Finite Group Databasebeta

← #10591173

Subgroups — #10591173

Order 576.

341 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 72 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×C3×#235C211yesmaximal, characteristic
2#64133C211yesmaximal, characteristic, complemented
3C2×#3719C211yesmaximal, characteristic, complemented
4C2×#3716C211yesmaximal, characteristic, complemented
5C3×#830C211yesmaximal, characteristic, complemented
6#63566C211yesmaximal, characteristic, complemented
7#63563C211yesmaximal, characteristic, complemented
8#5239—31nomaximal
9#4733—31nomaximal
10C22×C32×C4C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×#67—21no—
13C3×#235—22no—
14C3×#235—22no—
15C2×C3×D12C2211yescharacteristic
16C2×C3×C4×D3—21no—
17C2×C3×Dic6C2211yescharacteristic
18#3757—21no—
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20#3750—21no—
21C2×C3×C3⋊C8C2211yescharacteristic
22C2×C3×C3⋊C8C2211yescharacteristic
23#3719—22no—
24#3716—22no—
25C2×C3×#41—31no—
26C2×#235D311yescharacteristic
27C2×#216—31no—
28C2×#214—31no—
29#924—31no—
30#924—31no—
31#924—61no—
32C2×Dic12—31no—
33C2×#204—31no—
34C3×C22⋊C8—31no—
35#835—31no—
36#832—31no—
37#830D311yescharacteristic, complemented
38#821—31no—
39#816—31no—
40C23×C32D411yescharacteristic
41C22×C3×D3—21no—
42C2×C32×C4—21no—
43C2×C32×C4—21no—
44C2×C32×C4D411yescharacteristic
45C2×C32×C4C2311yescharacteristic
46C2×C3⋊Dic3—21no—
47C2×C3⋊Dic3—21no—
48#634—21no—
49#634—21no—
50C3×#67—24no—
51C2×C3×Dic3—21no—
52C3×D12—21no—
53C3×D12D412yes—
54C3×C4×D3—24no—
55C3×Dic6—21no—
56C3×Dic6D412yes—
57C3×C3⋊C8—21no—
58C3×C3⋊C8—21no—
59#452—91noSylow
60C3×#41—62no—
61C3×#41—62no—
62C2×C3×Q8—31no—
63C2×C3×D4—31no—
64C2×C3×D4—61no—
65C22×C3×C4—21no—
66C22×C3×C4—61no—
67C22×C3×C4C2×D311yescharacteristic
68C22×C3×C4C2×D311yescharacteristic
69C2×#67—21no—
70#235—22no—
71#235—22no—
72C2×D12C2×D311yescharacteristic
73C2×C4×D3—21no—
74C2×Dic6—31no—
75C2×Dic6—31no—
76C2×Dic6—61no—
77C2×Dic6C2×D311yescharacteristic
78C2×C3×C8—31no—
79C2×C3×C8—31no—
80#217—62no—
81#217—61no—
82#217—61no—
83#216—62no—
84#214—62no—
85C4⋊Dic3—31no—
86C4⋊Dic3—31no—
87C4⋊Dic3—61no—
88Dic3⋊C4—62no—
89Dic3⋊C4—61no—
90Dic3⋊C4—61no—
91C2×C3⋊C8C2×D311yescharacteristic
92C2×C3⋊C8C2×D311yescharacteristic
93Dic12—62no—
94#204—62no—
95C22×C32—21no—
96C22×C32—21no—
97C22×C32—21no—
98C22×C32C2×D411yescharacteristic
99C2×C3×D3—22no—
100C2×C3×D3—41no—
101C2×C3×D3—41no—
102C32×C4—21no—
103C32×C4C2×D411yescharacteristic
104C32×C4C2×D411yescharacteristic
105C3⋊Dic3—41no—
106C3⋊Dic3—41no—
107C3×Dic3—22no—
108C2×#41—31no—
109C2×Q16—91no—
110C2×SD16—91no—
111C22⋊Q8—91no—
112#103—91no—
113#102—91no—
114C22⋊C8—31no—
115C23×C3—21no—
116C23×C3—61no—
117C23×C3D1211yescharacteristic
118C23×C3#6711yescharacteristic
119C22×D3—21no—
120C3×Q8—32no—
121C3×Q8—61no—
122C3×D4—32no—
123C3×D4—64no—
124C3×D4—61no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—64no—
134C2×C3×C4—61no—
135C2×C3×C4D1211yescharacteristic
136C2×C3×C4#6711yescharacteristic
137C2×C3×C4C22×D311yescharacteristic
