Finite Group Databasebeta

← #13175332

Subgroups — #13175332

Order 1728.

1684 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C42×D3C211yesmaximal, characteristic, complemented
4#11723515C211yesmaximal, characteristic
5#11723513C212yesmaximal, complemented
6C4×#6290C212yesmaximal, complemented
7#11723454C211yesmaximal, characteristic, complemented
8#11723439C211yesmaximal, characteristic, complemented
9#11723438C212yesmaximal, complemented
10#11723437C212yesmaximal, complemented
11#11723426C211yesmaximal, characteristic, complemented
12#10595950—31nomaximal
13#10593974—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×C4×D3C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6290C2212yes—
21C33×C42C2211yescharacteristic
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23#88857C2212yes—
24C4×C3⋊(C3⋊Dic3)C2212yes—
25C3×#3752C2212yes—
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C32×C4×Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29C32×C4×Dic3C2211yescharacteristic
30#88807C2212yes—
31#88807C2212yes—
32#88806C2212yes—
33#88803C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C2×C32×C42—31no—
38#64139—31no—
39#64138—32no—
40C4×#638—32no—
41#64133—31no—
42#64085—32no—
43#64084—31no—
44C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
45#64081—32no—
46#64079—31no—
47C4×#634—31no—
48C4×#634—34no—
49C4×#634—64no—
50C3×C4×D12—34no—
51C3×C42×D3D314yescomplemented
52#63908—34no—
53#63905—38no—
54C4×#625—38no—
55#63869—34no—
56#63860—34no—
57#63859—38no—
58#63857—34no—
59#63855—38no—
60C22×C3×C3⋊D3C2312yes—
61C22×C32×D3C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67#6308—22no—
68C2×C3×C3⋊Dic3C2312yes—
69C3×C4⋊(C3⋊D3)—22no—
70C3×C4×C3⋊D3—24no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×C4×D3—22no—
75C32×C4×D3—22no—
76C32×C4×D3—22no—
77#6290—24no—
78#5557—91no—
79#5479—94no—
80C22×C32×C4—31no—
81C22×C32×C4—31no—
82C22×C32×C4—31no—
83C2×#638—32no—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D312yes—
86C2×#634—31no—
87C2×#634—34no—
88C2×#634—64no—
89C2×C3×D12—34no—
90C2×C3×C4×D3—38no—
91C2×C3×C4×D3C2×D314yes—
92C2×C3×C4×D3C2×D314yes—
93C2×C3×C4×D3C2×D314yes—
94C2×#625—38no—
95C32×C42—24no—
96C32×C42—31no—
97C32×C42—31no—
98C32×C42—31no—
99C32×C42C2×D314yes—
100C32×C42C2×D311yescharacteristic
101#3757—32no—
102#3752—32no—
103#3752—32no—
104#3752—32no—
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—34no—
108C4⋊(C3⋊Dic3)—64no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750—32no—
111#3750—38no—
112#3750—32no—
113#3750—32no—
114#3750—32no—
115#3750—68no—
116C4×C3⋊Dic3—32no—
117C4×C3⋊Dic3—38no—
118C4×C3⋊Dic3—68no—
119C3×(C2×D3)⋊C4—38no—
120C3×C4⋊Dic3—34no—
121C3×C4×Dic3C2×D314yes—
122C3×C4×Dic3C2×D314yes—
123C3×C4×Dic3C2×D314yes—
124#3724—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131C22×C33C2411yescharacteristic
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—22no—
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C2×C32×D3#4112yes—
138C33×C4—21no—
139C33×C4#4112yes—
140C33×C4#4112yes—
141C3⋊(C3⋊Dic3)—22no—
142C3⋊(C3⋊Dic3)—22no—
143C3×C3⋊Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C32×Dic3—21no—
147C3×C4×D4—91no—
148C2×C3×C42—34no—
149#938—91no—
150#936—91no—
151#930—94no—
152#929—98no—
153C4×#67—98no—
154C4×#67—92no—
155#926—92no—
156#924—94no—
157#897—91no—
158#890—91no—
159#883—92no—
160#879—92no—
161#876—98no—
162#875—94no—
163C4×D12—34no—
164#872—98no—
165C42×D3C3⋊D311yescharacteristic, complemented
