Finite Group Databasebeta

← #13175142

Subgroups — #13175142

Order 1728.

1472 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723910C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic
3C32×C4×Dic6C211yesmaximal, characteristic
4#11723447C212yesmaximal
5#11723443C212yesmaximal
6#11723400C214yesmaximal
7#11723399C212yesmaximal
8Dic6×C3⋊Dic3C212yesmaximal
9#10596041—31nomaximal
10#10593860—34nomaximal
11C2×#6308C2212yes—
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C42C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2212yes—
16C4⋊(C3⋊(C3⋊Dic3))C2212yes—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750C2212yes—
19C3×C4×C3⋊Dic3C2212yes—
20C32×C4⋊Dic3C2211yescharacteristic
21C32×Dic3⋊C4C2212yes—
22C32×C4×Dic3C2212yes—
23#88806C2214yes—
24Dic3⋊(C3⋊Dic3)C2214yes—
25Dic3⋊(C3⋊Dic3)C2214yes—
26Dic3×C3⋊Dic3C2214yes—
27C32×C4×Q8—31no—
28Q8×C3⋊Dic3—32no—
29#64099—34no—
30#64099—32no—
31#64078—32no—
32#64078—31no—
33#64078—34no—
34#64078—64no—
35C4×#634—32no—
36C4×#634D311yescharacteristic
37C3×C4×Dic6—34no—
38C3×C4×Dic6D314yes—
39#63869—38no—
40#63866—38no—
41#63845—316no—
42#63844—38no—
43Dic3×Dic6—38no—
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)C2314yes—
48#6308—24no—
49C2×C3×C3⋊Dic3C2312yes—
50C2×C3×C3⋊Dic3C2312yes—
51C3×#634—22no—
52C2×C32×Dic3C2312yes—
53C2×C32×Dic3C2312yes—
54C32×Dic6Q814yescomplemented
55#5570—94no—
56#5567—91no—
57C2×C32×Q8—31no—
58C2×#634—32no—
59C2×#634—38no—
60C2×#634—68no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—34no—
63C2×C3×Dic6C2×D314yes—
64C32×C4⋊C4—32no—
65C32×C4⋊C4—31no—
66C32×C42—24no—
67C32×C42—32no—
68C32×C42C2×D314yes—
69C32×C42C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)—32no—
71C4⋊(C3⋊Dic3)—32no—
72C4⋊(C3⋊Dic3)—38no—
73C4⋊(C3⋊Dic3)—38no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—34no—
76C4⋊(C3⋊Dic3)—68no—
77C4⋊(C3⋊Dic3)—68no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79#3750—34no—
80#3750C2×D312yes—
81C4×C3⋊Dic3—34no—
82C4×C3⋊Dic3C2×D312yes—
83C3×C4⋊Dic3—34no—
84C3×C4⋊Dic3C2×D314yes—
85C3×Dic3⋊C4—38no—
86C3×Dic3⋊C4C2×D318yes—
87C3×C4×Dic3—38no—
88C3×C4×Dic3C2×D318yes—
89#3724—316no—
90Dic3⋊Dic3—316no—
91Dic3⋊Dic3—316no—
92Dic32—316no—
93C22×C33C2411yescharacteristic
94C33×C4—21no—
95C33×C4C2×Q812yes—
96C33×C4#4112yes—
97C3⋊(C3⋊Dic3)—24no—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—22no—
100C32×Dic3—22no—
101C32×Dic3C2×Q814yes—
102C3×C4×Q8—34no—
103C3×C4×Q8—91no—
104Q8×Dic3—98no—
105#944—92no—
106#890—98no—
107#890—916no—
108#889—94no—
109#887—92no—
110#870—92no—
111#869—98no—
112#869—94no—
113#869—91no—
114#869—184no—
115C4×Dic6—34no—
116C4×Dic6—98no—
117C4×Dic6—92no—
118C4×Dic6C3⋊D311yescharacteristic
119C32×Q8—34no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—32no—
124C2×C32×C4—32no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—316no—
133C2×C3⋊Dic3—616no—
134C2×C3⋊Dic3C22×D312yes—
135C2×C3⋊Dic3C22×D312yes—
136#634—22no—
137#634—64no—
138#634—616no—
139#634—1216no—
140C2×C3×Dic3—38no—
141C2×C3×Dic3—38no—
142C2×C3×Dic3C22×D318yes—
143C2×C3×Dic3C22×D318yes—
144C3×Dic6—68no—
145C3×Dic6Dic6116yescomplemented
146#557—271noSylow
147C2×C33C22×Q811yescharacteristic
148C2×C33C2×#4111yescharacteristic
149C2×C332−1+411yescharacteristic
150C2×C3×Q8—34no—
151C2×C3×Q8—91no—
152C2×Dic6—34no—
153C2×Dic6—98no—
154C2×Dic6—92no—
155C2×Dic6—188no—
156C2×Dic6C2×C3⋊D311yescharacteristic
157C3×C4⋊C4—38no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—92no—
161C3×C42—24no—
162C3×C42—38no—
163C3×C42—92no—
164C3×C42D3214yes—
165C3×C42C2×C3⋊D311yescharacteristic
166C4⋊Dic3—34no—
167C4⋊Dic3—916no—
168C4⋊Dic3—916no—
169C4⋊Dic3—98no—
170C4⋊Dic3—98no—
171C4⋊Dic3—92no—
172C4⋊Dic3—92no—