138C2×C3×C4C22×D311yescharacteristic
139#67—24no—
140C2×Dic3—21no—
141C2×Dic3—62no—
142C2×Dic3—62no—
143C2×Dic3—61no—
144C2×Dic3—61no—
145C2×Dic3—61no—
146C2×Dic3—61no—
147D12—21no—
148D12#6712yes—
149C4×D3—24no—
150Dic6—21no—
151Dic6—61no—
152Dic6—61no—
153Dic6—61no—
154Dic6—61no—
155Dic6—121no—
156Dic6—121no—
157Dic6#6712yes—
158C3×C8—61no—
159C3×C8—61no—
160C3⋊C8—21no—
161C3⋊C8—21no—
162C2×C32—41no—
163C2×C32#12011yescharacteristic
164C2×C32#13511yescharacteristic
165C2×C32#13711yescharacteristic
166C3×D3—42no—
167#41—62no—
168#41—62no—
169C2×Q8—31no—
170C2×Q8—91no—
171C2×D4—31no—
172C2×D4—61no—
173C22×C4—61no—
174C22×C4D3211yescharacteristic
175Q16—182no—
176SD16—182no—
177C2×C8—31no—
178C2×C8—31no—
179C4⋊C4—91no—
180C4⋊C4—181no—
181C22⋊C4—181no—
182C22×C3—22no—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—62no—
193C22×C3—121no—
194C22×C3—121no—
195C22×C3C2×D1211yescharacteristic
196C22×C3C2×#6711yescharacteristic
197C2×D3—22no—
198C2×D3—41no—
199C2×D3—41no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—41no—
205C3×C4—62no—
206C3×C4D3×D411yescharacteristic
207C3×C4D3×D411yescharacteristic
208C3×C4C2×#6711yescharacteristic
209C3×C4C2×#6711yescharacteristic
210Dic3—22no—
211Dic3—122no—
212Dic3—122no—
213Dic3—121no—
214Dic3—121no—
215Dic3—121no—
216Dic3—121no—
217C32#45211yescharacteristic, complemented, Sylow
218C23—61no—
219C23#62611yescharacteristic
220Q8—32no—
221Q8—61no—
222Q8—181no—
223Q8—181no—
224D4—32no—
225D4—64no—
226D4—61no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—64no—
230C2×C4—61no—
231C2×C4—181no—
232C2×C4—181no—
233C2×C4#62611yescharacteristic
234C2×C4C2×D3211yescharacteristic
235C8—61no—
236C8—61no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—42no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—122no—
244C2×C3#88211yescharacteristic
245C2×C3#90411yescharacteristic
246C2×C3#90911yescharacteristic
247C2×C3#95011yescharacteristic
248C2×C3#95111yescharacteristic
249C2×C3#95311yescharacteristic
250D3—42no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—62no—
255C22—121no—
256C22—121no—
257C22C2×#62611yescharacteristic
258C4—21no—
259C4—62no—
260C4—361no—
261C4—361no—
262C4D3×#6711yescharacteristic
263C4D3×#6711yescharacteristic
264C3—21no—
265C3#473311yescharacteristic, complemented
266C3#523911yescharacteristic, complemented
267C2—41no—
268C2—122no—
269C2#6382911yescharacteristic
270C2#6383111yescharacteristic
271C2#6397711yescharacteristic
subgroupG/H
centerC22C2×#626
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC22×C32×C4C22
socleC22×C32C2×D4
prime radicalC2×C3×#235C2
hypercenterC22×C4D32
hypocenterC32#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #64133C2semidirect
3. C2×#3719C2semidirect
4. C2×#3716C2semidirect
5. C3×#830C2semidirect
6. #63566C2semidirect
7. #63563C2semidirect
37. #830D3semidirect
217. C32#452semidirect
265. C3#4733semidirect
266. C3#5239semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
#641332
C2×#37192
C2×#37162
C3×#8302
#635662
#635632
#52393
#47333