Finite Group Databasebeta

← #13175364

Subgroups — #13175364

Order 1728.

1600 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723916C211yesmaximal, characteristic, complemented
2C3×#64081C211yesmaximal, characteristic, complemented
3C32×#869C211yesmaximal, characteristic, complemented
4#11723526C211yesmaximal, characteristic
5#11723504C212yesmaximal, complemented
6#11723494C211yesmaximal, characteristic, complemented
7#11723477C212yesmaximal, complemented
8#11723431C211yesmaximal, characteristic, complemented
9#11723411C212yesmaximal, complemented
10#11723406C211yesmaximal, characteristic, complemented
11#11723392C212yesmaximal, complemented
12#10596204—31nomaximal
13#10593877—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×Dic6C2212yes—
18C2×#6292C2212yes—
19C2×#6289C2212yes—
20C33×C42C2211yescharacteristic
21#88859C2212yes—
22#88859C2212yes—
23C3×#3752C2211yescharacteristic
24C3×#3752C2211yescharacteristic
25C3×#3750C2211yescharacteristic
26C3×#3750C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4⋊Dic3C2212yes—
30C32×C4⋊Dic3C2211yescharacteristic
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4⋊Q8—31no—
38#64156—32no—
39#64155—32no—
40#64107—32no—
41#64106—31no—
42#64104—31no—
43#64103—32no—
44#64102—31no—
45#64096—31no—
46#64084—31no—
47#64084—34no—
48#64084—64no—
49#64081D311yescharacteristic, complemented
50C3×#872—34no—
51C3×#869D314yescomplemented
52#63909—34no—
53#63907—38no—
54#63894—34no—
55#63888—38no—
56#63857—34no—
57#63854—38no—
58#63848—38no—
59#63844—34no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C4⋊(C3⋊(C3⋊D3))—22no—
67#6308—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×C4×C3⋊D3—22no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2312yes—
74C32×Dic6—42no—
75#6292—42no—
76#6289—42no—
77#5733—94no—
78#5534—91no—
79C2×C32×Q8—32no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—32no—
83C2×C4⋊(C3⋊D3)—64no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×#634—31no—
87C2×#634—34no—
88C2×#634—64no—
89C2×C3×C4×D3—34no—
90C2×C3×Dic6C2×D318yes—
91C2×#626—38no—
92C2×#624—38no—
93C32×C4⋊C4—31no—
94C32×C4⋊C4—31no—
95C32×C4⋊C4—32no—
96C32×C42—24no—
97C32×C42C2×D314yes—
98C32×C42C2×D311yescharacteristic
99#3752—32no—
100#3752—32no—
101#3752—32no—
102#3752—38no—
103#3752—38no—
104#3752—32no—
105#3752—32no—
106#3752—68no—
107#3752—68no—
108#3752C2×D311yescharacteristic
109#3752C2×D311yescharacteristic
110C4⋊(C3⋊Dic3)—32no—
111#3750—32no—
112#3750C2×D311yescharacteristic
113#3750C2×D311yescharacteristic
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×(C2×D3)⋊C4—34no—
117C3×(C2×D3)⋊C4—34no—
118C3×C4⋊Dic3C2×D314yes—
119C3×C4⋊Dic3C2×D318yes—
120C3×C4⋊Dic3C2×D314yes—
121C3×Dic3⋊C4—34no—
122C3×Dic3⋊C4—34no—
123C3×C4×Dic3—34no—
124#3724—38no—
125Dic3⋊Dic3—38no—
126#3722—38no—
127#3722—38no—
128(C2×D3)⋊Dic3—38no—
129Dic32—38no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C33×C4—21no—
138C33×C4—21no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—22no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C32×Dic3—22no—
145C32×Dic3—22no—
146C3×C4⋊Q8—34no—
147C3×#117—91no—
148#947—98no—
149#946—98no—
150#930—91no—
151#924—91no—
152#898—98no—
153#898—92no—
154#897—94no—
155#896—92no—
156#895—94no—
157#894—98no—
158#893—94no—
159#887—94no—
160#885—92no—
161#883—92no—
162#875—94no—
