Finite Group Databasebeta

← #13176523

Subgroups — #13176523

Order 1728.

2348 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×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64101C211yesmaximal, characteristic, complemented
4C32×#892C211yesmaximal, characteristic, complemented
5#11723539C212yesmaximal
6#11723503C211yesmaximal, characteristic, complemented
7C4×#6289C212yesmaximal, complemented
8#11723464C212yesmaximal, complemented
9#11723459C212yesmaximal, complemented
10#11723424C211yesmaximal, characteristic, complemented
11#11723404C211yesmaximal, characteristic, complemented
12C3⋊D3×#117—31nomaximal
13#10594602—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×C4⋊C4C2211yescharacteristic, complemented
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
26#88857C2212yescomplemented
27C3×#3752C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29C3×C4×C3⋊Dic3C2212yescomplemented
30C32×(C2×D3)⋊C4C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic, complemented
32C32×C4×Dic3C2212yescomplemented
33#88806C2212yescomplemented
34#88803C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yescomplemented
37Dic3×C3⋊Dic3C2212yescomplemented
38C22×C4×C3⋊D3—31no—
39C2×C4×D32—34no—
40C32×#117—31no—
41#64135—31no—
42#64101—31no—
43#64101D311yescharacteristic, complemented
44C4⋊C4×C3⋊D3—31no—
45C4⋊C4×C3⋊D3—31no—
46C4⋊C4×C3⋊D3—34no—
47C4⋊C4×C3⋊D3—64no—
48C22⋊C4×C3⋊D3—32no—
49#64086—32no—
50#64081—32no—
51C42×C3⋊D3—32no—
52C3×#892—34no—
53C3×#892D314yescomplemented
54#63921—38no—
55#63900—34no—
56C4×#624—38no—
57#63877—38no—
58#63877—38no—
59#63841—34no—
60#63841—34no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2311yescharacteristic
63C22×C32×D3C2311yescharacteristic
64C2×D3×C3⋊D3—22no—
65C2×D3×C3⋊D3—22no—
66C2×D3×C3⋊D3C2×C414yes—
67C2×C33×C4C2312yes—
68C2×C33×C4C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4×C3⋊(C3⋊D3)—24no—
72C4×C3⋊(C3⋊D3)C2×C414yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4×C3⋊D3C2×C414yescomplemented
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2311yescharacteristic
78C32×C4×D3C2×C414yescomplemented
79#6289—24no—
80#6289C2×C414yes—
81Dic3×C3⋊D3C2×C414yescomplemented
82D3×C3⋊Dic3C2×C414yescomplemented
83D3×#117—94no—
84D3×#117—91no—
85C23×C3⋊D3—31no—
86C22×D32—34no—
87C22×C32×C4—31no—
88C22×C3⋊Dic3—31no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—31no—
91C2×C4×C3⋊D3—34no—
92C2×C4×C3⋊D3—31no—
93C2×C4×C3⋊D3—38no—
94C2×C4×C3⋊D3—32no—
95C2×C4×C3⋊D3—38no—
96C2×C4×C3⋊D3—32no—
97C2×C4×C3⋊D3—64no—
98C2×C4×C3⋊D3—68no—
99C2×C4×C3⋊D3C2×D311yescharacteristic
100C2×C3×C4×D3—34no—
101C2×C3×C4×D3C2×D314yes—
102C2×#624—34no—
103C2×#624—38no—
104C2×D3×Dic3—34no—
105C2×D3×Dic3—34no—
106C4×D32—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—32no—
114#3752—32no—
115#3752C2×D312yes—
116C4⋊(C3⋊Dic3)—31no—
117C4⋊(C3⋊Dic3)—34no—
118C4⋊(C3⋊Dic3)—64no—
119C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
120#3750—32no—
121#3750—38no—
122#3750—32no—
123#3750—68no—
124C4×C3⋊Dic3—32no—
125C4×C3⋊Dic3—32no—
126C4×C3⋊Dic3C2×D312yescomplemented
127C3×(C2×D3)⋊C4—38no—
128C3×(C2×D3)⋊C4C2×D318yes—
129C3×C4⋊Dic3—34no—
130C3×C4⋊Dic3C2×D314yescomplemented
