Finite Group Databasebeta

← #27181682

Subgroups — #27181682

Order 12928.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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#20265730C211yesmaximal, characteristic
2C2×#15692246C211yesmaximal, characteristic
3#20265585C212yesmaximal, complemented
4#20265584C212yesmaximal, complemented
5#20265578C212yesmaximal, complemented
6#20265577C212yesmaximal, complemented
7#20265220C211yesmaximal, characteristic
8#20265219C211yesmaximal, characteristic
9#20265217C211yesmaximal, characteristic
10M16×Dic101C211yesmaximal, characteristic
11#428×C101C211yesmaximal, characteristic
12C2×#13108058C2211yescharacteristic
13#15692254C2211yescharacteristic
14#15692254C2211yescharacteristic
15#15692246—21no—
16#15692246—21no—
17#15692246C2211yescharacteristic
18#15692246C2211yescharacteristic
19#15692246C2212yes—
20#15692246C2212yes—
21#15692232—21no—
22#15692232—21no—
23C4×Dic202—21no—
24C4×Dic202—21no—
25#15692214C2211yescharacteristic
26C2×C4⋊Dic101—21no—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×#13108053—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692189C2212yes—
31#15692188C2212yes—
32#15692188C2212yes—
33#15692187C2212yes—
34#15692186C2212yes—
35#15692186C2212yes—
36#15692165C2212yes—
37C8×Dic101C2212yes—
38C2×M16×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#107×C101C2212yes—
42#106×C101C2212yes—
43C2×Dic202—21no—
44C2×Dic202—21no—
45C22×Dic101—21no—
46C22×Dic101D411yescharacteristic
47C22×Dic101D411yescharacteristic
48#13108058Q814yescomplemented
49C2×#11717579C2312yes—
50C4⋊Dic101—21no—
51C4⋊Dic101—22no—
52C4⋊Dic101—22no—
53C4⋊Dic101—21no—
54C4⋊Dic101—21no—
55C4⋊Dic101C2311yescharacteristic
56C4⋊Dic101C2312yes—
57C4⋊Dic101C2311yescharacteristic
58#13108053—22no—
59#13108053—22no—
60#13108053—21no—
61#13108053—21no—
62#13108053—21no—
63#13108053—21no—
64C22⋊Dic101—21no—
65C4×Dic101—21no—
66C4×Dic101D411yescharacteristic
67C4×Dic101D411yescharacteristic
68C4×Dic101C2311yescharacteristic
69C4×Dic101C2311yescharacteristic
70C22×C4×C101—21no—
71C22×C4×C101C2311yescharacteristic
72M16×C101Q814yescomplemented
73C2×C8×C101C2312yes—
74C4⋊C4×C101—21no—
75C4⋊C4×C101C2311yescharacteristic
76C4⋊C4×C101C2312yes—
77C4⋊C4×C101C2311yescharacteristic
78C22⋊C4×C101—21no—
79C42×C101—21no—
80C23×C101C2×D411yescharacteristic
81C2×C4×C101—21no—
82C2×C4×C101—21no—
83C2×C4×C101—22no—
84C2×C4×C101—42no—
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2×Q812yes—
87C2×C4×C101C2×Q812yes—
88C2×C4×C101C2411yescharacteristic
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—22no—
92C2×Dic101—41no—
93C2×Dic101—41no—
94C2×Dic101—42no—
95C2×Dic101C2×D411yescharacteristic
96C2×Dic101C2×D411yescharacteristic
97C2×Dic101C2×D411yescharacteristic
98C2×Dic101C2×D411yescharacteristic
99Dic202—42no—
100Dic202—42no—
101C8×C101C2×Q814yes—
102#11717579C2×Q814yes—
103C22×C101—21no—
104C22×C101—21no—
105C22×C101C4⋊Q812yes—
106C22×C101C22×D411yescharacteristic
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—42no—
110C4×C101C4⋊Q812yes—
111C4×C101C22×Q811yescharacteristic
112C4×C101C22×Q811yescharacteristic
113Dic101—21no—
114Dic101—21no—
115Dic101—21no—
116Dic101—21no—