166#870—94no—
167C4×Dic6—94no—
168C4×Dic6—91no—
169C4×Dic6—184no—
170C23×C32—31no—
171C22×C3⋊D3C22×D312yes—
172C22×C3×D3—38no—
173C22×C3×D3C22×D314yes—
174C2×C32×C4—24no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—31no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—62no—
181C2×C32×C4—62no—
182C2×C32×C4—62no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189#638—64no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—32no—
192C2×C3⋊Dic3—38no—
193C2×C3⋊Dic3—38no—
194C2×C3⋊Dic3—68no—
195C2×C3⋊Dic3—68no—
196C2×C3⋊Dic3C22×D312yes—
197C4⋊(C3⋊D3)—22no—
198C4×C3⋊D3—24no—
199#634—616no—
200#634—68no—
201#634—62no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3C22×D314yes—
204C2×C3×Dic3C22×D314yes—
205C2×C3×Dic3C22×D314yes—
206C3×D12—68no—
207C3×C4×D3—28no—
208C3×C4×D3—28no—
209C3×C4×D3—28no—
210C3×C4×D3—616no—
211#625—616no—
212#529—271noSylow
213C2×C33C2×#4111yescharacteristic
214C2×C33C2×#4111yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C3×C3⋊D3—42no—
217C32×D3—22no—
218C2×C3×D4—91no—
219C22×C3×C4—34no—
220C22×C3×C4—34no—
221C22×C3×C4—34no—
222C22×C3×C4—92no—
223C2×#67—98no—
224C2×#67—92no—
225C2×D12—34no—
226C2×C4×D3—38no—
227C2×C4×D3C2×C3⋊D311yescharacteristic
228C2×C4×D3C2×C3⋊D311yescharacteristic
229C2×C4×D3C2×C3⋊D311yescharacteristic
230C2×Dic6—94no—
231C2×Dic6—91no—
232C2×Dic6—184no—
233C3×C4⋊C4—91no—
234C3×C22⋊C4—92no—
235C3×C42—24no—
236C3×C42—34no—
237C3×C42—34no—
238C3×C42—34no—
239C3×C42D3214yes—
240C3×C42C2×C3⋊D311yescharacteristic
241#217—98no—
242#217—92no—
243#217—92no—
244#217—92no—
245(C2×D3)⋊C4—38no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—98no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—34no—
251C4⋊Dic3—94no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—92no—
255C4⋊Dic3—184no—
256Dic3⋊C4—98no—
257Dic3⋊C4—98no—
258Dic3⋊C4—98no—
259Dic3⋊C4—98no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—188no—
263C4×Dic3—98no—
264C4×Dic3—92no—
265C4×Dic3—188no—
266C4×Dic3C2×C3⋊D311yescharacteristic
267C4×Dic3C2×C3⋊D311yescharacteristic
268C4×Dic3C2×C3⋊D311yescharacteristic
269C22×C32—24no—
270C22×C32—32no—
271C22×C32—61no—
272C22×C32—61no—
273C22×C32C23×D314yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—68no—
281C2×C3×D3—68no—
282C2×C3×D3—68no—
283C2×C3×D3#23518yes—
284C32×C4—21no—
285C32×C4—28no—
286C32×C4—28no—
287C32×C4—24no—
288C32×C4—44no—
289C32×C4—61no—
290C32×C4—61no—
291C32×C4—61no—
292C32×C4#23518yes—
293C32×C4#23518yes—
294C32×C4#23512yes—
295C32×C4#23712yes—
296C3⋊Dic3—22no—
297C3⋊Dic3—616no—
298C3⋊Dic3—68no—
299C3⋊Dic3—62no—
300C3⋊Dic3—62no—
301C3⋊Dic3—68no—
302C3⋊Dic3—128no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—24no—
306C3×Dic3—68no—
307#126—272no—
308#125—271no—
309#124—271no—
310#123—272no—
311C22⋊Q8—271no—
312#121—271no—
313C4×Q8—271no—
314C4×D4—91no—
315C4×D4—272no—
316#117—272no—
317C2×C42—31no—
318C33#52911yescharacteristic, complemented, Sylow
319C23×C3—34no—
320C23×C3—92no—
321C22×D3—38no—
322C22×D3C22×C3⋊D311yescharacteristic
323C3×D4—182no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—34no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—68no—
331C2×C3×C4—68no—
332C2×C3×C4—68no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C2×D3214yes—
338C2×C3×C4C22×C3⋊D311yescharacteristic
339C2×C3×C4C22×C3⋊D311yescharacteristic
340C2×C3×C4C22×C3⋊D311yescharacteristic
341#67—1816no—
342#67—184no—
343C2×Dic3—38no—
344C2×Dic3—98no—
