Finite Group Databasebeta

← #13175169

Subgroups — #13175169

Order 1728.

2024 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89021C212yesmaximal, complemented
2#11723910C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×C4×D12C211yesmaximal, characteristic, complemented
5#11723523C214yesmaximal, complemented
6#11723522C212yesmaximal, complemented
7C3⋊D3×C4⋊Dic3C212yesmaximal, complemented
8#11723426C212yesmaximal, complemented
9#10596022—31nomaximal
10#10593895—34nomaximal
11C2×#6308C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3C2212yes—
16C2×#6290C2214yes—
17C2×Dic3×C3⋊D3C2214yes—
18#89021—28no—
19C33×C42C2211yescharacteristic
20C4⋊(C3⋊(C3⋊Dic3))C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2212yes—
22C3×#3752C2212yes—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C32×(C2×D3)⋊C4C2212yes—
25C32×C4⋊Dic3C2211yescharacteristic
26#88807C2214yes—
27(C2×D3)⋊(C3⋊Dic3)C2214yes—
28C2×#3828—32no—
29C2×#3795—38no—
30C32×C4×D4—31no—
31#64147—32no—
32#64133—32no—
33C4⋊C4×C3⋊D3—32no—
34#64092—34no—
35C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
36#64078—31no—
37#64078—34no—
38#64078—64no—
39C3×C4×D12—34no—
40C3×C4×D12D314yescomplemented
41#63908—38no—
42#63891—316no—
43D3×C4⋊Dic3—38no—
44#63857—38no—
45C22×C3×C3⋊D3C2312yes—
46C22×C32×D3C2312yes—
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2314yes—
51#6308—24no—
52C2×C3×C3⋊Dic3C2312yes—
53C3×C4⋊(C3⋊D3)D414yescomplemented
54C3×C4×C3⋊D3—24no—
55C2×C32×Dic3C2312yes—
56C32×D12—22no—
57C32×C4×D3—24no—
58#6290—28no—
59Dic3×C3⋊D3—28no—
60#5555—91no—
61#5551—94no—
62C2×C32×D4—31no—
63C22×C32×C4—32no—
64C2×#638—34no—
65#3828—68no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—34no—
68C2×C4×C3⋊D3C2×D312yes—
69C2×#634—32no—
70C2×#634—38no—
71C2×#634—68no—
72C2×C3×D12—34no—
73C2×C3×D12C2×D314yes—
74C2×C3×C4×D3—38no—
75C2×C3×C4×D3C2×D318yes—
76C2×#625—316no—
77C2×D3×Dic3—316no—
78#3795—632no—
79C32×C4⋊C4—31no—
80C32×C22⋊C4—32no—
81C32×C42—24no—
82C32×C42C2×D314yes—
83C32×C42C2×D311yescharacteristic
84#3757—34no—
85#3752C2×D312yes—
86C4⋊(C3⋊Dic3)—32no—
87C4⋊(C3⋊Dic3)—32no—
88C4⋊(C3⋊Dic3)—38no—
89C4⋊(C3⋊Dic3)—38no—
90C4⋊(C3⋊Dic3)—68no—
91C4⋊(C3⋊Dic3)—68no—
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93#3750—34no—
94C3×(C2×D3)⋊C4—38no—
95C3×(C2×D3)⋊C4C2×D318yes—
96C3×C4⋊Dic3—34no—
97C3×C4⋊Dic3C2×D314yes—
98Dic3⋊Dic3—316no—
99(C2×D3)⋊Dic3—316no—
100C22×C33C2411yescharacteristic
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3C2×D414yes—
104C2×C32×D3—22no—
105C2×C32×D3—22no—
106C2×C32×D3—22no—
107C33×C4—21no—
108C33×C4C2×D412yes—
109C33×C4#4112yescomplemented
110C3⋊(C3⋊Dic3)—24no—
111C3×C3⋊Dic3—22no—
112C32×Dic3—22no—
113C2×#237—92no—
114C2×#235—98no—
115C3×C4×D4—34no—
116C3×C4×D4—91no—
117#938—98no—
118#930—92no—
119C2×C4⋊Dic3—92no—
120#924—98no—
121#897—92no—
122D3×C4⋊C4—98no—
123#886—94no—
124#883—916no—
125C4×D12—34no—
126C4×D12C3⋊D311yescharacteristic, complemented
127#869—94no—
128#869—91no—
129#869—184no—
130C23×C32—32no—
