Finite Group Databasebeta

← #27180665

Subgroups — #27180665

Order 12928.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692285C211yesmaximal, characteristic
2C22×Dic404C211yesmaximal, characteristic, complemented
3#20265748C211yesmaximal, characteristic
4#20265654C211yesmaximal, characteristic, complemented
5#20265398C214yesmaximal, complemented
6#20265386C214yesmaximal, complemented
7#20265203C211yesmaximal, characteristic, complemented
8#20265185C211yesmaximal, characteristic, complemented
9#416×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11C22×Dic202C2211yescharacteristic
12#15692285—24no—
13C2×Dic404—24no—
14C2×Dic404C2214yes—
15C2×#13108064C2212yes—
16C2×#13108062C2212yes—
17C2×#13108058C2211yescharacteristic
18#15692253C2212yes—
19#15692242C2212yes—
20#15692236—22no—
21#15692234—22no—
22C4×Dic202—22no—
23#15692220C2214yes—
24#15692214C2211yescharacteristic
25#15692211—22no—
26#15692184C2214yes—
27#15692178C2214yes—
28C2×#41×C101C2211yescharacteristic
29C22×C8×C101C2211yescharacteristic
30#117×C101C2211yescharacteristic
31#103×C101C2212yes—
32#102×C101C2212yes—
33C2×Dic202—21no—
34C2×Dic202—22no—
35C2×Dic202—22no—
36C2×Dic202—22no—
37C2×Dic202—21no—
38C2×Dic202D414yes—
39C2×Dic202C2312yes—
40C2×Dic202C2312yes—
41C22×Dic101—21no—
42C22×Dic101—21no—
43Dic404—28no—
44#13108064—24no—
45#13108062—24no—
46#13108058—22no—
47C2×#11717579C2312yes—
48C4⋊Dic101D412yes—
49#13108053—24no—
50#13108053—24no—
51C22⋊Dic101—24no—
52C22⋊Dic101D412yes—
53C4×Dic101—24no—
54C4×Dic101C2312yes—
55#41×C101—22no—
56#41×C101—22no—
57C2×Q8×C101C2311yescharacteristic
58C2×D4×C101—21no—
59C2×D4×C101C2311yescharacteristic
60C22×C4×C101—21no—
61C22×C4×C101C2311yescharacteristic
62C2×C8×C101—22no—
63C2×C8×C101C2312yes—
64C4⋊C4×C101C2312yes—
65C22⋊C4×C101—21no—
66C42×C101—21no—
67C23×C101—21no—
68C23×C101C2×D411yescharacteristic
69Q8×C101—21no—
70Q8×C101—21no—
71D4×C101—21no—
72D4×C101—21no—
73D4×C101—42no—
74C2×C4×C101—22no—
75C2×C4×C101—21no—
76C2×C4×C101—42no—
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D412yes—
79C2×C4×C101#4112yes—
80C2×C4×C101C2411yescharacteristic
81C2×Dic101—24no—
82C2×Dic101—22no—
83C2×Dic101—22no—
84C2×Dic101—42no—
85C2×Dic101C2×D414yes—
86Dic202—24no—
87Dic202—22no—
88Dic202—22no—
89Dic202—22no—
90Dic202—42no—
91Dic202C2×D414yes—
92C8×C101—22no—
93#11717579—22no—
94C22×C101—22no—
95C22×C101—41no—
96C22×C101—42no—
97C22×C101#12112yes—
98C22×C101C22×D411yescharacteristic
99C4×C101—41no—
100C4×C101—42no—
101C4×C101#12112yes—
102C4×C101C22×D411yescharacteristic
103C4×C101C2×#4111yescharacteristic
104Dic101—24no—
105Dic101—24no—
106Dic101—42no—
107C2×C101—22no—
108C2×C101—81no—
109C2×C101C2×#12111yescharacteristic
110C2×C101#57812yes—
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#424—1011no—
120#419—1011no—
121#416D10111yescharacteristic
122C2×#41C2×D10111yescharacteristic
123C22×Q8—1011no—
124C22×Q8—1011no—
125#137—2024no—
126C2×Q16—1012no—
127C2×Q16—1014no—
128C2×Q16—2024no—
129C2×SD16—1012no—
130C2×M16—1011no—
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⋊C8—1014no—
142#103—1014no—
143#103C2×D10112yes—
144#102C2×D10112yes—
145#41—22no—
146#41—22no—
147C2×Q8—1014no—
148C2×Q8—1012no—
149C2×Q8—1012no—
150C2×Q8—2022no—
151C2×Q8—2022no—
152C2×Q8—2021no—
153C2×Q8—2021no—
154C2×Q8—2022no—
155C2×Q8C22×D10111yescharacteristic
156C2×D4—21no—
157C2×D4C22×D10111yescharacteristic
158C22×C4—21no—
159C22×C4—2021no—
160C22×C4—2021no—
161C22×C4C22×D10111yescharacteristic
162Q16—2028no—
163Q16—2024no—
164SD16—2024no—
165M16—2022no—
166C2×C8—22no—
167C2×C8—1012no—
168C2×C8C22×D10112yes—
169C4⋊C4—1012no—
170C4⋊C4—2024no—
171C4⋊C4—2024no—
172C4⋊C4C22×D10112yes—
173C22⋊C4—21no—
174C22⋊C4—1012no—
175C22⋊C4—2024no—
176C42—21no—
177C42—1012no—
178C42—2024no—
179C23—21no—
180C23D4×D10111yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—1014no—
184Q8—4042no—
185Q8—2022no—
186Q8—2022no—
187Q8—2022no—
188Q8—2024no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—1014no—
196C2×C4—4042no—
197C2×C4—2022no—
198C2×C4—2022no—
199C2×C4—2024no—
200C2×C4D4×D10112yes—
201C2×C4D4×D10111yescharacteristic
202C2×C4#1310807412yes—
203C2×C4C23×D10111yescharacteristic
204C8—22no—
205C8—2022no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1569222812yes—
210C22C2×D4×D10111yescharacteristic
211C4—42no—
212C4—41no—
213C4—4042no—
214C4—2024no—
215C4—2024no—
216C4#1569222812yes—
217C4C2×D4×D10111yescharacteristic
218C4C2×#1310807411yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1569222811yescharacteristic
222C2#2026599112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
prime radicalC2×#41×C101C22
hypercenter#416D101
hypocenterC101#3044
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic404C2semidirect
4. #20265654C2semidirect
5. #20265398C2semidirect
6. #20265386C2semidirect
7. #20265203C2semidirect
8. #20265185C2semidirect
112. C101#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#156922852
C22×Dic4042
#202657482
#202656542
#202653982
#202653862
#202652032
#202651852
#416×C1012
#3044101