Finite Group Databasebeta

← #27180658

Subgroups — #27180658

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×#15692285C211yesmaximal, characteristic
2C22×#13108064C211yesmaximal, characteristic, complemented
3#20265654C211yesmaximal, characteristic, complemented
4#20265396C214yesmaximal, complemented
5#20265384C214yesmaximal, complemented
6#20265204C211yesmaximal, characteristic, complemented
7#20265186C211yesmaximal, characteristic, complemented
8#20265173C211yesmaximal, characteristic
9#536×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11#15692285—24no—
12C2×#13108064—24no—
13C2×#13108064C2212yes—
14C2×#13108064C2214yes—
15C2×#13108062C2212yes—
16C2×#13108058C2211yescharacteristic
17C22×#11717579C2211yescharacteristic
18C4×Dic202C2214yes—
19#15692220—22no—
20#15692214C2211yescharacteristic
21#15692211—22no—
22C4⋊#11717579C2214yes—
23#15692179C2212yes—
24#15692179C2214yes—
25#15692176C2212yes—
26C2×#41×C101C2211yescharacteristic
27C22×Q8×C101C2211yescharacteristic
28C4⋊Q8×C101C2212yes—
29#124×C101C2212yes—
30#123×C101—22no—
31C22⋊Q8×C101—22no—
32#117×C101C2211yescharacteristic
33C2×Dic202—21no—
34C2×Dic202—22no—
35C2×Dic202D414yes—
36C2×Dic202C2312yes—
37C22×Dic101—21no—
38#13108064—28no—
39#13108064—24no—
40#13108062—24no—
41#13108058—22no—
42C2×#11717579—22no—
43C2×#11717579C2312yes—
44C2×#11717579C2312yes—
45C4⋊Dic101C2312yes—
46#13108053—24no—
47C22⋊Dic101—21no—
48C4×Dic101—24no—
49C4×Dic101—21no—
50#41×C101—22no—
51#41×C101—22no—
52C2×Q8×C101—21no—
53C2×Q8×C101—22no—
54C2×Q8×C101—22no—
55C2×Q8×C101C2312yes—
56C2×Q8×C101C2311yescharacteristic
57C2×D4×C101—21no—
58C2×D4×C101C2311yescharacteristic
59C22×C4×C101—21no—
60C22×C4×C101—21no—
61C22×C4×C101C2311yescharacteristic
62C4⋊C4×C101—24no—
63C4⋊C4×C101D412yes—
64C22⋊C4×C101—24no—
65C22⋊C4×C101D412yes—
66C42×C101C2312yes—
67C23×C101—21no—
68C23×C101C2×D411yescharacteristic
69Q8×C101—22no—
70Q8×C101—21no—
71Q8×C101—22no—
72Q8×C101—21no—
73Q8×C101—42no—
74D4×C101—21no—
75D4×C101—21no—
76D4×C101—42no—
77C2×C4×C101—22no—
78C2×C4×C101—21no—
79C2×C4×C101—42no—
80C2×C4×C101—42no—
81C2×C4×C101C2×D412yes—
82C2×C4×C101C2×D414yes—
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101#4112yes—
85C2×C4×C101C2411yescharacteristic
86C2×Dic101—24no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89Dic202—24no—
90Dic202—22no—
91Dic202C2×D414yes—
92#11717579—22no—
93#11717579—22no—
94C22×C101—22no—
95C22×C101—41no—
96C22×C101—42no—
97C22×C101#12112yes—
98C22×C101C22×D411yescharacteristic
99C4×C101—24no—
100C4×C101—41no—
101C4×C101—42no—
102C4×C101#12112yes—
103C4×C101C22×D411yescharacteristic
104C4×C101C2×#4111yescharacteristic
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#536D10111yescharacteristic
116#519—1011no—
117#464—1014no—
118#462—1014no—
119#424—1011no—
120#419—1011no—
121#416—1011no—
122C2×#41C2×D10111yescharacteristic
123C22×Q8—1011no—
124C22×Q8C2×D10111yescharacteristic
125#137—2024no—
126C2×Q16—1014no—
127C2×Q16—1012no—
128C2×Q16—2024no—
129C2×SD16—1012no—
130C2×M16—1011no—
131C22×C8—1011no—
132C4⋊Q8C2×D10112yes—
133#124C2×D10112yes—
134#123—22no—
135C22⋊Q8—22no—
136C4×Q8—1014no—
137C4×Q8—2022no—
138#117—1011no—
139#117—2022no—
140#117C2×D10111yescharacteristic
141C4⋊C8—1014no—
142#103—1012no—
143#103—1014no—
144#102—1012no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—1014no—
151C2×Q8—1012no—
152C2×Q8—2021no—
153C2×Q8—2022no—
154C2×Q8C22×D10112yes—
155C2×Q8C22×D10111yescharacteristic
156C2×D4—21no—
157C2×D4C22×D10111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2021no—
161C22×C4C22×D10111yescharacteristic
162Q16—2028no—
163Q16—2024no—
164SD16—2024no—
165M16—2022no—
166C2×C8—1012no—
167C2×C8—1012no—
168C2×C8—2022no—
169C4⋊C4—24no—
170C4⋊C4—1012no—
171C4⋊C4—2024no—
172C4⋊C4C22⋊D10112yes—
173C22⋊C4—24no—
174C22⋊C4—2021no—
175C22⋊C4C22⋊D10112yes—
176C42—2021no—
177C42—2024no—
178C42C22×D10112yes—
179C23—21no—
180C23C2×C22⋊D10111yescharacteristic
181Q8—22no—
182Q8—21no—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—1014no—
187Q8—2022no—
188Q8—2024no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2022no—
197C2×C4—2022no—
198C2×C4—2024no—
199C2×C4D4×D10112yes—
200C2×C4C2×C22⋊D10114yes—
201C2×C4C2×C22⋊D10111yescharacteristic
202C2×C4#1310807312yes—
203C2×C4C23×D10111yescharacteristic
204C8—2022no—
205C8—2022no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1569222612yes—
210C22C22×C22⋊D10111yescharacteristic
211C4—24no—
212C4—41no—
213C4—42no—
214C4—4042no—
215C4—2024no—
216C4#1569222612yes—
217C4C2×D4×D10111yescharacteristic
218C4C2×#1310807311yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1569222611yescharacteristic
222C2#2026599212yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#536×C101C2
socleC22×C101C22×D4
prime radicalC2×#41×C101C22
hypercenter#536D101
hypocenterC101#3044
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#13108064C2semidirect
3. #20265654C2semidirect
4. #20265396C2semidirect
5. #20265384C2semidirect
6. #20265204C2semidirect
7. #20265186C2semidirect
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×#131080642
#202656542
#202653962
#202653842
#202652042
#202651862
#202651732
#536×C1012
#3044101