Finite Group Databasebeta

← #13177236

Subgroups — #13177236

Order 1728.

1876 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723971C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic, complemented
3C32×#924C211yesmaximal, characteristic, complemented
4#11723636C212yesmaximal, complemented
5#11723622C212yesmaximal, complemented
6#11723504C212yesmaximal, complemented
7#11723486C212yesmaximal, complemented
8#11723408C212yesmaximal, complemented
9#11723392C212yesmaximal, complemented
10#10596115—31nomaximal
11#10595038—34nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C2×#6312C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic, complemented
18C2×#6292C2212yes—
19C2×#6292C2212yes—
20C2×#6289C2212yescomplemented
21#88859C2212yescomplemented
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
23C3×#3752C2212yes—
24C3×#3752C2212yes—
25C32×#217C2212yes—
26C32×C4⋊Dic3C2211yescharacteristic, complemented
27C32×Dic3⋊C4C2212yescomplemented
28#88806C2212yescomplemented
29#88804C2212yescomplemented
30#88803C2212yes—
31#88803C2212yes—
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×C22⋊Q8—31no—
34#64156—32no—
35#64139—31no—
36#64139—34no—
37#64139—64no—
38C2×#3752D311yescharacteristic, complemented
39#64107—32no—
40#64104—32no—
41#64103—32no—
42#64091—32no—
43#64090—32no—
44C2×C3×(C2×D3)⋊C4—34no—
45C3×#924D314yescomplemented
46#63977—38no—
47#63952—38no—
48#63910—38no—
49#63907—38no—
50#63861—38no—
51#63854—38no—
52C23×C33C2311yescharacteristic
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C3×C3⋊D3—22no—
55C22×C3×C3⋊D3—41no—
56C22×C3×C3⋊D3—41no—
57C22×C3×C3⋊D3C2312yes—
58C2×C33×C4—41no—
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61#6312—24no—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C4⋊(C3⋊(C3⋊D3))—41no—
64C2×C3×C3⋊Dic3—41no—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68C32×Dic6—41no—
69#6292—42no—
70#6292—42no—
71#6289—42no—
72#5644—94no—
73#5494—91no—
74C23×C3⋊D3C2×D311yescharacteristic
75C23×C3×D3—34no—
76C2×C32×Q8—31no—
77C22×C32×C4—24no—
78C22×C32×C4C2×D314yes—
79C22×C32×C4C2×D311yescharacteristic
80C2×#638—32no—
81C2×#638—38no—
82C2×#638—68no—
83C22×C3⋊Dic3C2×D311yescharacteristic
84C2×C4⋊(C3⋊D3)—32no—
85C2×C4⋊(C3⋊D3)—32no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4⋊(C3⋊D3)—34no—
88C2×C4⋊(C3⋊D3)—64no—
89C2×C4×C3⋊D3—32no—
90C22×C3×Dic3—34no—
91C2×C3×Dic6C2×D314yescomplemented
92C2×#626—38no—
93C2×#626—38no—
94C2×#624—38no—
95C32×C4⋊C4—31no—
96C32×C4⋊C4—32no—
97C32×C22⋊C4—32no—
98#3752—32no—
99#3752—32no—
100#3752—32no—
101#3752—38no—
102#3752—32no—
103#3752—68no—
104#3752C2×D312yes—
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—34no—
108C4⋊(C3⋊Dic3)—64no—
109#3750—32no—
110C4×C3⋊Dic3—32no—
111C3×#217C2×D318yes—
112C3×(C2×D3)⋊C4—38no—
113C3×(C2×D3)⋊C4—38no—
114C3×C4⋊Dic3C2×D314yescomplemented
115C3×Dic3⋊C4C2×D318yescomplemented
116#3724—38no—
117#3722—38no—
118(C2×D3)⋊Dic3—38no—
119(C2×D3)⋊Dic3—38no—
120Dic32—38no—
121C22×C33—21no—
122C22×C33—21no—
123C22×C33#4112yes—
124C22×C33C2411yescharacteristic
125C2×C3⋊(C3⋊D3)—22no—
126C2×C3⋊(C3⋊D3)—22no—
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—44no—
132C33×C4—21no—
133C33×C4—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C32×Dic3—22no—
137C32×Dic3—22no—
138C3×C22⋊Q8—34no—
139C2×C3×C22⋊C4—91no—
140#953—92no—
141#947—98no—
142#930—94no—
143#930—91no—
144#930—184no—
145C4×#67—92no—
146C2×(C2×D3)⋊C4—34no—
147#924C3⋊D311yescharacteristic, complemented
148#898—98no—
149#895—98no—
150#894—98no—
151#885—92no—
152#884—92no—
153#883—92no—
154#882—98no—
155#67⋊C4—98no—
156#879—92no—
157C23×C32—24no—
158C23×C32C22×D314yes—
159C23×C32C22×D311yescharacteristic
160C22×C3⋊D3—22no—
161C22×C3⋊D3—32no—
162C22×C3⋊D3—38no—
163C22×C3⋊D3—41no—
164C22×C3⋊D3—41no—
