Finite Group Databasebeta

← #13175396

Subgroups — #13175396

Order 1728.

2332 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89026C211yesmaximal, characteristic
2#11723913C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×C4×D12C211yesmaximal, characteristic, complemented
5C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
6D3×#3752C212yesmaximal, complemented
7#11723507C212yesmaximal, complemented
8#11723506C212yesmaximal, complemented
9#11723501C211yesmaximal, characteristic, complemented
10#11723499C211yesmaximal, characteristic, complemented
11#11723413C211yesmaximal, characteristic, complemented
12#10596016—31nomaximal
13#10593996—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yescomplemented
18C2×C32×D12C2211yescharacteristic
19C2×C32×C4×D3C2212yescomplemented
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yescomplemented
22C2×D3×C3⋊Dic3C2212yescomplemented
23#89026C418yescomplemented
24C33×C42C2211yescharacteristic, complemented
25#88859C2211yescharacteristic
26#88859C2211yescharacteristic
27#88857C2211yescharacteristic, complemented
28#88857C2211yescharacteristic, complemented
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
30C3×#3752C2212yescomplemented
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32C32×(C2×D3)⋊C4C2212yescomplemented
33C32×C4⋊Dic3C2211yescharacteristic, complemented
34#88804C2212yes—
35#88803C2212yescomplemented
36(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
37Dic3×C3⋊Dic3C2212yescomplemented
38C2×D4×C3⋊D3—31no—
39C2×#3802—34no—
40C32×C4×D4—31no—
41D4×C3⋊Dic3—31no—
42C4×#638—32no—
43C2×#3752—32no—
44#64102—31no—
45#64101—31no—
46#64090—32no—
47C22⋊C4×C3⋊D3—32no—
48C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
49#64081—31no—
50#64081—34no—
51#64081—64no—
52C3×C4×D12—34no—
53C3×C4×D12D314yescomplemented
54D3×(C2×D3)⋊C4—38no—
55D3×(C2×D3)⋊C4—38no—
56#63898—34no—
57#63898—34no—
58#63879—38no—
59#63879—38no—
60#63849—34no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2312yes—
64C2×D3×C3⋊D3—42no—
65C2×D3×C3⋊D3—42no—
66C2×D3×C3⋊D3C2×C418yescomplemented
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C4×C3⋊(C3⋊D3)C2×C414yescomplemented
74C2×C3×C3⋊Dic3C2312yes—
75C3×C4⋊(C3⋊D3)C2×C414yescomplemented
76C3×C4×C3⋊D3—42no—
77C2×C32×Dic3C2312yes—
78C32×D12C2×C414yescomplemented
79C32×C4×D3—42no—
80#6290C2×C418yescomplemented
81Dic3×C3⋊D3—42no—
82D3×C3⋊Dic3—42no—
83#5545—94no—
84#5545—91no—
85C23×C3⋊D3—32no—
86C22×D32—38no—
87C2×C32×D4—31no—
88C22×C32×C4—32no—
89C2×#638—32no—
90C22×C3⋊Dic3—32no—
91D4×C3⋊D3—38no—
92C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—34no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—64no—
97C2×C4×C3⋊D3C2×D312yescomplemented
98C2×C3×D12—34no—
99C2×C3×D12C2×D314yes—
100C2×C3×C4×D3—38no—