117Dic101—41no—
118Dic101—41no—
119Dic101—42no—
120C2×C101—22no—
121C2×C101C2×C4⋊Q811yescharacteristic
122C2×C101#57511yescharacteristic
123C2×C101#57511yescharacteristic
124#3166—1011nomaximal, Sylow
125C101#316611yescharacteristic, complemented, Sylow
126#533—1011no—
127C2×#125—1011no—
128#501—1012no—
129#501—1012no—
130#499—1012no—
131#499—1012no—
132#428—1011no—
133#428—1011no—
134#428—1011no—
135#428D10111yescharacteristic
136C4×M16—1011no—
137C2×M16—1011no—
138C2×M16C2×D10111yescharacteristic
139C4⋊Q8—1011no—
140C4⋊Q8—1011no—
141#125—1012no—
142#125—1012no—
143#125—1011no—
144#125—1011no—
145#125—2021no—
146#125—2021no—
147C22⋊Q8—2021no—
148C22⋊Q8—2021no—
149C4×Q8—2021no—
150C4×Q8—2021no—
151#117—1011no—
152#117C2×D10111yescharacteristic
153C2×C4⋊C4—1011no—
154C2×C4⋊C4—2021no—
155C2×C4⋊C4—2021no—
156C2×C4⋊C4C2×D10111yescharacteristic
157C2×C42—1011no—
158#107—1012no—
159#107—1012no—
160#107—1012no—
161#107C2×D10112yes—
162#106—1012no—
163#106—1012no—
164#106—1012no—
165#106C2×D10112yes—
166#97—1012no—
167C4×C8—1012no—
168C2×Q8—2021no—
169C2×Q8—2021no—
170C22×C4—21no—
171C22×C4—1011no—
172C22×C4—1011no—
173C22×C4—2021no—
174C22×C4C22×D10111yescharacteristic
175M16—1014no—
176M16Dic20214yescomplemented
177C2×C8—1012no—
178C2×C8C22×D10112yes—
179C4⋊C4—21no—
180C4⋊C4—1012no—
181C4⋊C4—1011no—
182C4⋊C4—1011no—
183C4⋊C4—2022no—
184C4⋊C4—2022no—
185C4⋊C4—2021no—
186C4⋊C4—2021no—
187C4⋊C4—2021no—
188C4⋊C4—2021no—
189C4⋊C4—2022no—
190C4⋊C4—2022no—
191C4⋊C4—2021no—
192C4⋊C4—2021no—
193C4⋊C4—2021no—
194C4⋊C4C22×D10111yescharacteristic
195C4⋊C4C22×D10112yes—
196C4⋊C4C22×D10111yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—2021no—
199C42—21no—
200C42—1011no—
201C42—1011no—
202C42—1011no—
203C42—1011no—
204C42—2021no—
205C23D4×D10111yescharacteristic
206Q8—4042no—
207Q8—4042no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—1011no—
213C2×C4—1011no—
214C2×C4—1011no—
215C2×C4—1011no—
216C2×C4—4042no—
217C2×C4—4041no—
218C2×C4—4041no—
219C2×C4—2021no—
220C2×C4—2021no—
221C2×C4—2022no—
222C2×C4D4×D10111yescharacteristic
223C2×C4Q8×D10112yes—
224C2×C4C2×Dic20212yes—
225C2×C4C23×D10111yescharacteristic
226C8—1014no—
227C8C2×Dic20214yes—
228C22—21no—
229C22—21no—
230C22#1569225412yes—
231C22C2×D4×D10111yescharacteristic
232C4—42no—
233C4—41no—
234C4—41no—
235C4—4041no—
236C4—4042no—
237C4—4041no—
238C4—2021no—
239C4—2021no—
240C4—2021no—
241C4—2021no—
242C4#1569225412yes—
243C4C2×Q8×D10111yescharacteristic
244C4C22×Dic20211yescharacteristic
245C2—22no—
246C2C2×#1569225411yescharacteristic
247C2#2026597311yescharacteristic
248C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radicalC23×C101C2×D4
hypercenter#428D101
hypocenterC101#3166
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
3. #20265585C2semidirect
4. #20265584C2semidirect
5. #20265578C2semidirect
6. #20265577C2semidirect
48. #13108058Q8semidirect
72. M16×C101Q8semidirect
125. C101#3166semidirect
176. M16Dic202semidirect

Maximal subgroups

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

subgroupindex
#202657302
C2×#156922462
#202655852
#202655842
#202655782
#202655772
#202652202
#202652192
#202652172
M16×Dic1012
#428×C1012
#3166101