Finite Group Databasebeta

← #13177487

Subgroups — #13177487

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×C3×D4×C3⋊D3C211yesmaximal, characteristic
2D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723626C212yesmaximal, complemented
5C3⋊D3×#217C212yesmaximal, complemented
6Dic3×#638C212yesmaximal, complemented
7C2×#88803C212yesmaximal, complemented
8#11723501C211yesmaximal, characteristic, complemented
9Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
10#11723437C211yesmaximal, characteristic, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10596016—31nomaximal
13#10595215—34nomaximal
14C23×C3×C3⋊D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2212yescomplemented
17C2×C3×#638C2212yes—
18C3×D4×C3⋊D3C418yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C22×C32×Dic3C2212yescomplemented
22C2×Dic3×C3⋊D3C2212yescomplemented
23C2×Dic3×C3⋊D3C2212yescomplemented
24#88864C2212yescomplemented
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
27C32×#217C2212yescomplemented
28C32×C4⋊Dic3C2211yescharacteristic, complemented
29C32×C4×Dic3C2211yescharacteristic, complemented
30#88807C2211yescharacteristic, complemented
31#88807C2211yescharacteristic, complemented
32#88803C2211yescharacteristic
33#88803C2212yes—
34#88803C2211yescharacteristic
35#88803C2212yescomplemented
36#88803C2212yescomplemented
37Dic3×C3⋊Dic3C2212yescomplemented
38C2×D4×C3⋊D3D311yescharacteristic
39C2×C3×D3×D4—34no—
40C32×C4×D4—31no—
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44C4×#638—32no—
45C2×#3752—32no—
46#64102—31no—
47#64101—31no—
48#64090—32no—
49C22⋊C4×C3⋊D3—32no—
50C4×C4⋊(C3⋊D3)—31no—
51#64081—31no—
52C3×D4×Dic3D314yescomplemented
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×C3×C3⋊D3—42no—
63C22×C3×C3⋊D3—42no—
64C22×C3×C3⋊D3C2×C418yescomplemented
65C22×C3×C3⋊D3C2311yescharacteristic
66C22×C3×C3⋊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—
73C3×#638C2×C418yescomplemented
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4⋊(C3⋊D3)C2×C414yescomplemented
76C3×C4×C3⋊D3C2×C414yescomplemented
77C2×C32×Dic3—42no—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81Dic3×C3⋊D3—42no—
82Dic3×C3⋊D3—42no—
83#5798—91no—
84#5545—94no—
85C23×C3⋊D3C2×D312yes—
86C23×C3×D3—38no—
87C2×C32×D4—24no—
88C2×C32×D4C2×D314yes—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4—32no—
91C2×#638C2×D312yes—
92C22×C3⋊Dic3—32no—
93C22×C3⋊Dic3—38no—
94C22×C3⋊Dic3—68no—
95D4×C3⋊D3Dic318yescomplemented
96C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
97C2×C4×C3⋊D3—32no—
98C2×C4×C3⋊D3—32no—
99C2×C4×C3⋊D3C2×D311yescharacteristic
100C2×C3×#67—38no—
101C22×C3×Dic3C2×D318yescomplemented
102C3×D3×D4—332no—
103C2×C3×D12—34no—
104C2×C3×C4×D3—34no—
105C2×D3×Dic3—38no—
106C2×D3×Dic3—38no—
107C32×C4⋊C4—31no—
108C32×C22⋊C4—32no—
109C32×C42—31no—
110#3757—32no—
111#3757—38no—
112#3757—68no—
113#3752—31no—
114#3752—31no—
115#3752—32no—
116#3752—32no—
117#3752—32no—
118C4⋊(C3⋊Dic3)—31no—
119C4⋊(C3⋊Dic3)—34no—
120C4⋊(C3⋊Dic3)—64no—
121#3750—31no—
122#3750—31no—
123C4×C3⋊Dic3—31no—
124C4×C3⋊Dic3—34no—
125C4×C3⋊Dic3—32no—
126C4×C3⋊Dic3—64no—
127C3×#217C2×D318yescomplemented
128C3×C4⋊Dic3C2×D314yescomplemented
129C3×C4×Dic3C2×D314yescomplemented
