Finite Group Databasebeta

← #13177378

Subgroups — #13177378

Order 1728.

2236 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89028C212yesmaximal, complemented
2#11723979C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C32×#940C211yesmaximal, characteristic, complemented
5#67×C3⋊Dic3C212yesmaximal, complemented
6#11723600C212yesmaximal, complemented
7#11723599C212yesmaximal, complemented
8#11723534C211yesmaximal, characteristic, complemented
9#11723531C211yesmaximal, characteristic, complemented
10D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
11C4×#6289C211yesmaximal, characteristic, complemented
12#10596187—31nomaximal
13#10595139—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yes—
18C2×C32×#67C2212yes—
19C2×C32×#67C2212yes—
20C2×C32×D12C2211yescharacteristic
21C2×#6293C2212yes—
22C2×#6291C2212yes—
23C2×#6289C2211yescharacteristic
24C2×#6289C2211yescharacteristic
25#89028—28no—
26C2×D3×C3⋊Dic3C2212yes—
27#88864C2212yes—
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29C3×#3757C2212yes—
30C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C3×C4×C3⋊Dic3C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33Dic3⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×#3830—32no—
39C2×#3804—38no—
40C32×#127—31no—
41#64147—31no—
42#64147—32no—
43#64147—34no—
44#64147—31no—
45#64147—64no—
46#64145—32no—
47D4×C3⋊Dic3—32no—
48D4×C3⋊Dic3—31no—
49D4×C3⋊Dic3D311yescharacteristic, complemented
50C42×C3⋊D3—31no—
51#64078—31no—
52C3×#940D314yescomplemented
53C3×D4×Dic3—34no—
54Dic3×#67—38no—
55#63959—38no—
56#63958—38no—
57#63918—34no—
58#63916—34no—
59Dic3×D12—34no—
60C4×#624—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C32×D3C2312yes—
64C33×D4—22no—
65C2×C33×C4C2311yescharacteristic
66#6312—24no—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4×C3⋊(C3⋊D3)—22no—
70C2×C3×C3⋊Dic3—24no—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C32×#67—24no—
75C32×#67—24no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2311yescharacteristic
78C32×D12—22no—
79#6293—24no—
80#6291—24no—
81#6289—22no—
82#6289D414yescomplemented
83D3×C3⋊Dic3—24no—
84#5716—94no—
85#5548—91no—
86C2×C32×D4—24no—
87C2×C32×D4—32no—
88C2×C32×D4—32no—
89C2×C32×D4—31no—
90C2×C32×D4C2×D314yes—
91C2×C32×D4C2×D311yescharacteristic
92C2×#638—32no—
93C2×#638—38no—
94C2×#638—32no—
95C2×#638—68no—
96C22×C3⋊Dic3—32no—
97C22×C3⋊Dic3C2×D312yes—
98#3830—68no—
99C2×C4×C3⋊D3—31no—
100C2×C4×C3⋊D3—31no—
101C2×C4×C3⋊D3—34no—
102C2×C4×C3⋊D3—31no—
103C2×C4×C3⋊D3—64no—
104C2×#634—32no—
105C2×C3×#67C2×D318yes—
106C2×C3×#67C2×D318yes—
107C22×C3×Dic3—38no—
108C2×C3×D12C2×D314yes—
109C2×#627—38no—
110C2×#626—38no—
111C2×#624—34no—
112C2×#624—34no—
113#3804—632no—
114C2×D3×Dic3—38no—
115C32×C42—31no—
116#3757—32no—
117#3757—38no—
118#3757—32no—
119#3757—32no—
120#3757—68no—
121#3757C2×D312yes—
122C4⋊(C3⋊Dic3)—31no—
123C4⋊(C3⋊Dic3)—34no—
124C4⋊(C3⋊Dic3)—32no—
125C4⋊(C3⋊Dic3)—64no—
126C4⋊(C3⋊Dic3)C2×D311yescharacteristic
127C4×C3⋊Dic3—31no—
128C4×C3⋊Dic3—31no—
129C4×C3⋊Dic3C2×D311yescharacteristic
130C3×#217—38no—
