Finite Group Databasebeta

← #27172216

Subgroups — #27172216

Order 12928.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13108063C211yesmaximal, characteristic
2C22×#13108061C211yesmaximal, characteristic
3C2×#15692253C211yesmaximal, characteristic, complemented
4C2×#15692178C211yesmaximal, characteristic, complemented
5#20265078C211yesmaximal, characteristic
6#20264931C211yesmaximal, characteristic, complemented
7C2×#103×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C2211yescharacteristic
9C22×D404C2211yescharacteristic
10C2×#13108063—22no—
11C2×#13108063—22no—
12C2×#13108063—22no—
13C2×#13108061—22no—
14C2×#13108061—22no—
15C2×#13108061—22no—
16C22×#11717579C2211yescharacteristic
17#15692253—22no—
18#15692253—22no—
19C2×#13108053—22no—
20C2×#13108051—22no—
21C2×C4×Dic101C2211yescharacteristic
22#15692178—22no—
23C22×Q8×C101C2211yescharacteristic
24C22×C8×C101C2211yescharacteristic
25C2×C4⋊C4×C101C2211yescharacteristic
26#103×C101—22no—
27C23×D101—21no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×Dic202—21no—
32C2×Dic202—21no—
33C2×Dic202D412yes—
34C2×D404—21no—
35C2×D404—21no—
36C2×D404—21no—
37C2×D404—21no—
38C2×D404—21no—
39C2×D404D412yes—
40C22×Dic101—21no—
41C22×Dic101D412yes—
42#13108063—28no—
43#13108061—28no—
44C2×#11717579—21no—
45C2×#11717579—21no—
46C2×#11717579D412yes—
47#13108053—28no—
48#13108051—44no—
49C4×Dic101—21no—
50C4×Dic101—21no—
51C2×Q8×C101—21no—
52C2×Q8×C101—21no—
53C2×Q8×C101—21no—
54C2×Q8×C101—21no—
55C2×Q8×C101—21no—
56C2×Q8×C101D412yes—
57C22×C4×C101—21no—
58C22×C4×C101—21no—
59C22×C4×C101C2311yescharacteristic
60C2×C8×C101—21no—
61C2×C8×C101—21no—
62C2×C8×C101D412yes—
63C4⋊C4×C101—21no—
64C4⋊C4×C101—21no—
65C23×C101C2×D411yescharacteristic
66C22×D101—22no—
67C22×D101—41no—
68C22×D101—41no—
69C22×D101—42no—
70C22×D101—42no—
71Q8×C101—22no—
72Q8×C101—22no—
73Q8×C101—41no—
74Q8×C101—41no—
75C2×C4×C101—22no—
76C2×C4×C101—41no—
77C2×C4×C101—41no—
78C2×C4×C101—41no—
79C2×C4×C101—42no—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2×D411yescharacteristic
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×D411yescharacteristic
85C2×C4×C101#4111yescharacteristic
86C2×Dic101—22no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—41no—
90C2×Dic101—41no—
91C2×Dic101SD1614yescomplemented
92D404—22no—
93D404—22no—
94D404—41no—
95D404—41no—
96Dic202—22no—
97Dic202—22no—
98Dic202—41no—
99Dic202—41no—
100C8×C101—22no—
101#11717579—22no—
102C22×C101#12011yescharacteristic
103C22×C101#12111yescharacteristic
104C22×C101#12711yescharacteristic
105C22×C101C2×SD1612yes—
106C22×C101#13712yes—
107C2×D101—42no—
108C2×D101—44no—
109C2×D101—82no—
110C2×D101—81no—
111C2×D101—81no—
112C4×C101—42no—
113C4×C101—81no—
114C4×C101#12011yescharacteristic
115C4×C101#12011yescharacteristic
116C4×C101#12111yescharacteristic
117C4×C101#12111yescharacteristic
118Dic101—24no—
119Dic101—42no—
120C2×C101#39211yescharacteristic
121C2×C101#44812yes—
122C2×C101#46112yes—
123C2×C101#49211yescharacteristic
124C2×C101#49711yescharacteristic
125D101—82no—
