Finite Group Databasebeta

← #13176527

Subgroups — #13176527

Order 1728.

2404 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723934C211yesmaximal, characteristic, complemented
3C3×#64102C211yesmaximal, characteristic, complemented
4C32×#892C211yesmaximal, characteristic, complemented
5#11723539C212yesmaximal, complemented
6C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
7C4×#6292C212yesmaximal, complemented
8#11723501C211yesmaximal, characteristic, complemented
9#11723494C211yesmaximal, characteristic, complemented
10#11723471C212yesmaximal, complemented
11#11723423C211yesmaximal, characteristic, complemented
12#10595992—31nomaximal
13#10594604—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yescomplemented
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6292C2212yes—
21C2×#6289C2212yescomplemented
22C2×Dic3×C3⋊D3C2212yescomplemented
23D3×C4⋊(C3⋊D3)C418yescomplemented
24C33×C4⋊C4C2211yescharacteristic, complemented
25#88859C2212yescomplemented
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
27C3×#3752C2212yescomplemented
28C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
29C32×(C2×D3)⋊C4C2212yes—
30C32×C4⋊Dic3C2211yescharacteristic, complemented
31C32×C4×Dic3C2212yescomplemented
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
33Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
34#88804C2212yescomplemented
35#88803C2212yescomplemented
36(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
37(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
38C22×C4⋊(C3⋊D3)—31no—
39C2×D3×D12—34no—
40C32×#117—31no—
41#64135—31no—
42#64102—31no—
43#64102—31no—
44#64102—31no—
45#64102—34no—
46#64102—64no—
47#64102D311yescharacteristic, complemented
48C22⋊C4×C3⋊D3—32no—
49C22⋊C4×C3⋊D3—32no—
50C4×C4⋊(C3⋊D3)—32no—
51C4×C4⋊(C3⋊D3)—32no—
52C3×#893—34no—
53C3×#892D314yescomplemented
54#63921—38no—
55D3×(C2×D3)⋊C4—38no—
56C4×#626—38no—
57#63898—34no—
58#63894—34no—
59#63880—38no—
60#63855—34no—
61C22×C3⋊(C3⋊D3)C2312yes—
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2311yescharacteristic
64C2×D3×C3⋊D3—42no—
65C2×D3×C3⋊D3—42no—
66C2×D3×C3⋊D3C2×C418yescomplemented
67C2×C33×C4C2312yes—
68C2×C33×C4C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4⋊(C3⋊(C3⋊D3))C2×C414yescomplemented
72C4×C3⋊(C3⋊D3)—42no—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4⋊(C3⋊D3)C2×C414yescomplemented
76C3×C4×C3⋊D3—42no—
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79C32×C4×D3C2×C414yescomplemented
80#6292C2×C418yescomplemented
81#6289—42no—
82Dic3×C3⋊D3—42no—
83#5545—91no—
84#5521—94no—
85C23×C3⋊D3—32no—
86C22×D32—38no—
87C22×C32×C4—31no—
88C2×C4⋊(C3⋊D3)—32no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4⋊(C3⋊D3)—34no—
91C2×C4⋊(C3⋊D3)—38no—
92C2×C4⋊(C3⋊D3)—64no—