131C3×C4×Dic3—38no—
132C3×C4×Dic3C2×D318yescomplemented
133#3724—38no—
134(C2×D3)⋊Dic3—38no—
135(C2×D3)⋊Dic3—38no—
136Dic32—38no—
137Dic32—38no—
138C22×C33C2411yescharacteristic
139C2×C3⋊(C3⋊D3)C22×C412yes—
140C2×C3⋊(C3⋊D3)#4112yes—
141C2×C3⋊(C3⋊D3)#4112yes—
142C2×C3×C3⋊D3—21no—
143C2×C3×C3⋊D3—21no—
144C2×C3×C3⋊D3C22×C412yes—
145C2×C32×D3—21no—
146C2×C32×D3—21no—
147C2×C32×D3C22×C412yes—
148D3×C3⋊D3—28no—
149C33×C4—22no—
150C33×C4C22×C412yes—
151C3⋊(C3⋊Dic3)—22no—
152C3⋊(C3⋊Dic3)C22×C412yes—
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3C22×C412yes—
155C32×Dic3—22no—
156C32×Dic3C22×C412yes—
157C22×C4×D3—94no—
158C22×C4×D3—91no—
159C3×#117—34no—
160C3×#117—91no—
161#926—94no—
162#926—91no—
163#892—34no—
164#892—94no—
165#892—91no—
166#892C3⋊D311yescharacteristic, complemented
167D3×C4⋊C4—94no—
168D3×C4⋊C4—94no—
169D3×C4⋊C4—91no—
170D3×C4⋊C4—91no—
171D3×C4⋊C4—184no—
172D3×C22⋊C4—98no—
173D3×C22⋊C4—92no—
174#877—98no—
175#877—92no—
176#872—98no—
177#872—92no—
178C42×D3—98no—
179C42×D3—92no—
180C23×C32—31no—
181C22×C3⋊D3—31no—
182C22×C3⋊D3—34no—
183C22×C3⋊D3—34no—
184C22×C3⋊D3—64no—
185C22×C3⋊D3—62no—
186C22×C3⋊D3—62no—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—34no—
189C22×C3×D3C22×D314yes—
190C2×D32—316no—
191C2×D32—68no—
192C2×D32—68no—
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—34no—
204C2×C3⋊Dic3—32no—
205C2×C3⋊Dic3—34no—
206C2×C3⋊Dic3—38no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—68no—
209C2×C3⋊Dic3C22×D312yes—
210C2×C3⋊Dic3C22×D311yescharacteristic
211C4×C3⋊D3—34no—
212C4×C3⋊D3—34no—
213C4×C3⋊D3—34no—
214C4×C3⋊D3—316no—
215C4×C3⋊D3—616no—
216C4×C3⋊D3—64no—
217C4×C3⋊D3—64no—
218C4×C3⋊D3—616no—
219C4×C3⋊D3—1216no—
220C4×C3⋊D3C4×D314yescomplemented
221C2×C3×Dic3—34no—
222C2×C3×Dic3—38no—
223C2×C3×Dic3C22×D318yes—
224C2×C3×Dic3C22×D314yes—
225C3×C4×D3—316no—
226C3×C4×D3C4×D3116yescomplemented
227#624—316no—
228#624—616no—
229D3×Dic3—316no—
230D3×Dic3—316no—
231C2×#117—271noSylow
232C2×C33C23×C411yescharacteristic
233C2×C33C2×#4111yescharacteristic
234C2×C33C2×#4111yescharacteristic
235C3⋊(C3⋊D3)#11714yescomplemented
236C3×C3⋊D3—22no—
237C32×D3—22no—
238C23×D3—94no—
239C23×D3—91no—
240C22×C3×C4—34no—
241C22×C3×C4—91no—
242C22×Dic3—94no—
243C22×Dic3—91no—
244C2×C4×D3—34no—
245C2×C4×D3—932no—
246C2×C4×D3—94no—
247C2×C4×D3—94no—
248C2×C4×D3—91no—
249C2×C4×D3—94no—
250C2×C4×D3—91no—
251C2×C4×D3—91no—
252C2×C4×D3—98no—
253C2×C4×D3—98no—
254C2×C4×D3—98no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—184no—
258C2×C4×D3—188no—
259C2×C4×D3C2×C3⋊D311yescharacteristic
260C3×C4⋊C4—24no—
261C3×C4⋊C4—34no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4D3214yescomplemented
264C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
265C3×C22⋊C4—38no—
266C3×C22⋊C4—92no—
267C3×C42—38no—
268C3×C42—92no—
269#217—98no—
270#217—92no—
271(C2×D3)⋊C4—38no—
272(C2×D3)⋊C4—98no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4C2×C3⋊D312yes—
275C4⋊Dic3—34no—
276C4⋊Dic3—94no—
277C4⋊Dic3—91no—
278C4⋊Dic3—184no—
279C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
280Dic3⋊C4—98no—
281Dic3⋊C4—98no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—188no—
