Finite Group Databasebeta

← #13177189

Subgroups — #13177189

Order 1728.

2408 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3C3×C4×#638C211yesmaximal, characteristic, complemented
4C32×#924C211yesmaximal, characteristic, complemented
5#11723628C212yesmaximal, complemented
6#11723623C212yesmaximal, complemented
7#11723494C211yesmaximal, characteristic, complemented
8#11723470C212yesmaximal, complemented
9#11723455C211yesmaximal, characteristic, complemented
10#11723431C211yesmaximal, characteristic, complemented
11#11723421C212yesmaximal, complemented
12#10596102—31nomaximal
13#10594834—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2212yes—
16#89076—24no—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C4×C3⋊(C3⋊D3)C2212yes—
19C2×#6308C2211yescharacteristic
20C2×C3×#638C2211yescharacteristic
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C32×Dic6C2211yescharacteristic
23C2×#6292C2212yes—
24C2×#6289C2212yes—
25C2×#6289C2212yes—
26#89018—24no—
27C3×#3757C2211yescharacteristic
28C3×#3752C2211yescharacteristic
29C3×#3750C2211yescharacteristic
30C3×C4×C3⋊Dic3C2211yescharacteristic
31C32×#217C2212yes—
32C32×C4⋊Dic3C2211yescharacteristic
33C32×Dic3⋊C4C2212yes—
34#88807C2212yes—
35#88807C2212yes—
36Dic3⋊(C3⋊Dic3)C2212yes—
37#88804C2212yes—
38#88803C2212yes—
39C2×#3832—31no—
40C2×#3828—31no—
41C2×#3828—34no—
42C2×#3828—64no—
43C2×#3799—34no—
44C32×C22⋊Q8—31no—
45#64153—31no—
46C4×#638D311yescharacteristic, complemented
47#64106—31no—
48#64103—32no—
49#64102—31no—
50C4⋊C4×C3⋊D3—32no—
51#64093—32no—
52#64092—32no—
53C3×C4×#67—34no—
54C3×#924D314yescomplemented
55#63975—38no—
56#63958—38no—
57#63894—34no—
58#63883—38no—
59#63866—34no—
60#63857—34no—
61#63853—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C2×C33×C4—22no—
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68#6312—24no—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C4⋊(C3⋊(C3⋊D3))D414yescomplemented
71C4×C3⋊(C3⋊D3)—24no—
72#6308—22no—
73C3×#638—22no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4×C3⋊D3—22no—
78C2×C32×Dic3C2312yes—
79C2×C32×Dic3C2312yes—
80C32×Dic6—22no—
81#6292—24no—
82#6289—24no—
83#6289—24no—
84#5631—94no—
85#5551—91no—
86C2×C32×Q8—31no—
87C22×C32×C4—24no—
88C22×C32×C4C2×D314yes—
89C22×C32×C4C2×D311yescharacteristic
90C2×#638—32no—
91C2×#638—38no—
92C2×#638—68no—
93C2×#638C2×D311yescharacteristic
94#3832—64no—
95#3828—64no—
96#3828—616no—
97#3828—632no—
98C2×C4⋊(C3⋊D3)—31no—
99C2×C4⋊(C3⋊D3)—34no—
100C2×C4⋊(C3⋊D3)—32no—
101C2×C4⋊(C3⋊D3)—64no—
102C2×C4×C3⋊D3—32no—
103C2×C4×C3⋊D3—38no—
104C2×C4×C3⋊D3—32no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—68no—
107C2×C4×C3⋊D3C2×D311yescharacteristic
108C2×#634—31no—
109C2×#634—34no—
110C2×#634—64no—
111C2×C3×#67—34no—
112C2×C3×C4×D3—34no—
113C2×C3×Dic6C2×D314yes—
114C2×#626—38no—
115C2×#624—38no—
116C2×#624—38no—
117#3799—616no—
118C32×C4⋊C4—32no—
119C32×C4⋊C4—31no—
120C32×C22⋊C4—32no—
121#3757C2×D311yescharacteristic
122#3752—32no—
123#3752—32no—
124#3752C2×D311yescharacteristic
