Finite Group Databasebeta

← #13177196

Subgroups — #13177196

Order 1728.

2284 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C32×#235C211yesmaximal, characteristic, complemented
3#11723967C211yesmaximal, characteristic, complemented
4#11723634C212yesmaximal, complemented
5#11723602C212yesmaximal, complemented
6#11723522C212yesmaximal, complemented
7C4×#6290C214yesmaximal, complemented
8#11723435C212yesmaximal, complemented
9#10596299—31nomaximal
10#10595022—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C4×C3⋊D3—24no—
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×#67C2212yes—
18C32×#235—24no—
19C2×C32×D12C2211yescharacteristic, complemented
20C2×C32×C4×D3C2212yescomplemented
21C2×C32×Dic6C2211yescharacteristic, complemented
22C2×#6290C2214yes—
23#88864C2212yes—
24C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
25C4×C3⋊(C3⋊Dic3)C2212yescomplemented
26#88806C2214yescomplemented
27#88803C2214yes—
28(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
29C2×C32×#41—31no—
30C22×C4×C3⋊D3D311yescharacteristic, complemented
31C2×C3×#235D314yescomplemented
32C22×C3×C4×D3—34no—
33#64155—32no—
34#64147—32no—
35#64146—32no—
36#64144—32no—
37C4×#638—34no—
38#64135—31no—
39#64135—34no—
40#64135—64no—
41#63976—38no—
42#63961—38no—
43#63908—38no—
44C4×#625—316no—
45#63856—38no—
46C23×C33C2311yescharacteristic
47C22×C3×C3⋊D3—22no—
48C22×C3×C3⋊D3—24no—
49C22×C3×C3⋊D3C2312yes—
50C22×C32×D3C2312yes—
51C2×C33×C4D414yescomplemented
52C2×C33×C4C2311yescharacteristic
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2314yes—
55C2×C3×C3⋊Dic3—22no—
56C2×C3×C3⋊Dic3C2312yes—
57C3×C4×C3⋊D3—24no—
58C3×C4×C3⋊D3—24no—
59C32×#67—24no—
60C2×C32×Dic3C2312yes—
61C32×D12—22no—
62C32×C4×D3—24no—
63C32×Dic6—22no—
64#6290—28no—
65#5828—94no—
66#5791—91no—
67C23×C3⋊D3C2×D311yescharacteristic
68C23×C3×D3—34no—
69C32×#41—64no—
70C2×C32×Q8—31no—
71C2×C32×D4—32no—
72C2×C32×D4—31no—
73C22×C32×C4—24no—
74C22×C32×C4—32no—
75C22×C32×C4C2×D314yes—
76C22×C32×C4C2×D311yescharacteristic
77C2×#638—34no—
78C22×C3⋊Dic3C2×D311yescharacteristic
79C2×C4×C3⋊D3—24no—
80C2×C4×C3⋊D3C2×D312yes—
81C2×C4×C3⋊D3C2×D312yes—
82C2×C3×#67C2×D318yes—
83C22×C3×Dic3—34no—
84C3×#235—216no—
85C2×C3×D12C2×D314yescomplemented
86C2×C3×C4×D3—38no—
87C2×C3×C4×D3—38no—
88C2×C3×C4×D3—616no—
89C2×C3×C4×D3C2×D318yescomplemented
90C2×C3×Dic6C2×D314yescomplemented
91C2×#625—316no—
92#3757—32no—
93#3757—38no—
94#3757—34no—
95#3757—68no—
96#3752—34no—
97C4⋊(C3⋊Dic3)—32no—
98C4⋊(C3⋊Dic3)—38no—
99C4⋊(C3⋊Dic3)—68no—
100#3750—34no—
101C4×C3⋊Dic3—32no—
102C4×C3⋊Dic3—38no—