131C3×C4⋊Dic3—34no—
132C3×C4×Dic3—34no—
133C3×C4×Dic3C2×D314yes—
134Dic3⋊Dic3—38no—
135(C2×D3)⋊Dic3—38no—
136(C2×D3)⋊Dic3—38no—
137Dic32—34no—
138Dic32—34no—
139C22×C33—22no—
140C22×C33—22no—
141C22×C33—22no—
142C22×C33C2411yescharacteristic
143C2×C3⋊(C3⋊D3)—21no—
144C2×C3⋊(C3⋊D3)—21no—
145C2×C3⋊(C3⋊D3)C2×D412yes—
146C2×C32×D3—22no—
147C2×C32×D3—22no—
148C2×C32×D3—22no—
149C33×C4#4112yescomplemented
150C3⋊(C3⋊Dic3)—22no—
151C3⋊(C3⋊Dic3)—21no—
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3C2×D412yes—
155C32×Dic3C2×D412yes—
156C32×Dic3#4112yescomplemented
157C2×#237—98no—
158C2×#235—92no—
159C3×#127—34no—
160C3×C4×D4—91no—
161#940C3⋊D311yescharacteristic, complemented
162#938—94no—
163#938—98no—
164#938—94no—
165#938—91no—
166#938—184no—
167#936—98no—
168D4×Dic3—34no—
169D4×Dic3—94no—
170D4×Dic3—98no—
171C4×#67—92no—
172#895—91no—
173#888—91no—
174#885—92no—
175#884—92no—
176C4×D12—91no—
177C42×D3—94no—
178C42×D3—91no—
179#869—94no—
180C23×C32—28no—
181C23×C32—32no—
182C23×C32C22×D318yes—
183C23×C32C22×D312yes—
184C22×C3⋊D3—31no—
185C22×C3⋊D3—34no—
186C22×C3⋊D3—64no—
187C22×C3×D3C22×D318yes—
188C32×D4—22no—
189C32×D4—28no—
190C32×D4—48no—
191C32×D4—64no—
192C32×D4—64no—
193C32×D4—62no—
194C2×C32×C4—24no—
195C2×C32×C4—31no—
196C2×C32×C4—31no—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D311yescharacteristic
199#638—64no—
200#638—64no—
201#638—616no—
202#638—1216no—
203C2×C3⋊Dic3—24no—
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—34no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—68no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3C22×D311yescharacteristic
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D311yescharacteristic
214C4×C3⋊D3—34no—
215C4×C3⋊D3—62no—
216C4×C3⋊D3—62no—
217C4×C3⋊D3—68no—
218C4×C3⋊D3—128no—
219#634—64no—
220C3×#67—216no—
221C3×#67—216no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—38no—
225C2×C3×Dic3—616no—
226C2×C3×Dic3C22×D314yes—
227C2×C3×Dic3C22×D314yes—
228C3×D12—28no—
229#627—616no—
230#626—616no—
231#624—316no—
232#624—68no—
233D3×Dic3—616no—
234#532—271noSylow
235C2×C33—42no—
236C2×C33C22×D411yescharacteristic
237C2×C33C2×#4111yescharacteristic
238C2×C33C2×#4111yescharacteristic
239C3⋊(C3⋊D3)—22no—
240C32×D3—42no—
241C2×C3×D4—24no—
242C2×C3×D4—38no—
243C2×C3×D4—38no—
244C2×C3×D4—34no—
245C2×C3×D4D3214yes—
246C2×C3×D4C2×C3⋊D311yescharacteristic
247C22×C3×C4—92no—
248C2×#67—98no—
249C2×#67—98no—
250C2×#67—92no—
251C2×#67—188no—
252C2×#67C2×C3⋊D312yes—
253C2×#67C2×C3⋊D312yes—
254C22×Dic3—38no—
255C22×Dic3—98no—
256#237—1832no—
257#235—188no—
258C2×D12—92no—
259C2×D12C2×C3⋊D311yescharacteristic
260C2×C4×D3—94no—
261C2×C4×D3—91no—
262C2×C4×D3—94no—
263C2×C4×D3—92no—
264C2×C4×D3—94no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—184no—
268C2×Dic6—98no—
269C2×Dic6—92no—
270C3×C4⋊C4—91no—
271C3×C22⋊C4—92no—
272C3×C42—34no—
273C3×C42—91no—
274#217—38no—
275#217—98no—
276#217—98no—
277#217—98no—
278#217—92no—
279#217—188no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282C4⋊Dic3—34no—
283C4⋊Dic3—98no—