131C22×C3⋊D3C22×D312yes—
132C22×C3×D3—38no—
133C22×C3×D3C22×D318yes—
134C32×D4—62no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—32no—
139C2×C32×C4—64no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146#638—68no—
147C2×C3⋊Dic3—34no—
148C2×C3⋊Dic3—316no—
149C2×C3⋊Dic3—616no—
150C2×C3⋊Dic3C22×D312yes—
151C4⋊(C3⋊D3)D1214yescomplemented
152C4×C3⋊D3—24no—
153C4×C3⋊D3—68no—
154#634—64no—
155#634—616no—
156#634—1216no—
157C2×C3×Dic3—38no—
158C2×C3×Dic3C22×D318yes—
159C3×D12—28no—
160C3×D12—316no—
161C3×C4×D3—216no—
162C3×C4×D3—616no—
163#625—632no—
164D3×Dic3—632no—
165#547—271noSylow
166C2×C33C22×D411yescharacteristic
167C2×C33C2×#4111yescharacteristic
168C2×C332−1+411yescharacteristic
169C3×C3⋊D3—24no—
170C32×D3—42no—
171C2×C3×D4—34no—
172C2×C3×D4—91no—
173C22×C3×C4—38no—
174C22×C3×C4—92no—
175C2×#67—916no—
176C2×#67—94no—
177C22×Dic3—94no—
178#237—188no—
179#235—1832no—
180C2×D12—34no—
181C2×D12C2×C3⋊D311yescharacteristic
182C2×C4×D3—38no—
183C2×C4×D3—916no—
184C2×C4×D3C2×C3⋊D312yes—
185C2×Dic6—98no—
186C2×Dic6—92no—
187C2×Dic6—188no—
188C3×C4⋊C4—34no—
189C3×C4⋊C4—91no—
190C3×C22⋊C4—38no—
191C3×C22⋊C4—92no—
192C3×C42—24no—
193C3×C42D3214yes—
194C3×C42C2×C3⋊D311yescharacteristic
195#217—916no—
196(C2×D3)⋊C4—38no—
197(C2×D3)⋊C4—94no—
198(C2×D3)⋊C4C2×C3⋊D312yes—
199C4⋊Dic3—34no—
200C4⋊Dic3—98no—
201C4⋊Dic3—98no—
202C4⋊Dic3—92no—
203C4⋊Dic3—92no—
204C4⋊Dic3—94no—
205C4⋊Dic3—188no—
206C4⋊Dic3—188no—
207C4⋊Dic3C2×C3⋊D311yescharacteristic
208Dic3⋊C4—916no—
209C22×C32—24no—
210C22×C32—62no—
211C22×C32—62no—
212C22×C32—62no—
213C22×C32C23×D314yes—
214C22×C32C23×D311yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—22no—
217C2×C3⋊D3C2×D1214yes—
218C2×C3×D3—28no—
219C2×C3×D3—28no—
220C2×C3×D3—28no—
221C2×C3×D3—316no—
222C2×C3×D3—68no—
223C2×C3×D3—68no—
224C32×C4—28no—
225C32×C4—24no—
226C32×C4—21no—
227C32×C4—28no—
228C32×C4—28no—
229C32×C4—62no—
230C32×C4C2×D1212yes—
231C32×C4#23518yescomplemented
232C32×C4D3×D418yes—
233C32×C4#23712yescomplemented
234C3⋊Dic3—22no—
235C3⋊Dic3—64no—
236C3⋊Dic3—616no—
237C3⋊Dic3—1216no—
238C3×Dic3—28no—
239C3×Dic3—68no—
240C2×#41—272no—
241C4⋊Q8—271no—
242#123—274no—
243C22⋊Q8—272no—
244#121—272no—
245C4×D4—31no—
246C4×D4—91no—
247C2×C4⋊C4—272no—
248C33#54711yescharacteristic, complemented, Sylow
249C23×C3—38no—
250C23×C3—92no—
251C22×D3—38no—
252C22×D3C22×C3⋊D312yes—
253C3×D4—68no—
254C3×D4—94no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—38no—
259C2×C3×C4—616no—
260C2×C3×C4—92no—
261C2×C3×C4—184no—
262C2×C3×C4C2×D3214yes—
263C2×C3×C4C2×D3214yes—
264C2×C3×C4C2×D3214yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266C2×C3×C4C22×C3⋊D311yescharacteristic
267C2×C3×C4C22×C3⋊D311yescharacteristic
268#67—188no—
269#67—1832no—
270C2×Dic3—38no—
271C2×Dic3—916no—
272C2×Dic3—94no—
273C2×Dic3—1816no—
274C2×Dic3—188no—
275C2×Dic3C22×C3⋊D312yes—
276D12—22no—
277D12—316no—
278C4×D3—24no—
279C4×D3—616no—
280C4×D3—1832no—
281Dic6—1816no—
282Dic6—184no—
283Dic6—3616no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—122no—
