Finite Group Databasebeta

← #13177434

Subgroups — #13177434

Order 1728.

2948 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic, complemented
3C3×#64145C211yesmaximal, characteristic, complemented
4C32×#940C211yesmaximal, characteristic, complemented
5#11723638C214yesmaximal, complemented
6#11723616C214yesmaximal, complemented
7#11723524C211yesmaximal, characteristic, complemented
8#11723452C211yesmaximal, characteristic, complemented
9#11723441C211yesmaximal, characteristic, complemented
10#10596186—31nomaximal
11#10595168—34nomaximal
12C23×C3⋊(C3⋊D3)C2212yes—
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2212yes—
15D4×C3⋊(C3⋊D3)—24no—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic, complemented
19C2×C32×#67C2214yescomplemented
20C2×C32×D12C2211yescharacteristic, complemented
21C2×#6291C2212yes—
22C2×#6291C2214yescomplemented
23C2×D3×C3⋊Dic3C2212yescomplemented
24#89015—24no—
25C3×#3757C2214yescomplemented
26C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
27C32×C4×Dic3C2211yescharacteristic, complemented
28#88806C2212yescomplemented
29#88804C2212yes—
30#88804C2214yescomplemented
31C2×#3830—31no—
32C2×D4×C3⋊D3—31no—
33C2×D4×C3⋊D3—34no—
34C2×D4×C3⋊D3—64no—
35C2×#3796—34no—
36C32×#127—31no—
37#64149—31no—
38#64148—34no—
39#64146—34no—
40#64145D311yescharacteristic, complemented
41#64084—31no—
42#64081—31no—
43C3×#940D314yescomplemented
44C3×#936—34no—
45#63981—316no—
46#63971—316no—
47#63911—34no—
48#63874—34no—
49#63864—34no—
50C23×C33C2312yes—
51C22×C3⋊(C3⋊D3)—24no—
52C22×C3⋊(C3⋊D3)—44no—
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C3⋊(C3⋊D3)C2311yescharacteristic
55C22×C32×D3C2312yes—
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58#6312—24no—
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C4⋊(C3⋊(C3⋊D3))—22no—
61C4×C3⋊(C3⋊D3)D414yescomplemented
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C3×#634—22no—
65C32×#67—44no—
66C2×C32×Dic3C2312yes—
67C32×D12—22no—
68#6291—24no—
69#6291—44no—
70D3×C3⋊Dic3—24no—
71#5715—94no—
72#5674—91no—
73C23×C3⋊D3—32no—
74C23×C3⋊D3—38no—
75C23×C3⋊D3—68no—
76C2×C32×D4—24no—
77C2×C32×D4—34no—
78C2×C32×D4—31no—
79C2×C32×D4C2×D314yes—
80C2×C32×D4C2×D311yescharacteristic
81C2×#638—32no—
82C2×#638—38no—
83C2×#638—32no—
84C2×#638—34no—
85C2×#638—68no—
86C22×C3⋊Dic3—32no—
87#3830—64no—
88D4×C3⋊D3—64no—
89D4×C3⋊D3—616no—
90D4×C3⋊D3—632no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C4⋊(C3⋊D3)—34no—
93C2×C4⋊(C3⋊D3)—64no—
94C2×C4×C3⋊D3—31no—
95C2×C4×C3⋊D3—34no—
96C2×C4×C3⋊D3—64no—
97C2×#634C2×D311yescharacteristic, complemented
98C2×C3×#67C2×D3116yescomplemented
99C2×C3×D12C2×D314yescomplemented
100C2×C3×Dic6—34no—
101C2×#626—38no—
102C2×#626—316no—
103C2×D3×Dic3—38no—
104#3796—616no—
105C32×C42—31no—
106#3757C2×D314yescomplemented
107#3752—32no—
108#3752—34no—
109#3750—32no—