125C4⋊(C3⋊Dic3)—32no—
126#3750—32no—
127#3750—32no—
128#3750C2×D311yescharacteristic
129C4×C3⋊Dic3C2×D311yescharacteristic
130C3×#217—34no—
131C3×#217C2×D318yes—
132C3×(C2×D3)⋊C4—34no—
133C3×C4⋊Dic3C2×D314yes—
134C3×Dic3⋊C4—34no—
135C3×Dic3⋊C4C2×D318yes—
136C3×C4×Dic3—34no—
137Dic3⋊Dic3—38no—
138Dic3⋊Dic3—38no—
139Dic3⋊Dic3—38no—
140#3722—38no—
141(C2×D3)⋊Dic3—38no—
142C22×C33—21no—
143C22×C33—21no—
144C22×C33—21no—
145C22×C33C2411yescharacteristic
146C2×C3⋊(C3⋊D3)—22no—
147C2×C3⋊(C3⋊D3)—22no—
148C2×C3⋊(C3⋊D3)C2×D414yes—
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—21no—
152C33×C4—21no—
153C33×C4C2×D412yes—
154C3⋊(C3⋊Dic3)—22no—
155C3×C3⋊Dic3—21no—
156C3×C3⋊Dic3—21no—
157C3×C3⋊Dic3#4112yescomplemented
158C32×Dic3—22no—
159C32×Dic3—22no—
160C2×#239—94no—
161C2×#235—94no—
162C2×#235—91no—
163C2×#235—91no—
164C2×#235—184no—
165C3×C22⋊Q8—34no—
166C3×C4×D4—91no—
167#944—94no—
168#938—92no—
169C4×#67—34no—
170#924—91no—
171#924C3⋊D311yescharacteristic, complemented
172#897—94no—
173#894—98no—
174#893—94no—
175D3×C4⋊C4—98no—
176D3×C4⋊C4—92no—
177#884—98no—
178#883—98no—
179#883—92no—
180#883—92no—
181C4×D12—91no—
182#869—91no—
183C23×C32—24no—
184C23×C32C22×D314yes—
185C23×C32C22×D311yescharacteristic
186C22×C3⋊D3—32no—
187C22×C3⋊D3—38no—
188C22×C3⋊D3—68no—
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3×D3—34no—
191C32×Q8—62no—
192C2×C32×C4—216no—
193C2×C32×C4—28no—
194C2×C32×C4—22no—
195C2×C32×C4—24no—
196C2×C32×C4—24no—
197C2×C32×C4—32no—
198C2×C32×C4—32no—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203#638—22no—
204#638—64no—
205#638—616no—
206#638—632no—
207C2×C3⋊Dic3—32no—
208C2×C3⋊Dic3—38no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3C22×D311yescharacteristic
211C2×C3⋊Dic3C22×D311yescharacteristic
212C2×C3⋊Dic3C22×D311yescharacteristic
213C4⋊(C3⋊D3)—34no—
214C4⋊(C3⋊D3)—316no—
215C4⋊(C3⋊D3)—616no—
216C4⋊(C3⋊D3)—64no—
217C4×C3⋊D3—22no—
218C4×C3⋊D3—64no—
219C4×C3⋊D3—64no—
220C4×C3⋊D3—616no—
221C4×C3⋊D3—632no—
222C4×C3⋊D3—64no—
223#634—616no—
224#634—68no—
225#634—62no—
226C3×#67—68no—
227C2×C3×Dic3—34no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3C22×D318yes—
231C2×C3×Dic3C22×D318yes—
232C3×C4×D3—68no—
233C3×Dic6—28no—
234#626—616no—
235#624—616no—
236#624—616no—
237#547—271noSylow
238C2×C33—41no—
239C2×C33C22×D411yescharacteristic
240C2×C33C2×#4111yescharacteristic
241C2×C332−1+411yescharacteristic
242C3⋊(C3⋊D3)—24no—
243C3×C3⋊D3—41no—
244C2×C3×Q8—34no—
245C2×C3×D4—91no—
246C22×C3×C4—24no—
247C22×C3×C4—91no—
248C22×C3×C4D3214yes—
249C22×C3×C4C2×C3⋊D311yescharacteristic
250C2×#67—34no—
251C2×#67—98no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—188no—
255#239—1816no—
256#235—184no—
257#235—1816no—
258#235—184no—
259#235—1832no—
260C2×D12—94no—
261C2×D12—91no—
262C2×D12—98no—
263C2×D12—184no—
264C2×C4×D3—34no—
265C2×C4×D3—98no—
266C2×C4×D3—98no—
267C2×C4×D3—98no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C2×C4×D3—188no—
272C2×Dic6—94no—
273C2×Dic6—91no—
274C2×Dic6—184no—
275C2×Dic6C2×C3⋊D311yescharacteristic