288C2×C32C22×D1211yescharacteristic
289C2×C32C2×#23514yes—
290C2×C32C2×D3×D414yes—
291C2×C32C2×#23711yescharacteristic
292C2×C32#101014yes—
293C2×C32#101011yescharacteristic
294C3⋊D3—24no—
295C3×D3—48no—
296C3×D3—616no—
297#41—548no—
298C2×Q8—272no—
299C2×D4—31no—
300C2×D4—91no—
301C2×D4—274no—
302C22×C4—32no—
303C22×C4—92no—
304C22×C4—274no—
305C4⋊C4—31no—
306C4⋊C4—91no—
307C4⋊C4—274no—
308C4⋊C4—272no—
309C4⋊C4—272no—
310C22⋊C4—32no—
311C22⋊C4—92no—
312C22⋊C4—274no—
313C42D3×C3⋊D311yescharacteristic
314C22×C3—24no—
315C22×C3—68no—
316C22×C3—68no—
317C22×C3—68no—
318C22×C3—94no—
319C22×C3—182no—
320C22×C3—182no—
321C22×C3C22×D3214yes—
322C22×C3C23×C3⋊D311yescharacteristic
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—316no—
327C2×D3—68no—
328C2×D3—68no—
329C3×C4—28no—
330C3×C4—24no—
331C3×C4—21no—
332C3×C4—28no—
333C3×C4—28no—
334C3×C4—68no—
335C3×C4—182no—
336C3×C4#379518yescomplemented
337C3×C4D3×D1218yes—
338C3×C4#382812yescomplemented
339C3×C4D4×C3⋊D312yes—
340Dic3—22no—
341Dic3—68no—
342Dic3—1816no—
343Dic3—184no—
344Dic3—3616no—
345C32—24no—
346C32#555114yescomplemented
347C32#555511yescharacteristic, complemented
348C23—32no—
349C23—92no—
350Q8—544no—
351D4—62no—
352D4—94no—
353D4—548no—
354C2×C4—32no—
355C2×C4—64no—
356C2×C4—92no—
357C2×C4—184no—
358C2×C4—274no—
359C2×C4—548no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C2×C4C2×D3×C3⋊D311yescharacteristic
362C2×C4C2×D3×C3⋊D311yescharacteristic
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—128no—
367C2×C3—184no—
368C2×C3C2×#379514yes—
369C2×C3#6429814yes—
370C2×C3C2×D3×D1214yes—
371C2×C3C2×#382811yescharacteristic
372C2×C3C2×D4×C3⋊D311yescharacteristic
373C2×C3#6436711yescharacteristic
374D3—42no—
375D3—616no—
376C22—62no—
377C22—62no—
378C22—62no—
379C22—94no—
380C22—182no—
381C22—182no—
382C22C22×D3×C3⋊D311yescharacteristic
383C4—21no—
384C4—62no—
385C4—182no—
386C4—544no—
387C4#8902112yescomplemented
388C4C3⋊D3×D1212yes—
389C3—24no—
390C3#1059389514yescomplemented
391C3#1059602211yescharacteristic, complemented
392C2—122no—
393C2—184no—
394C2#1172456311yescharacteristic
395C2C2×#8902111yescharacteristic
396C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89021C2semidirect
2. #11723910C2semidirect
3. C3×C4×C4⋊(C3⋊D3)C2semidirect
4. C32×C4×D12C2semidirect
5. #11723523C2semidirect
6. #11723522C2semidirect
7. C3⋊D3×C4⋊Dic3C2semidirect
8. #11723426C2semidirect
35. C4×C4⋊(C3⋊D3)D3semidirect
40. C3×C4×D12D3semidirect
53. C3×C4⋊(C3⋊D3)D4semidirect
109. C33×C4#41semidirect
126. C4×D12C3⋊D3semidirect
151. C4⋊(C3⋊D3)D12semidirect
231. C32×C4#235semidirect
233. C32×C4#237semidirect
248. C33#547semidirect
336. C3×C4#3795semidirect
338. C3×C4#3828semidirect
346. C32#5551semidirect
347. C32#5555semidirect
387. C4#89021semidirect
390. C3#10593895semidirect
391. C3#10596022semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
#117239102
C3×C4×C4⋊(C3⋊D3)2
C32×C4×D122
#117235232
#117235222
C3⋊D3×C4⋊Dic32
#117234262
#105960223
#105938953