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C4×C3⋊D3—32no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—38no—
97C2×C4×C3⋊D3—32no—
98C2×C4×C3⋊D3—68no—
99C2×C4×C3⋊D3C2×D312yescomplemented
100C2×C3×D12—34no—
101C2×C3×C4×D3—38no—
102C2×C3×C4×D3C2×D314yes—
103C2×#626—38no—
104C2×#624—38no—
105C2×D3×Dic3—38no—
106D3×D12—332no—
107C32×C4⋊C4—24no—
108C32×C4⋊C4—31no—
109C32×C4⋊C4C2×D314yescomplemented
110C32×C4⋊C4C2×D311yescharacteristic, complemented
111C32×C22⋊C4—32no—
112C32×C42—32no—
113#3757—31no—
114#3757—31no—
115#3752—32no—
116#3752—32no—
117#3752—38no—
118#3752—32no—
119#3752—68no—
120#3752C2×D312yescomplemented
121C4⋊(C3⋊Dic3)—31no—
122C4⋊(C3⋊Dic3)—31no—
123C4×C3⋊Dic3—31no—
124C4×C3⋊Dic3—34no—
125C4×C3⋊Dic3—64no—
126C4×C3⋊Dic3C2×D311yescharacteristic, complemented
127C3×(C2×D3)⋊C4—38no—
128C3×(C2×D3)⋊C4C2×D318yes—
129C3×C4⋊Dic3C2×D314yescomplemented
130C3×C4×Dic3—34no—
131C3×C4×Dic3C2×D318yescomplemented
132Dic3⋊Dic3—34no—
133Dic3⋊Dic3—34no—
134#3722—38no—
135(C2×D3)⋊Dic3—34no—
136(C2×D3)⋊Dic3—34no—
137(C2×D3)⋊Dic3—38no—
138C22×C33C2411yescharacteristic
139C2×C3⋊(C3⋊D3)—22no—
140C2×C3⋊(C3⋊D3)—22no—
141C2×C3⋊(C3⋊D3)C22×C414yes—
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3C22×C414yes—
145C2×C32×D3—21no—
146C2×C32×D3—21no—
147C2×C32×D3C22×C412yes—
148D3×C3⋊D3—48no—
149C33×C4—22no—
150C33×C4C22×C412yes—
151C3⋊(C3⋊Dic3)—21no—
152C3⋊(C3⋊Dic3)—21no—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3—21no—
155C32×Dic3—22no—
156C32×Dic3C22×C412yes—
157C2×D3×D4—91no—
158C22×D12—94no—
159C3×C4×D4—91no—
160C3×#117—34no—
161D4×Dic3—91no—
162C4×#67—92no—
163C2×(C2×D3)⋊C4—92no—
164#926—94no—
165#893—34no—
166#893—94no—
167#893—94no—
168#893—94no—
169#893—91no—
170#893—184no—
171#892C3⋊D311yescharacteristic, complemented
172#67⋊C4—92no—
173D3×C22⋊C4—98no—
174D3×C22⋊C4—98no—
175D3×C22⋊C4—92no—
176C4×D12—98no—
177C4×D12—98no—
178C4×D12—91no—
179#872—91no—
180C23×C32—31no—
181C22×C3⋊D3—32no—
182C22×C3⋊D3—38no—
183C22×C3⋊D3—38no—
184C22×C3⋊D3—68no—
185C22×C3⋊D3—122no—
186C22×C3⋊D3—122no—
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—38no—
189C22×C3×D3C22×D314yes—
190C2×D32—332no—
191C2×D32—128no—
192C2×D32—128no—
193C2×C32×C4—24no—
194C2×C32×C4—28no—
195C2×C32×C4—31no—
196C2×C32×C4—34no—
197C2×C32×C4—32no—
198C2×C32×C4C22×D318yes—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D312yes—
201C2×C32×C4C22×D311yescharacteristic
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—34no—
205C2×C3⋊Dic3—34no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3C22×D311yescharacteristic
209C2×C3⋊Dic3C22×D311yescharacteristic
210C4⋊(C3⋊D3)—38no—
211C4⋊(C3⋊D3)—34no—
212C4⋊(C3⋊D3)—316no—
213C4⋊(C3⋊D3)—616no—
214C4⋊(C3⋊D3)C4×D314yescomplemented
215C4×C3⋊D3—42no—
216C4×C3⋊D3—122no—
217C4×C3⋊D3—122no—
218C4×C3⋊D3—128no—