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×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3—21no—
145C2×C3×C3⋊D3—21no—
146C2×C3×C3⋊D3—48no—
147C2×C3×C3⋊D3C22×C412yes—
148C2×C3×C3⋊D3C22×C414yes—
149C33×C4C22×C412yes—
150C3⋊(C3⋊Dic3)—21no—
151C3⋊(C3⋊Dic3)—21no—
152C3⋊(C3⋊Dic3)—22no—
153C3×C3⋊Dic3C22×C412yes—
154C32×Dic3—21no—
155C32×Dic3—22no—
156C32×Dic3—21no—
157C22×C3×D4—91no—
158C2×D3×D4—34no—
159C3×C4×D4—34no—
160C2×#217—92no—
161C2×#217—92no—
162D4×Dic3—94no—
163D4×Dic3—91no—
164D4×Dic3—92no—
165D4×Dic3—92no—
166D4×Dic3—91no—
167D4×Dic3—91no—
168D4×Dic3—184no—
169D4×Dic3C3⋊D311yescharacteristic, complemented
170C4×#67—98no—
171C2×(C2×D3)⋊C4—98no—
172#926—91no—
173#926—91no—
174#893—94no—
175#892—94no—
176#67⋊C4—98no—
177D3×C22⋊C4—98no—
178C4×D12—94no—
179#872—94no—
180C23×C32—28no—
181C23×C32C22×D318yes—
182C23×C32C22×D312yes—
183C22×C3⋊D3—42no—
184C22×C3⋊D3—42no—
185C22×C3⋊D3C2×Dic318yescomplemented
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—34no—
189C22×C3×D3—38no—
190C22×C3×D3—332no—
191C22×C3×D3—128no—
192C22×C3×D3—128no—
193C32×D4—216no—
194C32×D4C4×D3116yescomplemented
195C32×D4C2×Dic314yescomplemented
196C2×C32×C4—24no—
197C2×C32×C4—32no—
198C2×C32×C4—31no—
199C2×C32×C4—31no—
200C2×C32×C4—122no—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D311yescharacteristic
203#638C2×Dic318yescomplemented
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—31no—
206C2×C3⋊Dic3—32no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—34no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—68no—
213C2×C3⋊Dic3—122no—
214C2×C3⋊Dic3—128no—
215C2×C3⋊Dic3—1216no—
216C2×C3⋊Dic3C22×D311yescharacteristic
217C4⋊(C3⋊D3)C2×Dic314yescomplemented
218C4×C3⋊D3—122no—
219C4×C3⋊D3—122no—
220C4×C3⋊D3C2×Dic314yescomplemented
221C3×#67—332no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—48no—
224C2×C3×Dic3C22×D318yes—
225C2×C3×Dic3C22×D314yes—
226C2×C3×Dic3C22×D314yes—
227C3×D12—316no—
228C3×C4×D3—316no—
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
236C3×C3⋊D3—22no—
237C3×C3⋊D3—24no—
238C24×C3—92no—
239C23×D3—38no—
240C2×C3×D4—24no—
241C2×C3×D4—92no—
242C2×C3×D4—98no—
243C2×C3×D4—91no—
244C2×C3×D4D3214yes—
245C2×C3×D4C2×C3⋊D311yescharacteristic
246C22×C3×C4—38no—
247C22×C3×C4—91no—
248C2×#67—38no—
249C22×Dic3—98no—
250C22×Dic3—92no—
251C22×Dic3—92no—
252C22×Dic3—92no—
253C22×Dic3—188no—
254C22×Dic3C2×C3⋊D312yescomplemented
255D3×D4—332no—
256C2×D12—34no—
257C2×C4×D3—34no—
258C2×C4×D3—98no—
259C2×C4×D3—98no—
260C3×C4⋊C4—34no—
261C3×C22⋊C4—38no—
262C3×C42—34no—
263#217—98no—
264#217—92no—
265#217—92no—
266#217—91no—
267#217—91no—
268#217—92no—
269#217—92no—
270#217—188no—
271#217C2×C3⋊D312yescomplemented
272(C2×D3)⋊C4—94no—
273(C2×D3)⋊C4—94no—
274(C2×D3)⋊C4—98no—
275(C2×D3)⋊C4—98no—
276(C2×D3)⋊C4—98no—
277C4⋊Dic3—94no—
278C4⋊Dic3—91no—
279C4⋊Dic3—91no—
280C4⋊Dic3—91no—
281C4⋊Dic3—184no—
282C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
283Dic3⋊C4—94no—
284Dic3⋊C4—94no—
285C4×Dic3—98no—
286C4×Dic3—94no—
287C4×Dic3—91no—
288C4×Dic3—92no—
289C4×Dic3—184no—
290C4×Dic3C2×C3⋊D311yescharacteristic, complemented
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—24no—
294C22×C32—216no—
295C22×C32—28no—
296C22×C32—28no—
297C22×C32—48no—