110C4×C3⋊Dic3C2×D311yescharacteristic, complemented
111C3×#217—316no—
112C3×C4×Dic3—34no—
113C3×C4×Dic3C2×D314yescomplemented
114#3724—38no—
115#3722—316no—
116#3722—38no—
117C22×C33—22no—
118C22×C33—24no—
119C22×C33C2411yescharacteristic
120C2×C3⋊(C3⋊D3)—22no—
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)—24no—
123C2×C3⋊(C3⋊D3)—48no—
124C2×C3⋊(C3⋊D3)C2×D412yes—
125C2×C32×D3—22no—
126C2×C32×D3—24no—
127C33×C4C2×D412yes—
128C3⋊(C3⋊Dic3)C2×D412yes—
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—22no—
131C32×Dic3—22no—
132C2×#239—91no—
133C2×#237—94no—
134C2×D3×D4—94no—
135C2×D3×D4—91no—
136C2×D3×D4—184no—
137C3×#127—34no—
138C3×#124—91no—
139#947—91no—
140#940—94no—
141#940C3⋊D311yescharacteristic, complemented
142#939—916no—
143#937—916no—
144#936—34no—
145#884—94no—
146#882—94no—
147#875—94no—
148#874—91no—
149#872—94no—
150#872—91no—
151C23×C32—28no—
152C23×C32—32no—
153C23×C32C22×D318yes—
154C23×C32C22×D312yes—
155C22×C3⋊D3—31no—
156C22×C3⋊D3—32no—
157C22×C3⋊D3—34no—
158C22×C3⋊D3—38no—
159C22×C3⋊D3—64no—
160C22×C3⋊D3—68no—
161C22×C3⋊D3—64no—
162C22×C3⋊D3—616no—
163C22×C3⋊D3—632no—
164C22×C3⋊D3—124no—
165C22×C3⋊D3—1216no—
166C22×C3⋊D3—1232no—
167C22×C3×D3C22×D318yes—
168C32×D4—216no—
169C32×D4—28no—
170C32×D4—22no—
171C32×D4—62no—
172C32×D4—124no—
173C2×C32×C4—24no—
174C2×C32×C4—32no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D311yescharacteristic
177#638—64no—
178#638—64no—
179#638—616no—
180#638—632no—
181#638—124no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—34no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—64no—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188C4⋊(C3⋊D3)—616no—
189C4⋊(C3⋊D3)—68no—
190C4⋊(C3⋊D3)—62no—
191C4×C3⋊D3—34no—
192C4×C3⋊D3—316no—
193C4×C3⋊D3—616no—
194#634—22no—
195C3×#67—416no—
196C2×C3×Dic3—38no—
197C2×C3×Dic3—38no—
198C2×C3×Dic3C22×D318yes—
199C3×D12—28no—
200C3×Dic6—68no—
201#626—616no—
202#626—1216no—
203D3×Dic3—616no—
204#535—271noSylow
205C2×C33—42no—
206C2×C33C22×D411yescharacteristic
207C2×C33H(5,𝔽2)12yes—
208C3⋊(C3⋊D3)—22no—
209C3⋊(C3⋊D3)—42no—
210C32×D3—42no—
211C23×D3—98no—
212C23×D3—92no—
213C23×D3—188no—
214C2×C3×Q8—91no—
215C2×C3×D4—24no—
216C2×C3×D4—316no—
217C2×C3×D4—34no—
218C2×C3×D4D3214yes—
219C2×C3×D4C2×C3⋊D311yescharacteristic
220C2×#67—98no—
221C2×#67—916no—
222C2×#67—98no—
223C2×#67—92no—
224C2×#67—188no—
225C2×#67C2×C3⋊D314yescomplemented
226C22×Dic3—98no—
227#239—184no—
228#237—1816no—
229D3×D4—1816no—
230D3×D4—184no—
231D3×D4—1832no—
232C2×D12—92no—
233C2×D12—94no—
234C2×D12—91no—
235C2×D12—94no—
236C2×D12—184no—
237C2×D12C2×C3⋊D311yescharacteristic, complemented
238C2×C4×D3—94no—
239C2×C4×D3—91no—
240C2×C4×D3—92no—
241C2×C4×D3—184no—
242C2×Dic6—34no—
243C3×C22⋊C4—94no—
244C3×C42—34no—
245C3×C42—91no—
246#217—316no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—916no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—94no—