103C4×C3⋊Dic3—68no—
104#3724—316no—
105(C2×D3)⋊Dic3—316no—
106(C2×D3)⋊Dic3—316no—
107C22×C33—22no—
108C22×C33C2×D412yes—
109C22×C33C2411yescharacteristic
110C2×C3×C3⋊D3—28no—
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3#4114yescomplemented
114C2×C32×D3—22no—
115C2×C32×D3—24no—
116C33×C4C2×D412yes—
117C33×C4C2×D412yes—
118C3⋊(C3⋊Dic3)—24no—
119C3×C3⋊Dic3—22no—
120C32×Dic3—22no—
121C2×C3×#41—34no—
122C23×C3×C4—91no—
123C2×#235C3⋊D311yescharacteristic, complemented
124C22×C4×D3—34no—
125#953—92no—
126#946—98no—
127#938—98no—
128#937—98no—
129#935—98no—
130#930—92no—
131#929—92no—
132C4×#67—916no—
133C4×#67—94no—
134#926—94no—
135#926—91no—
136#926—184no—
137#924—92no—
138C23×C32—24no—
139C23×C32—32no—
140C23×C32C22×D314yes—
141C23×C32C22×D311yescharacteristic
142C22×C3⋊D3—22no—
143C22×C3⋊D3—24no—
144C22×C3⋊D3C22×D312yes—
145C22×C3×D3—38no—
146C22×C3×D3—616no—
147C22×C3×D3—68no—
148C22×C3×D3C22×D318yes—
149C32×Q8—62no—
150C32×D4—64no—
151C32×D4—62no—
152C2×C32×C4—216no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—32no—
156C2×C32×C4—64no—
157C2×C32×C4#67116yescomplemented
158C2×C32×C4#6714yescomplemented
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163#638—68no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—34no—
166C2×C3⋊Dic3—316no—
167C2×C3⋊Dic3—616no—
168C2×C3⋊Dic3C22×D312yes—
169C4×C3⋊D3—24no—
170C4×C3⋊D3—24no—
171C3×#67—216no—
172C2×C3×Dic3—38no—
173C2×C3×Dic3—68no—
174C2×C3×Dic3C22×D318yes—
175C3×D12—28no—
176C3×C4×D3—216no—
177C3×C4×D3—616no—
178C3×C4×D3—616no—
179C3×Dic6—28no—
180#625—632no—
181#525—271noSylow
182C2×C33—22no—
183C2×C33C22×D411yescharacteristic
184C2×C33C2×#4112yes—
185C3×C3⋊D3—24no—
186C32×D3—42no—
187C24×C3—91no—
188C23×D3—34no—
189C3×#41—616no—
190C2×C3×Q8—34no—
191C2×C3×D4—38no—
192C2×C3×D4—34no—
193C22×C3×C4—24no—
194C22×C3×C4—38no—
195C22×C3×C4—91no—
196C22×C3×C4—92no—
197C22×C3×C4—92no—
198C22×C3×C4—184no—
199C22×C3×C4D3214yes—
200C22×C3×C4C2×C3⋊D311yescharacteristic
201C2×#67—916no—
202C2×#67—94no—
203C2×#67C2×C3⋊D312yes—
204C22×Dic3—34no—
205#235—24no—
206C2×D12C2×C3⋊D311yescharacteristic, complemented
207C2×C4×D3—38no—
208C2×C4×D3—38no—
209C2×C4×D3—616no—
210C2×C4×D3C2×C3⋊D312yescomplemented
211C2×Dic6C2×C3⋊D311yescharacteristic, complemented
212#217—916no—
213#217—98no—
214#217—92no—
215#217—94no—
216#217—188no—
217(C2×D3)⋊C4—916no—
218(C2×D3)⋊C4—94no—
219C4⋊Dic3—98no—
220C4⋊Dic3—92no—
221C4⋊Dic3—188no—
222Dic3⋊C4—916no—
223Dic3⋊C4—94no—
224C4×Dic3—98no—
225C4×Dic3—92no—
226C4×Dic3—188no—
227C22×C32—22no—
228C22×C32—24no—
229C22×C32—28no—
230C22×C32—28no—
