Finite Group Databasebeta

← #13177381

Subgroups — #13177381

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×#89031C212yesmaximal, complemented
2#11723979C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C32×D4×Dic3C211yesmaximal, characteristic, complemented
5#11723623C212yesmaximal, complemented
6Dic3×#638C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8#11723535C211yesmaximal, characteristic, complemented
9#11723532C211yesmaximal, characteristic, complemented
10Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C4×#6289C211yesmaximal, characteristic, complemented
12#10596019—31nomaximal
13#10595139—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×#638C2212yes—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C22×C32×Dic3C2212yes—
21C2×#6293C2212yes—
22C2×#6292C2212yes—
23C2×#6289C2211yescharacteristic
24C2×#6289C2211yescharacteristic
25#89031—28no—
26C2×Dic3×C3⋊D3C2212yes—
27#88864C2212yes—
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30C32×#217C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33#88807C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×#3828—32no—
39C2×#3804—38no—
40C32×C4×D4—31no—
41#64149D311yescharacteristic, complemented
42#64147—31no—
43#64147—34no—
44#64147—64no—
45C4×#638—32no—
46#64104—31no—
47#64097—31no—
48#64094—32no—
49#64093—32no—
50C4×C4⋊(C3⋊D3)—31no—
51C42×C3⋊D3—31no—
52C3×#940—34no—
53C3×D4×Dic3D314yescomplemented
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×C3×C3⋊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—
70C3×#638—24no—
71C3×#638—24no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4⋊(C3⋊D3)—22no—
75C2×C32×Dic3—24no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79#6293—24no—
80#6292—24no—
81#6289—22no—
82#6289D414yescomplemented
83Dic3×C3⋊D3—24no—
84#5716—91no—
85#5548—94no—
86C2×C32×D4—24no—
87C2×C32×D4C2×D314yes—
88C2×C32×D4C2×D311yescharacteristic
89C22×C32×C4—32no—
90C2×#638—32no—
91C2×#638—38no—
92C2×#638—68no—
93C2×#638C2×D312yes—
94C2×#638C2×D312yes—
95#3828—68no—
96C2×C4⋊(C3⋊D3)—32no—
97C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
98C2×C4×C3⋊D3—31no—
99C2×C4×C3⋊D3—32no—
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×#67—38no—
106C2×C3×#67—38no—
107C22×C3×Dic3C2×D318yes—
108C2×C3×D12—34no—
109C2×#627—38no—
110C2×#626—38no—
111C2×#624—34no—
112C2×#624—34no—
113#3804—632no—
114C2×D3×Dic3—38no—
115C32×C4⋊C4—31no—
116C32×C22⋊C4—32no—
117C32×C42—31no—
118#3757—32no—
119#3757—38no—
120#3757—68no—
121#3752—32no—
122#3752—32no—
123C4⋊(C3⋊Dic3)—31no—
124C4⋊(C3⋊Dic3)—34no—
125C4⋊(C3⋊Dic3)—64no—
126#3750—32no—
127C4×C3⋊Dic3—31no—
128C4×C3⋊Dic3—31no—
129C4×C3⋊Dic3C2×D311yescharacteristic
130C3×#217C2×D318yes—
131C3×C4⋊Dic3C2×D314yes—
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×C3×C3⋊D3—22no—
147C2×C3×C3⋊D3—22no—
148C2×C3×C3⋊D3—22no—
149C33×C4#4112yescomplemented
150C3⋊(C3⋊Dic3)—22no—
151C3⋊(C3⋊Dic3)—21no—
152C3×C3⋊Dic3C2×D412yes—
153C3×C3⋊Dic3#4112yescomplemented
154C32×Dic3—22no—
155C32×Dic3—21no—
156C32×Dic3C2×D412yes—
157C2×#237—92no—
158C2×#235—98no—
159C3×#127—91no—
160C3×C4×D4—34no—
161#940—34no—
162#938—94no—
163#938—91no—
164#938—92no—
165#938—91no—
166#938—184no—
167#936—92no—
168D4×Dic3—92no—
169D4×Dic3—91no—
170D4×Dic3C3⋊D311yescharacteristic, complemented
171C4×#67—98no—
172#895—94no—
173#888—94no—
174#885—98no—
175#884—98no—
176C4×D12—94no—
177C42×D3—94no—
178C42×D3—91no—
179#869—91no—
180C23×C32—28no—
181C23×C32C22×D318yes—
182C23×C32C22×D312yes—
183C22×C3⋊D3—31no—
184C22×C3⋊D3—34no—
185C22×C3⋊D3—64no—
