Finite Group Databasebeta

← #13175187

Subgroups — #13175187

Order 1728.

2292 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89022C212yesmaximal, complemented
2#11723915C211yesmaximal, characteristic, complemented
3C3×#64078C211yesmaximal, characteristic, complemented
4C32×C42×D3C211yesmaximal, characteristic, complemented
5#11723516C214yesmaximal, complemented
6#11723484C214yesmaximal, complemented
7#11723441C212yesmaximal, complemented
8#10595948—31nomaximal
9#10593970—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
11C2×C4⋊(C3⋊(C3⋊D3))C2214yes—
12C2×C3×#634C2212yes—
13C2×C32×C4×D3C2212yes—
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6291C2214yes—
16C2×#6289C2214yes—
17#89022—28no—
18C33×C42C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2214yes—
20C32×C4×Dic3C2211yescharacteristic
21C32×C4×Dic3C2212yes—
22#88804C2218yes—
23C2×#3828—32no—
24C2×#3799—38no—
25C2×C32×C42—31no—
26#64139—34no—
27#64084—32no—
28#64083—31no—
29#64083—34no—
30#64083—64no—
31C4×C4⋊(C3⋊D3)—34no—
32#64078D311yescharacteristic, complemented
33C3×C42×D3D314yescomplemented
34C3×#869—34no—
35#63909—316no—
36#63894—316no—
37#63864—38no—
38C22×C3⋊(C3⋊D3)C2314yes—
39C22×C32×D3C2311yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C33×C4C2312yes—
42C4⋊(C3⋊(C3⋊D3))—24no—
43C4⋊(C3⋊(C3⋊D3))—28no—
44C2×C3×C3⋊Dic3C2314yes—
45C3×#634—24no—
46C2×C32×Dic3C2312yes—
47C2×C32×Dic3C2311yescharacteristic
48C32×C4×D3—24no—
49C32×C4×D3D414yescomplemented
50#6291—28no—
51#6289—28no—
52#5725—91no—
53#5477—94no—
54C22×C32×C4—31no—
55C22×C32×C4—32no—
56C2×#638—34no—
57#3828—68no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—32no—
60C2×C4⋊(C3⋊D3)—38no—
61C2×C4⋊(C3⋊D3)—316no—
62C2×C4⋊(C3⋊D3)—616no—
63C2×C4⋊(C3⋊D3)—68no—
64C2×C4×C3⋊D3—34no—
65C2×#634C2×D312yes—
66C2×C3×C4×D3C2×D318yes—
67C2×C3×C4×D3C2×D314yes—
68C2×C3×Dic6—38no—
69C2×#626—316no—
70C2×#624—316no—
71#3799—632no—
72C32×C42—24no—
73C32×C42—32no—
74C32×C42—31no—
75C32×C42C2×D314yes—
76C32×C42C2×D311yescharacteristic
77#3752—38no—
78C4⋊(C3⋊Dic3)C2×D314yes—
79C3×C4⋊Dic3—316no—
80C3×C4×Dic3C2×D314yes—
81C3×C4×Dic3C2×D318yes—
82#3722—332no—
83C22×C33C2411yescharacteristic
84C2×C3⋊(C3⋊D3)—24no—
85C2×C3⋊(C3⋊D3)—28no—
86C2×C32×D3—22no—
87C2×C32×D3C2×D412yes—
88C33×C4C2×D412yes—
89C33×C4#4114yescomplemented
90C3×C3⋊Dic3—24no—
91C32×Dic3—22no—
92C32×Dic3C2×D412yes—
93C2×#239—92no—
94C2×#235—98no—
95C3×C4⋊Q8—91no—
96C2×C3×C42—34no—
97#947—92no—
98#930—916no—
99#895—94no—
100#893—94no—
101#875—98no—
102#874—94no—
103#874—91no—
104#874—184no—
105C4×D12—916no—
106C42×D3C3⋊D311yescharacteristic, complemented
107#869—34no—
108C23×C32—31no—
109C22×C3⋊D3—34no—
110C22×C3⋊D3—316no—
