Finite Group Databasebeta

← #27170158

Subgroups — #27170158

Order 12928.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265802C211yesmaximal, characteristic
2#20265513C211yesmaximal, characteristic, complemented
3#20265510C211yesmaximal, characteristic, complemented
4#20265114C211yesmaximal, characteristic
5#20264913C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic
7#328×C101C211yesmaximal, characteristic, complemented
8#15692235—21no—
9#15692234C2211yescharacteristic
10#15692233—21no—
11#15692233—21no—
12#15692232—21no—
13#15692231—21no—
14C2×C4⋊Dic101—21no—
15C2×C4⋊Dic101C2211yescharacteristic
16C2×#13108051—21no—
17C2×#13108051C2211yescharacteristic
18C22⋊C4×D101—21no—
19#15692189—21no—
20#15692188—21no—
21#15692180—21no—
22#15692179—21no—
23C22⋊#11717579C2211yescharacteristic
24C22⋊Q8×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202D411yescharacteristic
30C22×Dic101—21no—
31C22×Dic101D411yescharacteristic
32C2×C4×D101—21no—
33C2×C4×D101—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101—42no—
39C4⋊Dic101D411yescharacteristic
40#13108053—21no—
41#13108053—21no—
42#13108053—21no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—42no—
49C2×Q8×C101D411yescharacteristic
50C22×C4×C101—21no—
51C22×C4×C101C2311yescharacteristic
52C2×C8×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101D411yescharacteristic
55C22⋊C4×C101—21no—
56C23×C101C2×D411yescharacteristic
57C22×D101—21no—
58C22×D101—21no—
59C22×D101—41no—
60C22×D101—41no—
61C22×D101Q1612yes—
62Q8×C101—41no—
63C2×C4×C101—22no—
64C2×C4×C101—21no—
65C2×C4×C101—21no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—41no—
75C2×Dic101—41no—
76C2×Dic101—41no—
77C2×Dic101—41no—
78C4×D101—81no—
79C4×D101—81no—
80Dic202—41no—
81Dic202—81no—
82C8×C101—41no—
83#11717579—41no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101#12011yescharacteristic
87C22×C101C2×Q1611yescharacteristic
88C22×C101#13711yescharacteristic
89C2×D101—22no—
90C2×D101—41no—
91C2×D101—41no—
92C2×D101—42no—
93C2×D101—81no—
94C4×C101—21no—
95C4×C101—21no—
96C4×C101—42no—
97C4×C101—41no—
98C4×C101—41no—
99Dic101—41no—
100Dic101—41no—
101Dic101—41no—
102Dic101—81no—
103C2×C101—21no—
104C2×C101—21no—
105C2×C101#45111yescharacteristic
106C2×C101#45311yescharacteristic
107C2×C101#45811yescharacteristic
108D101—42no—
109#1614—1011nomaximal, Sylow
110C101#161411yescharacteristic, complemented, Sylow
111#543—1011no—
112#484—1011no—
113#484—1011no—
114#399—1011no—
115#328—1011no—
116#328D10111yescharacteristic, complemented
117#323—1011no—
118C22⋊Q8—1011no—
119C22⋊Q8—2021no—
120C22⋊Q8—2021no—
121C22⋊Q8—2021no—
122C22⋊Q8—2021no—
123C22⋊Q8—2021no—
124C22⋊Q8C2×D10111yescharacteristic
125C2×C4⋊C4—1011no—
126C2×C4⋊C4—2021no—
127C2×C22⋊C4—1011no—
128C2×C22⋊C4—2021no—
129C2×C22⋊C4—2021no—
130#107—2021no—
131#107—2021no—
132#103—2021no—
133#103—2021no—
134C22⋊C8—1011no—
135C22⋊C8C2×D10111yescharacteristic
136#95C2×D10111yescharacteristic
137C24—1011no—
138C2×Q8—1011no—
139C2×Q8—2021no—
140C2×Q8D40411yescharacteristic
141C22×C4—21no—
142C22×C4—1011no—
143C22×C4—2021no—
144C22×C4—2021no—
145C22×C4—2021no—
146C22×C4C22×D10111yescharacteristic
147C2×C8—21no—
148C2×C8—2021no—
149C4⋊C4—21no—
150C4⋊C4—1011no—
151C4⋊C4—4042no—
152C4⋊C4—2021no—
153C4⋊C4—2021no—
154C4⋊C4—2021no—
155C4⋊C4—2021no—
156C4⋊C4—2021no—
157C4⋊C4—2021no—
158C4⋊C4D40411yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—4042no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C22⋊C4—2021no—
165C22⋊C4—2021no—
166C23—1012no—
167C23—4041no—
168C23—4041no—
169C23—2021no—
170C23—2021no—
171C23D4×D10111yescharacteristic
172Q8—41no—
173Q8—8081no—
174Q8—4041no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—8081no—
182C2×C4—8081no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—4041no—
186C2×C4—4041no—
187C2×C4—2021no—
188C2×C4—2021no—
189C2×C4—2021no—
190C2×C4D4×D10111yescharacteristic
191C2×C4C2×D40411yescharacteristic
192C8—41no—
193C8—4041no—
194C22—21no—
195C22—21no—
196C22—8081no—
197C22—4042no—
198C22—4041no—
199C22—4041no—
200C22—2022no—
201C22#1569222511yescharacteristic
202C22Q16×D10111yescharacteristic
203C22#1569228411yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—8081no—
210C4—4041no—
211C4—4041no—
212C4—4041no—
213C2—21no—
214C2—21no—
215C2—4042no—
216C2#2026533411yescharacteristic
217C2#2026534611yescharacteristic
218C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#328×C101C2
socleC22×C101#120
prime radicalC23×D101D4
hypercenter#328D101
hypocenterC101#1614
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20265513C2semidirect
3. #20265510C2semidirect
5. #20264913C2semidirect
7. #328×C101C2semidirect
110. C101#1614semidirect
116. #328D101semidirect

Maximal subgroups

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

subgroupindex
#202658022
#202655132
#202655102
#202651142
#202649132
#202649002
#328×C1012
#1614101