186C22×C3⋊D3C22×D312yes—
187C22×C3×D3—38no—
188C32×D4—22no—
189C32×D4—28no—
190C32×D4—48no—
191C2×C32×C4—24no—
192C2×C32×C4—31no—
193C2×C32×C4—31no—
194C2×C32×C4—32no—
195C2×C32×C4—64no—
196C2×C32×C4C22×D314yes—
197C2×C32×C4C22×D311yescharacteristic
198#638—24no—
199#638—24no—
200#638—64no—
201#638—616no—
202#638—1216no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—32no—
205C2×C3⋊Dic3—34no—
206C2×C3⋊Dic3—38no—
207C2×C3⋊Dic3—68no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D311yescharacteristic
211C4⋊(C3⋊D3)—22no—
212C4⋊(C3⋊D3)—64no—
213C4×C3⋊D3—34no—
214C4×C3⋊D3—64no—
215C4×C3⋊D3—62no—
216C4×C3⋊D3—62no—
217C4×C3⋊D3—68no—
218C4×C3⋊D3—128no—
219#634—64no—
220C3×#67—616no—
221C3×#67—616no—
222C2×C3×Dic3—216no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3C22×D314yes—
226C2×C3×Dic3C22×D318yes—
227C2×C3×Dic3C22×D314yes—
228C3×D12—68no—
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—
240C3×C3⋊D3—42no—
241C2×C3×D4—24no—
242C2×C3×D4—92no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C2×C3×D4D3214yes—
246C2×C3×D4C2×C3⋊D311yescharacteristic
247C22×C3×C4—38no—
248C2×#67—38no—
249C2×#67—38no—
250C2×#67—98no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—188no—
254C22×Dic3—92no—
255C22×Dic3C2×C3⋊D312yes—
256#237—188no—
257#235—1832no—
258C2×D12—34no—
259C2×D12—98no—
260C2×C4×D3—94no—
261C2×C4×D3—91no—
262C2×C4×D3—94no—
263C2×C4×D3—98no—
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—34no—
271C3×C22⋊C4—38no—
272C3×C42—34no—
273C3×C42—91no—
274#217—98no—
275#217—92no—
276#217—92no—
277#217—92no—
278#217—188no—
279#217C2×C3⋊D312yes—
280(C2×D3)⋊C4—98no—
281(C2×D3)⋊C4—98no—
282C4⋊Dic3—94no—
283C4⋊Dic3—91no—
284C4⋊Dic3—92no—
285C4⋊Dic3—184no—
286C4⋊Dic3C2×C3⋊D311yescharacteristic
287Dic3⋊C4—98no—
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×C32C23×D314yes—
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
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—68no—
319C2×C3×D3—68no—
320C2×C3×D3—68no—
321C32×C4—28no—
322C32×C4—32no—
323C32×C4—62no—
324C32×C4—61no—
325C32×C4#23518yescomplemented
326C32×C4#23712yescomplemented
327C3⋊Dic3—62no—
328C3⋊Dic3—61no—
329C3⋊Dic3—64no—
330C3⋊Dic3—68no—
331C3⋊Dic3—124no—
332C3⋊Dic3—128no—
333C3⋊Dic3D3×D412yes—
334C3⋊Dic3#23712yescomplemented
335C3×Dic3—28no—
336C3×Dic3—24no—
337C3×Dic3—38no—
338C3×Dic3—38no—
339C3×Dic3D3×D418yes—
340C2×#41—272no—
341C4⋊Q8—271no—
342#127—91no—
343#124—272no—
344#121—272no—
345#121—271no—
346#121—271no—
347C4×D4—31no—
348C4×D4—271no—
349C4×D4—272no—
350C2×C42—271no—
351C33#53211yescharacteristic, complemented, Sylow
352C23×C3—28no—
353C23×C3—92no—
354C23×C3C2×D3218yes—
355C23×C3C22×C3⋊D312yes—
356C22×D3—38no—
357C22×D3—94no—
358C22×D3—91no—
359C22×D3—184no—
360C3×D4—28no—
361C3×D4—22no—
362C3×D4—48no—
363C3×D4—184no—
364C3×D4—184no—
365C3×D4—182no—
366C2×C3×C4—24no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—38no—
370C2×C3×C4—616no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4C2×D3214yes—
374C2×C3×C4C22×C3⋊D311yescharacteristic
375#67—616no—
376#67—616no—
377#67—184no—
378#67—184no—
379#67—1816no—
380#67—3616no—
381C2×Dic3—24no—
382C2×Dic3—34no—
383C2×Dic3—34no—
384C2×Dic3—94no—
385C2×Dic3—98no—
386C2×Dic3—91no—
387C2×Dic3—92no—
388C2×Dic3—188no—
389C2×Dic3—184no—
390C2×Dic3—184no—
391C2×Dic3C22×C3⋊D311yescharacteristic
392C2×Dic3C22×C3⋊D312yes—
393C2×Dic3C22×C3⋊D311yescharacteristic
394D12—68no—
395D12—1816no—
396C4×D3—916no—
397C4×D3—94no—
398C4×D3—182no—
399C4×D3—188no—
400C4×D3—1816no—
401C4×D3—188no—
