Finite Group Databasebeta

← #13175390

Subgroups — #13175390

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64081C211yesmaximal, characteristic, complemented
4C32×#872C211yesmaximal, characteristic, complemented
5#11723539C212yesmaximal
6#11723470C212yesmaximal, complemented
7C4×#6289C211yesmaximal, characteristic, complemented
8#11723466C212yesmaximal, complemented
9#11723460C212yesmaximal, complemented
10#11723437C211yesmaximal, characteristic, complemented
11#11723423C211yesmaximal, characteristic, complemented
12C3⋊D3×#117—31nomaximal
13#10593992—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6289C2212yes—
20C2×#6289C2211yescharacteristic
21C2×Dic3×C3⋊D3C2211yescharacteristic
22C2×D3×C3⋊Dic3C2211yescharacteristic
23C4×D3×C3⋊D3C418yescomplemented
24C33×C42C2211yescharacteristic, complemented
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
26C4×C3⋊(C3⋊Dic3)C2212yescomplemented
27C3×#3752C2212yes—
28C3×#3750C2212yescomplemented
29C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
30C32×(C2×D3)⋊C4C2212yes—
31C32×Dic3⋊C4C2212yescomplemented
32C32×C4×Dic3C2211yescharacteristic, complemented
33#88807C2212yescomplemented
34Dic3⋊(C3⋊Dic3)C2212yescomplemented
35#88803C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37Dic3×C3⋊Dic3C2212yescomplemented
38C22×C4×C3⋊D3—31no—
39C2×C4×D32—34no—
40C32×#117—31no—
41#64135—31no—
42#64101—32no—
43C4⋊C4×C3⋊D3—32no—
44C22⋊C4×C3⋊D3—32no—
45#64086—32no—
46#64081—31no—
47#64081D311yescharacteristic, complemented
48C42×C3⋊D3—31no—
49C42×C3⋊D3—31no—
50C42×C3⋊D3—34no—
51C42×C3⋊D3—64no—
52C3×#872—34no—
53C3×#872D314yescomplemented
54#63921—38no—
55#63883—38no—
56C4×#624—34no—
57#63878—38no—
58#63878—38no—
59#63855—34no—
60#63855—34no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2311yescharacteristic
63C22×C32×D3C2311yescharacteristic
64C2×D3×C3⋊D3—24no—
65C2×D3×C3⋊D3C2×C414yes—
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C4×C3⋊(C3⋊D3)—24no—
71C4×C3⋊(C3⋊D3)C2×C414yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4×C3⋊D3C2×C414yescomplemented
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2311yescharacteristic
77C32×C4×D3C2×C414yescomplemented
78#6289—24no—
79#6289C2×C414yes—
80Dic3×C3⋊D3C2×C414yescomplemented
81D3×C3⋊Dic3C2×C414yescomplemented
82D3×#117—94no—
83D3×#117—91no—
84C23×C3⋊D3—31no—
85C22×D32—34no—
86C22×C32×C4—31no—
87C22×C3⋊Dic3—31no—
88C2×C4×C3⋊D3—31no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—34no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—38no—
93C2×C4×C3⋊D3—32no—
