Finite Group Databasebeta

← #13161228

Subgroups — #13161228

Order 1728.

708 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 168 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×Q8×C3⋊Dic3C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722056C211yesmaximal, characteristic
6#11722052C211yesmaximal, characteristic
7#11722051C211yesmaximal, characteristic
8#10592498—31nomaximal
9#10591545—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Q8×C3⋊Dic3D311yescharacteristic
22#64099—31no—
23C3×Q8×Dic3—34no—
24#63869—34no—
25C32×#103—31no—
26#63662—31no—
27#63662—34no—
28#63662—64no—
29#63644—31no—
30#63641—31no—
31#63634—31no—
32C3×#835D314yes—
33#63576—34no—
34#63573—34no—
35Dic3⋊(C3⋊C8)—34no—
36C33×Q8—21no—
37C33×Q8Q812yes—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C32×Dic3—21no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C32×C3⋊C8—21no—
47#5022—91no—
48#4799—94no—
49C2×C32×Q8—24no—
50C2×C32×Q8C2×D314yes—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×C8—31no—
53C32×C4⋊C4—31no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)—31no—
56C4⋊(C3⋊Dic3)—34no—
57C4⋊(C3⋊Dic3)—61no—
58C4⋊(C3⋊Dic3)—64no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—61no—
61C4×C3⋊Dic3—21no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—64no—
68C3×C4⋊Dic3C2×D314yes—
69C3×C4×Dic3—34no—
70C3×C4×Dic3—64no—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—64no—
73Dic3⋊Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—22no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—41no—
83C3×C4×Q8—91no—
84Q8×Dic3—34no—
85#890—94no—
86#870—91no—
87C3×#103—34no—
88#835—94no—
89#835—91no—
90#835—184no—
91#835C3⋊D311yescharacteristic
92#817—94no—
93Dic3⋊C8—94no—
94#808—91no—
95#807—94no—
96#807—91no—
97C4⋊(C3⋊C8)—91no—
98C32×Q8—21no—
99C32×Q8—28no—
100C32×Q8—24no—
101C32×Q8—44no—
102C32×Q8Dic618yes—
103C32×Q8Dic612yes—
104C2×C32×C4—21no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4—44no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115C2×C3⋊Dic3#6711yescharacteristic
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3×Dic3—24no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—64no—
121C32×C8—61no—
122C3⋊(C3⋊C8)—61no—
123C3⋊(C3⋊C8)—64no—
124C3⋊(C3⋊C8)—124no—
125C3×C3⋊C8—24no—
126#479—271noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C3×Q8—24no—
131C2×C3×Q8D3214yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C2×C3×C8—34no—
134C3×C4⋊C4—34no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C42—91no—
138C3×C42—181no—
139C4⋊Dic3—34no—
140C4⋊Dic3—64no—
141C4⋊Dic3—94no—
142C4⋊Dic3—91no—
143C4⋊Dic3—184no—
144C4⋊Dic3—184no—
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—184no—
149C4×Dic3—34no—
150C4×Dic3—64no—
151C2×C3⋊C8—94no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—184no—
154C2×C3⋊C8C2×C3⋊D311yescharacteristic
155C22×C32—24no—
156C22×C32D3×D414yes—
157C22×C32C2×#6711yescharacteristic
158C32×C4—22no—
159C32×C4—24no—
160C32×C4—24no—
161C32×C4—28no—
162C32×C4—48no—
163C32×C4—121no—
164C32×C4C2×Dic614yes—
165C32×C4C2×Dic611yescharacteristic
166C32×C4#23511yescharacteristic
167C32×C4#23714yes—
168C3⋊Dic3—21no—
169C3⋊Dic3—21no—
170C3⋊Dic3—41no—
171C3⋊Dic3—121no—
172C3⋊Dic3—124no—
173C3⋊Dic3—244no—
174C3×Dic3—44no—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C3×Dic3—124no—
178#125—271no—
179C4×Q8—91no—
180#107—271no—
181#106—271no—
182C4⋊C8—271no—
183#103—31no—
184#103—271no—
185C33#47911yescharacteristic, complemented, Sylow
186C3×Q8—24no—
187C3×Q8—21no—
188C3×Q8—28no—
189C3×Q8—44no—
190C3×Q8#62718yes—
191C3×Q8#63412yes—
192C2×C3×C4—24no—
193C2×C3×C4—21no—
194C2×C3×C4—24no—
195C2×C3×C4—44no—
196C2×C3×C4—64no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—181no—
200C2×C3×C4C2×D3214yes—
201C2×C3×C4C22×C3⋊D311yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—34no—
204C2×Dic3—34no—
205C2×Dic3—64no—
206C2×Dic3—184no—
207C2×Dic3—181no—
208C2×Dic3—364no—
209C3×C8—64no—
210C3⋊C8—21no—
211C3⋊C8—184no—
212C3⋊C8—181no—
213C3⋊C8—364no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32#87814yes—
218C2×C32#91614yes—
219C2×C32#91814yes—
220C2×C32#92411yescharacteristic
221C2×C32#94211yescharacteristic
222C2×C32#95011yescharacteristic
223C2×Q8D3×C3⋊D311yescharacteristic
224C2×C8—31no—
225C2×C8—271no—
226C4⋊C4—31no—
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—271no—
230C4⋊C4—541no—
231C4⋊C4—541no—
232C42—91no—
233C42—181no—
234C22×C3—24no—
235C22×C3D3×#6714yes—
236C22×C3D4×C3⋊D311yescharacteristic
237C3×C4—22no—
238C3×C4—28no—
239C3×C4—24no—
240C3×C4—24no—
241C3×C4—48no—
242C3×C4—124no—
243C3×C4—181no—
244C3×C4—181no—
245C3×C4—361no—
246C3×C4#380414yes—
247C3×C4C2×#62714yes—
248C3×C4C2×#63411yescharacteristic
249C3×C4#383011yescharacteristic
250Dic3—41no—
251Dic3—64no—
252Dic3—64no—
253Dic3—124no—
254Dic3—361no—
255Dic3—364no—
256Dic3—724no—
257C32—24no—
258C32#479914yescomplemented
259C32#502211yescharacteristic, complemented
260Q8—21no—
261Q8#629312yes—
262C2×C4—21no—
263C2×C4—61no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—181no—
267C2×C4—541no—
268C2×C4C2×D3×C3⋊D311yescharacteristic
269C8—61no—
270C8—541no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3#6394714yes—
275C2×C3#6394814yes—
276C2×C3#6396414yes—
277C2×C3#6408711yescharacteristic
278C2×C3#6412511yescharacteristic
279C2×C3#6412711yescharacteristic
280C22C3⋊D3×#6711yescharacteristic
281C4—22no—
282C4—121no—
283C4—181no—
284C4—181no—
285C4—361no—
286C4—1081no—
287C4#8902811yescharacteristic
288C4C2×#629311yescharacteristic
289C3—24no—
290C3#1059154514yescomplemented
291C3#1059249811yescharacteristic, complemented
292C2#1172358311yescharacteristic
293C2#1172358611yescharacteristic
294C2#1172364111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#479
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
185. C33#479semidirect
258. C32#4799semidirect
259. C32#5022semidirect
290. C3#10591545semidirect
291. C3#10592498semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
#117234472
#117222052
C32×#8352
#117220562
#117220522
#117220512
#105924983
#105915453