Finite Group Databasebeta

← #13159088

Subgroups — #13159088

Order 1728.

720 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64106C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic
8#10592587—31nomaximal
9#10590448—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21#64106D311yescharacteristic
22#64101—31no—
23C3×#897—34no—
24#63841—34no—
25C32×#107—31no—
26#63647—31no—
27#63643—31no—
28#63637—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32C3×#807D314yescomplemented
33#63566—34no—
34#63565—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3—21no—
43C3×C4×C3⋊D3—21no—
44C3×#634—21no—
45C3×#634—21no—
46C2×C32×Dic3—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5045—91no—
50#4888—94no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×#634C2×D311yescharacteristic
53C2×C3×C4×D3—34no—
54C2×C3×Dic6—34no—
55C2×C32×C8—31no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—31no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60#3752—21no—
61#3752—61no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66C4×C3⋊Dic3—61no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C3×(C2×D3)⋊C4—64no—
71C3×C4⋊Dic3—64no—
72C3×C4⋊Dic3C2×D314yes—
73C2×C3×C3⋊C8C2×D314yes—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87C3×C22⋊Q8—91no—
88#926—91no—
89#897—34no—
90#892—94no—
91C3×#107—34no—
92#835—91no—
93#835—91no—
94#830—91no—
95(C2×D3)⋊C8—94no—
96#816—94no—
97#810—94no—
98#808—94no—
99#808—91no—
100#808—184no—
101#807C3⋊D311yescharacteristic, complemented
102C22×C3⋊D3#6711yescharacteristic
103C22×C3×D3—34no—
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
116C4×C3⋊D3—21no—
117C4×C3⋊D3—21no—
118#634—21no—
119#634—21no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—64no—
123C3×C4×D3—64no—
124C3×C4×D3—64no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127C32×C8—61no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—61no—
130C3⋊(C3⋊C8)—64no—
131C3×C3⋊C8—24no—
132#485—271noSylow
133C2×C33#12311yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—41no—
137C2×C3×Q8—91no—
138C22×C3×C4—91no—
139C2×C4×D3—34no—
140C2×Dic6—34no—
141C2×C3×C8—34no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4D3214yes—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147C3×C22⋊C4—181no—
148#217—181no—
149(C2×D3)⋊C4—64no—
150(C2×D3)⋊C4—184no—
151C4⋊Dic3—64no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156C4×Dic3—181no—
157C4×Dic3—184no—
158C2×C3⋊C8—94no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—184no—
161C2×C3⋊C8C2×C3⋊D311yescharacteristic
162C22×C32—24no—
163C22×C32D3×D414yes—
164C22×C32C2×#6711yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—41no—
174C32×C4—44no—
175C32×C4—84no—
176C32×C4—121no—
177C32×C4#23514yes—
178C32×C4#23711yescharacteristic
179C32×C4#23711yescharacteristic
180C32×C4#23914yes—
181C3⋊Dic3—21no—
182C3⋊Dic3—41no—
183C3⋊Dic3—121no—
184C3⋊Dic3—124no—
185C3⋊Dic3—244no—
186C3×Dic3—44no—
187C3×Dic3—64no—
188C3×Dic3—124no—
189C22⋊Q8—91no—
190#117—271no—
191#107—31no—
192#106—271no—
193#103—271no—
194#103—271no—
195C22⋊C8—271no—
196C33#48511yescharacteristic, complemented, Sylow
197C23×C3—91no—
198C22×D3—34no—
199C3×Q8—181no—
200C3×Q8—181no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—44no—
205C2×C3×C4—64no—
206C2×C3×C4—91no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—34no—
214C2×Dic3—64no—
215C2×Dic3—184no—
216C2×Dic3—181no—
217C2×Dic3—364no—
218C4×D3—64no—
219C4×D3—64no—
220Dic6—64no—
221Dic6—64no—
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#89414yes—
231C2×C32#91214yes—
232C2×C32#91814yes—
233C2×C32#93511yescharacteristic
234C2×C32#94211yescharacteristic
235C2×C32#95011yescharacteristic
236C3⋊D3—41no—
237C3×D3—124no—
238C2×Q8—91no—
239C22×C4—91no—
240C2×C8—31no—
241C2×C8—271no—
242C4⋊C4—31no—
243C4⋊C4—181no—
244C4⋊C4—271no—
245C4⋊C4D3×C3⋊D311yescharacteristic
246C22⋊C4—181no—
247C22⋊C4—541no—
248C42—541no—
249C22×C3—24no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3D3×#6714yes—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—64no—
256C2×D3—64no—
257C2×D3—64no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—41no—
261C3×C4—44no—
262C3×C4—84no—
263C3×C4—124no—
264C3×C4—181no—
265C3×C4—361no—
266C3×C4#379514yes—
267C3×C4#379614yes—
268C3×C4#382811yescharacteristic
269C3×C4#383211yescharacteristic
270Dic3—41no—
271Dic3—64no—
272Dic3—124no—
273Dic3—361no—
274Dic3—364no—
275Dic3—724no—
276C32—24no—
277C32#488814yescomplemented
278C32#504511yescharacteristic, complemented
279C23—91no—
280Q8—181no—
281Q8—181no—
282C2×C4—21no—
283C2×C4—61no—
284C2×C4—91no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—541no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—61no—
291C8—541no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—361no—
296C2×C3#6388414yes—
297C2×C3#6393314yes—
298C2×C3#6394814yes—
299C2×C3#6410311yescharacteristic
300C2×C3#6412111yescharacteristic
301C2×C3#6412711yescharacteristic
302D3—124no—
303C22—181no—
304C22—181no—
305C22—181no—
306C22C3⋊D3×#6711yescharacteristic
307C4—41no—
308C4—121no—
309C4—181no—
310C4—361no—
311C4—1081no—
312C4#8901711yescharacteristic
313C4#8902111yescharacteristic
314C3—24no—
315C3#1059044814yescomplemented
316C3#1059258711yescharacteristic, complemented
317C2—361no—
318C2#1172348811yescharacteristic
319C2#1172355911yescharacteristic
320C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#485
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
3. #11722178C2semidirect
4. C32×#807C2semidirect
5. #11722040C2semidirect
6. #11722039C2semidirect
32. C3×#807D3semidirect
101. #807C3⋊D3semidirect
196. C33#485semidirect
277. C32#4888semidirect
278. C32#5045semidirect
315. C3#10590448semidirect
316. C3#10592587semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
#117234042
#117221782
C32×#8072
#117220402
#117220392
#117220232
#105925873
#105904483