94C2×C4×C3⋊D3—38no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—64no—
97C2×C4×C3⋊D3—68no—
98C2×C4×C3⋊D3C2×D311yescharacteristic
99C2×C3×C4×D3—34no—
100C2×C3×C4×D3C2×D314yes—
101C2×#624—34no—
102C2×#624—38no—
103C2×D3×Dic3—34no—
104C2×D3×Dic3—34no—
105C4×D32—332no—
106C32×C4⋊C4—32no—
107C32×C22⋊C4—32no—
108C32×C42—24no—
109C32×C42—31no—
110C32×C42C2×D314yescomplemented
111C32×C42C2×D311yescharacteristic, complemented
112#3757—32no—
113#3752—32no—
114#3752C2×D312yes—
115C4⋊(C3⋊Dic3)—32no—
116#3750—32no—
117#3750C2×D312yescomplemented
118C4×C3⋊Dic3—31no—
119C4×C3⋊Dic3—32no—
120C4×C3⋊Dic3—34no—
121C4×C3⋊Dic3—38no—
122C4×C3⋊Dic3—32no—
123C4×C3⋊Dic3—64no—
124C4×C3⋊Dic3—68no—
125C4×C3⋊Dic3C2×D311yescharacteristic, complemented
126C3×(C2×D3)⋊C4—38no—
127C3×(C2×D3)⋊C4C2×D318yes—
128C3×Dic3⋊C4—38no—
129C3×Dic3⋊C4C2×D318yescomplemented
130C3×C4×Dic3—34no—
131C3×C4×Dic3C2×D314yescomplemented
132Dic3⋊Dic3—38no—
133Dic3⋊Dic3—38no—
134(C2×D3)⋊Dic3—38no—
135(C2×D3)⋊Dic3—38no—
136Dic32—38no—
137C22×C33C2411yescharacteristic
138C2×C3⋊(C3⋊D3)C22×C412yes—
139C2×C3⋊(C3⋊D3)#4114yes—
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3C22×C412yes—
142C2×C32×D3—22no—
143C2×C32×D3C22×C412yes—
144D3×C3⋊D3—28no—
145C33×C4—22no—
146C33×C4C22×C412yes—
147C3⋊(C3⋊Dic3)—22no—
148C3⋊(C3⋊Dic3)C22×C412yes—
149C3×C3⋊Dic3—22no—
150C3×C3⋊Dic3C22×C412yes—
151C32×Dic3—22no—
152C32×Dic3C22×C412yes—
153C22×C4×D3—94no—
154C22×C4×D3—91no—
155C3×#117—34no—
156C3×#117—91no—
157#926—94no—
158#926—91no—
159#892—98no—
160#892—92no—
161D3×C4⋊C4—98no—
162D3×C4⋊C4—92no—
163D3×C22⋊C4—98no—
164D3×C22⋊C4—92no—
165#877—98no—
166#877—92no—
167#872—34no—
168#872—94no—
169#872—91no—
170#872C3⋊D311yescharacteristic, complemented
171C42×D3—94no—
172C42×D3—94no—
173C42×D3—91no—
174C42×D3—91no—
175C42×D3—184no—
176C23×C32—31no—
177C22×C3⋊D3—31no—
178C22×C3⋊D3—34no—
179C22×C3⋊D3—34no—
180C22×C3⋊D3—64no—
181C22×C3⋊D3—64no—
182C22×C3⋊D3C22×D311yescharacteristic
183C22×C3×D3—34no—
184C22×C3×D3C22×D314yes—
185C2×D32—316no—
186C2×D32—616no—
187C2×C32×C4—28no—
188C2×C32×C4—24no—
189C2×C32×C4—31no—
190C2×C32×C4—34no—
191C2×C32×C4—32no—
192C2×C32×C4C22×D318yes—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C3⋊Dic3—31no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—34no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—64no—
202C2×C3⋊Dic3—68no—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3—34no—
206C4×C3⋊D3—34no—
207C4×C3⋊D3—34no—
208C4×C3⋊D3—316no—
209C4×C3⋊D3—616no—
210C4×C3⋊D3—64no—
211C4×C3⋊D3—64no—