276C3×C4⋊C4—38no—
277C3×C4⋊C4—34no—
278C3×C4⋊C4—91no—
279C3×C22⋊C4—38no—
280C3×C22⋊C4—91no—
281C3×C22⋊C4—91no—
282C3×C42—91no—
283#217—34no—
284#217—92no—
285#217C2×C3⋊D312yes—
286(C2×D3)⋊C4—34no—
287(C2×D3)⋊C4—98no—
288(C2×D3)⋊C4—98no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—98no—
291C4⋊Dic3—92no—
292C4⋊Dic3—92no—
293C4⋊Dic3C2×C3⋊D311yescharacteristic
294Dic3⋊C4—34no—
295Dic3⋊C4—98no—
296Dic3⋊C4—98no—
297Dic3⋊C4—92no—
298Dic3⋊C4C2×C3⋊D312yes—
299C4×Dic3—34no—
300C22×C32—28no—
301C22×C32—21no—
302C22×C32—24no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—24no—
306C22×C32—24no—
307C22×C32—24no—
308C22×C32—44no—
309C22×C32—44no—
310C22×C32C23×D314yes—
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—21no—
313C2×C3⋊D3—21no—
314C2×C3⋊D3—21no—
315C2×C3⋊D3—34no—
316C2×C3⋊D3—316no—
317C2×C3⋊D3—616no—
318C2×C3⋊D3—62no—
319C2×C3⋊D3—62no—
320C2×C3⋊D3—68no—
321C2×C3⋊D3—68no—
322C2×C3⋊D3—128no—
323C2×C3⋊D3—128no—
324C2×C3×D3—64no—
325C2×C3×D3—64no—
326C2×C3×D3—64no—
327C32×C4—28no—
328C32×C4—24no—
329C32×C4—21no—
330C32×C4—28no—
331C32×C4—62no—
332C32×C4—62no—
333C32×C4D3×D418yes—
334C32×C4D3×D412yes—
335C3⋊Dic3—21no—
336C3⋊Dic3—21no—
337C3⋊Dic3—616no—
338C3⋊Dic3—68no—
339C3⋊Dic3—62no—
340C3⋊Dic3#23512yescomplemented
341C3×Dic3—28no—
342C3×Dic3—28no—
343C3×Dic3—38no—
344C3×Dic3—64no—
345C3×Dic3—64no—
346C2×#41—271no—
347C2×#41—271no—
348C4⋊Q8—271no—
349#123—272no—
350#123—272no—
351C22⋊Q8—31no—
352C22⋊Q8—271no—
353#121—272no—
354C4×D4—91no—
355C4×D4—271no—
356C2×C4⋊C4—272no—
357C33#54711yescharacteristic, complemented, Sylow
358C23×C3—24no—
359C23×C3—91no—
360C23×C3C2×D3214yes—
361C23×C3C22×C3⋊D311yescharacteristic
362C22×D3—34no—
363C22×D3—98no—
364C22×D3—92no—
365C22×D3—188no—
366C3×Q8—68no—
367C3×D4—182no—
368C2×C3×C4—216no—
369C2×C3×C4—28no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—38no—
374C2×C3×C4—38no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—182no—
379C2×C3×C4C2×D3214yes—
380C2×C3×C4C2×D3214yes—
381C2×C3×C4C22×C3⋊D311yescharacteristic
382C2×C3×C4C22×C3⋊D311yescharacteristic
383#67—68no—
384#67—184no—
385#67—1816no—
386#67—184no—
387#67—1832no—
388C2×Dic3—34no—
389C2×Dic3—34no—
390C2×Dic3—34no—
391C2×Dic3—98no—
392C2×Dic3—92no—
393C2×Dic3—188no—
394C2×Dic3C22×C3⋊D312yes—
395C2×Dic3C22×C3⋊D312yes—
396D12—916no—
397D12—94no—
398D12—1816no—
399D12—1816no—
400C4×D3—68no—
401C4×D3—184no—
402C4×D3—184no—
403C4×D3—1816no—
404C4×D3—1816no—
405C4×D3—1816no—
406C4×D3—184no—
407C4×D3—1832no—
408Dic6—22no—
409Dic6—1816no—
410Dic6—188no—
411Dic6—182no—
412C2×C32—24no—
413C2×C32—24no—
414C2×C32—24no—
415C2×C32—48no—
416C2×C32—41no—
417C2×C32—44no—
418C2×C32C2×#23511yescharacteristic
419C2×C32C2×D3×D414yes—
420C2×C32C2×D3×D411yescharacteristic
421C2×C32#100714yes—
422C2×C32D3×#4114yes—
423C2×C32#101011yescharacteristic
424C3⋊D3—41no—
425C3⋊D3—64no—
426C3⋊D3—616no—
427C3⋊D3—1216no—
428C3×D3—124no—
429#41—544no—
430#41—544no—
431C2×Q8—31no—
432C2×Q8—271no—
433C2×D4—91no—
434C2×D4—272no—
435C2×D4—272no—
436C2×D4—271no—
437C22×C4—91no—
