Finite Group Databasebeta

← #13177490

Subgroups — #13177490

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×C32×D3×D4C211yesmaximal, characteristic
2D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#67⋊(C3⋊Dic3)C212yesmaximal, complemented
5#67×C3⋊Dic3C212yesmaximal, complemented
6D3×#3757C212yesmaximal, complemented
7C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
8#11723499C211yesmaximal, characteristic, complemented
9D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
10#11723424C211yesmaximal, characteristic, complemented
11#11723423C211yesmaximal, characteristic, complemented
12#10596269—31nomaximal
13#10595215—34nomaximal
14C23×C32×D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2212yescomplemented
17C22×C3×C3⋊Dic3C2212yescomplemented
18C2×C32×#67C2212yes—
19C32×D3×D4C418yescomplemented
20C2×C32×D12C2211yescharacteristic
21C2×C32×C4×D3C2211yescharacteristic
22C2×D3×C3⋊Dic3C2212yescomplemented
23C2×D3×C3⋊Dic3C2212yescomplemented
24#88864C2212yescomplemented
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
27C3×#3757C2212yescomplemented
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
30Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
35(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
36(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
37Dic3×C3⋊Dic3C2212yescomplemented
38C22×C32×D4—31no—
39C2×C3×D3×D4D314yes—
40C2×#3757—32no—
41C2×#3757—32no—
42D4×C3⋊Dic3—31no—
43D4×C3⋊Dic3—32no—
44D4×C3⋊Dic3—32no—
45D4×C3⋊Dic3—31no—
46D4×C3⋊Dic3—34no—
47D4×C3⋊Dic3—31no—
48D4×C3⋊Dic3—64no—
49D4×C3⋊Dic3D311yescharacteristic, complemented
50#64135—31no—
51#64135—31no—
52C3×D4×Dic3—34no—
53#67⋊Dic3—38no—
54Dic3×#67—38no—
55D3×#217—38no—
56C2×(C2×D3)⋊Dic3—38no—
57#63898—34no—
58Dic3×D12—34no—
59#63855—34no—
60#63841—34no—
61C23×C33C2312yes—
62C22×C32×D3—42no—
63C22×C32×D3—42no—
64C22×C32×D3C2×C418yescomplemented
65C22×C32×D3C2311yescharacteristic
66C22×C32×D3C2312yes—
67C33×D4C2×C414yescomplemented
68C2×C33×C4C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)—42no—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C2×C3×C3⋊Dic3—42no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C32×#67C2×C418yescomplemented
78C2×C32×Dic3C2311yescharacteristic
79C32×D12C2×C414yescomplemented
80C32×C4×D3C2×C414yescomplemented
81D3×C3⋊Dic3—42no—
82D3×C3⋊Dic3—42no—
83#5798—94no—
84#5545—91no—
85C24×C32—32no—
86C23×C3×D3C2×D318yes—
87C2×C32×D4—24no—
88C2×C32×D4—32no—
89C2×C32×D4—38no—
90C2×C32×D4—31no—
91C2×C32×D4C2×D311yescharacteristic
92C2×C32×D4C2×D314yes—
93C22×C32×C4—31no—
94C22×C3⋊Dic3—32no—
95C22×C3⋊Dic3—38no—
96C22×C3⋊Dic3—32no—
97C22×C3⋊Dic3—32no—
98C22×C3⋊Dic3—68no—
99C22×C3⋊Dic3C2×D312yescomplemented
100C2×C3×#67C2×D318yes—
101C22×C3×Dic3—38no—
