Finite Group Databasebeta

← #27172233

Subgroups — #27172233

Order 12928.

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

normal subgroups only

This group has 370 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×Dic404C211yesmaximal, characteristic
2C22×#13108064C211yesmaximal, characteristic
3C2×#15692253C211yesmaximal, characteristic
4C2×#15692178C211yesmaximal, characteristic
5#20265082C211yesmaximal, characteristic
6#20264931C211yesmaximal, characteristic
7C2×#103×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C22×Dic202C2211yescharacteristic
10C2×Dic404—22no—
11C2×Dic404—22no—
12C2×Dic404—22no—
13C2×#13108064—22no—
14C2×#13108064—22no—
15C2×#13108064—22no—
16C22×#11717579C2211yescharacteristic
17#15692253—22no—
18#15692253—22no—
19C2×#13108053—22no—
20C2×C4×Dic101C2211yescharacteristic
21#15692178—22no—
22#15692162—22no—
23C22×Q8×C101C2211yescharacteristic
24C22×C8×C101C2211yescharacteristic
25C2×C4⋊C4×C101C2211yescharacteristic
26#103×C101—22no—
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×Dic202—21no—
32C2×Dic202—21no—
33C2×Dic202—21no—
34C2×Dic202—21no—
35C2×Dic202—21no—
36C2×Dic202—21no—
37C2×Dic202D412yes—
38C2×Dic202D412yes—
39C22×Dic101—21no—
40C22×Dic101—21no—
41C22×Dic101D412yes—
42Dic404—28no—
43#13108064—28no—
44C2×#11717579—21no—
45C2×#11717579—21no—
46C2×#11717579D412yes—
47#13108053—28no—
48C4×Dic101—21no—
49C4×Dic101—21no—
50C2×Q8×C101—21no—
51C2×Q8×C101—21no—
52C2×Q8×C101—21no—
53C2×Q8×C101—21no—
54C2×Q8×C101—21no—
55C2×Q8×C101D412yes—
56C22×C4×C101—21no—
57C22×C4×C101—21no—
58C22×C4×C101C2311yescharacteristic
59C2×C8×C101—21no—
60C2×C8×C101—21no—
61C2×C8×C101D412yes—
62C4⋊C4×C101—21no—
63C4⋊C4×C101—21no—
64C23×C101C2×D411yescharacteristic
65Q8×C101—22no—
66Q8×C101—22no—
67Q8×C101—41no—
68Q8×C101—41no—
69C2×C4×C101—22no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101—42no—
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101#4111yescharacteristic
80C2×Dic101—22no—
81C2×Dic101—22no—
82C2×Dic101—22no—
83C2×Dic101—22no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101—41no—
87C2×Dic101—41no—
88C2×Dic101Q1614yescomplemented
89Dic202—22no—
90Dic202—22no—
91Dic202—22no—
92Dic202—22no—
93Dic202—41no—
94Dic202—41no—
95Dic202—41no—
96Dic202—41no—
97C8×C101—22no—
98#11717579—22no—
99C22×C101#12011yescharacteristic
100C22×C101#12111yescharacteristic
101C22×C101#12711yescharacteristic
102C22×C101C2×Q1612yes—
103C22×C101#13712yes—
104C4×C101—42no—
105C4×C101—81no—
106C4×C101#12011yescharacteristic
107C4×C101#12011yescharacteristic
108C4×C101#12111yescharacteristic
109C4×C101#12111yescharacteristic
110Dic101—24no—
111Dic101—42no—
112Dic101—42no—
113C2×C101#39211yescharacteristic
114C2×C101#45112yes—
115C2×C101#46212yes—
116C2×C101#49411yescharacteristic
117C2×C101#49711yescharacteristic
118#2016—1011nomaximal, Sylow
119C101#201611yescharacteristic, complemented, Sylow
120C22×Q16—1011no—
121C22×Q16—1011no—
122C2×C4⋊Q8—1011no—
123C2×#103—1011no—
124C2×#103D10111yescharacteristic
125#391—1011no—
126#336—1011no—
127C22×Q8—1011no—
128C22×Q8—1011no—
129C22×Q8C2×D10111yescharacteristic
130C2×Q16—2022no—
131C2×Q16—2022no—
132C2×Q16—2022no—
133C2×Q16—2022no—
134C2×Q16—2022no—
135C2×Q16—2022no—
136C22×C8—1011no—
137C22×C8C2×D10111yescharacteristic
138C4⋊Q8—2022no—
139C4⋊Q8—2022no—
140C2×C4⋊C4—2022no—
141C2×C4⋊C4C2×D10111yescharacteristic
142C2×C42—1011no—
143#103—22no—
144#103—2022no—
145#95—2022no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—1012no—
152C2×Q8—1012no—
153C2×Q8—2021no—
154C2×Q8—2021no—
155C2×Q8—2021no—
156C2×Q8—2021no—
157C2×Q8—2021no—
158C2×Q8—2021no—
159C2×Q8—2021no—
160C2×Q8—2021no—
161C2×Q8—2021no—
162C2×Q8—2021no—
163C2×Q8D40412yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—1012no—
167C22×C4—2021no—
168C22×C4—2021no—
169C22×C4C22×D10111yescharacteristic
170Q16—2028no—
171Q16—2028no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1012no—
175C2×C8—2021no—
176C2×C8—2021no—
177C2×C8C22⋊D10112yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—2028no—
181C42—2021no—
182C42—2021no—
183C23D4×D10111yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8—4041no—
189Q8—4041no—
190Q8—4041no—
191Q8—4041no—
192Q8—2022no—
193Q8—2022no—
194Q8—2022no—
195Q8—2022no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—1014no—
202C2×C4—4041no—
203C2×C4—4041no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—2022no—
207C2×C4—2022no—
208C2×C4—2022no—
209C2×C4—2022no—
210C2×C4D4×D10111yescharacteristic
211C2×C4D4×D10111yescharacteristic
212C2×C4D4×D10111yescharacteristic
213C2×C4C2×C22⋊D10111yescharacteristic
214C2×C4C2×D40411yescharacteristic
215C2×C4#1310807411yescharacteristic
216C8—22no—
217C8—2022no—
218C22#1569222511yescharacteristic
219C22#1569222811yescharacteristic
220C22#1569225111yescharacteristic
221C22Q16×D10112yes—
222C22#1569228412yes—
223C4—42no—
224C4—81no—
225C4—4042no—
226C4—4042no—
227C4—2024no—
228C4#1569222311yescharacteristic
229C4#1569222511yescharacteristic
230C4#1569222811yescharacteristic
231C4#1569222911yescharacteristic
232C2#2026508611yescharacteristic
233C2#2026533412yes—
234C2#2026538612yes—
235C2#2026555511yescharacteristic
236C2#2026556811yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
FittingC2×#103×C101C2
socle, prime radicalC23×C101C2×D4
hypercenterC2×#103D101
hypocenterC101#2016
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
88. C2×Dic101Q16semidirect
119. C101#2016semidirect

Maximal subgroups

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

subgroupindex
C22×Dic4042
C22×#131080642
C2×#156922532
C2×#156921782
#202650822
#202649312
C2×#103×C1012
#2016101