284C4⋊Dic3—94no—
285C4⋊Dic3—91no—
286C4⋊Dic3—184no—
287Dic3⋊C4—92no—
288C4×Dic3—34no—
289C4×Dic3—94no—
290C4×Dic3—94no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3C2×C3⋊D311yescharacteristic
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—24no—
298C22×C32—28no—
299C22×C32—28no—
300C22×C32—28no—
301C22×C32—48no—
302C22×C32—48no—
303C22×C32—48no—
304C22×C32—62no—
305C22×C32—62no—
306C22×C32—62no—
307C22×C32C23×D314yes—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—32no—
310C2×C3⋊D3—38no—
311C2×C3⋊D3—68no—
312C2×C3⋊D3—61no—
313C2×C3⋊D3—61no—
314C2×C3⋊D3—64no—
315C2×C3⋊D3—64no—
316C2×C3⋊D3—124no—
317C2×C3⋊D3—124no—
318C2×C3×D3—28no—
319C2×C3×D3—28no—
320C2×C3×D3—28no—
321C32×C4—28no—
322C32×C4—32no—
323C32×C4—32no—
324C32×C4#23512yescomplemented
325C32×C4#23718yescomplemented
326C3⋊Dic3—22no—
327C3⋊Dic3—21no—
328C3⋊Dic3—61no—
329C3⋊Dic3—62no—
330C3⋊Dic3—68no—
331C3⋊Dic3—64no—
332C3⋊Dic3—124no—
333C3⋊Dic3—128no—
334C3⋊Dic3D3×D412yes—
335C3×Dic3—38no—
336C3×Dic3—68no—
337C3×Dic3—64no—
338C3×Dic3D3×D418yes—
339C3×Dic3#23718yescomplemented
340C2×#41—272no—
341C4⋊Q8—271no—
342#127—31no—
343#124—272no—
344#121—272no—
345#121—271no—
346#121—271no—
347C4×D4—91no—
348C4×D4—271no—
349C4×D4—272no—
350C2×C42—271no—
351C33#53211yescharacteristic, complemented, Sylow
352C23×C3—28no—
353C23×C3—38no—
354C23×C3C2×D3218yes—
355C23×C3C22×C3⋊D312yes—
356C22×D3—94no—
357C22×D3—91no—
358C22×D3—184no—
359C22×D3C22×C3⋊D312yes—
360C3×D4—28no—
361C3×D4—22no—
362C3×D4—48no—
363C3×D4—616no—
364C3×D4—616no—
365C3×D4—68no—
366C2×C3×C4—24no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—184no—
373C2×C3×C4C2×D3214yes—
374C2×C3×C4C22×C3⋊D311yescharacteristic
375#67—24no—
376#67—24no—
377#67—1816no—
378#67—184no—
379#67—1816no—
380#67—3616no—
381C2×Dic3—34no—
382C2×Dic3—34no—
383C2×Dic3—38no—
384C2×Dic3—616no—
385C2×Dic3—94no—
386C2×Dic3—98no—
387C2×Dic3—91no—
388C2×Dic3—92no—
389C2×Dic3—188no—
390C2×Dic3—184no—
391C2×Dic3—1816no—
392C2×Dic3C22×C3⋊D311yescharacteristic
393C2×Dic3C22×C3⋊D311yescharacteristic
394D12—22no—
395D12—184no—
396C4×D3—916no—
397C4×D3—94no—
398C4×D3—188no—
399C4×D3—184no—
400C4×D3—188no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—368no—
404Dic6—1816no—
405Dic6—184no—
406C2×C32—24no—
407C2×C32—24no—
408C2×C32—24no—
409C2×C32—42no—
410C2×C32—48no—
411C2×C32—88no—
412C2×C32—122no—
413C2×C32C2×#23511yescharacteristic
414C2×C32C2×D3×D414yes—
415C2×C32C2×D3×D411yescharacteristic
416C2×C32C2×#23714yes—
417C2×C32C2×#23714yes—
418C2×C32D3×#4111yescharacteristic
419C3⋊D3—62no—
420C3⋊D3—68no—
421C3⋊D3—128no—
422C3×D3—48no—
423#41—548no—
424C2×Q8—272no—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—31no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4D3×C3⋊D311yescharacteristic
431C22×C4—92no—
432C22×C4—271no—
433C22×C4—271no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—91no—
437C4⋊C4—272no—
438C4⋊C4—271no—
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C42—31no—
444C42—91no—
445C42—271no—
446C42—271no—
447C22×C3—28no—
448C22×C3—28no—