219C4×C3⋊D3—1216no—
220C4×C3⋊D3—122no—
221C2×C3×Dic3—34no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3C22×D318yes—
224C2×C3×Dic3C22×D314yes—
225C3×D12—316no—
226C3×C4×D3—128no—
227C3×C4×D3C4×D3116yescomplemented
228#626—332no—
229#624—128no—
230D3×Dic3—128no—
231#518—271noSylow
232C2×C33C23×C411yescharacteristic
233C2×C33H(5,𝔽2)11yescharacteristic
234C2×C33H(5,𝔽2)11yescharacteristic
235C3⋊(C3⋊D3)—24no—
236C3×C3⋊D3—24no—
237C32×D3—22no—
238C23×D3—98no—
239C23×D3—92no—
240C2×C3×D4—91no—
241C22×C3×C4—34no—
242C22×C3×C4—92no—
243C2×#67—92no—
244C22×Dic3—92no—
245D3×D4—98no—
246C2×D12—34no—
247C2×D12—932no—
248C2×D12—98no—
249C2×D12—94no—
250C2×D12—91no—
251C2×D12—184no—
252C2×C4×D3—38no—
253C2×C4×D3—98no—
254C2×C4×D3—98no—
255C2×C4×D3—98no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—188no—
259C2×C4×D3C2×C3⋊D311yescharacteristic
260C3×C4⋊C4—24no—
261C3×C4⋊C4—34no—
262C3×C4⋊C4D3214yescomplemented
263C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
264C3×C22⋊C4—38no—
265C3×C22⋊C4—92no—
266C3×C42—38no—
267C3×C42—91no—
268#217—94no—
269#217—94no—
270#217—92no—
271(C2×D3)⋊C4—38no—
272(C2×D3)⋊C4—98no—
273(C2×D3)⋊C4—98no—
274(C2×D3)⋊C4—91no—
275(C2×D3)⋊C4—91no—
276(C2×D3)⋊C4—98no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—188no—
280(C2×D3)⋊C4C2×C3⋊D312yes—
281C4⋊Dic3—94no—
282C4⋊Dic3—94no—
283C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
284Dic3⋊C4—91no—
285Dic3⋊C4—91no—
286C4×Dic3—34no—
287C4×Dic3—94no—
288C4×Dic3—91no—
289C4×Dic3—184no—
290C4×Dic3C2×C3⋊D312yescomplemented
291C22×C32—24no—
292C22×C32—32no—
293C22×C32—61no—
294C22×C32—61no—
295C22×C32C23×D314yes—
296C22×C32C23×D311yescharacteristic
297C2×C3⋊D3—22no—
298C2×C3⋊D3—22no—
299C2×C3⋊D3—34no—
300C2×C3⋊D3—316no—
301C2×C3⋊D3—616no—
302C2×C3⋊D3—62no—
303C2×C3⋊D3—62no—
304C2×C3⋊D3—68no—
305C2×C3⋊D3—68no—
306C2×C3⋊D3—128no—
307C2×C3⋊D3—128no—
308C2×C3⋊D3—128no—
309C2×C3⋊D3C2×C4×D314yes—
310C2×C3×D3—24no—
311C2×C3×D3—24no—
312C2×C3×D3—316no—
313C2×C3×D3—68no—
314C2×C3×D3—68no—
315C2×C3×D3C2×C4×D318yes—
316D32—1232no—
317C32×C4—22no—
318C32×C4—28no—
319C32×C4—28no—
320C32×C4—32no—
321C32×C4—48no—
322C32×C4—62no—
323C32×C4C2×C4×D318yes—
324C32×C4C2×C4×D312yes—
325C3⋊Dic3—21no—
326C3⋊Dic3—21no—
327C3⋊Dic3—68no—
328C3⋊Dic3—61no—
329C3⋊Dic3—64no—
330C3⋊Dic3—61no—
331C3⋊Dic3—64no—
332C3⋊Dic3—124no—
333C3×Dic3—28no—
334C3×Dic3—64no—
335C3×Dic3—64no—
336C3×Dic3C2×C4×D318yes—
337C22×D4—271no—
338C4×D4—91no—
339C4×D4—272no—
340C4×D4—272no—
341C4×D4—271no—
342C4×D4—271no—
343C4×D4—271no—
344#117—31no—
345#117—271no—
346C2×C22⋊C4—272no—
347C2×C22⋊C4—272no—
348C33#51811yescharacteristic, complemented, Sylow
349C23×C3—34no—
350C23×C3—92no—
351C22×D3—38no—
352C22×D3—932no—
353C22×D3—98no—
354C22×D3—92no—
355C22×D3—98no—
356C22×D3—188no—
357C22×D3—362no—
