Finite Group Databasebeta

← #13159539

Subgroups — #13159539

Order 1728.

852 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 196 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×#3743C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3#11723404C211yesmaximal, characteristic
4#11722187C211yesmaximal, characteristic, complemented
5C32×#818C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722054C211yesmaximal, characteristic, complemented
8#10592580—31nomaximal
9#10590683—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×Dic3×C3⋊D3—21no—
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#3743C414yescomplemented
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20#88803—21no—
21Dic3×C3⋊Dic3—21no—
22C2×#3743D311yescharacteristic
23#64101—31no—
24C4⋊C4×C3⋊D3—31no—
25C2×C3×ΓL(1,𝔽25)—34no—
26D3×C4⋊Dic3—34no—
27#63841—34no—
28C32×#107—31no—
29#63644—31no—
30#63644—34no—
31#63644—64no—
32#63635—31no—
33#63634—31no—
34C3×#818D314yescomplemented
35#63575—34no—
36#63573—34no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3C2×C412yes—
43C3×C4×C3⋊D3Q812yes—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46Dic3×C3⋊D3—42no—
47C33×C8C2×C412yescomplemented
48C3×C3⋊(C3⋊C8)C2×C412yescomplemented
49#5121—91no—
50#4881—94no—
51C2×C4×C3⋊D3—61no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—34no—
54C2×D3×Dic3—64no—
55C2×C32×C8—24no—
56C2×C32×C8C2×D314yes—
57C2×C32×C8C2×D311yescharacteristic
58C32×C4⋊C4—31no—
59C32×C4⋊C4—31no—
60#3752—61no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)—64no—
67#3750—61no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70#3743Dic314yescomplemented
71C3×C4⋊Dic3C2×D314yes—
72C3×C4⋊Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74C3×ΓL(1,𝔽25)—316no—
75Dic3⋊Dic3—64no—
76(C2×D3)⋊Dic3—64no—
77Dic32—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3C4⋊C412yes—
82C33×C4C22×C411yescharacteristic
83C33×C4C2×Q811yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3⋊(C3⋊Dic3)—41no—
86C3×C3⋊Dic3C4⋊C412yes—
87C32×Dic3—41no—
88C32×Dic3—41no—
89C2×C3×M16—91no—
90#926—91no—
91C2×C4⋊Dic3—91no—
92C2×ΓL(1,𝔽25)—34no—
93#892—94no—
94D3×C4⋊C4—94no—
95C3×#107—34no—
96#818—91no—
97#818C3⋊D311yescharacteristic, complemented
98#817—94no—
99#817—91no—
100#817—91no—
101#817—184no—
102#808—94no—
103#807—94no—
104C22×C3⋊D3D1211yescharacteristic
105C22×C3×D3—34no—
106C2×C32×C4—24no—
107C2×C32×C4—61no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3D1211yescharacteristic
118C4×C3⋊D3—122no—
119C4×C3⋊D3Dic612yes—
120C4×C3⋊D3C2×Dic312yes—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3—34no—
124C3×C4×D3—38no—
125C3×C4×D3—38no—
126D3×Dic3—128no—
127C32×C8—28no—
128C32×C8C4×D318yescomplemented
129C32×C8C2×Dic312yescomplemented
130C3⋊(C3⋊C8)C2×Dic312yescomplemented
131C3×C3⋊C8—38no—
132#428—271noSylow
133C2×C33C2×C4⋊C411yescharacteristic
134C2×C33AGL(1,ℤ/8)11yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—22no—
137C22×C3×C4—91no—
138C22×Dic3—181no—
139C2×C4×D3—34no—
140C2×C4×D3—184no—
141C3×M16—94no—
142C2×C3×C8—24no—
143C2×C3×C8—91no—
144C2×C3×C8D3214yes—
145C2×C3×C8C2×C3⋊D311yescharacteristic
146C3×C4⋊C4—34no—
147C3×C4⋊C4—34no—
148#217—181no—
149(C2×D3)⋊C4—184no—
150C4⋊Dic3—94no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—184no—
156C4⋊Dic3—181no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158C4⋊Dic3C2×C3⋊D311yescharacteristic
159Dic3⋊C4—184no—
160C4×Dic3—181no—
161C4×Dic3—184no—
162C2×C3⋊C8—34no—
163ΓL(1,𝔽25)—316no—
164C22×C32—24no—
165C22×C32C2×D1211yescharacteristic
166C22×C32D3×D414yes—
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3⋊D3C4⋊Dic312yes—
