Finite Group Databasebeta

← #13175338

Subgroups — #13175338

Order 1728.

1720 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723916C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C4×Dic6C211yesmaximal, characteristic, complemented
4#11723517C211yesmaximal, characteristic
5#11723512C212yesmaximal, complemented
6#11723504C212yesmaximal, complemented
7#11723484C211yesmaximal, characteristic, complemented
8#11723439C211yesmaximal, characteristic, complemented
9#11723417C212yesmaximal, complemented
10#11723399C211yesmaximal, characteristic, complemented
11#11723392C212yesmaximal, complemented
12#10596056—31nomaximal
13#10593978—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yes—
18C2×C32×Dic6C2211yescharacteristic
19C2×#6292C2212yes—
20C2×#6292C2212yes—
21C2×#6289C2212yes—
22C33×C42C2211yescharacteristic
23#88859C2212yes—
24#88859C2212yes—
25C3×#3752C2212yes—
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2211yescharacteristic
29C32×Dic3⋊C4C2211yescharacteristic
30C32×C4×Dic3C2211yescharacteristic
31C32×C4×Dic3C2211yescharacteristic
32#88806C2212yes—
33#88804C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×Q8—31no—
38#64156—32no—
39#64103—32no—
40#64103—32no—
41#64102—32no—
42#64084—31no—
43#64084—31no—
44#64084—34no—
45#64084—64no—
46#64083—31no—
47C4×C4⋊(C3⋊D3)—31no—
48C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
49C4×#634—31no—
50C3×C4×D12—34no—
51C3×C4×Dic6D314yescomplemented
52#63911—34no—
53#63907—38no—
54#63904—38no—
55#63894—34no—
56#63860—34no—
57#63854—38no—
58#63850—38no—
59#63844—34no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3×C3⋊D3C2312yes—
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⋊Dic3C2312yes—
69C3×C4⋊(C3⋊D3)—22no—
70C3×C4×C3⋊D3—42no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—22no—
76#6292—24no—
77#6292—42no—
78#6289—42no—
79#5585—94no—
80#5559—91no—
81C2×C32×Q8—31no—
82C2×C4⋊(C3⋊D3)—32no—
83C2×C4⋊(C3⋊D3)—31no—
84C2×C4⋊(C3⋊D3)—34no—
85C2×C4⋊(C3⋊D3)—32no—
86C2×C4⋊(C3⋊D3)—64no—
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
88C2×C4×C3⋊D3—32no—
89C2×C4×C3⋊D3C2×D312yes—
90C2×#634—31no—
91C2×#634—34no—
92C2×#634—64no—
93C2×C3×D12—34no—
94C2×C3×C4×D3—38no—
95C2×C3×Dic6C2×D314yes—
96C2×#626—38no—
97C2×#626—38no—
98C2×#624—38no—
99C32×C4⋊C4—31no—
100C32×C4⋊C4—31no—
101C32×C4⋊C4—31no—
102C32×C42—24no—
103C32×C42—31no—
104C32×C42—31no—
105C32×C42C2×D314yes—
106C32×C42C2×D311yescharacteristic
107#3752—32no—
108#3752—32no—
109#3752—32no—
110#3752—32no—
111#3752—38no—
112#3752—38no—
113#3752—32no—
114#3752—68no—
115#3752—68no—
116#3752C2×D312yes—
117C4⋊(C3⋊Dic3)C2×D311yescharacteristic
118#3750—32no—
119C4×C3⋊Dic3—32no—
120C3×(C2×D3)⋊C4—38no—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3C2×D314yes—
123C3×Dic3⋊C4C2×D314yes—
124C3×Dic3⋊C4C2×D314yes—
125C3×C4×Dic3C2×D314yes—
126C3×C4×Dic3C2×D314yes—
127#3724—38no—
128#3722—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C33×C4—21no—
140C33×C4—21no—
141C33×C4#4112yes—
142C3⋊(C3⋊Dic3)—22no—
143C3×C3⋊Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C32×Dic3—21no—
147C32×Dic3#4112yes—
148C3×C4×Q8—34no—
149C3×C4×D4—91no—
150#947—98no—
151#940—91no—
152#936—91no—
153#929—92no—
154C4×#67—92no—
155#894—98no—
156#894—98no—
157#893—98no—
158#893—91no—
159#887—91no—
160#885—92no—
161#883—92no—
162#875—94no—
163#875—94no—
164#875—91no—
165#875—184no—
166#874—94no—
167C4×D12—34no—
168C4×D12—94no—
169C4×Dic6—94no—
170C4×Dic6C3⋊D311yescharacteristic, complemented
171C22×C3⋊D3—32no—