298C22×C32—48no—
299C22×C32C2×C4×D3116yes—
300C22×C32C22×Dic314yes—
301C22×C32C23×D314yes—
302C22×C32C23×D311yescharacteristic
303C2×C3⋊D3—22no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—21no—
306C2×C3⋊D3—21no—
307C2×C3⋊D3—48no—
308C2×C3⋊D3C22×Dic312yes—
309C2×C3⋊D3C22×Dic314yes—
310C2×C3×D3—38no—
311C2×C3×D3—316no—
312C2×C3×D3—68no—
313C2×C3×D3—64no—
314C2×C3×D3—68no—
315C2×C3×D3—64no—
316C2×C3×D3—1232no—
317C32×C4—28no—
318C32×C4—61no—
319C32×C4—62no—
320C32×C4—61no—
321C32×C4C2×C4×D318yes—
322C32×C4C22×Dic312yes—
323C3⋊Dic3—68no—
324C3⋊Dic3—61no—
325C3⋊Dic3—62no—
326C3⋊Dic3—61no—
327C3⋊Dic3—64no—
328C3⋊Dic3—64no—
329C3⋊Dic3—68no—
330C3⋊Dic3—124no—
331C3⋊Dic3—128no—
332C3⋊Dic3C22×Dic312yes—
333C3×Dic3—28no—
334C3×Dic3—24no—
335C3×Dic3—24no—
336C3×Dic3—38no—
337C22×D4—91no—
338C4×D4—31no—
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—91no—
351C23×C3—92no—
352C23×C3—98no—
353C23×C3—362no—
354C23×C3—362no—
355C23×C3C2×D3218yes—
356C23×C3C22×C3⋊D312yes—
357C22×D3—34no—
358C22×D3—38no—
359C22×D3—332no—
360C22×D3—128no—
361C22×D3—128no—
362C3×D4—216no—
363C3×D4—98no—
364C3×D4—94no—
365C3×D4D3×Dic3116yescomplemented
366C3×D4C4×C3⋊D314yescomplemented
367C2×C3×C4—24no—
368C2×C3×C4—38no—
369C2×C3×C4—34no—
370C2×C3×C4—34no—
371C2×C3×C4—91no—
372C2×C3×C4—94no—
373C2×C3×C4—128no—
374C2×C3×C4C2×D3214yes—
375C2×C3×C4C22×C3⋊D311yescharacteristic
376#67—332no—
377C2×Dic3—34no—
378C2×Dic3—42no—
379C2×Dic3—94no—
380C2×Dic3—94no—
381C2×Dic3—98no—
382C2×Dic3—91no—
383C2×Dic3—92no—
384C2×Dic3—91no—
385C2×Dic3—184no—
386C2×Dic3—184no—
387C2×Dic3—188no—
388C2×Dic3—362no—
389C2×Dic3—362no—
390C2×Dic3—368no—
391C2×Dic3—362no—
392C2×Dic3—3616no—
393C2×Dic3C22×C3⋊D312yes—
394C2×Dic3C22×C3⋊D311yescharacteristic
395C2×Dic3C22×C3⋊D311yescharacteristic
396D12—316no—
397C4×D3—316no—
398C4×D3—368no—
399C4×D3—368no—
400C2×C32—24no—
401C2×C32—24no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—216no—
405C2×C32—416no—
406C2×C32C22×C4×D314yes—
407C2×C32#100414yes—
408C2×C32#100411yescharacteristic
409C2×C32#100411yescharacteristic
410C2×C32#100914yes—
411C2×C32C23×Dic311yescharacteristic
412C3⋊D3—22no—
413C3⋊D3—24no—
414C3×D3—616no—
415C3×D3—68no—
416C24—92no—
417C2×D4—92no—
418C2×D4—98no—
419C2×D4—91no—
420C2×D4D3×C3⋊D311yescharacteristic
421C22×C4—32no—
422C22×C4—91no—
423C22×C4—272no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—31no—
427C4⋊C4—271no—
428C4⋊C4—271no—
429C4⋊C4—271no—
430C22⋊C4—32no—
431C22⋊C4—271no—
432C22⋊C4—271no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C42—31no—
438C42—272no—
439C42—271no—
440C22×C3—28no—
441C22×C3—28no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3—216no—
446C22×C3—48no—
447C22×C3—48no—
448C22×C3—92no—
449C22×C3—94no—
450C22×C3—182no—
451C22×C3—181no—
452C22×C3—182no—
453C22×C3—181no—
454C22×C3—368no—
455C22×C3C2×D3×Dic3116yes—
456C22×C3C2×C4×C3⋊D314yes—
457C22×C3C22×D3214yes—
458C22×C3C23×C3⋊D311yescharacteristic
459C2×D3—38no—
460C2×D3—316no—
461C2×D3—68no—
462C2×D3—64no—
463C2×D3—68no—
464C2×D3—64no—
465C2×D3—1232no—
466C3×C4—28no—
467C3×C4—68no—
468C3×C4—64no—
469C3×C4—64no—
470C3×C4—92no—
471C3×C4C2×D3×Dic318yes—
472C3×C4C2×C4×C3⋊D312yes—
473Dic3—21no—
474Dic3—22no—
475Dic3—21no—
476Dic3—38no—
477Dic3—188no—