102C3×D3×D4Dic3132yescomplemented
103C2×C3×D12C2×D314yes—
104C2×C3×C4×D3C2×D314yes—
105C2×D3×Dic3—38no—
106C2×D3×Dic3—38no—
107#3757—32no—
108#3757—38no—
109#3757—32no—
110#3757—31no—
111#3757—31no—
112#3757—32no—
113#3757—32no—
114#3757—68no—
115#3757C2×D312yescomplemented
116C4⋊(C3⋊Dic3)—31no—
117C4⋊(C3⋊Dic3)—34no—
118C4⋊(C3⋊Dic3)—31no—
119C4⋊(C3⋊Dic3)—31no—
120C4⋊(C3⋊Dic3)—64no—
121C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
122C4×C3⋊Dic3—31no—
123C4×C3⋊Dic3—34no—
124C4×C3⋊Dic3—32no—
125C4×C3⋊Dic3—64no—
126C4×C3⋊Dic3C2×D311yescharacteristic, complemented
127C3×#217—38no—
128C3×C4⋊Dic3—34no—
129C3×C4×Dic3—34no—
130Dic3⋊Dic3—34no—
131Dic3⋊Dic3—34no—
132(C2×D3)⋊Dic3—38no—
133(C2×D3)⋊Dic3—34no—
134(C2×D3)⋊Dic3—34no—
135(C2×D3)⋊Dic3—38no—
136(C2×D3)⋊Dic3—38no—
137Dic32—38no—
138C22×C33—22no—
139C22×C33—22no—
140C22×C33C22×C414yes—
141C22×C33C2411yescharacteristic
142C2×C32×D3—22no—
143C2×C32×D3—22no—
144C2×C32×D3—21no—
145C2×C32×D3—21no—
146C2×C32×D3—48no—
147C2×C32×D3C22×C412yes—
148C2×C32×D3C22×C414yes—
149C33×C4C22×C412yes—
150C3⋊(C3⋊Dic3)—21no—
151C3⋊(C3⋊Dic3)—21no—
152C3⋊(C3⋊Dic3)—22no—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—21no—
156C32×Dic3C22×C412yes—
157C22×C3×D4—34no—
158C2×D3×D4C3⋊D311yescharacteristic
159C3×C4×D4—91no—
160C2×#217—98no—
161C2×#217—98no—
162D4×Dic3—34no—
163D4×Dic3—94no—
164D4×Dic3—94no—
165D4×Dic3—98no—
166D4×Dic3—98no—
167D4×Dic3—94no—
168D4×Dic3—91no—
169D4×Dic3—184no—
170C4×#67—92no—
171C2×(C2×D3)⋊C4—92no—
172#926—94no—
173#926—94no—
174#893—91no—
175#892—91no—
176#67⋊C4—92no—
177D3×C22⋊C4—92no—
178C4×D12—91no—
179#872—91no—
180C23×C32—28no—
181C23×C32—31no—
182C23×C32—32no—
183C23×C32—38no—
184C23×C32—122no—
185C23×C32—122no—
186C23×C32C22×D318yes—
187C23×C32C22×D312yes—
188C22×C3×D3—48no—
189C22×C3×D3—48no—
190C22×C3×D3C2×Dic3132yescomplemented
191C22×C3×D3C22×D314yes—
192C22×C3×D3C22×D318yes—
193C32×D4—216no—
194C32×D4—38no—
195C32×D4—34no—
196C32×D4C4×D314yescomplemented
197C32×D4C2×Dic3116yescomplemented
198C2×C32×C4—24no—
199C2×C32×C4—31no—
200C2×C32×C4—34no—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D311yescharacteristic
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—38no—
207C2×C3⋊Dic3—34no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—42no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—68no—
213C2×C3⋊Dic3—122no—
214C2×C3⋊Dic3—122no—
215C2×C3⋊Dic3—122no—
216C2×C3⋊Dic3—128no—
217C2×C3⋊Dic3—1216no—
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D311yescharacteristic
220C2×C3⋊Dic3C22×D311yescharacteristic
221C3×#67C2×Dic3132yescomplemented
222C2×C3×Dic3—38no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3—128no—
226C2×C3×Dic3C22×D314yes—
227C3×D12C2×Dic3116yescomplemented
228C3×C4×D3C2×Dic3116yescomplemented