165C22×C3⋊D3—68no—
166C22×C3⋊D3C22×D312yes—
167C22×C3×D3—38no—
168C22×C3×D3—68no—
169C22×C3×D3—124no—
170C22×C3×D3—124no—
171C32×Q8—121no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—32no—
175C2×C32×C4—32no—
176C2×C32×C4—41no—
177C2×C32×C4—44no—
178C2×C32×C4—48no—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183#638—64no—
184#638—616no—
185#638—1216no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—41no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3C22×D312yes—
191C4⋊(C3⋊D3)—121no—
192C4⋊(C3⋊D3)—124no—
193C4⋊(C3⋊D3)—128no—
194C4⋊(C3⋊D3)—122no—
195C4⋊(C3⋊D3)—122no—
196C4×C3⋊D3—122no—
197C2×C3×Dic3—38no—
198C2×C3×Dic3—124no—
199C2×C3×Dic3C22×D318yes—
200C2×C3×Dic3C22×D318yes—
201C3×Dic6—44no—
202#626—128no—
203#626—128no—
204#624—128no—
205#538—271noSylow
206C2×C33—22no—
207C2×C33C2×#4111yescharacteristic
208C2×C33H(5,𝔽2)11yescharacteristic
209C2×C33H(5,𝔽2)11yescharacteristic
210C3⋊(C3⋊D3)—42no—
211C3×C3⋊D3—42no—
212C24×C3—91no—
213C23×D3—34no—
214C2×C3×Q8—34no—
215C22×C3×C4—24no—
216C22×C3×C4—91no—
217C22×C3×C4D3214yes—
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C2×#67—98no—
220C2×#67—92no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67—188no—
224C22×Dic3—34no—
225C2×D12—98no—
226C2×D12—98no—
227C2×D12—94no—
228C2×D12—91no—
229C2×D12—184no—
230C2×C4×D3—98no—
231C2×C4×D3—92no—
232C2×Dic6C2×C3⋊D311yescharacteristic, complemented
233C3×C4⋊C4—34no—
234C3×C4⋊C4—38no—
235C3×C22⋊C4—38no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238#217—92no—
239#217—92no—
240#217C2×C3⋊D312yes—
241(C2×D3)⋊C4—38no—
242(C2×D3)⋊C4—38no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—188no—
250C4⋊Dic3—94no—
251C4⋊Dic3—91no—
252C4⋊Dic3—184no—
253C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
254Dic3⋊C4—98no—
255Dic3⋊C4—92no—
256Dic3⋊C4C2×C3⋊D312yescomplemented
257C4×Dic3—98no—
258C4×Dic3—92no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—24no—
262C22×C32—28no—
263C22×C32—24no—
264C22×C32—24no—
265C22×C32—44no—
266C22×C32—44no—
267C22×C32#23512yes—
268C22×C32#23918yes—
269C22×C32C23×D314yes—
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—44no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—68no—
280C2×C3⋊D3—68no—
281C2×C3⋊D3—128no—
282C2×C3⋊D3—128no—
283C2×C3⋊D3—128no—
284C2×C3×D3—68no—
285C2×C3×D3—68no—
286C2×C3×D3—68no—
287C2×C3×D3—1216no—
288C32×C4—28no—
289C32×C4—24no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—24no—
293C32×C4—44no—
294C32×C4—62no—
295C32×C4—62no—
296C3⋊Dic3—22no—
297C3⋊Dic3—62no—
298C3⋊Dic3—68no—
299C3⋊Dic3—128no—
300C3×Dic3—28no—
301C3×Dic3—28no—
302C3×Dic3—68no—
303#126—272no—
304#124—272no—
305#123—272no—
306C22⋊Q8—31no—
307#121—272no—
308#121—271no—
309#120—272no—
310C4×D4—272no—
311C2×C22⋊C4—91no—
312C33#53811yescharacteristic, complemented, Sylow
313C23×C3—24no—
314C23×C3—92no—
315C23×C3—182no—
316C23×C3—361no—
317C23×C3—361no—
318C23×C3C2×D3214yes—
319C23×C3C22×C3⋊D311yescharacteristic
320C22×D3—38no—
321C22×D3—68no—
322C22×D3—98no—
323C22×D3—92no—
324C22×D3—124no—
325C22×D3—124no—
326C22×D3—188no—
327C3×Q8—124no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—38no—
331C2×C3×C4—38no—
332C2×C3×C4—44no—
333C2×C3×C4—41no—
334C2×C3×C4—48no—
335C2×C3×C4—92no—
336C2×C3×C4—361no—
337C2×C3×C4C2×D3214yes—
338C2×C3×C4C2×D3214yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340C2×C3×C4C22×C3⋊D311yescharacteristic
341#67—1816no—
342#67—184no—
343#67—3616no—
344#67—362no—
345#67—362no—
346C2×Dic3—38no—
347C2×Dic3—98no—
348C2×Dic3—92no—
349C2×Dic3—124no—
350C2×Dic3—188no—
351C2×Dic3C22×C3⋊D312yes—
352C2×Dic3C22×C3⋊D312yes—
353D12—364no—
354D12—361no—
355D12—368no—
356D12—368no—
357D12—368no—
358C4×D3—362no—
359C4×D3—368no—
360Dic6—41no—
361C2×C32—22no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32—24no—