163#875—91no—
164#875—184no—
165C4×D12—91no—
166#872—34no—
167#869C3⋊D311yescharacteristic, complemented
168C4×Dic6—91no—
169C22×C3⋊D3—32no—
170C22×C3⋊D3—38no—
171C22×C3⋊D3—68no—
172C22×C3⋊D3C22×D311yescharacteristic
173C22×C3×D3—34no—
174C32×Q8—122no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—32no—
179C2×C32×C4—32no—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—68no—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C4⋊(C3⋊D3)—62no—
193C4⋊(C3⋊D3)—68no—
194C4⋊(C3⋊D3)—128no—
195C4⋊(C3⋊D3)—122no—
196C4×C3⋊D3—22no—
197C4×C3⋊D3—122no—
198#634—62no—
199#634—68no—
200#634—128no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3C22×D318yes—
205C2×C3×Dic3C22×D318yes—
206C3×C4×D3—68no—
207C3×Dic6—48no—
208#626—128no—
209#624—128no—
210#542—271noSylow
211C2×C33C2×#4111yescharacteristic
212C2×C33H(5,𝔽2)11yescharacteristic
213C2×C332−1+411yescharacteristic
214C3⋊(C3⋊D3)—42no—
215C3×C3⋊D3—41no—
216C2×C3×Q8—38no—
217C22×C3×C4—91no—
218C2×#67—92no—
219C2×D12—94no—
220C2×D12—91no—
221C2×D12—98no—
222C2×D12—184no—
223C2×C4×D3—34no—
224C2×C4×D3—98no—
225C2×C4×D3—92no—
226C2×Dic6—94no—
227C2×Dic6—91no—
228C2×Dic6—184no—
229C2×Dic6C2×C3⋊D312yes—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—91no—
235C3×C22⋊C4—91no—
236C3×C22⋊C4—91no—
237C3×C42—24no—
238C3×C42—91no—
239C3×C42D3214yes—
240C3×C42C2×C3⋊D311yescharacteristic
241#217—92no—
242(C2×D3)⋊C4—34no—
243(C2×D3)⋊C4—34no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—98no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—188no—
254(C2×D3)⋊C4—188no—
255C4⋊Dic3—98no—
256C4⋊Dic3C2×C3⋊D311yescharacteristic
257C4⋊Dic3C2×C3⋊D311yescharacteristic
258C4⋊Dic3C2×C3⋊D312yes—
259Dic3⋊C4—34no—
260Dic3⋊C4—34no—
261Dic3⋊C4—98no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264C4×Dic3—34no—
265C4×Dic3—98no—
266C4×Dic3—92no—
267C22×C32—24no—
268C22×C32C23×D314yes—
269C22×C32C23×D311yescharacteristic
270C2×C3⋊D3—21no—
271C2×C3⋊D3—21no—
272C2×C3⋊D3—21no—
273C2×C3⋊D3—62no—
274C2×C3⋊D3—62no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—68no—
277C2×C3⋊D3—68no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—128no—
280C2×C3⋊D3—128no—
281C2×C3⋊D3—128no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—28no—
288C32×C4—24no—
289C32×C4—24no—
290C32×C4—44no—
291C32×C4—44no—
292C32×C4—62no—
293C32×C4—62no—
294C32×C4#23512yes—
295C32×C4#23918yes—
296C3⋊Dic3—21no—
297C3⋊Dic3—21no—
298C3⋊Dic3—21no—
299C3⋊Dic3—62no—
300C3⋊Dic3—68no—
301C3⋊Dic3—128no—
302C3×Dic3—28no—
303C3×Dic3—28no—
304C3×Dic3—64no—
305C3×Dic3—64no—
306C3×Dic3—64no—
307C4⋊Q8—31no—
308#126—272no—
309#124—272no—
310#124—271no—
311#123—272no—
312C22⋊Q8—271no—
313C22⋊Q8—272no—
314#121—271no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—91no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C22×D3—34no—
321C22×D3—98no—
322C22×D3—92no—
323C22×D3—188no—
324C3×Q8—128no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—38no—
329C2×C3×C4—38no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—182no—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338C2×C3×C4C22×C3⋊D311yescharacteristic
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—362no—
341C2×Dic3—34no—
342C2×Dic3—34no—
343C2×Dic3—34no—
344C2×Dic3—98no—
345C2×Dic3—92no—
346C2×Dic3—188no—
347C2×Dic3C22×C3⋊D312yes—
348C2×Dic3C22×C3⋊D312yes—
349D12—188no—