285C4×Dic3—38no—
286C4×Dic3—98no—
287C4×Dic3—98no—
288C4×Dic3—92no—
289C4×Dic3—92no—
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—21no—
298C2×C3⋊D3—21no—
299C2×C3⋊D3—32no—
300C2×C3⋊D3—32no—
301C2×C3⋊D3—32no—
302C2×C3⋊D3—38no—
303C2×C3⋊D3—38no—
304C2×C3⋊D3—38no—
305C2×C3⋊D3—68no—
306C2×C3⋊D3—68no—
307C2×C3⋊D3—68no—
308C2×C3⋊D3—68no—
309C2×C3⋊D3C2×C4×D312yes—
310C2×C3×D3—24no—
311C2×C3×D3—24no—
312C2×C3×D3—38no—
313C2×C3×D3—64no—
314C2×C3×D3—64no—
315C2×C3×D3C2×C4×D318yes—
316D32—632no—
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—22no—
326C3⋊Dic3—32no—
327C3⋊Dic3—38no—
328C3⋊Dic3—68no—
329C3⋊Dic3—62no—
330C3⋊Dic3—68no—
331C3⋊Dic3—128no—
332C3⋊Dic3C2×C4×D312yes—
333C3×Dic3—28no—
334C3×Dic3—38no—
335C3×Dic3—68no—
336C3×Dic3C2×C4×D318yes—
337C23×C4—271no—
338#117—31no—
339#117—91no—
340#117—272no—
341#117—272no—
342#117—271no—
343#117—271no—
344C2×C4⋊C4—271no—
345C2×C4⋊C4—271no—
346C2×C22⋊C4—272no—
347C2×C42—272no—
348C33C2×#11711yescharacteristic, complemented, Sylow
349C23×C3—34no—
350C23×C3—91no—
351C22×D3—34no—
352C22×D3—916no—
353C22×D3—94no—
354C22×D3—91no—
355C22×D3—94no—
356C22×D3—184no—
357C22×D3—182no—
358C22×D3—182no—
359C22×D3—188no—
360C22×D3—188no—
361C22×D3C22×C3⋊D311yescharacteristic
362C2×C3×C4—24no—
363C2×C3×C4—28no—
364C2×C3×C4—34no—
365C2×C3×C4—316no—
366C2×C3×C4—38no—
367C2×C3×C4—91no—
368C2×C3×C4—94no—
369C2×C3×C4—92no—
370C2×C3×C4C2×D3218yes—
371C2×C3×C4C2×D3214yes—
372C2×C3×C4C22×C3⋊D312yes—
373C2×C3×C4C22×C3⋊D311yescharacteristic
374C2×Dic3—34no—
375C2×Dic3—38no—
376C2×Dic3—916no—
377C2×Dic3—94no—
378C2×Dic3—91no—
379C2×Dic3—94no—
380C2×Dic3—98no—
381C2×Dic3—92no—
382C2×Dic3—184no—
383C2×Dic3—188no—
384C2×Dic3C22×C3⋊D312yes—
385C2×Dic3C22×C3⋊D311yescharacteristic
386C4×D3—316no—
387C4×D3—916no—
388C4×D3—916no—
389C4×D3—94no—
390C4×D3—916no—
391C4×D3—94no—
392C4×D3—94no—
393C4×D3—1816no—
394C4×D3—184no—
395C4×D3—1816no—
396C4×D3—184no—
397C4×D3—1816no—
398C4×D3—3616no—
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×C32D3×#4114yes—
407C2×C32D3×#4114yes—
408C2×C32D3×#4111yescharacteristic
409C2×C32D3×#4111yescharacteristic
410C3⋊D3—22no—
411C3⋊D3—34no—
412C3⋊D3—316no—
413C3⋊D3—616no—
414C3×D3—28no—
415C3×D3—68no—
416C24—271no—
417C22×C4—31no—
418C22×C4—91no—
419C22×C4—271no—
420C22×C4—278no—
421C22×C4—272no—
422C22×C4—271no—
423C22×C4—271no—
424C22×C4—271no—
425C22×C4—272no—
426C4⋊C4—31no—
427C4⋊C4—91no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—271no—
431C4⋊C4D3×C3⋊D311yescharacteristic, complemented
432C22⋊C4—32no—
433C22⋊C4—92no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C42—32no—
437C42—92no—
438C42—272no—
439C42—272no—
440C22×C3—24no—
441C22×C3—38no—
442C22×C3—64no—
443C22×C3—64no—
444C22×C3—92no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3C22×D3214yes—
448C22×C3C23×C3⋊D311yescharacteristic
449C2×D3—21no—
450C2×D3—21no—
451C2×D3—38no—
452C2×D3—64no—
453C2×D3—64no—
454C2×D3—98no—
455C2×D3—98no—
456C2×D3—98no—
457C2×D3—92no—
458C2×D3—92no—
459C2×D3—92no—
460C2×D3—188no—
461C2×D3—188no—
462C2×D3—188no—
463C2×D3—188no—
464C2×D3—1832no—
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—92no—
473C3×C4—182no—