229D3×Dic3—128no—
230D3×Dic3—128no—
231#518—271noSylow
232C2×C33—24no—
233C2×C33C23×C411yescharacteristic
234C2×C33H(5,𝔽2)11yescharacteristic
235C2×C33H(5,𝔽2)11yescharacteristic
236C32×D3—22no—
237C32×D3—24no—
238C24×C3—38no—
239C23×D3C2×C3⋊D312yes—
240C2×C3×D4—24no—
241C2×C3×D4—38no—
242C2×C3×D4—332no—
243C2×C3×D4—34no—
244C2×C3×D4D3214yes—
245C2×C3×D4C2×C3⋊D311yescharacteristic
246C22×C3×C4—34no—
247C22×C3×C4—92no—
248C2×#67C2×C3⋊D312yes—
249C22×Dic3—38no—
250C22×Dic3—98no—
251C22×Dic3—98no—
252C22×Dic3—98no—
253C22×Dic3—92no—
254C22×Dic3—188no—
255D3×D4C3⋊Dic318yescomplemented
256C2×D12C2×C3⋊D311yescharacteristic
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C2×C4×D3C2×C3⋊D311yescharacteristic
260C3×C4⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C42—91no—
263#217—38no—
264#217—94no—
265#217—94no—
266#217—98no—
267#217—98no—
268#217—98no—
269#217—98no—
270#217—92no—
271#217—188no—
272(C2×D3)⋊C4—91no—
273(C2×D3)⋊C4—91no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277C4⋊Dic3—34no—
278C4⋊Dic3—94no—
279C4⋊Dic3—94no—
280C4⋊Dic3—94no—
281C4⋊Dic3—91no—
282C4⋊Dic3—184no—
283Dic3⋊C4—91no—
284Dic3⋊C4—91no—
285C4×Dic3—34no—
286C4×Dic3—98no—
287C4×Dic3—94no—
288C4×Dic3—91no—
289C4×Dic3—92no—
290C4×Dic3—184no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—24no—
294C22×C32—216no—
295C22×C32—28no—
296C22×C32—28no—
297C22×C32—32no—
298C22×C32—34no—
299C22×C32—48no—
300C22×C32—48no—
301C22×C32—62no—
302C22×C32—61no—
303C22×C32—62no—
304C22×C32—61no—
305C22×C32—128no—
306C22×C32C2×C4×D314yes—
307C22×C32C22×Dic3116yes—
308C22×C32C23×D314yes—
309C22×C32C23×D311yescharacteristic
310C2×C3×D3—28no—
311C2×C3×D3—28no—
312C2×C3×D3—24no—
313C2×C3×D3—24no—
314C2×C3×D3—432no—
315C2×C3×D3C22×Dic318yes—
316C2×C3×D3C22×Dic3116yes—
317C32×C4—28no—
318C32×C4—32no—
319C32×C4C2×C4×D312yes—
320C32×C4C22×Dic318yes—
321C3⋊Dic3—21no—
322C3⋊Dic3—22no—
323C3⋊Dic3—21no—
324C3⋊Dic3—68no—
325C3⋊Dic3—64no—
326C3⋊Dic3—61no—
327C3⋊Dic3—61no—
328C3⋊Dic3—62no—
329C3⋊Dic3—64no—
330C3⋊Dic3—68no—
331C3⋊Dic3—124no—
332C3⋊Dic3—128no—
333C3×Dic3—64no—
334C3×Dic3—68no—
335C3×Dic3—64no—
336C3×Dic3C22×Dic318yes—
337C22×D4—31no—
338C4×D4—91no—
339C4×D4—271no—
340C4×D4—272no—
341C4×D4—271no—
342C4×D4—272no—
343C4×D4—271no—
344#117—271no—
345#117—271no—
346C2×C22⋊C4—272no—
347C2×C22⋊C4—272no—
348C33#51811yescharacteristic, complemented, Sylow
349C23×C3—28no—
350C23×C3—34no—
351C23×C3—38no—
352C23×C3—332no—
353C23×C3—128no—
354C23×C3—128no—
355C23×C3C2×D3218yes—
356C23×C3C22×C3⋊D312yes—
357C22×D3—42no—
358C22×D3—42no—
359C22×D3C2×C3⋊Dic318yescomplemented
360C22×D3C22×C3⋊D311yescharacteristic
361C22×D3C22×C3⋊D312yes—
362C3×D4—216no—
363C3×D4—332no—
364C3×D4—316no—
365C3×D4D3×Dic3116yescomplemented
366C3×D4C2×C3⋊Dic314yescomplemented