402C4×D3—182no—
403C4×D3—368no—
404Dic6—184no—
405Dic6—1816no—
406C2×C32—24no—
407C2×C32—24no—
408C2×C32—24no—
409C2×C32—42no—
410C2×C32—48no—
411C2×C32—88no—
412C2×C32C2×#23514yes—
413C2×C32C2×D3×D414yes—
414C2×C32C2×D3×D411yescharacteristic
415C2×C32C2×#23711yescharacteristic
416C2×C32C2×#23711yescharacteristic
417C2×C32D3×#4114yes—
418C3⋊D3—42no—
419C3⋊D3—62no—
420C3⋊D3—68no—
421C3⋊D3—128no—
422C3×D3—128no—
423#41—548no—
424C2×Q8—272no—
425C2×D4—92no—
426C2×D4—92no—
427C2×D4—91no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4D3×C3⋊D311yescharacteristic
431C22×C4—32no—
432C22×C4—271no—
433C22×C4—271no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—31no—
437C4⋊C4—272no—
438C4⋊C4—271no—
439C22⋊C4—32no—
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—182no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3C22×D3214yes—
461C22×C3C23×C3⋊D311yescharacteristic
462C2×D3—68no—
463C2×D3—68no—
464C2×D3—68no—
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—68no—
477C3×C4—64no—
478C3×C4—92no—
479C3×C4—92no—
480C3×C4#380418yescomplemented
481C3×C4#382812yescomplemented
482Dic3—22no—
483Dic3—21no—
484Dic3—38no—
485Dic3—38no—
486Dic3—188no—
487Dic3—184no—
488Dic3—181no—
489Dic3—182no—
490Dic3—364no—
491Dic3—368no—
492Dic3D4×C3⋊D312yes—
493C32—24no—
494C32#554814yescomplemented
495C32#571611yescharacteristic, complemented
496C23—92no—
497C23—271no—
498C23C2×D3×C3⋊D312yes—
499Q8—544no—
500D4—22no—
501D4—184no—
502D4—184no—
503D4—182no—
504D4—544no—
505D4—544no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—32no—
509C2×C4—64no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—274no—
513C2×C4—271no—
514C2×C4—272no—
515C2×C4—544no—
516C2×C4—542no—
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—362no—
526C2×C3D32×D414yes—
527C2×C3D3×#23714yes—
528C2×C3C2×#380414yes—
529C2×C3C2×#382811yescharacteristic
530C2×C3C2×D4×C3⋊D311yescharacteristic
531C2×C3#41×C3⋊D311yescharacteristic
532D3—128no—
533D3—188no—
534D3—182no—
535D3—368no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—182no—
540C22—182no—
541C22—182no—
542C22—272no—
543C22—541no—
544C22—541no—
545C22C22×D3×C3⋊D311yescharacteristic
546C4—32no—
547C4—62no—
548C4—61no—
549C4—92no—
550C4—92no—
551C4—542no—
552C4—541no—
553C4#8903112yescomplemented
554C3—24no—
555C3#1059513914yescomplemented
556C3#1059601911yescharacteristic, complemented
557C2—42no—
558C2—362no—
559C2—542no—
560C2D3×D4×C3⋊D311yescharacteristic
561C2C3⋊D3×#23711yescharacteristic
562C2C2×#8903111yescharacteristic
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×#89031C2semidirect
2. #11723979C2semidirect
3. C3×#64149C2semidirect
4. C32×D4×Dic3C2semidirect
5. #11723623C2semidirect
6. Dic3×#638C2semidirect
7. #11723591C2semidirect
8. #11723535C2semidirect
9. #11723532C2semidirect
10. Dic3×C4⋊(C3⋊D3)C2semidirect
11. C4×#6289C2semidirect
41. #64149D3semidirect
53. C3×D4×Dic3D3semidirect
82. #6289D4semidirect
149. C33×C4#41semidirect
153. C3×C3⋊Dic3#41semidirect
170. D4×Dic3C3⋊D3semidirect
325. C32×C4#235semidirect
326. C32×C4#237semidirect
334. C3⋊Dic3#237semidirect
351. C33#532semidirect
480. C3×C4#3804semidirect
481. C3×C4#3828semidirect
494. C32#5548semidirect
495. C32#5716semidirect
553. C4#89031semidirect
555. C3#10595139semidirect
556. C3#10596019semidirect

Maximal subgroups

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

subgroupindex
C2×#890312
#117239792
C3×#641492
C32×D4×Dic32
#117236232
Dic3×#6382
#117235912
#117235352
#117235322
Dic3×C4⋊(C3⋊D3)2
C4×#62892
#105960193
#105951393