170C2×C3×D3—38no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—121no—
176C32×C4—121no—
177C32×C4C2×Dic611yescharacteristic
178C32×C4C2×C4×D314yes—
179C32×C4D3×Q814yes—
180C32×C4C22×Dic311yescharacteristic
181C3⋊Dic3—121no—
182C3⋊Dic3—121no—
183C3⋊Dic3—124no—
184C3⋊Dic3—124no—
185C3⋊Dic3—244no—
186C3⋊Dic3—244no—
187C3⋊Dic3C4⋊Dic312yes—
188C3×Dic3—38no—
189C3×Dic3—44no—
190C3×Dic3—44no—
191C2×M16—91no—
192#117—271no—
193C2×C4⋊C4—271no—
194#107—31no—
195#107—271no—
196#106—271no—
197#106—271no—
198C33#42811yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—34no—
201C2×C3×C4—24no—
202C2×C3×C4—64no—
203C2×C3×C4—64no—
204C2×C3×C4—91no—
205C2×C3×C4—92no—
206C2×C3×C4—92no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—34no—
212C2×Dic3—184no—
213C2×Dic3—184no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217C2×Dic3—364no—
218C2×Dic3—362no—
219C4×D3—38no—
220C4×D3—38no—
221C4×D3—368no—
222C3×C8—28no—
223C3×C8—92no—
224C3×C8D3×Dic318yescomplemented
225C3×C8C4×C3⋊D312yescomplemented
226C3⋊C8—38no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32D3×C4⋊C414yes—
231C2×C32#90811yescharacteristic
232C2×C32#90911yescharacteristic
233C2×C32#91114yes—
234C2×C32#91814yes—
235C2×C32C2×C4⋊Dic311yescharacteristic
236C3⋊D3—22no—
237C3×D3—68no—
238C22×C4—91no—
239C22×C4—541no—
240M16—94no—
241C2×C8—91no—
242C2×C8D3×C3⋊D311yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—271no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C22⋊C4—541no—
249C42—541no—
250C22×C3—24no—
251C22×C3—92no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3D3×D1214yes—
255C22×C3D4×C3⋊D311yescharacteristic
256C2×D3—38no—
257C2×D3—64no—
258C2×D3—64no—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—92no—
262C3×C4—124no—
263C3×C4—124no—
264C3×C4D3×Dic614yes—
265C3×C4C2×D3×Dic314yes—
266C3×C4C2×C4×C3⋊D311yescharacteristic
267C3×C4Q8×C3⋊D311yescharacteristic
268Dic3—38no—
269Dic3—41no—
270Dic3—41no—
271Dic3—364no—
272Dic3—364no—
273Dic3—361no—
274Dic3—361no—
275Dic3—724no—
276Dic3—724no—
277C32—24no—
278C32#488114yescomplemented
279C32#512111yescharacteristic, complemented
280C23—91no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—541no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—92no—
291C8Dic3×C3⋊D312yescomplemented
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—182no—
296C2×C3#6379514yes—
297C2×C3#6380114yes—
298C2×C3D3×C4⋊Dic314yes—
299C2×C3C4⋊C4×C3⋊D311yescharacteristic
300C2×C3#6412011yescharacteristic
301C2×C3#6412711yescharacteristic
302D3—68no—
303C22—92no—
304C22—181no—
305C22—181no—
306C22C3⋊D3×D1211yescharacteristic
307C4—92no—
308C4—121no—
309C4—121no—
310C4—1081no—
311C4—1081no—
312C4C3⋊D3×Dic611yescharacteristic
313C4C2×Dic3×C3⋊D311yescharacteristic
314C3—24no—
315C3#1059068314yescomplemented
316C3#1059258011yescharacteristic, complemented
317C2—182no—
318C2#1172332011yescharacteristic
319C2#1172332911yescharacteristic
320C2C3⋊D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #11722187C2semidirect
5. C32×#818C2semidirect
6. #11722057C2semidirect
7. #11722054C2semidirect
16. C3×#3743C4semidirect
34. C3×#818D3semidirect
47. C33×C8C2×C4semidirect
48. C3×C3⋊(C3⋊C8)C2×C4semidirect
70. #3743Dic3semidirect
97. #818C3⋊D3semidirect
128. C32×C8C4×D3semidirect
129. C32×C8C2×Dic3semidirect
130. C3⋊(C3⋊C8)C2×Dic3semidirect
198. C33#428semidirect
224. C3×C8D3×Dic3semidirect
225. C3×C8C4×C3⋊D3semidirect
278. C32#4881semidirect
279. C32#5121semidirect
291. C8Dic3×C3⋊D3semidirect
315. C3#10590683semidirect
316. C3#10592580semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C3⋊D3×C4⋊Dic32
#117234042
#117221872
C32×#8182
#117220572
#117220542
#105925803
#105906833