Finite Group Databasebeta

← #27180663

Subgroups — #27180663

Order 12928.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15692288C211yesmaximal, characteristic
2C22×Dic404C211yesmaximal, characteristic, complemented
3#20265753C211yesmaximal, characteristic
4#20265654C211yesmaximal, characteristic, complemented
5#20265400C214yesmaximal, complemented
6#20265388C214yesmaximal, complemented
7#20265187C211yesmaximal, characteristic, complemented
8#20265174C211yesmaximal, characteristic
9#424×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic
11C22×Dic202C2211yescharacteristic
12C22×Dic202C2211yescharacteristic
13#15692288—24no—
14C2×Dic404—24no—
15C2×Dic404C2212yes—
16C2×Dic404C2214yes—
17C2×#13108063C2212yes—
18#15692255C2212yes—
19#15692243C2212yes—
20#15692239—22no—
21#15692235—22no—
22C4×Dic202C2214yes—
23#15692220—22no—
24#15692214C2211yescharacteristic
25#15692211—22no—
26#15692180C2212yes—
27#15692180C2214yes—
28#15692177C2212yes—
29C2×M16×C101C2211yescharacteristic
30C22×C8×C101C2211yescharacteristic
31#117×C101C2211yescharacteristic
32C4⋊C8×C101C2214yes—
33#13108074—22no—
34#13108074—22no—
35C2×Dic202—21no—
36C2×Dic202—22no—
37C2×Dic202—22no—
38C2×Dic202—22no—
39C2×Dic202—21no—
40C2×Dic202D414yes—
41C2×Dic202C2312yes—
42C2×Dic202C2311yescharacteristic
43C2×Dic202C2312yes—
44C2×D404C2311yescharacteristic
45C2×C22⋊D101—21no—
46C22×Dic101—21no—
47C22×Dic101—21no—
48C2×C4×D101—21no—
49Dic404—24no—
50Dic404—28no—
51#13108063—24no—
52C4⋊Dic101—24no—
53C4⋊Dic101C2312yes—
54#13108053—24no—
55C22⋊Dic101—21no—
56#13108051—24no—
57C4×Dic101—24no—
58C4×Dic101—21no—
59C22×C4×C101C2311yescharacteristic
60M16×C101—22no—
61C2×C8×C101—22no—
62C2×C8×C101C2312yes—
63C2×C8×C101C2312yes—
64C4⋊C4×C101D412yes—
65C22⋊C4×C101D412yes—
66C42×C101C2312yes—
67C23×C101C2×D411yescharacteristic
68C22×D101—21no—
69C2×C4×C101C2×D412yes—
70C2×C4×C101C2×D414yes—
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101#4112yes—
73C2×C4×C101C2411yescharacteristic
74C22⋊D101—42no—
75C2×Dic101—24no—
76C2×Dic101—22no—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—21no—
80C2×Dic101—42no—
81D404—21no—
82D404—21no—
83C4×D101—42no—
84Dic202—24no—
85Dic202—22no—
86Dic202—22no—
87Dic202—21no—
88Dic202—22no—
89Dic202—21no—
90Dic202—42no—
91Dic202C2×D414yes—
92C8×C101—22no—
93C8×C101—22no—
94C22×C101—22no—
95C22×C101#12112yes—
96C22×C101C22×D411yescharacteristic
97C2×D101—41no—
98C2×D101—42no—
99C4×C101—24no—
100C4×C101#12112yes—
101C4×C101C22×D411yescharacteristic
102C4×C101C2×#4111yescharacteristic
103Dic101—24no—
104Dic101—42no—
105Dic101—41no—
106Dic101—42no—
107C2×C101—22no—
108C2×C101C2×#12111yescharacteristic
109C2×C101#57812yes—
110D101—81no—
111#3044—1011nomaximal, Sylow
112C101#304411yescharacteristic, complemented, Sylow
113C2×#137—1011no—
114C22×Q16—1011no—
115#536—1011no—
116#519—1011no—
117#464—1014no—
118#462—1014no—
119#424D10111yescharacteristic, complemented
120#419—1011no—
121#416—1011no—
122C2×#41—1011no—
123C22×Q8—1011no—
124C22×Q8—1011no—
125#137—2024no—
126C2×Q16—1014no—
127C2×Q16—1012no—
128C2×Q16—2024no—
129C2×SD16—1012no—
130C2×M16C2×D10111yescharacteristic
131C22×C8C2×D10111yescharacteristic
132C4⋊Q8—1012no—
133#124—1012no—
134#123—2022no—
135C22⋊Q8—2022no—
136C4×Q8—1014no—
137C4×Q8—2022no—
138#117—1011no—
139#117—2022no—
140#117C2×D10111yescharacteristic
141C4⋊C8C2×D10114yes—
142#103—1012no—
143#103—1014no—
144#102—1012no—
145#41—2022no—
146#41—2022no—
147C2×Q8—1014no—
148C2×Q8—1012no—
149C2×Q8—1012no—
150C2×Q8—1011no—
151C2×Q8—2022no—
152C2×Q8—2022no—
153C2×Q8—2021no—
154C2×Q8—2021no—
155C2×Q8—2022no—
156C2×D4—1011no—
157C2×D4—2021no—
158C22×C4—2021no—
159C22×C4—2021no—
160C22×C4—2021no—
161C22×C4C22×D10111yescharacteristic
162Q16—2028no—
163Q16—2024no—
164SD16—2024no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D10112yes—
168C2×C8C22×D10112yes—
169C4⋊C4—1012no—
170C4⋊C4—2024no—
171C4⋊C4—2024no—
172C4⋊C4D40412yes—
173C22⋊C4—2021no—
174C22⋊C4—2024no—
175C22⋊C4D40412yes—
176C42—2021no—
177C42—2024no—
178C42C22×D10112yes—
179C23—2021no—
180C23C2×D40411yescharacteristic
181Q8—1014no—
182Q8—4042no—
183Q8—2022no—
184Q8—2021no—
185Q8—2022no—
186Q8—2021no—
187Q8—2022no—
188Q8—2024no—
189D4—4042no—
190D4—2021no—
191D4—2021no—
192C2×C4—4042no—
193C2×C4—4042no—
194C2×C4—2022no—
195C2×C4—2022no—
196C2×C4—2021no—
197C2×C4—2022no—
198C2×C4—2024no—
199C2×C4D4×D10112yes—
200C2×C4C2×D40414yes—
201C2×C4C2×D40411yescharacteristic
202C2×C4#1310807512yes—
203C2×C4C23×D10111yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—4042no—
208C22—4041no—
209C22#1569223012yes—
210C22C22×D40411yescharacteristic
211C4—24no—
212C4—4042no—
213C4—4041no—
214C4—4042no—
215C4—2024no—
216C4#1569223012yes—
217C4C2×D4×D10111yescharacteristic
218C4C2×#1310807511yescharacteristic
219C2—22no—
220C2—8081no—
221C2C2×#1569223011yescharacteristic
222C2#2026599412yes—
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#424×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108074C22
hypercenter#424D101
hypocenterC101#3044
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic404C2semidirect
4. #20265654C2semidirect
5. #20265400C2semidirect
6. #20265388C2semidirect
7. #20265187C2semidirect
9. #424×C101C2semidirect
112. C101#3044semidirect
119. #424D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922882
C22×Dic4042
#202657532
#202656542
#202654002
#202653882
#202651872
#202651742
#424×C1012
#3044101