Finite Group Databasebeta

← #10591624

Subgroups — #10591624

Order 576.

221 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 60 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×#947C211yesmaximal, characteristic
2#63918C211yesmaximal, characteristic, complemented
3#63662C211yesmaximal, characteristic, complemented
4C3×#835C211yesmaximal, characteristic, complemented
5#63563C211yesmaximal, characteristic
6#63562C211yesmaximal, characteristic
7#63554C211yesmaximal, characteristic, complemented
8#5055—31nomaximal
9#4794—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×D12C2211yescharacteristic
12C2×#627—21no—
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×(C2×D3)⋊C4—21no—
16C3×(C2×D3)⋊C4—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21C3×#124—31no—
22#947D311yescharacteristic
23#888—31no—
24#869—31no—
25C3×#103—31no—
26#835—31no—
27#835—31no—
28#835—61no—
29#835D311yescharacteristic, complemented
30#832—31no—
31#832—31no—
32#821—31no—
33#815—31no—
34#809—31no—
35#803—31no—
36C22×C3×D3—21no—
37C32×Q8—21no—
38C32×Q8—21no—
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3D411yescharacteristic
44C2×C3×Dic3D411yescharacteristic
45C3×D12—21no—
46C3×D12—21no—
47#627—42no—
48C3⋊(C3⋊C8)—21no—
49C3×C3⋊C8—21no—
50#489—91noSylow
51C2×C3×Q8—21no—
52C2×C3×Q8C2×D311yescharacteristic
53C2×C3×Q8C2×D311yescharacteristic
54C2×C3×D4—31no—
55C2×D12C2×D311yescharacteristic
56C2×Dic6—61no—
57C2×Dic6—61no—
58C2×C3×C8—31no—
59C3×C4⋊C4—31no—
60C3×C22⋊C4—61no—
61C3×C22⋊C4—61no—
62C3×C42—31no—
63(C2×D3)⋊C4—21no—
64(C2×D3)⋊C4—21no—
65C4⋊Dic3—31no—
66C4⋊Dic3—31no—
67C4⋊Dic3—61no—
68C4⋊Dic3—61no—
69C4⋊Dic3C2×D311yescharacteristic
70Dic3⋊C4—61no—
71C4×Dic3C2×D311yescharacteristic
72C2×C3⋊C8—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8C2×D311yescharacteristic
76C22×C32C2×D411yescharacteristic
77C2×C3×D3—41no—
78C2×C3×D3—41no—
79C2×C3×D3—41no—
80C32×C4—41no—
81C32×C4#4111yescharacteristic
82C32×C4#4111yescharacteristic
83C3⋊Dic3—41no—
84C3×Dic3—21no—
85C3×Dic3—21no—
86C3×Dic3—41no—
87C4⋊Q8—91no—
88#124—31no—
89#103—31no—
90#103—91no—
91#102—91no—
92#102—91no—
93#97—91no—
94C23×C3—61no—
95C22×D3—21no—
96C3×Q8—22no—
97C3×Q8—21no—
98C3×Q8—21no—
99C3×Q8—21no—
100C3×Q8—21no—
101C3×Q8—41no—
102C3×D4—61no—
103C3×D4—61no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—41no—
110C2×C3×C4—61no—
111C2×C3×C4C22×D311yescharacteristic
112C2×C3×C4C22×D311yescharacteristic
113C2×Dic3—21no—
114C2×Dic3—61no—
115C2×Dic3—61no—
116C2×Dic3—121no—
117C2×Dic3#6711yescharacteristic
118C2×Dic3#6711yescharacteristic
119D12—21no—
120D12—21no—
121Dic6—122no—
122Dic6—122no—
123C3×C8—61no—
124C3⋊C8—21no—
125C3⋊C8—61no—
126C3⋊C8—61no—
127C3⋊C8—121no—
128C2×C32#12411yescharacteristic
129C2×C32AGL(1,ℤ/8)11yescharacteristic
130C2×C32#13711yescharacteristic
131C3×D3—81no—
132C2×Q8—181no—
133C2×Q8D3211yescharacteristic
134C2×D4—31no—
135C2×C8—31no—
136C2×C8—91no—
137C4⋊C4—31no—
138C4⋊C4—91no—
139C4⋊C4—181no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C42—31no—
143C22×C3—21no—
144C22×C3—121no—
145C22×C3—121no—
146C22×C3—121no—
147C22×C3D3×D411yescharacteristic
148C22×C3C2×#6711yescharacteristic
149C2×D3—41no—
150C2×D3—41no—
151C2×D3—41no—
152C3×C4—21no—
153C3×C4—21no—
154C3×C4—42no—
155C3×C4—41no—
156C3×C4—41no—
157C3×C4—61no—
158C3×C4—61no—
159C3×C4—121no—
160C3×C4#23511yescharacteristic
161C3×C4#23711yescharacteristic
162C3×C4#23711yescharacteristic
163C3×C4#23711yescharacteristic
164Dic3—21no—
165Dic3—21no—
166Dic3—41no—
167Dic3—121no—
168Dic3—121no—
169Dic3—241no—
170C32#48911yescharacteristic, complemented, Sylow
171C23—61no—
172Q8—21no—
173Q8—21no—
174Q8—362no—
175D4—61no—
176D4—61no—
177C2×C4—21no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—61no—
181C2×C4—181no—
182C2×C4C2×D3211yescharacteristic
183C8—61no—
184C8—181no—
185C2×C3—21no—
186C2×C3—21no—
187C2×C3—21no—
188C2×C3—241no—
189C2×C3#88511yescharacteristic
190C2×C3#91411yescharacteristic
191C2×C3#91811yescharacteristic
192C2×C3#93611yescharacteristic
193C2×C3#94911yescharacteristic
194C2×C3#95111yescharacteristic
195D3—81no—
196C22—121no—
197C22—121no—
198C22—121no—
199C22D3×#6711yescharacteristic
200C4—41no—
201C4—61no—
202C4—61no—
203C4—121no—
204C4—361no—
205C4#380411yescharacteristic
206C4#380511yescharacteristic
207C3—21no—
208C3#479411yescharacteristic, complemented
209C3#505511yescharacteristic, complemented
210C2—241no—
211C2#6393911yescharacteristic
212C2#6394311yescharacteristic
213C2#6395911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
hypercenterC2×Q8D32
hypocenterC32#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #63918C2semidirect
3. #63662C2semidirect
4. C3×#835C2semidirect
7. #63554C2semidirect
29. #835D3semidirect
170. C32#489semidirect
208. C3#4794semidirect
209. C3#5055semidirect

Maximal subgroups

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

subgroupindex
C3×#9472
#639182
#636622
C3×#8352
#635632
#635622
#635542
#50553
#47943