474C3×C4C4×D3218yes—
475C3×C4C2×C4×C3⋊D312yes—
476Dic3—22no—
477Dic3—38no—
478Dic3—68no—
479Dic3—98no—
480Dic3—92no—
481Dic3—188no—
482Dic3—188no—
483Dic3—182no—
484Dic3—368no—
485Dic3C2×C4×C3⋊D312yes—
486C32—24no—
487C32D3×#11714yescomplemented
488C32D3×#11711yescharacteristic, complemented
489C23—31no—
490C23—91no—
491C23—274no—
492C23—271no—
493C23—542no—
494C23—542no—
495C2×C4—31no—
496C2×C4—34no—
497C2×C4—32no—
498C2×C4—91no—
499C2×C4—94no—
500C2×C4—92no—
501C2×C4—274no—
502C2×C4—274no—
503C2×C4—274no—
504C2×C4—271no—
505C2×C4—274no—
506C2×C4—272no—
507C2×C4—544no—
508C2×C4—544no—
509C2×C4C2×D3×C3⋊D312yes—
510C2×C4C2×D3×C3⋊D311yescharacteristic
511C2×C3—24no—
512C2×C3—24no—
513C2×C3—24no—
514C2×C3—68no—
515C2×C3—182no—
516C2×C3C2×C4×D3214yes—
517C2×C3#6431214yes—
518C2×C3#6431914yes—
519C2×C3C22×C4×C3⋊D311yescharacteristic
520C2×C3#41×C3⋊D311yescharacteristic
521C2×C3#41×C3⋊D311yescharacteristic
522D3—22no—
523D3—68no—
524D3—916no—
525D3—94no—
526D3—1816no—
527C22—32no—
528C22—61no—
529C22—61no—
530C22—92no—
531C22—181no—
532C22—181no—
533C22—272no—
534C22—272no—
535C22—272no—
536C22—548no—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—22no—
539C4—32no—
540C4—62no—
541C4—92no—
542C4—182no—
543C4—272no—
544C4—542no—
545C4C4×D3×C3⋊D312yes—
546C3—24no—
547C3#1059460214yescomplemented
548C3C3⋊D3×#11711yescharacteristic, complemented
549C2—62no—
550C2—182no—
551C2—274no—
552C2C2×C4×D3×C3⋊D311yescharacteristic
553C2#1172458611yescharacteristic
554C2#1172459611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C4⋊C4C22
socle, cosocleC22×C33C24
prime radicalC22×D3×C3⋊D3C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
3. C3×#64101C2semidirect
4. C32×#892C2semidirect
6. #11723503C2semidirect
7. C4×#6289C2semidirect
8. #11723464C2semidirect
9. #11723459C2semidirect
10. #11723424C2semidirect
11. #11723404C2semidirect
23. C4×D3×C3⋊D3C4semidirect
24. C33×C4⋊C4C22semidirect
25. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
26. #88857C22semidirect
28. C3×C4⋊(C3⋊Dic3)C22semidirect
29. C3×C4×C3⋊Dic3C22semidirect
31. C32×C4⋊Dic3C22semidirect
32. C32×C4×Dic3C22semidirect
33. #88806C22semidirect
36. Dic3×C3⋊Dic3C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
43. #64101D3semidirect
53. C3×#892D3semidirect
75. C3×C4×C3⋊D3C2×C4semidirect
78. C32×C4×D3C2×C4semidirect
81. Dic3×C3⋊D3C2×C4semidirect
82. D3×C3⋊Dic3C2×C4semidirect
109. C32×C4⋊C4C2×D3semidirect
110. C32×C4⋊C4C2×D3semidirect
119. C4⋊(C3⋊Dic3)C2×D3semidirect
126. C4×C3⋊Dic3C2×D3semidirect
130. C3×C4⋊Dic3C2×D3semidirect
132. C3×C4×Dic3C2×D3semidirect
166. #892C3⋊D3semidirect
220. C4×C3⋊D3C4×D3semidirect
226. C3×C4×D3C4×D3semidirect
235. C3⋊(C3⋊D3)#117semidirect
263. C3×C4⋊C4D32semidirect
264. C3×C4⋊C4C2×C3⋊D3semidirect
279. C4⋊Dic3C2×C3⋊D3semidirect
290. C4×Dic3C2×C3⋊D3semidirect
348. C33C2×#117semidirect
399. C4×D3C4×C3⋊D3semidirect
431. C4⋊C4D3×C3⋊D3semidirect
487. C32D3×#117semidirect
488. C32D3×#117semidirect
547. C3#10594602semidirect
548. 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
C4⋊C4×C3⋊(C3⋊D3)2
C3×#641012
C32×#8922
#117235392
#117235032
C4×#62892
#117234642
#117234592
#117234242
#117234042
C3⋊D3×#1173
#105946023