Finite Group Databasebeta

← #11751153

Subgroups — #11751153

Order 960.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C3×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×#6718C211yesmaximal, characteristic, complemented
3#91813C211yesmaximal, characteristic
4C2×#6710C211yesmaximal, characteristic
5#91119C211yesmaximal, characteristic, complemented
6#91094C211yesmaximal, characteristic, complemented
7#91092C211yesmaximal, characteristic, complemented
8#65341—31nomaximal, Hall
9C2×C3×C5×Q8C2211yescharacteristic
10C22×C3×C4×C5C2211yescharacteristic
11C2×Dic30C2211yescharacteristic
12C3×C5×C4⋊C4—21no—
13C3×C5×C4⋊C4C2211yescharacteristic
14C3×C5×C22⋊C4—21no—
15#6719—21no—
16#6718—22no—
17C4⋊Dic15C2211yescharacteristic
18#6712—21no—
19#6710—22no—
20C2×C3⋊#168C2211yescharacteristic
21C2×C3⋊#168C2211yescharacteristic
22#5052—51nomaximal, Hall
23C5×C22⋊Q8D311yescharacteristic
24C2×#689—31no—
25#4070—31no—
26C2×#681—31no—
27#3967—31no—
28#3942—31no—
29#3940—31no—
30C23×C3×C5D411yescharacteristic
31C3×C5×Q8—21no—
32C3×C5×Q8—21no—
33C2×C3×C4×C5—21no—
34C2×C3×C4×C5—21no—
35C2×C3×C4×C5—21no—
36C2×C3×C4×C5—21no—
37C2×C3×C4×C5D411yescharacteristic
38C2×C3×C4×C5C2311yescharacteristic
39C2×Dic15—21no—
40C2×Dic15—21no—
41Dic30—21no—
42Dic30—21no—
43C3⋊#168—21no—
44C3⋊#168D412yes—
45C3×C22⋊Q8D511yescharacteristic
46C2×#216—51no—
47#924—51no—
48C2×#208—51no—
49#835—51no—
50#810—51no—
51#808—51no—
52C2×C5×Q8C2×D311yescharacteristic
53C22×C4×C5C2×D311yescharacteristic
54C2×Dic10—31no—
55C5×C4⋊C4—21no—
56C5×C4⋊C4C2×D311yescharacteristic
57C5×C22⋊C4—21no—
58C22⋊Dic5—61no—
59#689—62no—
60C4⋊Dic5—31no—
61#683—61no—
62#681—62no—
63C2×#168—31no—
64C2×#168—31no—
65#470—151noSylow
66C22×C3×C5—21no—
67C22×C3×C5—21no—
68C22×C3×C5—21no—
69C22×C3×C5C2×D411yescharacteristic
70C3×C4×C5—21no—
71C3×C4×C5—41no—
72C3×C4×C5—41no—
73C3×C4×C5C2×D411yescharacteristic
74C3×C4×C5#4111yescharacteristic
75Dic15—41no—
76Dic15—41no—
77C2×C3×Q8C2×D511yescharacteristic
78C22×C3×C4C2×D511yescharacteristic
79C2×Dic6—51no—
80C3×C4⋊C4—21no—
81C3×C4⋊C4C2×D511yescharacteristic
82C3×C22⋊C4—21no—
83#217—101no—
84#216—102no—
85C4⋊Dic3—51no—
86Dic3⋊C4—101no—
87#208—102no—
88C2×C3⋊C8—51no—
89C2×C3⋊C8—51no—
90C23×C5#6711yescharacteristic
91C5×Q8—21no—
92C5×Q8—21no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5#6711yescharacteristic
98C2×C4×C5C22×D311yescharacteristic
99C2×Dic5—61no—
100C2×Dic5—61no—
101Dic10—61no—
102Dic10—61no—
103#168—32no—
104#168—61no—
105C2×Q16—151no—
106C2×M16—151no—
107C22⋊Q8—151no—
108C22⋊Q8D1511yescharacteristic
109#106—151no—
110#103—151no—
111#103—151no—
112C2×C3×C5—41no—
113C2×C3×C5#12111yescharacteristic
114C2×C3×C5#13711yescharacteristic
115C2×C3×C5#13711yescharacteristic
116C23×C3C22⋊D511yescharacteristic
117C3×Q8—21no—
118C3×Q8—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4C22⋊D511yescharacteristic
124C2×C3×C4C22×D511yescharacteristic
125C2×Dic3—101no—
126C2×Dic3—101no—
127Dic6—101no—
128Dic6—101no—
129C3⋊C8—52no—
130C3⋊C8—101no—
131C22×C5—21no—
132C22×C5—21no—
133C22×C5—21no—
134C22×C5C2×#6711yescharacteristic
135C4×C5—21no—
136C4×C5—41no—
137C4×C5—41no—
138C4×C5D3×D411yescharacteristic
139C4×C5#23711yescharacteristic
140Dic5—121no—
141Dic5—121no—
142C2×Q8—151no—
143C2×Q8C2×D1511yescharacteristic
144C22×C4C2×D1511yescharacteristic
145Q16—302no—
146M16—302no—
147C2×C8—151no—
148C2×C8—151no—
149C4⋊C4—21no—
150C4⋊C4—151no—
151C4⋊C4—301no—
152C4⋊C4C2×D1511yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—301no—
155C3×C5#47011yescharacteristic, complemented, Hall
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3C2×C22⋊D511yescharacteristic
160C3×C4—21no—
161C3×C4—41no—
162C3×C4—41no—
163C3×C4D4×D511yescharacteristic
164C3×C4#71111yescharacteristic
165Dic3—201no—
166Dic3—201no—
167C2×C5—41no—
168C2×C5#93911yescharacteristic
169C2×C5#94211yescharacteristic
170C2×C5#95111yescharacteristic
171C23#122011yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—301no—
175Q8—301no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—301no—
181C2×C4—301no—
182C2×C4#122011yescharacteristic
183C2×C4C22×D1511yescharacteristic
184C8—152no—
185C8—301no—
186C2×C3—41no—
187C2×C3#408511yescharacteristic
188C2×C3#408811yescharacteristic
189C2×C3#409711yescharacteristic
190C5#505211yescharacteristic, complemented, Sylow
191C22—21no—
192C22—21no—
193C22—21no—
194C22C2×#122011yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—601no—
199C4—601no—
200C4D4×D1511yescharacteristic
201C4#681911yescharacteristic
202C3#6534111yescharacteristic, complemented, Sylow
203C2—41no—
204C2#9182811yescharacteristic
205C2#9183111yescharacteristic
206C2#9184011yescharacteristic
subgroupG/H
centerC22C2×#1220
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C22×D15
FittingC3×C5×C22⋊Q8C2
socleC22×C3×C5C2×D4
prime radicalC23×C3×C5D4
hypercenterC22⋊Q8D15
hypocenterC3×C5#470
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#6718C2semidirect
5. #91119C2semidirect
6. #91094C2semidirect
7. #91092C2semidirect
155. C3×C5#470semidirect
190. C5#5052semidirect
202. C3#65341semidirect

Maximal subgroups

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

subgroupindex
C3×C5×C22⋊Q82
C2×#67182
#918132
C2×#67102
#911192
#910942
#910922
#653413
#50525