212C4×C3⋊D3—616no—
213C4×C3⋊D3—632no—
214C4×C3⋊D3C4×D314yescomplemented
215C2×C3×Dic3—34no—
216C2×C3×Dic3—38no—
217C2×C3×Dic3C22×D318yes—
218C2×C3×Dic3C22×D314yes—
219C3×C4×D3—316no—
220C3×C4×D3C4×D3116yescomplemented
221#624—316no—
222#624—616no—
223D3×Dic3—316no—
224D3×Dic3—316no—
225C2×#117—271noSylow
226C2×C33C23×C411yescharacteristic
227C2×C33C2×#4112yes—
228C3⋊(C3⋊D3)#11714yescomplemented
229C3×C3⋊D3—22no—
230C32×D3—22no—
231C23×D3—94no—
232C23×D3—91no—
233C22×C3×C4—34no—
234C22×C3×C4—91no—
235C22×Dic3—94no—
236C22×Dic3—91no—
237C2×C4×D3—34no—
238C2×C4×D3—932no—
239C2×C4×D3—94no—
240C2×C4×D3—94no—
241C2×C4×D3—91no—
242C2×C4×D3—94no—
243C2×C4×D3—91no—
244C2×C4×D3—91no—
245C2×C4×D3—98no—
246C2×C4×D3—98no—
247C2×C4×D3—98no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C2×C4×D3—184no—
251C2×C4×D3—188no—
252C2×C4×D3C2×C3⋊D311yescharacteristic
253C3×C4⋊C4—38no—
254C3×C4⋊C4—92no—
255C3×C22⋊C4—38no—
256C3×C22⋊C4—92no—
257C3×C42—24no—
258C3×C42—34no—
259C3×C42—91no—
260C3×C42D3214yescomplemented
261C3×C42C2×C3⋊D311yescharacteristic, complemented
262#217—98no—
263#217—92no—
264(C2×D3)⋊C4—38no—
265(C2×D3)⋊C4—98no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4C2×C3⋊D312yes—
268C4⋊Dic3—98no—
269C4⋊Dic3—92no—
270Dic3⋊C4—38no—
271Dic3⋊C4—98no—
272Dic3⋊C4—92no—
273Dic3⋊C4C2×C3⋊D312yescomplemented
274C4×Dic3—34no—
275C4×Dic3—98no—
276C4×Dic3—94no—
277C4×Dic3—98no—
278C4×Dic3—91no—
279C4×Dic3—92no—
280C4×Dic3—92no—
281C4×Dic3—188no—
282C4×Dic3—184no—
283C4×Dic3C2×C3⋊D311yescharacteristic, complemented
284C22×C32—24no—
285C22×C32—32no—
286C22×C32—62no—
287C22×C32C23×D314yes—
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—22no—
290C2×C3⋊D3—32no—
291C2×C3⋊D3—34no—
292C2×C3⋊D3—38no—
293C2×C3⋊D3—316no—
294C2×C3⋊D3—68no—
295C2×C3⋊D3—616no—
296C2×C3⋊D3—68no—
297C2×C3⋊D3C2×C4×D312yes—
298C2×C3×D3—28no—
299C2×C3×D3—38no—
300C2×C3×D3—68no—
301C2×C3×D3C2×C4×D318yes—
302D32—632no—
303C32×C4—216no—
304C32×C4—28no—
305C32×C4—22no—
306C32×C4—28no—
307C32×C4—32no—
308C32×C4—62no—
309C32×C4C2×C4×D318yes—
310C32×C4C2×C4×D312yes—
311C3⋊Dic3—22no—
312C3⋊Dic3—32no—
313C3⋊Dic3—38no—
314C3⋊Dic3—68no—
315C3⋊Dic3—616no—
316C3⋊Dic3—68no—
317C3⋊Dic3—62no—
318C3⋊Dic3C2×C4×D312yes—
319C3×Dic3—28no—
320C3×Dic3—38no—
321C3×Dic3—68no—
322C3×Dic3C2×C4×D318yes—
323C23×C4—271no—
324#117—31no—
325#117—91no—
326#117—271no—
327#117—271no—
328#117—272no—
329#117—272no—
330C2×C4⋊C4—272no—
331C2×C22⋊C4—272no—
332C2×C42—271no—
333C2×C42—271no—
334C33C2×#11711yescharacteristic, complemented, Sylow
335C23×C3—34no—