367C2×C3×C4—24no—
368C2×C3×C4—34no—
369C2×C3×C4—316no—
370C2×C3×C4—92no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4—362no—
374C2×C3×C4C2×D3214yes—
375C2×C3×C4C22×C3⋊D311yescharacteristic
376#67C2×C3⋊Dic318yescomplemented
377C2×Dic3—34no—
378C2×Dic3—34no—
379C2×Dic3—38no—
380C2×Dic3—94no—
381C2×Dic3—94no—
382C2×Dic3—98no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—92no—
386C2×Dic3—128no—
387C2×Dic3—184no—
388C2×Dic3—184no—
389C2×Dic3—188no—
390C2×Dic3—368no—
391C2×Dic3—368no—
392C2×Dic3—368no—
393C2×Dic3—362no—
394C2×Dic3—3616no—
395C2×Dic3C22×C3⋊D311yescharacteristic
396D12C2×C3⋊Dic314yescomplemented
397C4×D3—362no—
398C4×D3—362no—
399C4×D3C2×C3⋊Dic314yescomplemented
400C2×C32—24no—
401C2×C32—24no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—216no—
405C2×C32—416no—
406C2×C32—64no—
407C2×C32—62no—
408C2×C32C22×C4×D311yescharacteristic
409C2×C32#100414yes—
410C2×C32#100414yes—
411C2×C32#100411yescharacteristic
412C2×C32#100911yescharacteristic
413C2×C32C23×Dic314yes—
414C3×D3—28no—
415C3×D3—216no—
416C24—32no—
417C2×D4—32no—
418C2×D4—38no—
419C2×D4—31no—
420C2×D4D3×C3⋊D311yescharacteristic
421C22×C4—31no—
422C22×C4—92no—
423C22×C4—272no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—91no—
427C4⋊C4—271no—
428C4⋊C4—271no—
429C4⋊C4—271no—
430C22⋊C4—92no—
431C22⋊C4—271no—
432C22⋊C4—271no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C42—91no—
438C42—272no—
439C42—271no—
440C22×C3—28no—
441C22×C3—28no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3—216no—
446C22×C3—38no—
447C22×C3—316no—
448C22×C3—48no—
449C22×C3—48no—
450C22×C3—68no—
451C22×C3—64no—
452C22×C3—68no—
453C22×C3—64no—
454C22×C3—1232no—
455C22×C3C2×D3×Dic3116yes—
456C22×C3C22×C3⋊Dic314yes—
457C22×C3C22×D3214yes—
458C22×C3C23×C3⋊D311yescharacteristic
459C2×D3—22no—
460C2×D3—22no—
461C2×D3—21no—
462C2×D3—21no—
463C2×D3—48no—
464C2×D3C22×C3⋊Dic312yes—
465C2×D3C22×C3⋊Dic314yes—
466C3×C4—28no—
467C3×C4—38no—
468C3×C4—181no—
469C3×C4—182no—
470C3×C4—181no—
471C3×C4C2×D3×Dic318yes—
472C3×C4C22×C3⋊Dic312yes—
473Dic3—64no—
474Dic3—68no—
475Dic3—64no—
476Dic3—188no—
477Dic3—184no—
478Dic3—184no—
479Dic3—188no—
480Dic3—181no—
481Dic3—182no—
482Dic3—181no—
483Dic3—368no—
484Dic3—364no—
485Dic3C22×C3⋊Dic312yes—
486C32—24no—
487C32#554511yescharacteristic, complemented
488C32#579814yescomplemented
489C23—31no—
490C23—32no—
491C23—38no—
492C23—122no—
493C23—122no—
494C23C2×D3×C3⋊D312yes—
495D4—38no—
496D4—34no—
497D4D3×C3⋊Dic314yescomplemented
498C2×C4—31no—
499C2×C4—34no—
500C2×C4—91no—
501C2×C4—91no—
502C2×C4—92no—
503C2×C4—271no—
504C2×C4—271no—
505C2×C4—272no—
506C2×C4—362no—
507C2×C4—1082no—
508C2×C4—1082no—
509C2×C4—1082no—
510C2×C4C2×D3×C3⋊D311yescharacteristic
511C2×C3—216no—
512C2×C3—24no—
513C2×C3—24no—