126#2013—1011nomaximal, Sylow
127C101#201311yescharacteristic, complemented, Sylow
128C22×SD16—1011no—
129C22×SD16—1011no—
130C2×C4⋊Q8—1011no—
131C2×#103—1011no—
132C2×#103D10111yescharacteristic, complemented
133#390—1011no—
134#336—1011no—
135C22×Q8—1011no—
136C22×Q8C2×D10111yescharacteristic
137C22×D4—1011no—
138C2×SD16—2022no—
139C2×SD16—2022no—
140C2×SD16—2022no—
141C2×SD16—2022no—
142C2×SD16—2022no—
143C2×SD16—2022no—
144C22×C8—1011no—
145C22×C8C2×D10111yescharacteristic
146C4⋊Q8—2022no—
147C4⋊Q8—2022no—
148C2×C4⋊C4—2022no—
149C2×C4⋊C4C2×D10111yescharacteristic
150C2×C22⋊C4—2022no—
151C2×C42—1011no—
152#103—22no—
153#103—2022no—
154C24—2021no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—1012no—
161C2×Q8—2021no—
162C2×Q8—2021no—
163C2×Q8—2021no—
164C2×Q8—2021no—
165C2×Q8—2021no—
166C2×Q8D40412yes—
167C2×D4—1012no—
168C2×D4—2021no—
169C2×D4—2021no—
170C2×D4—2021no—
171C2×D4—2021no—
172C2×D4—2021no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—1012no—
176C22×C4—2021no—
177C22×C4C22×D10111yescharacteristic
178SD16—2028no—
179SD16—2028no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—1012no—
183C2×C8—2021no—
184C2×C8—2021no—
185C2×C8C22⋊D10112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2028no—
189C22⋊C4—4044no—
190C42—2021no—
191C42—2021no—
192C23—4041no—
193C23—4041no—
194C23—4042no—
195C23—4042no—
196C23—2022no—
197C23D4×D10111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—4041no—
203Q8—4041no—
204Q8—2022no—
205Q8—2022no—
206D4—4041no—
207D4—4041no—
208D4—2022no—
209D4—2022no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—1014no—
216C2×C4—4041no—
217C2×C4—4041no—
218C2×C4—2022no—
219C2×C4—2022no—
220C2×C4—2022no—
221C2×C4D4×D10111yescharacteristic
222C2×C4D4×D10111yescharacteristic
223C2×C4D4×D10111yescharacteristic
224C2×C4C2×C22⋊D10111yescharacteristic
225C2×C4C2×D40411yescharacteristic
226C2×C4#1310807411yescharacteristic
227C8—22no—
228C8—2022no—
229C22—8081no—
230C22—8081no—
231C22—8082no—
232C22—4042no—
233C22—4044no—
234C22#1569222511yescharacteristic
235C22#1569222811yescharacteristic
236C22#1569225111yescharacteristic
237C22SD16×D10112yes—
238C22#1569228712yes—
239C4—42no—
240C4—81no—
241C4—4042no—
242C4—2024no—
243C4#1569222311yescharacteristic
244C4#1569222511yescharacteristic
245C4#1569222811yescharacteristic
246C4#1569222911yescharacteristic
247C2—8082no—
248C2#2026508611yescharacteristic
249C2#2026531712yes—
250C2#2026538112yes—
251C2#2026554811yescharacteristic
252C2#2026557011yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
FittingC2×#103×C101C2
socleC23×C101C2×D4
prime radicalC22×D404C22
hypercenterC2×#103D101
hypocenterC101#2013
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#15692253C2semidirect
4. C2×#15692178C2semidirect
6. #20264931C2semidirect
7. C2×#103×C101C2semidirect
91. C2×Dic101SD16semidirect
127. C101#2013semidirect
132. C2×#103D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080632
C22×#131080612
C2×#156922532
C2×#156921782
#202650782
#202649312
C2×#103×C1012
#2013101