—24no—
62C2×#11717579C2312yes—
63C4⋊Dic101—21no—
64C4⋊Dic101—21no—
65C4⋊Dic101C2311yescharacteristic
66#13108053—21no—
67#13108053—22no—
68#13108053—22no—
69#13108053—21no—
70C22⋊Dic101—21no—
71#13108051—21no—
72C4×Dic101—21no—
73C4×Dic101—21no—
74C4×Dic101—21no—
75C4×Dic101D411yescharacteristic
76C4×Dic101D411yescharacteristic
77C4×Dic101C2311yescharacteristic
78C2×Q8×C101C2312yes—
79C2×D4×C101C2311yescharacteristic
80C22×C4×C101—21no—
81SD16×C101—24no—
82C2×C8×C101C2312yes—
83C4⋊C4×C101—21no—
84C4⋊C4×C101—21no—
85C4⋊C4×C101C2311yescharacteristic
86C22⋊C4×C101—21no—
87C42×C101C2311yescharacteristic
88C23×C101—21no—
89C22×D101C2×D411yescharacteristic
90Q8×C101—22no—
91Q8×C101—22no—
92D4×C101—21no—
93D4×C101C2×D412yes—
94C2×C4×C101—21no—
95C2×C4×C101—22no—
96C2×C4×C101—42no—
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2411yescharacteristic
100C22⋊D101—24no—
101C2×Dic101—21no—
102C2×Dic101—24no—
103C2×Dic101—21no—
104C2×Dic101—22no—
105C2×Dic101—42no—
106C2×Dic101C2×D411yescharacteristic
107C2×Dic101C2×D411yescharacteristic
108C2×Dic101C2×D411yescharacteristic
109C4×D101—22no—
110C4×D101—22no—
111C4×D101—44no—
112C4×D101C2×D412yes—
113C4×D101#4112yes—
114Dic202—21no—
115Dic202—22no—
116Dic202—22no—
117Dic202—44no—
118Dic202C2×D412yes—
119C8×C101—22no—
120#11717579—22no—
121C22×C101—22no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101C22×D411yescharacteristic
125C2×D101—21no—
126C2×D101—21no—
127C2×D101#12112yes—
128C4×C101—21no—
129C4×C101—42no—
130C4×C101—41no—
131C4×C101#13712yescomplemented
132C4×C101C22×D411yescharacteristic
133C4×C101C2×#4111yescharacteristic
134Dic101—22no—
135Dic101—21no—
136Dic101—42no—
137Dic101—41no—
138Dic101#12112yes—
139Dic101#13712yescomplemented
140C2×C101—42no—
141C2×C101C2×#12111yescharacteristic
142C2×C101C2×#13711yescharacteristic
143C2×C101C2×#13711yescharacteristic
144D101—22no—
145#3039—1011nomaximal, Sylow
146C101#303911yescharacteristic, complemented, Sylow
147C2×#137—1012no—
148C2×C4⋊Q8—1011no—
149C4×#41—1011no—
150#462—1011no—
151#462—1011no—
152#462—1012no—
153#461—1012no—
154#461—1011no—
155#461D10111yescharacteristic, complemented
156#423—1011no—
157#419—1012no—
158C2×#41—1011no—
159C22×Q8—1012no—
160#137—2028no—
161C2×Q16—1012no—
162C2×Q16—1012no—
163C2×SD16—1012no—
164C2×SD16C2×D10112yes—
165C2×M16—1012no—
166C4⋊Q8—1012no—
167C4⋊Q8—1011no—
168C4⋊Q8—2021no—
169C4⋊Q8—2021no—
170C4⋊Q8C2×D10111yescharacteristic
171C4×Q8—1011no—
172C4×Q8—1011no—
173C4×D4—1011no—
174C4×D4—2021no—
175C4×D4—2021no—
176C4×D4C2×D10111yescharacteristic
177#117—1011no—
178#117—2021no—
179C2×C4⋊C4—2021no—
180C2×C4⋊C4—2021no—
181C2×C42—1011no—
182C2×C42—2021no—
183C4⋊C8—1012no—
184C4⋊C8—1011no—
185C4⋊C8C2×D10111yescharacteristic
186#103—1012no—
187#103—1012no—
188#103—1012no—
189#103C2×D10112yes—
190#41—1014no—
191#41—2022no—
192C2×Q8—1011no—
193C2×Q8—1012no—
194C2×Q8—2022no—
195C2×Q8—2024no—
196C2×Q8—2024no—
197C2×Q8C22×D10112yes—
198C2×D4—2021no—
199C2×D4C22×D10111yescharacteristic
200C22×C4—21no—
201C22×C4—1011no—
202C22×C4—1011no—
203C22×C4—1011no—
204C22×C4—2022no—
205C22×C4—2021no—
206C22×C4—2021no—
207Q16—2024no—
208Q16—2024no—
209SD16—24no—
210SD16—2024no—
211M16—2024no—
212C2×C8—1012no—
213C2×C8C22×D10112yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—1011no—
217C4⋊C4—2022no—
218C4⋊C4—2022no—
219C4⋊C4—2021no—
220C4⋊C4—2021no—
221C4⋊C4—2021no—
222C4⋊C4—2021no—
223C4⋊C4C22×D10111yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—2021no—
226C22⋊C4—2021no—
227C42—1011no—
228C42—1011no—
229C42—1011no—
230C42—2021no—
231C42—2021no—
232C42—2021no—
233C42C22×D10111yescharacteristic
234C23—21no—
235C23—1011no—
236Q8—22no—
237Q8—22no—
238Q8—1012no—
239Q8—4044no—
240Q8—2022no—
241Q8—2022no—
242Q8—2021no—
243D4—21no—
244D4—2024no—
245D4D4×D10112yes—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—1011no—
250C2×C4—1011no—
251C2×C4—1012no—
252C2×C4—1012no—
253C2×C4—1011no—
254C2×C4—4044no—
255C2×C4—4042no—
256C2×C4—2022no—
257C2×C4—2024no—
258C2×C4—2021no—
259C2×C4—2022no—
260C2×C4—2021no—
261C2×C4—2022no—
262C2×C4D4×D10111yescharacteristic
263C2×C4D4×D10111yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—22no—
266C8—2022no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—1012no—
271C22—2021no—
272C22—2021no—
273C22C2×D4×D10111yescharacteristic
274C4—21no—
275C4—42no—
276C4—41no—
277C4—1012no—
278C4—1012no—
279C4—4042no—
280C4—4041no—
281C4—2022no—
282C4—2021no—
283C4#1569228412yescomplemented
284C4C2×D4×D10111yescharacteristic
285C4#41×D10111yescharacteristic
286C2—42no—
287C2—2022no—
288C2#121×D10111yescharacteristic
289C2#137×D10111yescharacteristic
290C2C2×#1569228411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#461×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108073C22
hypercenter#461D101
hypocenterC101#3039
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D101C2semidirect
4. #20265388C2semidirect
5. #20265386C2semidirect
6. #20265384C2semidirect
7. #20265382C2semidirect
8. #20265380C2semidirect
9. #20265199C2semidirect
10. #20265184C2semidirect
11. #461×C101C2semidirect
131. C4×C101#137semidirect
139. Dic101#137semidirect
146. C101#3039semidirect
155. #461D101semidirect
283. C4#15692284semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
C4⋊Q8×D1012
C4×#131080732
#202653882
#202653862
#202653842
#202653822
#202653802
#202651992
#202651842
#461×C1012
#3039101