358C22×D3—362no—
359C22×D3—368no—
360C22×D3—368no—
361C22×D3C22×C3⋊D311yescharacteristic
362C3×D4—94no—
363C2×C3×C4—24no—
364C2×C3×C4—28no—
365C2×C3×C4—34no—
366C2×C3×C4—316no—
367C2×C3×C4—38no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—362no—
371C2×C3×C4C2×D3218yes—
372C2×C3×C4C2×D3214yes—
373C2×C3×C4C22×C3⋊D312yes—
374C2×C3×C4C22×C3⋊D311yescharacteristic
375#67—98no—
376C2×Dic3—34no—
377C2×Dic3—34no—
378C2×Dic3—94no—
379C2×Dic3—94no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—184no—
383C2×Dic3—184no—
384C2×Dic3—362no—
385C2×Dic3C22×C3⋊D312yes—
386C2×Dic3C22×C3⋊D311yescharacteristic
387D12—316no—
388D12—932no—
389D12—916no—
390D12—94no—
391D12—1816no—
392C4×D3—128no—
393C4×D3—362no—
394C4×D3—368no—
395C4×D3—368no—
396C4×D3—362no—
397C4×D3—3616no—
398C4×D3—368no—
399C4×D3C4×C3⋊D314yescomplemented
400C2×C32—24no—
401C2×C32—24no—
402C2×C32—24no—
403C2×C32—62no—
404C2×C32C22×C4×D314yes—
405C2×C32C22×C4×D311yescharacteristic
406C2×C32#100411yescharacteristic
407C2×C32#100914yes—
408C2×C32#100914yes—
409C2×C32#100911yescharacteristic
410C3⋊D3—24no—
411C3⋊D3—64no—
412C3⋊D3—616no—
413C3⋊D3—1216no—
414C3×D3—28no—
415C3×D3—616no—
416C24—272no—
417C2×D4—91no—
418C2×D4—278no—
419C2×D4—272no—
420C2×D4—271no—
421C22×C4—31no—
422C22×C4—92no—
423C22×C4—272no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—31no—
427C4⋊C4—271no—
428C4⋊C4—271no—
429C4⋊C4D3×C3⋊D311yescharacteristic, complemented
430C22⋊C4—32no—
431C22⋊C4—92no—
432C22⋊C4—272no—
433C22⋊C4—272no—
434C22⋊C4—271no—
435C22⋊C4—271no—
436C22⋊C4—272no—
437C42—32no—
438C42—91no—
439C42—271no—
440C22×C3—24no—
441C22×C3—38no—
442C22×C3—64no—
443C22×C3—64no—
444C22×C3—94no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3C22×D3214yes—
448C22×C3C23×C3⋊D311yescharacteristic
449C2×D3—21no—
450C2×D3—21no—
451C2×D3—316no—
452C2×D3—68no—
453C2×D3—68no—
454C2×D3—916no—
455C2×D3—94no—
456C2×D3—1816no—
457C2×D3—188no—
458C2×D3—188no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—368no—
462C2×D3—368no—
463C2×D3—368no—
464C2×D3—3632no—
465C2×D3C2×C4×C3⋊D312yes—
466C3×C4—28no—
467C3×C4—22no—
468C3×C4—28no—
469C3×C4—38no—
470C3×C4—48no—
471C3×C4—68no—
472C3×C4—181no—
473C3×C4—181no—
474C3×C4C4×D3218yes—
475C3×C4C2×C4×C3⋊D312yes—
476Dic3—22no—
477Dic3—64no—
478Dic3—64no—
479Dic3—188no—
480Dic3—184no—
481Dic3—184no—
482Dic3—181no—
483Dic3—181no—
484Dic3—364no—
485Dic3C2×C4×C3⋊D312yes—
486C32—24no—
487C32#552114yescomplemented
488C32#554511yescharacteristic, complemented
489C23—31no—
490C23—92no—
491C23—278no—
492C23—272no—
493C23—1082no—
494C23—1082no—
495D4—94no—
496D4—278no—
497D4—274no—
498C2×C4—31no—
499C2×C4—34no—
500C2×C4—32no—
501C2×C4—91no—
502C2×C4—91no—
503C2×C4—271no—
504C2×C4—271no—
505C2×C4—362no—
506C2×C4—1082no—
507C2×C4—1082no—
508C2×C4—1082no—