172C22×C3⋊D3—38no—
173C22×C3⋊D3—68no—
174C22×C3⋊D3C22×D312yes—
175C22×C3×D3—38no—
176C32×Q8—62no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—31no—
181C2×C32×C4—31no—
182C2×C32×C4—31no—
183C2×C32×C4—31no—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—38no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D312yes—
194C4⋊(C3⋊D3)—22no—
195C4⋊(C3⋊D3)—64no—
196C4⋊(C3⋊D3)—62no—
197C4⋊(C3⋊D3)—68no—
198C4⋊(C3⋊D3)—128no—
199C4⋊(C3⋊D3)—122no—
200C4×C3⋊D3—42no—
201C4×C3⋊D3—122no—
202#634—62no—
203#634—68no—
204#634—128no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3C22×D314yes—
207C2×C3×Dic3C22×D314yes—
208C2×C3×Dic3C22×D314yes—
209C2×C3×Dic3C22×D314yes—
210C3×D12—68no—
211C3×C4×D3—128no—
212C3×Dic6—28no—
213#626—616no—
214#626—128no—
215#624—128no—
216#559—271noSylow
217C2×C33C2×#4111yescharacteristic
218C2×C33C2×#4111yescharacteristic
219C2×C33H(5,𝔽2)11yescharacteristic
220C3⋊(C3⋊D3)—42no—
221C3×C3⋊D3—42no—
222C2×C3×Q8—34no—
223C2×C3×D4—91no—
224C22×C3×C4—92no—
225C2×#67—92no—
226C2×#67—92no—
227C2×D12—34no—
228C2×D12—98no—
229C2×D12—94no—
230C2×D12—91no—
231C2×D12—98no—
232C2×D12—184no—
233C2×C4×D3—38no—
234C2×C4×D3—98no—
235C2×C4×D3—92no—
236C2×Dic6—94no—
237C2×Dic6—91no—
238C2×Dic6—184no—
239C2×Dic6C2×C3⋊D311yescharacteristic
240C3×C4⋊C4—34no—
241C3×C4⋊C4—34no—
242C3×C4⋊C4—34no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—92no—
245C3×C42—24no—
246C3×C42—34no—
247C3×C42—34no—
248C3×C42D3214yes—
249C3×C42C2×C3⋊D311yescharacteristic
250#217—92no—
251#217—92no—
252(C2×D3)⋊C4—38no—
253(C2×D3)⋊C4—98no—
254(C2×D3)⋊C4—98no—
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—98no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—188no—
262(C2×D3)⋊C4—188no—
263C4⋊Dic3—34no—
264C4⋊Dic3C2×C3⋊D311yescharacteristic
265Dic3⋊C4—98no—
266Dic3⋊C4—92no—
267Dic3⋊C4C2×C3⋊D311yescharacteristic
268Dic3⋊C4C2×C3⋊D311yescharacteristic
269C4×Dic3—98no—
270C4×Dic3—92no—
271C4×Dic3C2×C3⋊D311yescharacteristic
272C4×Dic3C2×C3⋊D311yescharacteristic
273C22×C32—24no—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—68no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—128no—
286C2×C3⋊D3—128no—
287C2×C3⋊D3—128no—
288C2×C3×D3—68no—
289C2×C3×D3—68no—
290C2×C3×D3—68no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—28no—
294C32×C4—24no—
295C32×C4—24no—
296C32×C4—32no—
297C32×C4—44no—
298C32×C4—44no—
299C32×C4—61no—
300C32×C4—61no—
301C32×C4—61no—
302C32×C4#23512yes—
303C32×C4#23918yes—
304C3⋊Dic3—22no—
305C3⋊Dic3—62no—
306C3⋊Dic3—68no—
307C3⋊Dic3—128no—
308C3×Dic3—24no—
309C3×Dic3—24no—
310C3×Dic3—24no—
311C3×Dic3—68no—
312C3×Dic3#23518yes—
313#127—271no—
314#124—271no—
315#124—272no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—31no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—271no—
323C4×D4—272no—
324C33#55911yescharacteristic, complemented, Sylow
325C23×C3—92no—
326C22×D3—38no—
327C22×D3—98no—
328C22×D3—92no—
329C22×D3—188no—
330C3×Q8—68no—
331C3×D4—182no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—34no—
336C2×C3×C4—34no—
337C2×C3×C4—34no—
338C2×C3×C4—34no—
339C2×C3×C4—92no—
340C2×C3×C4—362no—
341C2×C3×C4C2×D3214yes—
342C2×C3×C4C2×D3214yes—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D311yescharacteristic
346C2×C3×C4C22×C3⋊D311yescharacteristic
347#67—184no—
348#67—362no—
349C2×Dic3—38no—
350C2×Dic3—98no—
351C2×Dic3—92no—
352C2×Dic3—188no—
353C2×Dic3C22×C3⋊D311yescharacteristic
354C2×Dic3C22×C3⋊D311yescharacteristic
355C2×Dic3C22×C3⋊D311yescharacteristic
356C2×Dic3C22×C3⋊D311yescharacteristic