101C2×C3×C4×D3C2×D318yescomplemented
102C2×#625—38no—
103C2×D3×Dic3—38no—
104C2×D3×Dic3—38no—
105#3802—332no—
106C32×C4⋊C4—31no—
107C32×C22⋊C4—32no—
108C32×C42—24no—
109C32×C42C2×D314yescomplemented
110C32×C42C2×D311yescharacteristic, complemented
111#3757—32no—
112#3752—32no—
113#3752—31no—
114#3752—31no—
115#3752—34no—
116#3752—34no—
117#3752—32no—
118#3752—64no—
119#3752—64no—
120#3752C2×D312yescomplemented
121C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
122#3750—31no—
123#3750—31no—
124#3750—34no—
125#3750—34no—
126#3750—64no—
127#3750—64no—
128C4×C3⋊Dic3—31no—
129C4×C3⋊Dic3—34no—
130C4×C3⋊Dic3—32no—
131C4×C3⋊Dic3—64no—
132C3×(C2×D3)⋊C4—38no—
133C3×(C2×D3)⋊C4C2×D318yescomplemented
134C3×C4⋊Dic3—34no—
135C3×C4⋊Dic3C2×D314yescomplemented
136#3722—38no—
137(C2×D3)⋊Dic3—38no—
138(C2×D3)⋊Dic3—38no—
139Dic32—38no—
140C22×C33C2411yescharacteristic
141C2×C3⋊(C3⋊D3)—21no—
142C2×C3⋊(C3⋊D3)—21no—
143C2×C3⋊(C3⋊D3)C22×C412yes—
144C2×C3×C3⋊D3—22no—
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3C22×C414yes—
147C2×C32×D3—22no—
148C2×C32×D3—22no—
149C2×C32×D3C22×C414yes—
150D3×C3⋊D3—48no—
151C33×C4—21no—
152C33×C4—21no—
153C33×C4C22×C412yes—
154C3⋊(C3⋊Dic3)—21no—
155C3⋊(C3⋊Dic3)—21no—
156C3⋊(C3⋊Dic3)C22×C412yes—
157C3×C3⋊Dic3—22no—
158C32×Dic3—22no—
159C2×D3×D4—94no—
160C2×D3×D4—91no—
161C3×C4×D4—34no—
162C3×C4×D4—91no—
163D4×Dic3—94no—
164D4×Dic3—91no—
165C4×#67—98no—
166C4×#67—92no—
167C2×(C2×D3)⋊C4—98no—
168C2×(C2×D3)⋊C4—92no—
169#893—94no—
170#893—91no—
171#892—94no—
172#892—91no—
173#67⋊C4—98no—
174#67⋊C4—92no—
175D3×C22⋊C4—98no—
176D3×C22⋊C4—92no—
177C4×D12—34no—
178C4×D12C3⋊D311yescharacteristic, complemented
179#872—94no—
180#872—91no—
181#872—184no—
182C23×C32—32no—
183C22×C3⋊D3—31no—
184C22×C3⋊D3—38no—
185C22×C3⋊D3—34no—
186C22×C3⋊D3—64no—
187C22×C3⋊D3—122no—
188C22×C3⋊D3—122no—
189C22×C3⋊D3C22×D312yes—
190C22×C3×D3—38no—
191C22×C3×D3C22×D318yes—
192C2×D32—332no—
193C2×D32—128no—
194C2×D32—128no—
195C32×D4—34no—
196C2×C32×C4—24no—
197C2×C32×C4—24no—
198C2×C32×C4—24no—
199C2×C32×C4—32no—
200C2×C32×C4—122no—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D314yes—
203C2×C32×C4C22×D314yes—
204C2×C32×C4C22×D311yescharacteristic
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D311yescharacteristic
207#638—38no—
208C2×C3⋊Dic3—31no—
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—31no—
211C2×C3⋊Dic3—34no—
212C2×C3⋊Dic3—34no—
213C2×C3⋊Dic3—34no—
214C2×C3⋊Dic3—64no—
215C2×C3⋊Dic3—64no—
216C2×C3⋊Dic3—64no—
217C2×C3⋊Dic3—122no—
218C2×C3⋊Dic3C22×D312yes—
219C4⋊(C3⋊D3)C4×D314yescomplemented
220C4×C3⋊D3—34no—
221C4×C3⋊D3—316no—
222C4×C3⋊D3—42no—
223C4×C3⋊D3—616no—
224C4×C3⋊D3—122no—
225C2×C3×Dic3—38no—
226C2×C3×Dic3C22×D318yes—
227C3×D12—316no—
228C3×D12C4×D3116yescomplemented
229C3×C4×D3—48no—
230C3×C4×D3—128no—