336C23×C3—91no—
337C22×D3—34no—
338C22×D3—916no—
339C22×D3—94no—
340C22×D3—91no—
341C22×D3—94no—
342C22×D3—184no—
343C22×D3—184no—
344C22×D3—1816no—
345C22×D3C22×C3⋊D311yescharacteristic
346C2×C3×C4—28no—
347C2×C3×C4—24no—
348C2×C3×C4—34no—
349C2×C3×C4—316no—
350C2×C3×C4—38no—
351C2×C3×C4—91no—
352C2×C3×C4—94no—
353C2×C3×C4—92no—
354C2×C3×C4C2×D3218yes—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C22×C3⋊D312yes—
357C2×C3×C4C22×C3⋊D311yescharacteristic
358C2×Dic3—34no—
359C2×Dic3—38no—
360C2×Dic3—916no—
361C2×Dic3—94no—
362C2×Dic3—91no—
363C2×Dic3—94no—
364C2×Dic3—98no—
365C2×Dic3—92no—
366C2×Dic3—184no—
367C2×Dic3—188no—
368C2×Dic3C22×C3⋊D312yes—
369C2×Dic3C22×C3⋊D311yescharacteristic
370C4×D3—316no—
371C4×D3—916no—
372C4×D3—916no—
373C4×D3—94no—
374C4×D3—916no—
375C4×D3—94no—
376C4×D3—94no—
377C4×D3—1816no—
378C4×D3—184no—
379C4×D3—1816no—
380C4×D3—1816no—
381C4×D3—184no—
382C4×D3—1832no—
383C4×D3C4×C3⋊D314yescomplemented
384C2×C32—24no—
385C2×C32—28no—
386C2×C32—62no—
387C2×C32C22×C4×D314yes—
388C2×C32C22×C4×D311yescharacteristic
389C2×C32D3×#4118yes—
390C2×C32D3×#4112yes—
391C3⋊D3—22no—
392C3⋊D3—34no—
393C3⋊D3—316no—
394C3⋊D3—616no—
395C3×D3—28no—
396C3×D3—68no—
397C24—271no—
398C22×C4—31no—
399C22×C4—91no—
400C22×C4—271no—
401C22×C4—278no—
402C22×C4—272no—
403C22×C4—271no—
404C22×C4—271no—
405C22×C4—271no—
406C22×C4—272no—
407C4⋊C4—32no—
408C4⋊C4—92no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C22⋊C4—32no—
412C22⋊C4—92no—
413C22⋊C4—272no—
414C22⋊C4—272no—
415C42—31no—
416C42—91no—
417C42—272no—
418C42—272no—
419C42—271no—
420C42D3×C3⋊D311yescharacteristic, complemented
421C22×C3—24no—
422C22×C3—38no—
423C22×C3—68no—
424C22×C3—92no—
425C22×C3—182no—
426C22×C3C22×D3214yes—
427C22×C3C23×C3⋊D311yescharacteristic
428C2×D3—22no—
429C2×D3—38no—
430C2×D3—68no—
431C2×D3—98no—
432C2×D3—916no—
433C2×D3—92no—
434C2×D3—94no—
435C2×D3—188no—
436C2×D3—1816no—
437C2×D3—188no—
438C2×D3—1832no—
439C2×D3C2×C4×C3⋊D312yes—
440C3×C4—216no—
441C3×C4—28no—
442C3×C4—22no—
443C3×C4—28no—
444C3×C4—38no—
445C3×C4—68no—
446C3×C4—92no—
447C3×C4—182no—
448C3×C4C4×D3218yes—
449C3×C4C2×C4×C3⋊D312yes—
450Dic3—22no—
451Dic3—38no—
452Dic3—68no—
453Dic3—98no—
454Dic3—92no—
455Dic3—188no—
456Dic3—1816no—
457Dic3—188no—
458Dic3—182no—
459Dic3C2×C4×C3⋊D312yes—
460C32—24no—
461C32D3×#11714yescomplemented
462C32D3×#11711yescharacteristic, complemented
463C23—31no—
464C23—91no—
465C23—274no—
466C23—271no—
467C23—544no—
468C2×C4—31no—
469C2×C4—34no—
470C2×C4—32no—
471C2×C4—91no—
472C2×C4—94no—
