Finite Group Databasebeta

← #13159099

Subgroups — #13159099

Order 1728.

792 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 160 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×#64097C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic
5#11722055C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592458—31nomaximal
9#10590463—34nomaximal
10C2×C3×#634—21no—
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21D4×C3⋊Dic3—31no—
22#64097D311yescharacteristic, complemented
23C3×#888—34no—
24Dic3×D12—34no—
25C32×#102—31no—
26#63659—31no—
27#63645—31no—
28#63641—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#809D314yes—
33#63575—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63562—34no—
36C22×C32×D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×#634—42no—
44C32×D12—21no—
45C32×D12Q812yescomplemented
46D3×C3⋊Dic3—42no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5087—91no—
50#4759—94no—
51C2×C32×D4—31no—
52C22×C3⋊Dic3—61no—
53C2×#634—21no—
54C2×C3×D12C2×D314yes—
55C2×C3×Dic6—64no—
56C2×D3×Dic3—64no—
57C2×C32×C8—31no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4C2×D314yes—
60C32×C4⋊C4C2×D311yescharacteristic
61#3757—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3—34no—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8C2×D314yes—
75(C2×D3)⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—22no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4—41no—
81C33×C4C2×Q811yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3D812yes—
87C3×C4⋊Q8—91no—
88D4×Dic3—94no—
89#888—34no—
90C4×D12—91no—
91C3×#102—34no—
92#832—94no—
93#818—94no—
94Dic3⋊C8—94no—
95#809—91no—
96#809C3⋊D311yescharacteristic
97#807—94no—
98#807—91no—
99#807—91no—
100#807—184no—
101C4⋊(C3⋊C8)—91no—
102C23×C32—61no—
103C22×C3×D3—24no—
104C32×D4—32no—
105C32×D4—61no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—44no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6711yescharacteristic
119#634—42no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—64no—
123C3×D12—24no—
124C3×D12Dic618yescomplemented
125C3×Dic6—128no—
126D3×Dic3—128no—
127C32×C8—61no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—24no—
132#474—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×D811yescharacteristic
135C2×C33#13711yescharacteristic
136C32×D3—42no—
137C2×C3×Q8—181no—
138C2×C3×D4—34no—
139C22×Dic3—184no—
140C2×D12C2×C3⋊D311yescharacteristic
141C2×C4×D3—181no—
142C2×Dic6—64no—
143C2×C3×C8—34no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—181no—
147C3×C4⋊C4D3214yes—
148C3×C4⋊C4C2×C3⋊D311yescharacteristic
149C3×C42—91no—
150#217—184no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—34no—
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156Dic3⋊C4—64no—
157C4×Dic3—34no—
158C2×C3⋊C8—94no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—184no—
161C2×C3⋊C8C2×C3⋊D311yescharacteristic
162C22×C32—24no—
163C22×C32—62no—
164C22×C32—121no—
165C22×C32—121no—
166C22×C32D3×D414yes—
167C22×C32C2×#6711yescharacteristic
168C2×C3×D3—28no—
169C2×C3×D3—44no—
170C2×C3×D3—44no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—41no—
174C32×C4—44no—
175C32×C4—84no—
176C32×C4C2×Dic614yes—
177C32×C4#23714yes—
178C32×C4D3×Q811yescharacteristic
179C32×C4#23911yescharacteristic
180C3⋊Dic3—21no—
181C3⋊Dic3—41no—
182C3⋊Dic3—121no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3⋊Dic3#21312yes—
186C3×Dic3—38no—
187C3×Dic3—64no—
188C3×Dic3—124no—
189C4⋊Q8—91no—
190C4×D4—271no—
191#107—271no—
192#107—271no—
193C4⋊C8—271no—
194#102—31no—
195#102—271no—
196C33#47411yescharacteristic, complemented, Sylow
197C23×C3—64no—
198C22×D3—21no—
199C3×Q8—362no—
200C3×D4—38no—
201C3×D4—64no—
202C2×C3×C4—24no—
203C2×C3×C4—21no—
204C2×C3×C4—24no—
205C2×C3×C4—44no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209C2×C3×C4C2×D3214yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—34no—
212C2×Dic3—34no—
213C2×Dic3—64no—
214C2×Dic3—184no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217C2×Dic3—368no—
218D12—21no—
219D12#63412yescomplemented
220C4×D3—362no—
221Dic6—128no—
222C3×C8—64no—
223C3⋊C8—21no—
224C3⋊C8—188no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—122no—
231C2×C32#87814yes—
232C2×C32D3×D814yes—
233C2×C32#91514yes—
234C2×C32C2×#21311yescharacteristic
235C2×C32#94211yescharacteristic
236C2×C32#94611yescharacteristic
237C3×D3—48no—
238C2×Q8—181no—
239C2×D4—31no—
240C22×C4—541no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C4⋊C4—271no—
246C4⋊C4D3×C3⋊D311yescharacteristic
247C22⋊C4—541no—
248C42—91no—
249C22×C3—24no—
250C22×C3—68no—
251C22×C3—124no—
252C22×C3—124no—
253C22×C3D3×#6714yes—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—22no—
256C2×D3—41no—
257C2×D3—41no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—41no—
261C3×C4—44no—
262C3×C4—84no—
263C3×C4—92no—
264C3×C4—181no—
265C3×C4—361no—
266C3×C4D3×Dic614yes—
267C3×C4#379614yes—
268C3×C4C2×#63411yescharacteristic
269C3×C4#383011yescharacteristic
270Dic3—38no—
271Dic3—64no—
272Dic3—124no—
273Dic3—361no—
274Dic3—364no—
275Dic3—724no—
276C32—24no—
277C32#475914yescomplemented
278C32#508711yescharacteristic, complemented
279C23—61no—
280Q8—362no—
281D4—32no—
282D4—61no—
283C2×C4—21no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—61no—
291C8—541no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—128no—
296C2×C3#6388714yes—
297C2×C3D3×#21314yes—
298C2×C3#6394614yes—
299C2×C3#6408711yescharacteristic
300C2×C3D8×C3⋊D311yescharacteristic
301C2×C3#6412411yescharacteristic
302D3—42no—
303C22—62no—
304C22—121no—
305C22—121no—
306C22C3⋊D3×#6711yescharacteristic
307C4—41no—
308C4—92no—
309C4—181no—
310C4—361no—
311C4—1081no—
312C4D3×#63411yescharacteristic
313C4#8901511yescharacteristic
314C3—24no—
315C3#1059046314yescomplemented
316C3#1059245811yescharacteristic, complemented
317C2—122no—
318C2#1172348711yescharacteristic
319C2C3⋊D3×#21311yescharacteristic
320C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64097C2semidirect
3. #11722177C2semidirect
5. #11722055C2semidirect
6. #11722051C2semidirect
22. #64097D3semidirect
45. C32×D12Q8semidirect
124. C3×D12Dic6semidirect
196. C33#474semidirect
219. D12#634semidirect
277. C32#4759semidirect
278. C32#5087semidirect
315. C3#10590463semidirect
316. C3#10592458semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
D12×C3⋊Dic32
#117221772
C32×#8092
#117220552
#117220512
#117220312
#105924583
#105904633