231C22×C32—48no—
232C22×C32—64no—
233C22×C32—62no—
234C22×C32C2×#6718yes—
235C22×C32C2×#6712yes—
236C22×C32C23×D314yes—
237C22×C32C23×D311yescharacteristic
238C2×C3⋊D3—28no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3#23514yescomplemented
242C2×C3×D3—216no—
243C2×C3×D3—28no—
244C2×C3×D3—316no—
245C2×C3×D3—632no—
246C2×C3×D3—68no—
247C2×C3×D3—68no—
248C32×C4—28no—
249C32×C4—28no—
250C32×C4—62no—
251C32×C4C2×#6718yes—
252C32×C4C2×#6718yes—
253C32×C4C2×#6712yes—
254C32×C4C2×#6712yes—
255C3⋊Dic3—22no—
256C3⋊Dic3—64no—
257C3⋊Dic3—616no—
258C3⋊Dic3—1216no—
259C3×Dic3—28no—
260C3×Dic3—68no—
261C2×#41—31no—
262C23×C4—91no—
263#123—272no—
264C22⋊Q8—272no—
265#121—272no—
266#120—272no—
267C4×D4—274no—
268#117—271no—
269C33#52511yescharacteristic, complemented, Sylow
270C23×C3—24no—
271C23×C3—38no—
272C23×C3—92no—
273C23×C3—184no—
274C23×C3—182no—
275C23×C3C2×D3214yes—
276C23×C3C22×C3⋊D311yescharacteristic
277C22×D3—38no—
278C22×D3—616no—
279C22×D3—68no—
280C22×D3C22×C3⋊D312yes—
281C3×Q8—68no—
282C3×D4—616no—
283C3×D4—68no—
284C2×C3×C4—216no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—38no—
288C2×C3×C4—616no—
289C2×C3×C4—92no—
290C2×C3×C4—182no—
291C2×C3×C4—184no—
292C2×C3×C4—184no—
293C2×C3×C4#625116yescomplemented
294C2×C3×C4#63814yescomplemented
295C2×C3×C4C2×D3214yes—
296C2×C3×C4C2×D3214yes—
297C2×C3×C4C22×C3⋊D311yescharacteristic
298C2×C3×C4C22×C3⋊D311yescharacteristic
299#67—24no—
300#67—188no—
301#67—1832no—
302C2×Dic3—38no—
303C2×Dic3—68no—
304C2×Dic3—916no—
305C2×Dic3—94no—
306C2×Dic3—1816no—
307C2×Dic3C22×C3⋊D312yes—
308D12—22no—
309C4×D3—24no—
310C4×D3—616no—
311C4×D3—616no—
312Dic6—22no—
313C2×C32—22no—
314C2×C32—28no—
315C2×C32—24no—
316C2×C32—28no—
317C2×C32—48no—
318C2×C32—122no—
319C2×C32C2×#23512yes—
320C2×C32D3×#4118yes—
321C2×C32C22×#6714yes—
322C2×C32C22×#6711yescharacteristic
323C3⋊D3—24no—
324C3×D3—48no—
325C3×D3—616no—
326C24—91no—
327#41—64no—
328C2×Q8—31no—
329C2×D4—32no—
330C2×D4—31no—
331C2×D4—274no—
332C22×C4—32no—
333C22×C4—91no—
334C22×C4—92no—
335C22×C4—92no—
336C22×C4—184no—
337C22×C4D3×C3⋊D311yescharacteristic
338C4⋊C4—274no—
339C4⋊C4—272no—
340C22⋊C4—274no—
341C22⋊C4—274no—
342C22⋊C4—272no—
343C42—272no—
344C22×C3—28no—
345C22×C3—22no—
346C22×C3—24no—
347C22×C3—28no—
348C22×C3—48no—
349C22×C3—616no—
350C22×C3—68no—
351C22×C3—94no—
352C22×C3—188no—
353C22×C3—182no—
354C22×C3—182no—
355C22×C3C2×#62518yes—
356C22×C3C2×#63812yes—
357C22×C3C22×D3214yes—
358C22×C3C23×C3⋊D311yescharacteristic
359C2×D3—24no—
360C2×D3—22no—
361C2×D3—316no—
362C2×D3—632no—
363C2×D3—68no—