251Dic3⋊C4—98no—
252Dic3⋊C4—92no—
253C4×Dic3—34no—
254C4×Dic3C2×C3⋊D311yescharacteristic, complemented
255C22×C32—216no—
256C22×C32—28no—
257C22×C32—22no—
258C22×C32—24no—
259C22×C32—24no—
260C22×C32—216no—
261C22×C32—416no—
262C22×C32—62no—
263C22×C32—64no—
264C22×C32C23×D314yes—
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—32no—
267C2×C3⋊D3—38no—
268C2×C3⋊D3—68no—
269C2×C3⋊D3—616no—
270C2×C3⋊D3—68no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—616no—
273C2×C3⋊D3—68no—
274C2×C3⋊D3—62no—
275C2×C3⋊D3—64no—
276C2×C3⋊D3—616no—
277C2×C3⋊D3—1216no—
278C2×C3⋊D3—1264no—
279C2×C3⋊D3—128no—
280C2×C3⋊D3—1232no—
281C2×C3×D3—28no—
282C2×C3×D3—216no—
283C32×C4—28no—
284C32×C4—62no—
285C32×C4D3×D418yes—
286C32×C4D3×D412yes—
287C3⋊Dic3—22no—
288C3⋊Dic3—22no—
289C3⋊Dic3—32no—
290C3⋊Dic3—38no—
291C3⋊Dic3—68no—
292C3×Dic3—28no—
293C3×Dic3—68no—
294C3×Dic3—68no—
295C2×#41—271no—
296C22×D4—271no—
297#127—31no—
298#127—271no—
299#124—91no—
300#124—271no—
301#121—274no—
302#120—274no—
303#117—271no—
304C33#53511yescharacteristic, complemented, Sylow
305C23×C3—28no—
306C23×C3—38no—
307C23×C3C2×D3218yes—
308C23×C3C22×C3⋊D312yes—
309C22×D3—94no—
310C22×D3—91no—
311C22×D3—98no—
312C22×D3—92no—
313C22×D3—184no—
314C22×D3—188no—
315C22×D3—1816no—
316C22×D3—184no—
317C22×D3—1832no—
318C22×D3—3616no—
319C22×D3—364no—
320C22×D3—3632no—
321C22×D3C22×C3⋊D312yes—
322C3×Q8—182no—
323C3×D4—216no—
324C3×D4—28no—
325C3×D4—22no—
326C3×D4—68no—
327C3×D4—1216no—
328C2×C3×C4—24no—
329C2×C3×C4—38no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4C2×D3214yes—
333C2×C3×C4C22×C3⋊D311yescharacteristic
334#67—44no—
335#67—1816no—
336#67—1816no—
337#67—184no—
338#67—1832no—
339#67—3616no—
340C2×Dic3—38no—
341C2×Dic3—38no—
342C2×Dic3—94no—
343C2×Dic3—91no—
344C2×Dic3—184no—
345C2×Dic3—1816no—
346C2×Dic3C22×C3⋊D312yes—
347D12—22no—
348D12—1816no—
349D12—184no—
350D12—188no—
351D12—182no—
352D12—364no—
353C4×D3—916no—
354C4×D3—94no—
355C4×D3—1816no—
356C4×D3—184no—
357Dic6—68no—
358C2×C32—28no—
359C2×C32—24no—
360C2×C32—416no—
361C2×C32—42no—
362C2×C32—48no—
363C2×C32—122no—
364C2×C32C2×D3×D414yes—
365C2×C32C2×D3×D411yescharacteristic
366C2×C32#100418yes—
367C2×C32#100912yes—
368C3⋊D3—616no—
369C3⋊D3—62no—
370C3⋊D3—68no—
371C3⋊D3—1216no—
372C3⋊D3—122no—
373C3⋊D3—128no—
374C3×D3—48no—
375C24—272no—
376#41—544no—
377C2×Q8—91no—
378C2×D4—34no—
379C2×D4—31no—
380C2×D4—272no—
381C2×D4—274no—
382C2×D4—272no—
383C2×D4—271no—
384C2×D4—544no—
385C2×D4D3×C3⋊D311yescharacteristic
386C22×C4—272no—
387C22×C4—271no—
388C4⋊C4—272no—
389C22⋊C4—94no—
390C22⋊C4—272no—
391C22⋊C4—274no—
392C42—31no—
393C42—91no—
394C22×C3—216no—
395C22×C3—216no—
396C22×C3—28no—
397C22×C3—22no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—416no—
401C22×C3—616no—
402C22×C3—68no—