449C22×C3—28no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—48no—
455C22×C3—48no—
456C22×C3—48no—
457C22×C3—68no—
458C22×C3—68no—
459C22×C3—68no—
460C22×C3C22×D3214yes—
461C22×C3C23×C3⋊D311yescharacteristic
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—98no—
466C2×D3—92no—
467C2×D3—188no—
468C2×D3—184no—
469C2×D3—184no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—364no—
473C2×D3—364no—
474C3×C4—28no—
475C3×C4—38no—
476C3×C4—38no—
477C3×C4—92no—
478C3×C4—182no—
479C3×C4—181no—
480C3×C4#380418yescomplemented
481C3×C4#383012yescomplemented
482Dic3—38no—
483Dic3—68no—
484Dic3—64no—
485Dic3—188no—
486Dic3—184no—
487Dic3—182no—
488Dic3—181no—
489Dic3—364no—
490Dic3—368no—
491Dic3D4×C3⋊D312yes—
492Dic3#383012yescomplemented
493C32—24no—
494C32#554811yescharacteristic, complemented
495C32#571614yescomplemented
496C23—32no—
497C23—271no—
498C23C2×D3×C3⋊D312yes—
499Q8—544no—
500D4—22no—
501D4—64no—
502D4—64no—
503D4—62no—
504D4—544no—
505D4—544no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—91no—
509C2×C4—91no—
510C2×C4—92no—
511C2×C4—184no—
512C2×C4—274no—
513C2×C4—271no—
514C2×C4—272no—
515C2×C4—542no—
516C2×C4—544no—
517C2×C4—542no—
518C2×C4C2×D3×C3⋊D311yescharacteristic
519C2×C3—24no—
520C2×C3—24no—
521C2×C3—24no—
522C2×C3—48no—
523C2×C3—42no—
524C2×C3—88no—
525C2×C3—128no—
526C2×C3D32×D414yes—
527C2×C3D3×#23714yes—
528C2×C3C2×#380414yes—
529C2×C3C2×D4×C3⋊D311yescharacteristic
530C2×C3C2×#383011yescharacteristic
531C2×C3C2×#383011yescharacteristic
532D3—42no—
533D3—188no—
534D3—182no—
535D3—368no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—62no—
540C22—62no—
541C22—62no—
542C22—272no—
543C22—541no—
544C22—541no—
545C22C22×D3×C3⋊D311yescharacteristic
546C4—32no—
547C4—32no—
548C4—92no—
549C4—182no—
550C4—181no—
551C4—542no—
552C4—541no—
553C4#8902812yescomplemented
554C3—24no—
555C3#1059513914yescomplemented
556C3#1059618711yescharacteristic, complemented
557C2—42no—
558C2—122no—
559C2—542no—
560C2D3×D4×C3⋊D311yescharacteristic
561C2D3×#383011yescharacteristic
562C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89028C2semidirect
2. #11723979C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. C32×#940C2semidirect
5. #67×C3⋊Dic3C2semidirect
6. #11723600C2semidirect
7. #11723599C2semidirect
8. #11723534C2semidirect
9. #11723531C2semidirect
10. D12×C3⋊Dic3C2semidirect
11. C4×#6289C2semidirect
49. D4×C3⋊Dic3D3semidirect
52. C3×#940D3semidirect
82. #6289D4semidirect
149. C33×C4#41semidirect
156. C32×Dic3#41semidirect
161. #940C3⋊D3semidirect
324. C32×C4#235semidirect
325. C32×C4#237semidirect
339. C3×Dic3#237semidirect
351. C33#532semidirect
480. C3×C4#3804semidirect
481. C3×C4#3830semidirect
492. Dic3#3830semidirect
494. C32#5548semidirect
495. C32#5716semidirect
553. C4#89028semidirect
555. C3#10595139semidirect
556. C3#10596187semidirect

Maximal subgroups

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

subgroupindex
C2×#890282
#117239792
C3×D4×C3⋊Dic32
C32×#9402
#67×C3⋊Dic32
#117236002
#117235992
#117235342
#117235312
D12×C3⋊Dic32
C4×#62892
#105961873
#105951393