473C2×C4—92no—
474C2×C4—274no—
475C2×C4—274no—
476C2×C4—274no—
477C2×C4—271no—
478C2×C4—274no—
479C2×C4—272no—
480C2×C4—544no—
481C2×C4—544no—
482C2×C4C2×D3×C3⋊D312yes—
483C2×C4C2×D3×C3⋊D311yescharacteristic
484C2×C3—24no—
485C2×C3—28no—
486C2×C3—68no—
487C2×C3—182no—
488C2×C3C2×C4×D3214yes—
489C2×C3#6430618yes—
490C2×C3C22×C4×C3⋊D311yescharacteristic
491C2×C3#41×C3⋊D312yes—
492D3—22no—
493D3—68no—
494D3—94no—
495D3—916no—
496D3—1816no—
497C22—32no—
498C22—62no—
499C22—92no—
500C22—182no—
501C22—272no—
502C22—274no—
503C22—548no—
504C22C22×D3×C3⋊D311yescharacteristic
505C4—22no—
506C4—32no—
507C4—62no—
508C4—92no—
509C4—182no—
510C4—272no—
511C4—542no—
512C4C4×D3×C3⋊D312yes—
513C3—24no—
514C3#1059399214yescomplemented
515C3C3⋊D3×#11711yescharacteristic, complemented
516C2—62no—
517C2—182no—
518C2—274no—
519C2C2×C4×D3×C3⋊D311yescharacteristic
520C2#1172457812yes—
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
socle, cosocleC22×C33C24
prime radicalC22×D3×C3⋊D3C22
hypercenterC42D3×C3⋊D3
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×C3⋊(C3⋊D3)C2semidirect
3. C3×#64081C2semidirect
4. C32×#872C2semidirect
6. #11723470C2semidirect
7. C4×#6289C2semidirect
8. #11723466C2semidirect
9. #11723460C2semidirect
10. #11723437C2semidirect
11. #11723423C2semidirect
23. C4×D3×C3⋊D3C4semidirect
24. C33×C42C22semidirect
25. C4×C3⋊(C3⋊Dic3)C22semidirect
26. C4×C3⋊(C3⋊Dic3)C22semidirect
28. C3×#3750C22semidirect
29. C3×C4×C3⋊Dic3C22semidirect
31. C32×Dic3⋊C4C22semidirect
32. C32×C4×Dic3C22semidirect
33. #88807C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
47. #64081D3semidirect
53. C3×#872D3semidirect
74. C3×C4×C3⋊D3C2×C4semidirect
77. C32×C4×D3C2×C4semidirect
80. Dic3×C3⋊D3C2×C4semidirect
81. D3×C3⋊Dic3C2×C4semidirect
110. C32×C42C2×D3semidirect
111. C32×C42C2×D3semidirect
117. #3750C2×D3semidirect
125. C4×C3⋊Dic3C2×D3semidirect
129. C3×Dic3⋊C4C2×D3semidirect
131. C3×C4×Dic3C2×D3semidirect
170. #872C3⋊D3semidirect
214. C4×C3⋊D3C4×D3semidirect
220. C3×C4×D3C4×D3semidirect
228. C3⋊(C3⋊D3)#117semidirect
260. C3×C42D32semidirect
261. C3×C42C2×C3⋊D3semidirect
273. Dic3⋊C4C2×C3⋊D3semidirect
283. C4×Dic3C2×C3⋊D3semidirect
334. C33C2×#117semidirect
383. C4×D3C4×C3⋊D3semidirect
420. C42D3×C3⋊D3semidirect
461. C32D3×#117semidirect
462. C32D3×#117semidirect
514. C3#10593992semidirect
515. C3C3⋊D3×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C42×C3⋊(C3⋊D3)2
C3×#640812
C32×#8722
#117235392
#117234702
C4×#62892
#117234662
#117234602
#117234372
#117234232
C3⋊D3×#1173
#105939923