111C22×C3⋊D3—616no—
112C22×C3×D3C22×D314yes—
113C2×C32×C4—28no—
114C2×C32×C4—24no—
115C2×C32×C4—31no—
116C2×C32×C4—34no—
117C2×C32×C4—32no—
118C2×C32×C4—64no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D318yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D312yes—
123#638—68no—
124C2×C3⋊Dic3C22×D314yes—
125C4⋊(C3⋊D3)—64no—
126C4⋊(C3⋊D3)—68no—
127C4⋊(C3⋊D3)—616no—
128C4⋊(C3⋊D3)—632no—
129C4⋊(C3⋊D3)—1232no—
130C4⋊(C3⋊D3)—1216no—
131C4×C3⋊D3—68no—
132#634—24no—
133C2×C3×Dic3—316no—
134C2×C3×Dic3C22×D318yes—
135C2×C3×Dic3C22×D314yes—
136C3×C4×D3—216no—
137C3×C4×D3D12116yescomplemented
138C3×Dic6—616no—
139#626—632no—
140#624—632no—
141#532—271noSylow
142C2×C33C22×D411yescharacteristic
143C2×C33C2×#4112yes—
144C3⋊(C3⋊D3)—44no—
145C32×D3—22no—
146C2×C3×Q8—92no—
147C22×C3×C4—34no—
148C22×C3×C4—38no—
149C2×#67—916no—
150#239—188no—
151#235—1832no—
152C2×D12—94no—
153C2×D12—916no—
154C2×D12—98no—
155C2×D12—94no—
156C2×D12—92no—
157C2×D12—1816no—
158C2×D12—188no—
159C2×C4×D3—916no—
160C2×C4×D3—94no—
161C2×C4×D3C2×C3⋊D312yes—
162C2×C4×D3C2×C3⋊D311yescharacteristic
163C2×Dic6—38no—
164C3×C4⋊C4—94no—
165C3×C42—24no—
166C3×C42—38no—
167C3×C42—34no—
168C3×C42D3214yes—
169C3×C42C2×C3⋊D311yescharacteristic
170(C2×D3)⋊C4—932no—
171(C2×D3)⋊C4—98no—
172C4⋊Dic3—316no—
173C4×Dic3C2×C3⋊D312yes—
174C4×Dic3C2×C3⋊D311yescharacteristic
175C22×C32—24no—
176C22×C32—32no—
177C22×C32—62no—
178C22×C32C23×D314yes—
179C22×C32C23×D311yescharacteristic
180C2×C3⋊D3—64no—
181C2×C3⋊D3—68no—
182C2×C3⋊D3—616no—
183C2×C3⋊D3—632no—
184C2×C3⋊D3—1232no—
185C2×C3⋊D3—1216no—
186C2×C3×D3—28no—
187C2×C3×D3C2×D1218yes—
188C32×C4—216no—
189C32×C4—28no—
190C32×C4—32no—
191C32×C4—62no—
192C32×C4C2×D1218yes—
193C32×C4#235116yescomplemented
194C32×C4D3×D412yes—
195C32×C4#23914yescomplemented
196C3⋊Dic3—24no—
197C3×Dic3—28no—
198C3×Dic3—616no—
199C3×Dic3C2×D1218yes—
200C2×#41—272no—
201C4⋊Q8—91no—
202#127—271no—
203#124—272no—
204#121—274no—
205C4×D4—274no—
206C2×C42—31no—
207C33#53211yescharacteristic, complemented, Sylow
208C23×C3—34no—
209C22×D3—916no—
210C22×D3—94no—
211C22×D3—1816no—
212C22×D3C22×C3⋊D311yescharacteristic
213C3×Q8—184no—
214C2×C3×C4—28no—
215C2×C3×C4—24no—
216C2×C3×C4—34no—
217C2×C3×C4—316no—
218C2×C3×C4—38no—
219C2×C3×C4—616no—
220C2×C3×C4—94no—
221C2×C3×C4C2×D3218yes—
222C2×C3×C4C2×D3214yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×C3×C4C22×C3⋊D312yes—
225#67—1832no—
226C2×Dic3—316no—
227C2×Dic3C22×C3⋊D312yes—
228C2×Dic3C22×C3⋊D311yescharacteristic
229D12—188no—
230D12—1832no—
231D12—1816no—
232D12—184no—
233D12—188no—
234D12—3632no—
235D12—3616no—
236C4×D3—24no—
237C4×D3—1832no—
238C4×D3—188no—
239C4×D3C4⋊(C3⋊D3)14yescomplemented