231#625—332no—
232D3×Dic3—128no—
233D3×Dic3—128no—
234#518—271noSylow
235C2×C33C23×C411yescharacteristic
236C2×C33H(5,𝔽2)11yescharacteristic
237C2×C33H(5,𝔽2)11yescharacteristic
238C3⋊(C3⋊D3)—22no—
239C3×C3⋊D3—24no—
240C32×D3—24no—
241C23×D3—98no—
242C23×D3—92no—
243C2×C3×D4—34no—
244C2×C3×D4—91no—
245C22×C3×C4—38no—
246C22×C3×C4—92no—
247C2×#67—98no—
248C2×#67—92no—
249C22×Dic3—98no—
250C22×Dic3—92no—
251D3×D4—932no—
252D3×D4—98no—
253C2×D12—34no—
254C2×D12C2×C3⋊D311yescharacteristic
255C2×C4×D3—38no—
256C2×C4×D3—94no—
257C2×C4×D3—91no—
258C2×C4×D3—98no—
259C2×C4×D3—92no—
260C2×C4×D3—184no—
261C2×C4×D3C2×C3⋊D312yescomplemented
262C3×C4⋊C4—34no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—38no—
265C3×C22⋊C4—92no—
266C3×C42—24no—
267C3×C42D3214yescomplemented
268C3×C42C2×C3⋊D311yescharacteristic, complemented
269#217—98no—
270#217—92no—
271(C2×D3)⋊C4—38no—
272(C2×D3)⋊C4—98no—
273(C2×D3)⋊C4—98no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—94no—
276(C2×D3)⋊C4—94no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—91no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—184no—
281(C2×D3)⋊C4—184no—
282(C2×D3)⋊C4C2×C3⋊D312yescomplemented
283C4⋊Dic3—34no—
284C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
285Dic3⋊C4—94no—
286Dic3⋊C4—94no—
287Dic3⋊C4—91no—
288Dic3⋊C4—91no—
289Dic3⋊C4—184no—
290Dic3⋊C4—184no—
291C4×Dic3—98no—
292C4×Dic3—94no—
293C4×Dic3—91no—
294C4×Dic3—92no—
295C4×Dic3—184no—
296C22×C32—24no—
297C22×C32—34no—
298C22×C32—62no—
299C22×C32—62no—
300C22×C32C23×D314yes—
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3—32no—
305C2×C3⋊D3—38no—
306C2×C3⋊D3—68no—
307C2×C3⋊D3—61no—
308C2×C3⋊D3—61no—
309C2×C3⋊D3—64no—
310C2×C3⋊D3—64no—
311C2×C3⋊D3—124no—
312C2×C3⋊D3—124no—
313C2×C3⋊D3—128no—
314C2×C3⋊D3C2×C4×D314yes—
315C2×C3×D3—28no—
316C2×C3×D3—28no—
317C2×C3×D3—316no—
318C2×C3×D3—68no—
319C2×C3×D3—68no—
320C2×C3×D3C2×C4×D3116yes—
321D32—1232no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—28no—
325C32×C4—24no—
326C32×C4—24no—
327C32×C4—44no—
328C32×C4—44no—
329C32×C4—62no—
330C32×C4C2×C4×D318yes—
331C32×C4C2×C4×D312yes—
332C3⋊Dic3—22no—
333C3⋊Dic3—32no—
334C3⋊Dic3—38no—
335C3⋊Dic3—68no—
336C3⋊Dic3—61no—
337C3⋊Dic3—61no—
338C3⋊Dic3—64no—
339C3⋊Dic3—64no—
340C3⋊Dic3—124no—
341C3⋊Dic3—124no—
342C3×Dic3—28no—
343C3×Dic3—68no—
344C22×D4—271no—
345C4×D4—31no—
346C4×D4—91no—
347C4×D4—272no—
348C4×D4—272no—
349C4×D4—271no—
350C4×D4—271no—
351#117—271no—
352#117—271no—
353C2×C22⋊C4—272no—
354C2×C22⋊C4—272no—
355C33#51811yescharacteristic, complemented, Sylow
356C23×C3—38no—
357C23×C3—92no—
358C22×D3—38no—
359C22×D3—932no—
360C22×D3—94no—
361C22×D3—91no—
362C22×D3—98no—
363C22×D3—184no—
364C22×D3—362no—
365C22×D3—362no—
366C22×D3—368no—
367C22×D3—368no—
368C22×D3C22×C3⋊D312yes—
369C3×D4—316no—