403C22×C3C22×D3214yes—
404C22×C3C23×C3⋊D311yescharacteristic
405C2×D3—22no—
406C2×D3—24no—
407C2×D3—98no—
408C2×D3—92no—
409C2×D3—188no—
410C2×D3—1816no—
411C2×D3—188no—
412C2×D3—182no—
413C2×D3—1816no—
414C2×D3—1816no—
415C2×D3—188no—
416C2×D3—182no—
417C2×D3—184no—
418C2×D3—3616no—
419C2×D3—3664no—
420C2×D3—368no—
421C2×D3—3632no—
422C3×C4—28no—
423C3×C4—68no—
424C3×C4—182no—
425C3×C4—182no—
426C3×C4#381118yes—
427C3×C4D4×C3⋊D312yes—
428Dic3—22no—
429Dic3—68no—
430Dic3—68no—
431Dic3—98no—
432Dic3—92no—
433Dic3—188no—
434C32—24no—
435C32#567411yescharacteristic, complemented
436C32#571514yescomplemented
437C23—32no—
438C23—271no—
439C23—272no—
440C23—544no—
441C23—1084no—
442C23C2×D3×C3⋊D312yes—
443Q8—182no—
444D4—22no—
445D4—62no—
446D4—124no—
447D4—544no—
448D4—544no—
449D4—542no—
450D4—1084no—
451C2×C4—32no—
452C2×C4—92no—
453C2×C4—92no—
454C2×C4—271no—
455C2×C4—274no—
456C2×C4—544no—
457C2×C4C2×D3×C3⋊D311yescharacteristic
458C2×C3—28no—
459C2×C3—24no—
460C2×C3—416no—
461C2×C3—42no—
462C2×C3—48no—
463C2×C3—128no—
464C2×C3#6431418yes—
465C2×C3C2×#381114yes—
466C2×C3C2×D4×C3⋊D311yescharacteristic
467C2×C3#6436112yes—
468D3—42no—
469D3—1816no—
470D3—182no—
471D3—188no—
472D3—3616no—
473D3—362no—
474D3—368no—
475C22—24no—
476C22—22no—
477C22—64no—
478C22—62no—
479C22—272no—
480C22—542no—
481C22—544no—
482C22—542no—
483C22—1088no—
484C22C22×D3×C3⋊D311yescharacteristic
485C4—62no—
486C4—182no—
487C4—182no—
488C4—272no—
489C4#8903912yes—
490C3—24no—
491C3#1059516814yescomplemented
492C3#1059618611yescharacteristic, complemented
493C2—42no—
494C2—122no—
495C2—542no—
496C2—1082no—
497C2#1172458812yes—
498C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊(C3⋊D3)C2semidirect
2. C2×#89015C2semidirect
3. C3×#64145C2semidirect
4. C32×#940C2semidirect
5. #11723638C2semidirect
6. #11723616C2semidirect
7. #11723524C2semidirect
8. #11723452C2semidirect
9. #11723441C2semidirect
18. C2×C3×#634C22semidirect
19. C2×C32×#67C22semidirect
20. C2×C32×D12C22semidirect
22. C2×#6291C22semidirect
23. C2×D3×C3⋊Dic3C22semidirect
25. C3×#3757C22semidirect
26. C3×C4×C3⋊Dic3C22semidirect
27. C32×C4×Dic3C22semidirect
28. #88806C22semidirect
30. #88804C22semidirect
40. #64145D3semidirect
43. C3×#940D3semidirect
61. C4×C3⋊(C3⋊D3)D4semidirect
97. C2×#634C2×D3semidirect
98. C2×C3×#67C2×D3semidirect
99. C2×C3×D12C2×D3semidirect
106. #3757C2×D3semidirect
110. C4×C3⋊Dic3C2×D3semidirect
113. C3×C4×Dic3C2×D3semidirect
141. #940C3⋊D3semidirect
225. C2×#67C2×C3⋊D3semidirect
237. C2×D12C2×C3⋊D3semidirect
254. C4×Dic3C2×C3⋊D3semidirect
304. C33#535semidirect
435. C32#5674semidirect
436. C32#5715semidirect
491. C3#10595168semidirect
492. C3#10596186semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×#890152
C3×#641452
C32×#9402
#117236382
#117236162
#117235242
#117234522
#117234412
#105961863
#105951683