365C2×C32—28no—
366C2×C32—48no—
367C2×C32C2×#23511yescharacteristic
368C2×C32#100414yes—
369C2×C32#100411yescharacteristic
370C2×C32#100411yescharacteristic
371C2×C32C2×#23914yes—
372C2×C32#100914yes—
373C3⋊D3—42no—
374C3⋊D3—122no—
375C3⋊D3—128no—
376C3⋊D3—248no—
377C3×D3—128no—
378C24—91no—
379C2×Q8—31no—
380C2×D4—272no—
381C2×D4—272no—
382C2×D4—272no—
383C2×D4—271no—
384C22×C4—91no—
385C22×C4—272no—
386C22×C4D3×C3⋊D311yescharacteristic
387C4⋊C4—32no—
388C4⋊C4—31no—
389C4⋊C4—272no—
390C4⋊C4—271no—
391C22⋊C4—32no—
392C22⋊C4—92no—
393C22⋊C4—92no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C42—272no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—24no—
404C22×C3—28no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—364no—
411C22×C3#379918yes—
412C22×C3#383212yes—
413C22×C3C22×D3214yes—
414C22×C3C23×C3⋊D311yescharacteristic
415C2×D3—68no—
416C2×D3—68no—
417C2×D3—68no—
418C2×D3—1216no—
419C2×D3—188no—
420C2×D3—188no—
421C2×D3—188no—
422C2×D3—182no—
423C2×D3—182no—
424C2×D3—182no—
425C2×D3—368no—
426C2×D3—368no—
427C2×D3—368no—
428C3×C4—28no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—44no—
434C3×C4—68no—
435C3×C4—68no—
436C3×C4—182no—
437Dic3—22no—
438Dic3—22no—
439Dic3—68no—
440Dic3—188no—
441Dic3—182no—
442Dic3—368no—
443C32—24no—
444C32#549411yescharacteristic, complemented
445C32#564414yescomplemented
446C23—92no—
447C23—182no—
448C23—272no—
449C23—361no—
450C23—361no—
451C23C2×D3×C3⋊D311yescharacteristic
452Q8—121no—
453D4—544no—
454D4—1081no—
455D4—1082no—
456D4—1082no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—41no—
460C2×C4—92no—
461C2×C4—272no—
462C2×C4—361no—
463C2×C4—1082no—
464C2×C4C2×D3×C3⋊D311yescharacteristic
465C2×C4C2×D3×C3⋊D311yescharacteristic
466C2×C3—22no—
467C2×C3—28no—
468C2×C3—24no—
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—48no—
472C2×C3—362no—
473C2×C3C2×#379914yes—
474C2×C3#6430714yes—
475C2×C3#6433014yes—
476C2×C3#6436111yescharacteristic
477C2×C3C2×#383211yescharacteristic
478C2×C3#6436611yescharacteristic
479D3—128no—
480D3—362no—
481D3—368no—
482D3—728no—
483C22—21no—
484C22—21no—
485C22—182no—
486C22—182no—
487C22—182no—
488C22—364no—
489C22—542no—
490C22—542no—
491C22—542no—
492C22#8901812yes—
493C22C22×D3×C3⋊D311yescharacteristic
494C4—21no—
495C4—21no—
496C4—62no—
497C4—62no—
498C4—182no—
499C4—542no—
500C3—24no—
501C3#1059503814yescomplemented
502C3#1059611511yescharacteristic, complemented
503C2—22no—
504C2—362no—
505C2—1082no—
506C2C2×#8901811yescharacteristic
507C2#1172458011yescharacteristic
508C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723971C2semidirect
2. C2×C3×#3752C2semidirect
3. C32×#924C2semidirect
4. #11723636C2semidirect
5. #11723622C2semidirect
6. #11723504C2semidirect
7. #11723486C2semidirect
8. #11723408C2semidirect
9. #11723392C2semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C2×C32×Dic6C22semidirect
20. C2×#6289C22semidirect
21. #88859C22semidirect
22. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
26. C32×C4⋊Dic3C22semidirect
27. C32×Dic3⋊C4C22semidirect
28. #88806C22semidirect
29. #88804C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
38. C2×#3752D3semidirect
45. C3×#924D3semidirect
91. C2×C3×Dic6C2×D3semidirect
114. C3×C4⋊Dic3C2×D3semidirect
115. C3×Dic3⋊C4C2×D3semidirect
147. #924C3⋊D3semidirect
232. C2×Dic6C2×C3⋊D3semidirect
253. C4⋊Dic3C2×C3⋊D3semidirect
256. Dic3⋊C4C2×C3⋊D3semidirect
312. C33#538semidirect
444. C32#5494semidirect
445. C32#5644semidirect
501. C3#10595038semidirect
502. C3#10596115semidirect

Maximal subgroups

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

subgroupindex
#117239712
C2×C3×#37522
C32×#9242
#117236362
#117236222
#117235042
#117234862
#117234082
#117233922
#105961153
#105950383