345C2×Dic3—98no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—188no—
349C2×Dic3—188no—
350C2×Dic3C22×C3⋊D311yescharacteristic
351C2×Dic3C22×C3⋊D311yescharacteristic
352C2×Dic3C22×C3⋊D311yescharacteristic
353D12—68no—
354C4×D3—22no—
355C4×D3—22no—
356C4×D3—22no—
357C4×D3—616no—
358Dic6—1816no—
359Dic6—188no—
360Dic6—182no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32—62no—
365C2×C32C2×#23514yes—
366C2×C32C2×#23514yes—
367C2×C32C2×#23514yes—
368C2×C32C2×#23511yescharacteristic
369C2×C32C2×#23711yescharacteristic
370C2×C32D3×#4111yescharacteristic
371C3⋊D3—42no—
372C3×D3—28no—
373C3×D3—128no—
374C2×Q8—271no—
375C2×D4—91no—
376C2×D4—272no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—92no—
381C4⋊C4—91no—
382C4⋊C4—272no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C4⋊C4—272no—
387C22⋊C4—92no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C42—31no—
393C42—31no—
394C42—31no—
395C42—272no—
396C42D3×C3⋊D311yescharacteristic
397C22×C3—24no—
398C22×C3—38no—
399C22×C3—64no—
400C22×C3—64no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3C22×D3214yes—
405C22×C3C23×C3⋊D311yescharacteristic
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—68no—
409C2×D3—68no—
410C2×D3—68no—
411C2×D3#382812yes—
412C3×C4—24no—
413C3×C4—21no—
414C3×C4—28no—
415C3×C4—28no—
416C3×C4—44no—
417C3×C4—64no—
418C3×C4—64no—
419C3×C4—64no—
420C3×C4—182no—
421C3×C4#379518yes—
422C3×C4#379818yes—
423C3×C4#382812yes—
424C3×C4#382812yes—
425Dic3—21no—
426Dic3—21no—
427Dic3—21no—
428Dic3—68no—
429Dic3—1816no—
430Dic3—188no—
431Dic3—188no—
432Dic3—182no—
433Dic3—182no—
434Dic3—368no—
435C32—24no—
436C32#547914yescomplemented
437C32#555711yescharacteristic, complemented
438C23—31no—
439C23—92no—
440Q8—542no—
441D4—182no—
442D4—544no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—92no—
450C2×C4—184no—
451C2×C4—272no—
452C2×C4—272no—
453C2×C4C2×D3×C3⋊D311yescharacteristic
454C2×C4C2×D3×C3⋊D311yescharacteristic
455C2×C4C2×D3×C3⋊D311yescharacteristic
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—68no—
460C2×C3—362no—
461C2×C3C2×#379514yes—
462C2×C3C2×#379814yes—
463C2×C3D3×#23514yes—
464C2×C3C2×#382811yescharacteristic
465C2×C3C2×#382811yescharacteristic
466C2×C3C2×#382811yescharacteristic
467D3—22no—
468D3—128no—
469C22—32no—
470C22—61no—
471C22—61no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22C22×D3×C3⋊D311yescharacteristic
476C4—21no—
477C4—61no—
478C4—61no—
479C4—61no—
480C4—182no—
481C4—542no—
482C4—542no—
483C4#8902012yes—
484C4#8902112yes—
485C3—24no—
486C3#1059397414yescomplemented
487C3#1059595011yescharacteristic, complemented
488C2—62no—
489C2—362no—
490C2C2×#8902011yescharacteristic
491C2C2×#8902111yescharacteristic
492C2D3×#382811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
prime radical#11723515C2
hypercenterC42D3×C3⋊D3
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C42×D3C2semidirect
5. #11723513C2semidirect
6. C4×#6290C2semidirect
7. #11723454C2semidirect
8. #11723439C2semidirect
9. #11723438C2semidirect
10. #11723437C2semidirect
11. #11723426C2semidirect
44. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×C42×D3D3semidirect
165. C42×D3C3⋊D3semidirect
318. C33#529semidirect
436. C32#5479semidirect
437. C32#5557semidirect
486. C3#10593974semidirect
487. C3#10595950semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×C4×C4⋊(C3⋊D3)2
C32×C42×D32
#117235152
#117235132
C4×#62902
#117234542
#117234392
#117234382
#117234372
#117234262
#105959503
#105939743