350D12—182no—
351D12—368no—
352D12—368no—
353C4×D3—68no—
354C4×D3—362no—
355C4×D3—368no—
356Dic6—42no—
357Dic6—188no—
358Dic6—182no—
359Dic6—368no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32C2×#23511yescharacteristic
364C2×C32#100414yes—
365C2×C32C2×#23914yes—
366C2×C32#100714yes—
367C2×C32#100911yescharacteristic
368C2×C32#101011yescharacteristic
369C3⋊D3—41no—
370C3⋊D3—122no—
371C3⋊D3—128no—
372C3⋊D3—248no—
373C3×D3—124no—
374C2×Q8—32no—
375C2×Q8—271no—
376C2×D4—272no—
377C2×D4—271no—
378C22×C4—91no—
379C22×C4—272no—
380C4⋊C4—31no—
381C4⋊C4—31no—
382C4⋊C4—32no—
383C4⋊C4—91no—
384C4⋊C4—91no—
385C4⋊C4—272no—
386C4⋊C4—272no—
387C22⋊C4—91no—
388C22⋊C4—91no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—91no—
395C42—272no—
396C42D3×C3⋊D311yescharacteristic
397C22×C3—24no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3C22×D3214yes—
402C22×C3C23×C3⋊D311yescharacteristic
403C2×D3—64no—
404C2×D3—64no—
405C2×D3—64no—
406C2×D3—188no—
407C2×D3—188no—
408C2×D3—188no—
409C2×D3—182no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—368no—
413C2×D3—368no—
414C2×D3—368no—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—28no—
420C3×C4—44no—
421C3×C4—44no—
422C3×C4—68no—
423C3×C4—68no—
424C3×C4—181no—
425C3×C4—181no—
426C3×C4—181no—
427C3×C4#379918yes—
428C3×C4#383212yes—
429Dic3—22no—
430Dic3—22no—
431Dic3—64no—
432Dic3—64no—
433Dic3—64no—
434Dic3—188no—
435Dic3—182no—
436Dic3—368no—
437C32—24no—
438C32#553411yescharacteristic, complemented
439C32#573314yescomplemented
440C23—91no—
441C23—272no—
442Q8—122no—
443Q8—542no—
444D4—542no—
445D4—1082no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—182no—
452C2×C4—272no—
453C2×C4—1082no—
454C2×C4C2×D3×C3⋊D311yescharacteristic
455C2×C4C2×D3×C3⋊D311yescharacteristic
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—361no—
461C2×C3#6429714yes—
462C2×C3C2×#379914yes—
463C2×C3#6430714yes—
464C2×C3#6436111yescharacteristic
465C2×C3C2×#383211yescharacteristic
466C2×C3#6436411yescharacteristic
467D3—124no—
468D3—368no—
469D3—362no—
470D3—728no—
471C22—181no—
472C22—181no—
473C22—181no—
474C22—542no—
475C22—542no—
476C22—542no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—21no—
479C4—21no—
480C4—62no—
481C4—62no—
482C4—181no—
483C4—181no—
484C4—181no—
485C4—542no—
486C4#8901812yes—
487C3—24no—
488C3#1059387714yescomplemented
489C3#1059620411yescharacteristic, complemented
490C2—361no—
491C2—1082no—
492C2#1172456611yescharacteristic
493C2C2×#8901811yescharacteristic
494C2#1172457911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723526C2
hypercenterC42D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723916C2semidirect
2. C3×#64081C2semidirect
3. C32×#869C2semidirect
5. #11723504C2semidirect
6. #11723494C2semidirect
7. #11723477C2semidirect
8. #11723431C2semidirect
9. #11723411C2semidirect
10. #11723406C2semidirect
11. #11723392C2semidirect
49. #64081D3semidirect
51. C3×#869D3semidirect
167. #869C3⋊D3semidirect
318. C33#542semidirect
438. C32#5534semidirect
439. C32#5733semidirect
488. C3#10593877semidirect
489. C3#10596204semidirect

Maximal subgroups

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

subgroupindex
#117239162
C3×#640812
C32×#8692
#117235262
#117235042
#117234942
#117234772
#117234312
#117234112
#117234062
#117233922
#105962043
#105938773