357D12—68no—
358D12—1816no—
359D12—188no—
360D12—182no—
361D12—368no—
362D12—368no—
363C4×D3—128no—
364C4×D3—362no—
365C4×D3—368no—
366Dic6—22no—
367Dic6—188no—
368Dic6—182no—
369Dic6—368no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32C2×#23514yes—
374C2×C32C2×#23511yescharacteristic
375C2×C32#100411yescharacteristic
376C2×C32C2×#23914yes—
377C2×C32D3×#4111yescharacteristic
378C2×C32#100914yes—
379C3⋊D3—42no—
380C3⋊D3—122no—
381C3⋊D3—128no—
382C3⋊D3—248no—
383C3×D3—128no—
384C2×Q8—31no—
385C2×Q8—271no—
386C2×D4—91no—
387C2×D4—272no—
388C2×D4—272no—
389C2×D4—271no—
390C22×C4—92no—
391C22×C4—272no—
392C4⋊C4—31no—
393C4⋊C4—31no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—272no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—31no—
404C42—31no—
405C42—272no—
406C42D3×C3⋊D311yescharacteristic
407C22×C3—24no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3C22×D3214yes—
412C22×C3C23×C3⋊D311yescharacteristic
413C2×D3—68no—
414C2×D3—68no—
415C2×D3—68no—
416C2×D3—188no—
417C2×D3—188no—
418C2×D3—188no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—368no—
423C2×D3—368no—
424C2×D3—368no—
425C3×C4—24no—
426C3×C4—24no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—28no—
430C3×C4—38no—
431C3×C4—44no—
432C3×C4—44no—
433C3×C4—64no—
434C3×C4—64no—
435C3×C4—64no—
436C3×C4—182no—
437C3×C4#379918yes—
438C3×C4#383212yes—
439Dic3—21no—
440Dic3—21no—
441Dic3—21no—
442Dic3—68no—
443Dic3—188no—
444Dic3—182no—
445Dic3—368no—
446Dic3#382812yes—
447C32—24no—
448C32#555911yescharacteristic, complemented
449C32#558514yescomplemented
450C23—92no—
451C23—272no—
452Q8—62no—
453Q8—542no—
454D4—182no—
455D4—544no—
456D4—542no—
457D4—1082no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—92no—
463C2×C4—272no—
464C2×C4—362no—
465C2×C4—1082no—
466C2×C4C2×D3×C3⋊D311yescharacteristic
467C2×C4C2×D3×C3⋊D311yescharacteristic
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—362no—
473C2×C3C2×#379914yes—
474C2×C3D3×#23514yes—
475C2×C3#6430714yes—
476C2×C3C2×#382811yescharacteristic
477C2×C3C2×#383211yescharacteristic
478C2×C3#6436611yescharacteristic
479D3—128no—
480D3—368no—
481D3—362no—
482D3—728no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22—542no—
487C22—542no—
488C22—542no—
489C22C22×D3×C3⋊D311yescharacteristic
490C4—21no—
491C4—21no—
492C4—32no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—182no—
497C4—542no—
498C4#8901812yes—
499C3—24no—
500C3#1059397814yescomplemented
501C3#1059605611yescharacteristic, complemented
502C2—362no—
503C2—1082no—
504C2C2×#8901811yescharacteristic
505C2D3×#382811yescharacteristic
506C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723517C2
hypercenterC42D3×C3⋊D3
hypocenterC33#559
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×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C4×Dic6C2semidirect
5. #11723512C2semidirect
6. #11723504C2semidirect
7. #11723484C2semidirect
8. #11723439C2semidirect
9. #11723417C2semidirect
10. #11723399C2semidirect
11. #11723392C2semidirect
48. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×C4×Dic6D3semidirect
170. C4×Dic6C3⋊D3semidirect
324. C33#559semidirect
448. C32#5559semidirect
449. C32#5585semidirect
500. C3#10593978semidirect
501. C3#10596056semidirect

Maximal subgroups

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

subgroupindex
#117239162
C3×C4×C4⋊(C3⋊D3)2
C32×C4×Dic62
#117235172
#117235122
#117235042
#117234842
#117234392
#117234172
#117233992
#117233922
#105960563
#105939783