438C22×C4—272no—
439C22×C4—272no—
440C22×C4—272no—
441C22×C4D3×C3⋊D311yescharacteristic
442C4⋊C4—32no—
443C4⋊C4—31no—
444C4⋊C4—91no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C4⋊C4—272no—
448C22⋊C4—32no—
449C22⋊C4—91no—
450C22⋊C4—91no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C42—91no—
454C22×C3—28no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—24no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C22×D3214yes—
468C22×C3C23×C3⋊D311yescharacteristic
469C2×D3—64no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—916no—
473C2×D3—94no—
474C2×D3—1816no—
475C2×D3—188no—
476C2×D3—188no—
477C2×D3—182no—
478C2×D3—182no—
479C2×D3—368no—
480C2×D3—368no—
481C3×C4—28no—
482C3×C4—24no—
483C3×C4—21no—
484C3×C4—28no—
485C3×C4—68no—
486C3×C4—68no—
487C3×C4—92no—
488C3×C4—181no—
489C3×C4—181no—
490C3×C4#381118yes—
491C3×C4D4×C3⋊D312yes—
492Dic3—22no—
493Dic3—22no—
494Dic3—38no—
495Dic3—64no—
496Dic3—64no—
497Dic3—1816no—
498Dic3—188no—
499Dic3—182no—
500C32—24no—
501C32#555111yescharacteristic, complemented
502C32#563114yescomplemented
503C23—91no—
504C23—272no—
505C23C2×D3×C3⋊D311yescharacteristic
506Q8—62no—
507Q8—542no—
508D4—182no—
509D4—274no—
510D4—544no—
511D4—544no—
512C2×C4—22no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—182no—
519C2×C4—272no—
520C2×C4—544no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4C2×D3×C3⋊D311yescharacteristic
524C2×C4C2×D3×C3⋊D311yescharacteristic
525C2×C3—24no—
526C2×C3—24no—
527C2×C3—24no—
528C2×C3—48no—
529C2×C3—41no—
530C2×C3—44no—
531C2×C3—361no—
532C2×C3#6429714yes—
533C2×C3D3×#23514yes—
534C2×C3C2×#381114yes—
535C2×C3C2×D4×C3⋊D311yescharacteristic
536C2×C3#6436411yescharacteristic
537C2×C3#41×C3⋊D311yescharacteristic
538D3—124no—
539D3—184no—
540D3—1816no—
541D3—3616no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—181no—
546C22—181no—
547C22—181no—
548C22—274no—
549C22—542no—
550C22—542no—
551C22C22×D3×C3⋊D311yescharacteristic
552C4—21no—
553C4—62no—
554C4—62no—
555C4—92no—
556C4—181no—
557C4—181no—
558C4—542no—
559C4#8903912yes—
560C3—24no—
561C3#1059483414yescomplemented
562C3#1059610211yescharacteristic, complemented
563C2—41no—
564C2—361no—
565C2—544no—
566C2#1172456611yescharacteristic
567C2C3⋊D3×#23511yescharacteristic
568C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C2×#89018C2semidirect
3. C3×C4×#638C2semidirect
4. C32×#924C2semidirect
5. #11723628C2semidirect
6. #11723623C2semidirect
7. #11723494C2semidirect
8. #11723470C2semidirect
9. #11723455C2semidirect
10. #11723431C2semidirect
11. #11723421C2semidirect
46. C4×#638D3semidirect
54. C3×#924D3semidirect
70. C4⋊(C3⋊(C3⋊D3))D4semidirect
157. C3×C3⋊Dic3#41semidirect
171. #924C3⋊D3semidirect
340. C3⋊Dic3#235semidirect
357. C33#547semidirect
501. C32#5551semidirect
502. C32#5631semidirect
561. C3#10594834semidirect
562. C3#10596102semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×#890182
C3×C4×#6382
C32×#9242
#117236282
#117236232
#117234942
#117234702
#117234552
#117234312
#117234212
#105961023
#105948343