370C3×D4—94no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—38no—
375C2×C3×C4—92no—
376C2×C3×C4—128no—
377C2×C3×C4—362no—
378C2×C3×C4C2×D3214yes—
379C2×C3×C4C2×D3214yes—
380C2×C3×C4C2×D3214yes—
381C2×C3×C4C22×C3⋊D311yescharacteristic
382C2×C3×C4C22×C3⋊D311yescharacteristic
383C2×C3×C4C22×C3⋊D311yescharacteristic
384#67—932no—
385#67—98no—
386C2×Dic3—38no—
387C2×Dic3—94no—
388C2×Dic3—91no—
389C2×Dic3—94no—
390C2×Dic3—94no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—184no—
394C2×Dic3—184no—
395C2×Dic3—184no—
396C2×Dic3—362no—
397C2×Dic3—368no—
398C2×Dic3C22×C3⋊D312yes—
399D12—316no—
400D12C4×C3⋊D314yescomplemented
401C4×D3—42no—
402C4×D3—916no—
403C4×D3—94no—
404C4×D3—128no—
405C4×D3—1816no—
406C4×D3—368no—
407C4×D3—362no—
408C2×C32—24no—
409C2×C32—24no—
410C2×C32—24no—
411C2×C32—64no—
412C2×C32C22×C4×D314yes—
413C2×C32C22×C4×D311yescharacteristic
414C2×C32#100414yes—
415C2×C32#100411yescharacteristic
416C2×C32#100914yes—
417C2×C32#100911yescharacteristic
418C3⋊D3—24no—
419C3⋊D3—62no—
420C3⋊D3—68no—
421C3⋊D3—128no—
422C3×D3—216no—
423C3×D3—616no—
424C24—272no—
425C2×D4—31no—
426C2×D4—91no—
427C2×D4—278no—
428C2×D4—272no—
429C22×C4—32no—
430C22×C4—92no—
431C22×C4—272no—
432C22×C4—272no—
433C22×C4—271no—
434C4⋊C4—31no—
435C4⋊C4—91no—
436C4⋊C4—271no—
437C4⋊C4—271no—
438C22⋊C4—32no—
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C22⋊C4—271no—
444C22⋊C4—271no—
445C42—272no—
446C42—271no—
447C42D3×C3⋊D311yescharacteristic, complemented
448C22×C3—24no—
449C22×C3—316no—
450C22×C3—68no—
451C22×C3—68no—
452C22×C3—94no—
453C22×C3—182no—
454C22×C3—182no—
455C22×C3C22×D3214yes—
456C22×C3C23×C3⋊D311yescharacteristic
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—316no—
460C2×D3—68no—
461C2×D3—68no—
462C2×D3—98no—
463C2×D3—92no—
464C2×D3—188no—
465C2×D3—184no—
466C2×D3—184no—
467C2×D3—181no—
468C2×D3—181no—
469C2×D3—364no—
470C2×D3—364no—
471C2×D3—368no—
472C2×D3—3632no—
473C2×D3C2×C4×C3⋊D314yes—
474C3×C4—24no—
475C3×C4—24no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—28no—
479C3×C4—44no—
480C3×C4—44no—
481C3×C4—68no—
482C3×C4—182no—
483C3×C4C4×D3218yes—
484C3×C4C2×C4×C3⋊D312yes—
485Dic3—22no—
486Dic3—68no—
487Dic3—98no—
488Dic3—92no—
489Dic3—188no—
490Dic3—184no—
491Dic3—184no—
492Dic3—181no—
493Dic3—181no—
494Dic3—364no—
495Dic3—364no—
496C32—24no—
497C32#554514yescomplemented
498C32#554511yescharacteristic, complemented
499C23—32no—
500C23—92no—
501C23—278no—
502C23—271no—
503C23—1082no—
504C23—1082no—
505D4—34no—
506D4—94no—
507D4—278no—
508C2×C4—32no—
509C2×C4—92no—
510C2×C4—122no—
511C2×C4—271no—
512C2×C4—274no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—362no—
516C2×C4—1082no—
517C2×C4—1082no—
518C2×C4C2×D3×C3⋊D311yescharacteristic
519C2×C4C2×D3×C3⋊D311yescharacteristic
