Finite Group Databasebeta

← #13159342

Subgroups — #13159342

Order 1728.

852 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C32×#892C211yesmaximal, characteristic
2D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×#88786C211yesmaximal, characteristic
4#11722178C211yesmaximal, characteristic, complemented
5C3×#63635C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722056C211yesmaximal, characteristic, complemented
8#10592694—31nomaximal
9#10590437—34nomaximal
10C2×C32×C4×D3C2211yescharacteristic
11C2×D3×C3⋊Dic3—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88786C414yescomplemented
22C32×#117—31no—
23C2×C4⋊(C3⋊Dic3)—31no—
24C2×#3748—31no—
25C3×#892D314yes—
26D3×C4⋊Dic3—34no—
27C2×#3711—34no—
28#63635—31no—
29#63635—34no—
30#63635—64no—
31#63635D311yescharacteristic, complemented
32#63634—31no—
33#63634—31no—
34C3×#808—34no—
35#63576—34no—
36#63575—34no—
37C22×C32×D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C32×C4×D3C2×C412yes—
45C32×C4×D3Q812yes—
46D3×C3⋊Dic3—42no—
47C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
48C3×C3⋊(C3⋊C8)C2×C412yescomplemented
49#4995—94no—
50#4861—91no—
51C22×C32×C4—31no—
52C22×C3⋊Dic3—61no—
53C2×C3×C4×D3C2×D314yes—
54C2×D3×Dic3—64no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—31no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C22⋊C4—61no—
60C32×C42—61no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—61no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3748—34no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—24no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—24no—
75C2×C3×C3⋊C8—34no—
76Dic3⋊Dic3—64no—
77#3711—316no—
78C22×C33C2×D411yescharacteristic
79C2×C32×D3—21no—
80C2×C32×D3—21no—
81C2×C32×D3C4⋊C412yes—
82C33×C4—41no—
83C33×C4C22×C411yescharacteristic
84C33×C4C2×Q811yescharacteristic
85C3⋊(C3⋊Dic3)—41no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88C32×Dic3C4⋊C412yes—
89C3×#117—34no—
90C2×C4⋊Dic3—94no—
91C2×#208—94no—
92C2×ΓL(1,𝔽25)—91no—
93#892C3⋊D311yescharacteristic
94D3×C4⋊C4—91no—
95C3×#106—91no—
96#818—91no—
97#808—34no—
98#808—94no—
99#808—91no—
100#808—184no—
101#807—94no—
102#807—94no—
103#807—91no—
104C23×C32—31no—
105C22×C3×D3#6714yes—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—31no—
110C2×C32×C4—32no—
111C2×C32×C4—32no—
112C2×C32×C4—44no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3—122no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3#6714yes—
124C3×C4×D3Dic618yes—
125C3×C4×D3C4×D318yes—
126D3×Dic3—128no—
127C3⋊(C3⋊C8)—32no—
128C3⋊(C3⋊C8)—38no—
129C3⋊(C3⋊C8)—68no—
130C3⋊(C3⋊C8)C4×D312yescomplemented
131C3×C3⋊C8—38no—
132#428—271noSylow
133C2×C33C2×C4⋊C411yescharacteristic
134C2×C33AGL(1,ℤ/8)11yescharacteristic
135C2×C33#13711yescharacteristic
136C32×D3—22no—
137C22×C3×C4—34no—
138C22×Dic3—184no—
139C2×C4×D3—181no—
140C2×C4×D3C2×C3⋊D311yescharacteristic
141C2×C3×C8—91no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4D3214yes—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147C3×C22⋊C4—64no—
148C3×C42—64no—
149(C2×D3)⋊C4—21no—
150C4⋊Dic3—34no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—184no—
154C4⋊Dic3—184no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—181no—
157C4×Dic3—21no—
158#208—916no—
159C2×C3⋊C8—34no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163ΓL(1,𝔽25)—94no—
164C22×C32—24no—
165C22×C32—32no—
166C22×C32—61no—
167C22×C32—61no—
168C22×C32D3×D411yescharacteristic
169C22×C32C2×#6714yes—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C2×C3×D3Dic3⋊C418yes—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—32no—