240Dic6—616no—
241C2×C32—28no—
242C2×C32—24no—
243C2×C32—62no—
244C2×C32C22×D1214yes—
245C2×C32C2×#23518yes—
246C2×C32C2×D3×D411yescharacteristic
247C2×C32C2×#23912yes—
248C3⋊D3—124no—
249C3⋊D3—1216no—
250C3⋊D3—2416no—
251C3×D3—28no—
252#41—548no—
253C2×Q8—92no—
254C2×D4—274no—
255C2×D4—274no—
256C2×D4—272no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—274no—
260C4⋊C4—94no—
261C22⋊C4—278no—
262C42—32no—
263C42—31no—
264C42D3×C3⋊D311yescharacteristic
265C22×C3—24no—
266C22×C3—38no—
267C22×C3—68no—
268C22×C3C22×D3214yes—
269C22×C3C23×C3⋊D311yescharacteristic
270C2×D3—22no—
271C2×D3—1832no—
272C2×D3—1816no—
273C2×D3—184no—
274C2×D3—188no—
275C2×D3—3632no—
276C2×D3—3616no—
277C2×D3C2×C4⋊(C3⋊D3)12yes—
278C3×C4—216no—
279C3×C4—28no—
280C3×C4—38no—
281C3×C4—68no—
282C3×C4—184no—
283C3×C4#3799116yescomplemented
284C3×C4D3×D1218yes—
285C3×C4C2×C4⋊(C3⋊D3)12yes—
286C3×C4#382814yescomplemented
287Dic3—22no—
288Dic3—616no—
289Dic3C2×C4⋊(C3⋊D3)12yes—
290C32—24no—
291C32#547714yescomplemented
292C32#572511yescharacteristic, complemented
293C23—31no—
294C23—274no—
295Q8—184no—
296D4—548no—
297D4—548no—
298D4—544no—
299C2×C4—31no—
300C2×C4—34no—
301C2×C4—32no—
302C2×C4—64no—
303C2×C4—94no—
304C2×C4—548no—
305C2×C4C2×D3×C3⋊D312yes—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C2×C3—28no—
308C2×C3—24no—
309C2×C3—68no—
310C2×C3C2×#379918yes—
311C2×C3C2×D3×D1214yes—
312C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
313C2×C3C2×#382812yes—
314D3—22no—
315D3—364no—
316D3—3616no—
317D3—7216no—
318C22—32no—
319C22—62no—
320C22—548no—
321C22—544no—
322C22C22×D3×C3⋊D311yescharacteristic
323C4—32no—
324C4—62no—
325C4—184no—
326C4#8902214yescomplemented
327C4D3×C4⋊(C3⋊D3)12yes—
328C3—24no—
329C3#1059397014yescomplemented
330C3#1059594811yescharacteristic, complemented
331C2—62no—
332C2—1084no—
333C2C2×#8902212yes—
334C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89022C2semidirect
2. #11723915C2semidirect
3. C3×#64078C2semidirect
4. C32×C42×D3C2semidirect
5. #11723516C2semidirect
6. #11723484C2semidirect
7. #11723441C2semidirect
32. #64078D3semidirect
33. C3×C42×D3D3semidirect
49. C32×C4×D3D4semidirect
89. C33×C4#41semidirect
106. C42×D3C3⋊D3semidirect
137. C3×C4×D3D12semidirect
193. C32×C4#235semidirect
195. C32×C4#239semidirect
207. C33#532semidirect
239. C4×D3C4⋊(C3⋊D3)semidirect
283. C3×C4#3799semidirect
286. C3×C4#3828semidirect
291. C32#5477semidirect
292. C32#5725semidirect
326. C4#89022semidirect
329. C3#10593970semidirect
330. C3#10595948semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
#117239152
C3×#640782
C32×C42×D32
#117235162
#117234842
#117234412
#105959483
#105939703