520C2×C4C2×D3×C3⋊D311yescharacteristic
521C2×C3—24no—
522C2×C3—24no—
523C2×C3—24no—
524C2×C3—616no—
525C2×C3—184no—
526C2×C3C2×C4×D3214yes—
527C2×C3#6430714yes—
528C2×C3#6430714yes—
529C2×C3C22×C4×C3⋊D311yescharacteristic
530C2×C3#6436111yescharacteristic
531C2×C3#6436611yescharacteristic
532D3—24no—
533D3—616no—
534D3—182no—
535D3—188no—
536D3—368no—
537C22—34no—
538C22—62no—
539C22—62no—
540C22—94no—
541C22—182no—
542C22—182no—
543C22—272no—
544C22—541no—
545C22—541no—
546C22—1088no—
547C22C22×D3×C3⋊D311yescharacteristic
548C4—21no—
549C4—21no—
550C4—62no—
551C4—182no—
552C4—272no—
553C4—541no—
554C4—541no—
555C4C4×D3×C3⋊D312yes—
556C3—24no—
557C3#1059399614yescomplemented
558C3#1059601611yescharacteristic, complemented
559C2—64no—
560C2—184no—
561C2—542no—
562C2C2×C4×D3×C3⋊D311yescharacteristic
563C2#1172457911yescharacteristic
564C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C42C22
socle, cosocleC22×C33C24
prime radicalC2×#89026C2
hypercenterC42D3×C3⋊D3
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11723913C2semidirect
3. C3×C4×C4⋊(C3⋊D3)C2semidirect
4. C32×C4×D12C2semidirect
5. C3⋊D3×(C2×D3)⋊C4C2semidirect
6. D3×#3752C2semidirect
7. #11723507C2semidirect
8. #11723506C2semidirect
9. #11723501C2semidirect
10. #11723499C2semidirect
11. #11723413C2semidirect
17. C2×C3×C4×C3⋊D3C22semidirect
19. C2×C32×C4×D3C22semidirect
21. C2×Dic3×C3⋊D3C22semidirect
22. C2×D3×C3⋊Dic3C22semidirect
23. #89026C4semidirect
24. C33×C42C22semidirect
27. #88857C22semidirect
28. #88857C22semidirect
29. C4×C3⋊(C3⋊Dic3)C22semidirect
30. C3×#3752C22semidirect
31. C3×C4⋊(C3⋊Dic3)C22semidirect
32. C32×(C2×D3)⋊C4C22semidirect
33. C32×C4⋊Dic3C22semidirect
35. #88803C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
48. C4×C4⋊(C3⋊D3)D3semidirect
53. C3×C4×D12D3semidirect
66. C2×D3×C3⋊D3C2×C4semidirect
73. C4×C3⋊(C3⋊D3)C2×C4semidirect
75. C3×C4⋊(C3⋊D3)C2×C4semidirect
78. C32×D12C2×C4semidirect
80. #6290C2×C4semidirect
97. C2×C4×C3⋊D3C2×D3semidirect
101. C2×C3×C4×D3C2×D3semidirect
109. C32×C42C2×D3semidirect
110. C32×C42C2×D3semidirect
120. #3752C2×D3semidirect
121. C4⋊(C3⋊Dic3)C2×D3semidirect
133. C3×(C2×D3)⋊C4C2×D3semidirect
135. C3×C4⋊Dic3C2×D3semidirect
178. C4×D12C3⋊D3semidirect
219. C4⋊(C3⋊D3)C4×D3semidirect
228. C3×D12C4×D3semidirect
261. C2×C4×D3C2×C3⋊D3semidirect
267. C3×C42D32semidirect
268. C3×C42C2×C3⋊D3semidirect
282. (C2×D3)⋊C4C2×C3⋊D3semidirect
284. C4⋊Dic3C2×C3⋊D3semidirect
355. C33#518semidirect
400. D12C4×C3⋊D3semidirect
447. C42D3×C3⋊D3semidirect
497. C32#5545semidirect
498. C32#5545semidirect
557. C3#10593996semidirect
558. C3#10596016semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239132
C3×C4×C4⋊(C3⋊D3)2
C32×C4×D122
C3⋊D3×(C2×D3)⋊C42
D3×#37522
#117235072
#117235062
#117235012
#117234992
#117234132
#105960163
#105939963