176C32×C4—41no—
177C32×C4—44no—
178C32×C4—84no—
179C32×C4—121no—
180C32×C4C2×Dic614yes—
181C32×C4C2×C4×D314yes—
182C32×C4C2×C4×D311yescharacteristic
183C32×C4D3×Q811yescharacteristic
184C3⋊Dic3—41no—
185C3⋊Dic3—121no—
186C3⋊Dic3—124no—
187C3⋊Dic3—244no—
188C3×Dic3—44no—
189C3×Dic3—124no—
190C3×Dic3Dic3⋊C418yes—
191C2×M16—271no—
192#117—31no—
193C2×C4⋊C4—271no—
194#107—271no—
195#107—271no—
196#106—91no—
197#106—271no—
198C33#42811yescharacteristic, complemented, Sylow
199C23×C3—34no—
200C22×D3#63811yescharacteristic
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—34no—
205C2×C3×C4—38no—
206C2×C3×C4—38no—
207C2×C3×C4—44no—
208C2×C3×C4—64no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—64no—
214C2×Dic3—184no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217C2×Dic3—368no—
218C2×Dic3#63811yescharacteristic
219C4×D3—362no—
220C4×D3#63412yes—
221C4×D3C4×C3⋊D312yes—
222C3×C8—92no—
223C3⋊C8—38no—
224C3⋊C8—98no—
225C3⋊C8—92no—
226C3⋊C8—188no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—62no—
231C2×C32D3×C4⋊C411yescharacteristic
232C2×C32#91411yescharacteristic
233C2×C32#91511yescharacteristic
234C2×C32C2×Dic3⋊C414yes—
235C2×C32#94914yes—
236C2×C32#95114yes—
237C3×D3—28no—
238C22×C4—31no—
239C22×C4—541no—
240M16—274no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—91no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C4⋊C4D3×C3⋊D311yescharacteristic
248C22⋊C4—61no—
249C42—61no—
250C22×C3—24no—
251C22×C3—38no—
252C22×C3—64no—
253C22×C3—64no—
254C22×C3D3×#6714yes—
255C22×C3C2×#63811yescharacteristic
256C2×D3—21no—
257C2×D3—21no—
258C2×D3#375012yes—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—38no—
262C3×C4—41no—
263C3×C4—44no—
264C3×C4—84no—
265C3×C4—124no—
266C3×C4—361no—
267C3×C4D3×Dic614yes—
268C3×C4C4×D3214yes—
269C3×C4C2×#63411yescharacteristic
270C3×C4C2×C4×C3⋊D311yescharacteristic
271Dic3—41no—
272Dic3—124no—
273Dic3—361no—
274Dic3—364no—
275Dic3—724no—
276Dic3#375012yes—
277C32—24no—
278C32#486111yescharacteristic, complemented
279C32#499514yescomplemented
280C23—31no—
281C2×C4—21no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—32no—
285C2×C4—61no—
286C2×C4—181no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—92no—
291C8—272no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—68no—
296C2×C3D3×Dic3⋊C414yes—
297C2×C3#6392914yes—
298C2×C3#6393914yes—
299C2×C3C2×#375011yescharacteristic
300C2×C3#6415811yescharacteristic
301C2×C3#6416011yescharacteristic
302D3—22no—
303C22—32no—
304C22—61no—
305C22—61no—
306C22D3×#63811yescharacteristic
307C4—32no—
308C4—41no—
309C4—121no—
310C4—361no—
311C4—1081no—
312C4D3×#63411yescharacteristic
313C4C4×D3×C3⋊D311yescharacteristic
314C3—24no—
315C3#1059043714yescomplemented
316C3#1059269411yescharacteristic, complemented
317C2—62no—
318C2D3×#375011yescharacteristic
319C2#1172355011yescharacteristic
320C2#1172356811yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#428
LCSG ▷ 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
4. #11722178C2semidirect
5. C3×#63635C2semidirect
6. #11722057C2semidirect
7. #11722056C2semidirect
21. #88786C4semidirect
31. #63635D3semidirect
47. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
48. C3×C3⋊(C3⋊C8)C2×C4semidirect
130. C3⋊(C3⋊C8)C4×D3semidirect
198. C33#428semidirect
278. C32#4861semidirect
279. C32#4995semidirect
315. C3#10590437semidirect
316. C3#10592694semidirect

Maximal subgroups

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

subgroupindex
C32×#8922
D3×C4⋊(C3⋊Dic3)2
C2×#887862
#117221782
C3×#636352
#117220572
#117220562
#105926943
#105904373