514C2×C3—24no—
515C2×C3—24no—
516C2×C3—416no—
517C2×C3—68no—
518C2×C3—616no—
519C2×C3#6431314yes—
520C2×C3#6431414yes—
521C2×C3C22×D3×Dic314yes—
522C2×C3#6436111yescharacteristic
523C2×C3#6436111yescharacteristic
524C2×C3C23×C3⋊Dic311yescharacteristic
525D3—22no—
526D3—24no—
527C22—22no—
528C22—22no—
529C22—32no—
530C22—34no—
531C22—62no—
532C22—61no—
533C22—62no—
534C22—61no—
535C22—128no—
536C22C2×D3×C3⋊Dic314yes—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—32no—
539C4—181no—
540C4—182no—
541C4—181no—
542C4—541no—
543C4—541no—
544C4—542no—
545C4C2×D3×C3⋊Dic312yes—
546C3—24no—
547C3#1059521514yescomplemented
548C3#1059626911yescharacteristic, complemented
549C2—24no—
550C2—62no—
551C2—64no—
552C2#1172458711yescharacteristic
553C2#1172458811yescharacteristic
554C2C22×D3×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×C32×D3×D4C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×C3⋊(C3⋊Dic3)C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #67⋊(C3⋊Dic3)C2semidirect
5. #67×C3⋊Dic3C2semidirect
6. D3×#3757C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. #11723499C2semidirect
9. D12×C3⋊Dic3C2semidirect
10. #11723424C2semidirect
11. #11723423C2semidirect
16. C22×C3⋊(C3⋊Dic3)C22semidirect
17. C22×C3×C3⋊Dic3C22semidirect
19. C32×D3×D4C4semidirect
22. C2×D3×C3⋊Dic3C22semidirect
23. C2×D3×C3⋊Dic3C22semidirect
24. #88864C22semidirect
25. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
26. C4×C3⋊(C3⋊Dic3)C22semidirect
27. C3×#3757C22semidirect
28. C3×C4⋊(C3⋊Dic3)C22semidirect
29. C3×C4×C3⋊Dic3C22semidirect
30. Dic3⋊(C3⋊Dic3)C22semidirect
31. Dic3⋊(C3⋊Dic3)C22semidirect
35. (C2×D3)⋊(C3⋊Dic3)C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
49. D4×C3⋊Dic3D3semidirect
64. C22×C32×D3C2×C4semidirect
67. C33×D4C2×C4semidirect
77. C32×#67C2×C4semidirect
79. C32×D12C2×C4semidirect
80. C32×C4×D3C2×C4semidirect
99. C22×C3⋊Dic3C2×D3semidirect
102. C3×D3×D4Dic3semidirect
115. #3757C2×D3semidirect
121. C4⋊(C3⋊Dic3)C2×D3semidirect
126. C4×C3⋊Dic3C2×D3semidirect
190. C22×C3×D3C2×Dic3semidirect
196. C32×D4C4×D3semidirect
197. C32×D4C2×Dic3semidirect
221. C3×#67C2×Dic3semidirect
227. C3×D12C2×Dic3semidirect
228. C3×C4×D3C2×Dic3semidirect
255. D3×D4C3⋊Dic3semidirect
348. C33#518semidirect
359. C22×D3C2×C3⋊Dic3semidirect
365. C3×D4D3×Dic3semidirect
366. C3×D4C2×C3⋊Dic3semidirect
376. #67C2×C3⋊Dic3semidirect
396. D12C2×C3⋊Dic3semidirect
399. C4×D3C2×C3⋊Dic3semidirect
487. C32#5545semidirect
488. C32#5798semidirect
497. D4D3×C3⋊Dic3semidirect
547. C3#10595215semidirect
548. C3#10596269semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
D4×C3⋊(C3⋊Dic3)2
C3×D4×C3⋊Dic32
#67⋊(C3⋊Dic3)2
#67×C3⋊Dic32
D3×#37572
C2×(C2×D3)⋊(C3⋊Dic3)2
#117234992
D12×C3⋊Dic32
#117234242
#117234232
#105962693
#105952153