364C2×D3—68no—
365C3×C4—28no—
366C3×C4—28no—
367C3×C4—68no—
368C3×C4—182no—
369C3×C4C2×#62518yes—
370C3×C4C2×#62518yes—
371C3×C4C2×#63812yes—
372C3×C4C2×#63812yes—
373Dic3—22no—
374Dic3—68no—
375Dic3—1816no—
376Dic3—184no—
377Dic3—3616no—
378C32—24no—
379C32#579111yescharacteristic, complemented
380C32#582814yescomplemented
381C23—32no—
382C23—92no—
383C23—184no—
384C23—182no—
385C23C2×D3×C3⋊D311yescharacteristic
386Q8—62no—
387D4—64no—
388D4—62no—
389D4—548no—
390C2×C4—32no—
391C2×C4—64no—
392C2×C4—92no—
393C2×C4—182no—
394C2×C4—184no—
395C2×C4—184no—
396C2×C4—274no—
397C2×C4#629014yescomplemented
398C2×C4C2×D3×C3⋊D311yescharacteristic
399C2×C4C2×D3×C3⋊D311yescharacteristic
400C2×C3—22no—
401C2×C3—28no—
402C2×C3—28no—
403C2×C3—24no—
404C2×C3—48no—
405C2×C3—128no—
406C2×C3—184no—
407C2×C3D3×#23518yes—
408C2×C3C22×#62514yes—
409C2×C3#41×C3⋊D312yes—
410C2×C3C22×#63811yescharacteristic
411D3—42no—
412D3—616no—
413C22—22no—
414C22—64no—
415C22—62no—
416C22—94no—
417C22—188no—
418C22—182no—
419C22—182no—
420C22C2×#629012yes—
421C22C22×D3×C3⋊D311yescharacteristic
422C4—62no—
423C4—182no—
424C4—544no—
425C4C2×#629012yes—
426C4C2×#629012yes—
427C3—24no—
428C3#1059502214yescomplemented
429C3#1059629911yescharacteristic, complemented
430C2—22no—
431C2—122no—
432C2—184no—
433C2C3⋊D3×#23512yes—
434C2C22×#629011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×C3⋊D3C2semidirect
2. C2×C32×#235C2semidirect
3. #11723967C2semidirect
4. #11723634C2semidirect
5. #11723602C2semidirect
6. #11723522C2semidirect
7. C4×#6290C2semidirect
8. #11723435C2semidirect
19. C2×C32×D12C22semidirect
20. C2×C32×C4×D3C22semidirect
21. C2×C32×Dic6C22semidirect
24. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
25. C4×C3⋊(C3⋊Dic3)C22semidirect
26. #88806C22semidirect
28. (C2×D3)⋊(C3⋊Dic3)C22semidirect
30. C22×C4×C3⋊D3D3semidirect
31. C2×C3×#235D3semidirect
51. C2×C33×C4D4semidirect
85. C2×C3×D12C2×D3semidirect
89. C2×C3×C4×D3C2×D3semidirect
90. C2×C3×Dic6C2×D3semidirect
113. C2×C3×C3⋊D3#41semidirect
123. C2×#235C3⋊D3semidirect
157. C2×C32×C4#67semidirect
158. C2×C32×C4#67semidirect
206. C2×D12C2×C3⋊D3semidirect
210. C2×C4×D3C2×C3⋊D3semidirect
211. C2×Dic6C2×C3⋊D3semidirect
241. C2×C3⋊D3#235semidirect
269. C33#525semidirect
293. C2×C3×C4#625semidirect
294. C2×C3×C4#638semidirect
379. C32#5791semidirect
380. C32#5828semidirect
397. C2×C4#6290semidirect
428. C3#10595022semidirect
429. C3#10596299semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C32×#2352
#117239672
#117236342
#117236022
#117235222
C4×#62902
#117234352
#105962993
#105950223