Finite Group Databasebeta

← #10591488

Subgroups — #10591488

Order 576.

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

normal subgroups only

This group has 225 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×#936C211yesmaximal, characteristic
2#63918C211yesmaximal, characteristic, complemented
3#63659C211yesmaximal, characteristic
4C3×#832C211yesmaximal, characteristic
5#63566C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63554C211yesmaximal, characteristic, complemented
8#5055—31nomaximal
9#4765—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×#627—21no—
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#217—21no—
16C3×#217—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21C3×#124—31no—
22#936D311yescharacteristic
23#888—31no—
24#869—31no—
25C3×#102—31no—
26#835—31no—
27#835—31no—
28#832—31no—
29#832—31no—
30#832—61no—
31#832D311yescharacteristic
32#816—31no—
33#815—31no—
34#810—31no—
35#803—31no—
36C23×C32—21no—
37C32×D4—21no—
38C32×D4—21no—
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3D411yescharacteristic
44C2×C3×Dic3D411yescharacteristic
45C3×Dic6—21no—
46C3×Dic6—21no—
47#627—42no—
48C3⋊(C3⋊C8)—21no—
49C3×C3⋊C8—21no—
50#489—91noSylow
51C2×C3×Q8—31no—
52C2×C3×D4—21no—
53C2×C3×D4C2×D311yescharacteristic
54C2×C3×D4C2×D311yescharacteristic
55C2×Dic6—61no—
56C2×Dic6—61no—
57C2×Dic6C2×D311yescharacteristic
58C2×C3×C8—31no—
59C3×C4⋊C4—31no—
60C3×C22⋊C4—61no—
61C3×C22⋊C4—61no—
62C3×C42—31no—
63#217—21no—
64#217—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×C32—41no—
77C22×C32—41no—
78C22×C32—41no—
79C22×C32C2×D411yescharacteristic
80C32×C4#4111yescharacteristic
81C32×C4#4111yescharacteristic
82C3⋊Dic3—41no—
83C3×Dic3—21no—
84C3×Dic3—21no—
85C3×Dic3—41no—
86C3×Dic3—41no—
87C4⋊Q8—91no—
88#124—31no—
89#103—91no—
90#103—91no—
91#102—31no—
92#102—91no—
93#97—91no—
94C23×C3—21no—
95C23×C3—21no—
96C23×C3—41no—
97C3×Q8—61no—
98C3×Q8—61no—
99C3×D4—22no—
100C3×D4—21no—
101C3×D4—21no—
102C3×D4—21no—
103C3×D4—21no—
104C3×D4—41no—
105C2×C3×C4—21no—
106C2×C3×C4—31no—
107C2×C3×C4—31no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4C22×D311yescharacteristic
111C2×C3×C4C22×D311yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—61no—
115C2×Dic3—61no—
116C2×Dic3—121no—
117C2×Dic3#6711yescharacteristic
118C2×Dic3#6711yescharacteristic
119Dic6—21no—
120Dic6—21no—
121Dic6—122no—
122Dic6—122no—
123C3×C8—61no—
124C3⋊C8—21no—
125C3⋊C8—61no—
126C3⋊C8—61no—
127C3⋊C8—121no—
128C2×C32—81no—
129C2×C32#12411yescharacteristic
130C2×C32AGL(1,ℤ/8)11yescharacteristic
131C2×C32#13711yescharacteristic
132C2×Q8—31no—
133C2×Q8—181no—
134C2×D4D3211yescharacteristic
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—42no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3—81no—
152C22×C3—81no—
153C22×C3D3×D411yescharacteristic
154C22×C3C2×#6711yescharacteristic
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—61no—
158C3×C4—61no—
159C3×C4—121no—
160C3×C4—121no—
161C3×C4#23511yescharacteristic
162C3×C4#23711yescharacteristic
163C3×C4#23711yescharacteristic
164C3×C4#23711yescharacteristic
165Dic3—21no—
166Dic3—21no—
167Dic3—41no—
168Dic3—41no—
169Dic3—121no—
170Dic3—121no—
171Dic3—241no—
172C32#48911yescharacteristic, complemented, Sylow
173C23—21no—
174Q8—61no—
175Q8—61no—
176Q8—362no—
177D4—21no—
178D4—21no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—181no—
184C2×C4C2×D3211yescharacteristic
185C8—61no—
186C8—181no—
187C2×C3—21no—
188C2×C3—21no—
189C2×C3—21no—
190C2×C3—82no—
191C2×C3—81no—
192C2×C3—81no—
193C2×C3#88511yescharacteristic
194C2×C3#91111yescharacteristic
195C2×C3#91511yescharacteristic
196C2×C3#93611yescharacteristic
197C2×C3#94911yescharacteristic
198C2×C3#95111yescharacteristic
199C22—41no—
200C22—41no—
201C22—41no—
202C22D3×#6711yescharacteristic
203C4—61no—
204C4—61no—
205C4—121no—
206C4—121no—
207C4—361no—
208C4#380411yescharacteristic
209C4#380511yescharacteristic
210C3—21no—
211C3#476511yescharacteristic, complemented
212C3#505511yescharacteristic, complemented
213C2—81no—
214C2#6392511yescharacteristic
215C2#6392911yescharacteristic
216C2#6395911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
Fitting, prime radicalC2×C32×D4C22
socleC22×C32C2×D4
hypercenterC2×D4D32
hypocenterC32#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #63918C2semidirect
5. #63566C2semidirect
6. #63565C2semidirect
7. #63554C2semidirect
172. C32#489semidirect
211. C3#4765semidirect
212. C3#5055semidirect

Maximal subgroups

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

subgroupindex
C3×#9362
#639182
#636592
C3×#8322
#635662
#635652
#635542
#50553
#47653