509C2×C4C2×D3×C3⋊D312yes—
510C2×C4C2×D3×C3⋊D311yescharacteristic
511C2×C3—24no—
512C2×C3—24no—
513C2×C3—24no—
514C2×C3—68no—
515C2×C3—184no—
516C2×C3C2×C4×D3214yes—
517C2×C3#6431414yes—
518C2×C3#6432014yes—
519C2×C3C22×C4×C3⋊D311yescharacteristic
520C2×C3#6436611yescharacteristic
521C2×C3#6436611yescharacteristic
522D3—22no—
523D3—616no—
524D3—1816no—
525D3—184no—
526D3—3616no—
527C22—32no—
528C22—61no—
529C22—61no—
530C22—94no—
531C22—182no—
532C22—182no—
533C22—274no—
534C22—542no—
535C22—542no—
536C22—1088no—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—22no—
539C4—32no—
540C4—62no—
541C4—181no—
542C4—181no—
543C4—541no—
544C4—541no—
545C4C4×D3×C3⋊D312yes—
546C3—24no—
547C3#1059460414yescomplemented
548C3#1059599211yescharacteristic, complemented
549C2—62no—
550C2—184no—
551C2—544no—
552C2C2×C4×D3×C3⋊D311yescharacteristic
553C2#1172458911yescharacteristic
554C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C4⋊C4C22
socle, cosocleC22×C33C24
prime radicalC2×D3×C4⋊(C3⋊D3)C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723934C2semidirect
3. C3×#64102C2semidirect
4. C32×#892C2semidirect
5. #11723539C2semidirect
6. C3⋊D3×(C2×D3)⋊C4C2semidirect
7. C4×#6292C2semidirect
8. #11723501C2semidirect
9. #11723494C2semidirect
10. #11723471C2semidirect
11. #11723423C2semidirect
16. C2×C4×C3⋊(C3⋊D3)C22semidirect
18. C2×C3×C4×C3⋊D3C22semidirect
21. C2×#6289C22semidirect
22. C2×Dic3×C3⋊D3C22semidirect
23. D3×C4⋊(C3⋊D3)C4semidirect
24. C33×C4⋊C4C22semidirect
25. #88859C22semidirect
26. C4×C3⋊(C3⋊Dic3)C22semidirect
27. C3×#3752C22semidirect
28. C3×C4×C3⋊Dic3C22semidirect
30. C32×C4⋊Dic3C22semidirect
31. C32×C4×Dic3C22semidirect
32. Dic3⋊(C3⋊Dic3)C22semidirect
33. Dic3⋊(C3⋊Dic3)C22semidirect
34. #88804C22semidirect
35. #88803C22semidirect
47. #64102D3semidirect
53. C3×#892D3semidirect
66. C2×D3×C3⋊D3C2×C4semidirect
71. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
75. C3×C4⋊(C3⋊D3)C2×C4semidirect
79. C32×C4×D3C2×C4semidirect
80. #6292C2×C4semidirect
99. C2×C4×C3⋊D3C2×D3semidirect
109. C32×C4⋊C4C2×D3semidirect
110. C32×C4⋊C4C2×D3semidirect
120. #3752C2×D3semidirect
126. C4×C3⋊Dic3C2×D3semidirect
129. C3×C4⋊Dic3C2×D3semidirect
131. C3×C4×Dic3C2×D3semidirect
171. #892C3⋊D3semidirect
214. C4⋊(C3⋊D3)C4×D3semidirect
227. C3×C4×D3C4×D3semidirect
262. C3×C4⋊C4D32semidirect
263. C3×C4⋊C4C2×C3⋊D3semidirect
283. C4⋊Dic3C2×C3⋊D3semidirect
290. C4×Dic3C2×C3⋊D3semidirect
348. C33#518semidirect
399. C4×D3C4×C3⋊D3semidirect
429. C4⋊C4D3×C3⋊D3semidirect
487. C32#5521semidirect
488. C32#5545semidirect
547. C3#10594604semidirect
548. C3#10595992semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
#117239342
C3×#641022
C32×#8922
#117235392
C3⋊D3×(C2×D3)⋊C42
C4×#62922
#117235012
#117234942
#117234712
#117234232
#105959923
#105946043