173C4⋊Dic3—188no—
174C4⋊Dic3—188no—
175C4⋊Dic3C2×C3⋊D311yescharacteristic
176Dic3⋊C4—38no—
177Dic3⋊C4—916no—
178Dic3⋊C4—94no—
179Dic3⋊C4—94no—
180Dic3⋊C4—94no—
181Dic3⋊C4C2×C3⋊D312yes—
182C4×Dic3—38no—
183C4×Dic3—916no—
184C4×Dic3—94no—
185C4×Dic3C2×C3⋊D312yes—
186C22×C32—24no—
187C22×C32C23×D314yes—
188C22×C32C23×D311yescharacteristic
189C32×C4—28no—
190C32×C4—24no—
191C32×C4—21no—
192C32×C4—28no—
193C32×C4—28no—
194C32×C4—34no—
195C32×C4—62no—
196C32×C4C2×Dic618yes—
197C32×C4#23512yes—
198C32×C4D3×Q812yes—
199C32×C4#23918yes—
200C3⋊Dic3—22no—
201C3⋊Dic3—22no—
202C3⋊Dic3—64no—
203C3⋊Dic3—616no—
204C3⋊Dic3—1216no—
205C3×Dic3—28no—
206C3×Dic3—68no—
207C3×Dic3—68no—
208C3×Dic3C2×Dic6116yes—
209C4⋊Q8—272no—
210C4⋊Q8—271no—
211#125—272no—
212#125—274no—
213C4×Q8—31no—
214C4×Q8—91no—
215C4×Q8—272no—
216C4×Q8—272no—
217C33#55711yescharacteristic, complemented, Sylow
218C3×Q8—316no—
219C3×Q8—182no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—38no—
224C2×C3×C4—38no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4C2×D3214yes—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×C3×C4C22×C3⋊D311yescharacteristic
232C2×C3×C4C22×C3⋊D311yescharacteristic
233C2×Dic3—38no—
234C2×Dic3—38no—
235C2×Dic3—916no—
236C2×Dic3—94no—
237C2×Dic3—1816no—
238C2×Dic3C22×C3⋊D312yes—
239C2×Dic3C22×C3⋊D312yes—
240Dic6—68no—
241Dic6—1816no—
242Dic6—184no—
243Dic6—3616no—
244Dic6#63414yescomplemented
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32C22×Dic614yes—
249C2×C32C2×#23511yescharacteristic
250C2×C32C2×D3×Q811yescharacteristic
251C2×C32C2×#23914yes—
252C2×C32#101014yes—
253C2×C32#101011yescharacteristic
254C2×Q8—31no—
255C2×Q8—91no—
256C2×Q8—272no—
257C4⋊C4—32no—
258C4⋊C4—31no—
259C4⋊C4—91no—
260C4⋊C4—92no—
261C4⋊C4—274no—
262C4⋊C4—274no—
263C4⋊C4—274no—
264C4⋊C4—272no—
265C4⋊C4—272no—
266C42—32no—
267C42—92no—
268C42—274no—
269C42D3×C3⋊D311yescharacteristic
270C22×C3—24no—
271C22×C3C22×D3214yes—
272C22×C3C23×C3⋊D311yescharacteristic
273C3×C4—28no—
274C3×C4—24no—
275C3×C4—21no—
276C3×C4—28no—
277C3×C4—28no—
278C3×C4—316no—
279C3×C4—68no—
280C3×C4—182no—
281C3×C4—182no—
282C3×C4D3×Dic618yes—
283C3×C4#379918yes—
284C3×C4C2×#63412yes—
285C3×C4#383212yes—
286Dic3—22no—
287Dic3—68no—
288Dic3—68no—
289Dic3—1816no—
290Dic3—184no—
291Dic3—3616no—
292Dic3C2×#63414yes—
293C32—24no—
294C32#556711yescharacteristic, complemented
295C32#557014yescomplemented
296Q8—34no—
297Q8—182no—
298Q8—544no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—92no—
302C2×C4—92no—
303C2×C4—274no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C2×C4C2×D3×C3⋊D311yescharacteristic
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3C2×D3×Dic614yes—
311C2×C3#6429814yes—
312C2×C3C2×#379914yes—
313C2×C3C22×#63411yescharacteristic
314C2×C3C2×#383211yescharacteristic
315C2×C3#6436711yescharacteristic
316C22C22×D3×C3⋊D311yescharacteristic
317C4—21no—
318C4—34no—
319C4—62no—
320C4—182no—
321C4—182no—
322C4—544no—
323C4D3×#63412yes—
324C4#8901812yes—
325C3—24no—
326C3#1059386014yescomplemented
327C3#1059604111yescharacteristic, complemented
328C2C2×D3×#63411yescharacteristic
329C2#1172456311yescharacteristic
330C2C2×#8901811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, prime radical, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
54. C32×Dic6Q8semidirect
145. C3×Dic6Dic6semidirect
217. C33#557semidirect
244. Dic6#634semidirect
294. C32#5567semidirect
295. C32#5570semidirect
326. C3#10593860semidirect
327. C3#10596041semidirect

Maximal subgroups

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

subgroupindex
#117239102
C3×C4×#6342
C32×C4×Dic62
#117234472
#117234432
#117234002
#117233992
Dic6×C3⋊Dic32
#105960413
#105938603