478Dic3—184no—
479Dic3—188no—
480Dic3—184no—
481Dic3—181no—
482Dic3—181no—
483Dic3—182no—
484Dic3—368no—
485Dic3—364no—
486C32—24no—
487C32#554514yescomplemented
488C32#579811yescharacteristic, complemented
489C23—91no—
490C23—92no—
491C23—98no—
492C23—362no—
493C23—362no—
494C23C2×D3×C3⋊D312yes—
495D4—98no—
496D4—94no—
497D4Dic3×C3⋊D314yescomplemented
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—32no—
501C2×C4—91no—
502C2×C4—94no—
503C2×C4—122no—
504C2×C4—271no—
505C2×C4—271no—
506C2×C4—272no—
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—182no—
518C2×C3—184no—
519C2×C3#6431314yes—
520C2×C3#6431414yes—
521C2×C3C22×D3×Dic314yes—
522C2×C3C22×C4×C3⋊D311yescharacteristic
523C2×C3#6436111yescharacteristic
524C2×C3#6436611yescharacteristic
525D3—68no—
526D3—616no—
527C22—22no—
528C22—22no—
529C22—92no—
530C22—94no—
531C22—182no—
532C22—181no—
533C22—182no—
534C22—181no—
535C22—368no—
536C22C2×Dic3×C3⋊D314yes—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—61no—
539C4—62no—
540C4—61no—
541C4—92no—
542C4—541no—
543C4—541no—
544C4—542no—
545C4C2×Dic3×C3⋊D312yes—
546C3—24no—
547C3#1059521514yescomplemented
548C3#1059601611yescharacteristic, complemented
549C2—24no—
550C2—182no—
551C2—184no—
552C2#1172458711yescharacteristic
553C2#1172458911yescharacteristic
554C2C22×Dic3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×C3×D4×C3⋊D3C2
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. C32×D4×Dic3C2semidirect
4. #11723626C2semidirect
5. C3⋊D3×#217C2semidirect
6. Dic3×#638C2semidirect
7. C2×#88803C2semidirect
8. #11723501C2semidirect
9. Dic3×C4⋊(C3⋊D3)C2semidirect
10. #11723437C2semidirect
11. #11723404C2semidirect
16. C22×C3⋊(C3⋊Dic3)C22semidirect
18. C3×D4×C3⋊D3C4semidirect
21. C22×C32×Dic3C22semidirect
22. C2×Dic3×C3⋊D3C22semidirect
23. C2×Dic3×C3⋊D3C22semidirect
24. #88864C22semidirect
25. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
26. C4×C3⋊(C3⋊Dic3)C22semidirect
27. C32×#217C22semidirect
28. C32×C4⋊Dic3C22semidirect
29. C32×C4×Dic3C22semidirect
30. #88807C22semidirect
31. #88807C22semidirect
35. #88803C22semidirect
36. #88803C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
52. C3×D4×Dic3D3semidirect
64. C22×C3×C3⋊D3C2×C4semidirect
67. C33×D4C2×C4semidirect
73. C3×#638C2×C4semidirect
75. C3×C4⋊(C3⋊D3)C2×C4semidirect
76. C3×C4×C3⋊D3C2×C4semidirect
95. D4×C3⋊D3Dic3semidirect
101. C22×C3×Dic3C2×D3semidirect
127. C3×#217C2×D3semidirect
128. C3×C4⋊Dic3C2×D3semidirect
129. C3×C4×Dic3C2×D3semidirect
169. D4×Dic3C3⋊D3semidirect
185. C22×C3⋊D3C2×Dic3semidirect
194. C32×D4C4×D3semidirect
195. C32×D4C2×Dic3semidirect
203. #638C2×Dic3semidirect
217. C4⋊(C3⋊D3)C2×Dic3semidirect
220. C4×C3⋊D3C2×Dic3semidirect
254. C22×Dic3C2×C3⋊D3semidirect
271. #217C2×C3⋊D3semidirect
282. C4⋊Dic3C2×C3⋊D3semidirect
290. C4×Dic3C2×C3⋊D3semidirect
348. C33#518semidirect
365. C3×D4D3×Dic3semidirect
366. C3×D4C4×C3⋊D3semidirect
487. C32#5545semidirect
488. C32#5798semidirect
497. D4Dic3×C3⋊D3semidirect
547. C3#10595215semidirect
548. C3#10596016semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
D4×C3⋊(C3⋊Dic3)2
C32×D4×Dic32
#117236262
C3⋊D3×#2172
Dic3×#6382
C2×#888032
#117235012
Dic3×C4⋊(C3⋊D3)2
#117234372
#117234042
#105960163
#105952153