Finite Group Databasebeta

← #13177521

Subgroups — #13177521

Order 1728.

2128 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89031C212yesmaximal, complemented
2#11723979C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C32×D4×Dic3C211yesmaximal, characteristic, complemented
5#11723640C212yesmaximal, complemented
6#11723628C212yesmaximal, complemented
7#11723621C212yesmaximal, complemented
8#11723534C211yesmaximal, characteristic, complemented
9C4×#6292C211yesmaximal, characteristic, complemented
10#11723503C211yesmaximal, characteristic, complemented
11C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#10596022—31nomaximal
13#10595136—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C22×C32×Dic3C2212yes—
20C2×#6293C2212yes—
21C2×#6292C2211yescharacteristic
22C2×#6289C2212yes—
23#89031—28no—
24C2×Dic3×C3⋊D3C2212yes—
25#88864C2212yes—
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
27C3×#3757C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C32×#217C2212yes—
30C32×C4⋊Dic3C2211yescharacteristic
31C32×C4×Dic3C2211yescharacteristic
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35Dic3⋊(C3⋊Dic3)C2211yescharacteristic
36#88804C2211yescharacteristic
37#88803C2211yescharacteristic
38C2×#3828—32no—
39C2×#3804—38no—
40C32×C4×D4—31no—
41#64147—31no—
42#64147—34no—
43#64147—64no—
44#64147D311yescharacteristic, complemented
45#64133—32no—
46C4⋊C4×C3⋊D3—31no—
47C4⋊C4×C3⋊D3—31no—
48#64092—32no—
49#64092—32no—
50C4×C4⋊(C3⋊D3)—31no—
51#64078—31no—
52C3×#938—34no—
53C3×D4×Dic3D314yescomplemented
54#63975—38no—
55#63969—38no—
56#63964—38no—
57#63918—34no—
58C4×#626—34no—
59#63900—34no—
60D3×C4⋊Dic3—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
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—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—22no—
74C2×C32×Dic3—24no—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2311yescharacteristic
78#6293—24no—
79#6292D414yescomplemented
80#6289—24no—
81Dic3×C3⋊D3—24no—
82#5611—91no—
83#5551—94no—
84C2×C32×D4—24no—
85C2×C32×D4C2×D314yes—
86C2×C32×D4C2×D311yescharacteristic
87C22×C32×C4—32no—
88C2×#638—32no—
89C2×#638—38no—
90C2×#638—68no—
91C2×#638C2×D312yes—
92#3828—68no—
93C2×C4⋊(C3⋊D3)—31no—
94C2×C4×C3⋊D3—32no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—34no—
97C2×C4×C3⋊D3—32no—
98C2×C4×C3⋊D3—64no—
99C2×C4×C3⋊D3C2×D311yescharacteristic
100C2×#634—32no—
101C2×C3×#67—38no—
102C22×C3×Dic3C2×D318yes—
103C2×C3×C4×D3—34no—
104C2×#627—38no—
105C2×#626—34no—
106C2×#624—38no—
107#3804—632no—
108C2×D3×Dic3—38no—
109C32×C4⋊C4—31no—
110C32×C22⋊C4—32no—
111C32×C42—31no—
112#3757—32no—
113#3757—38no—
114#3757—68no—
115#3757C2×D312yes—
116#3752—31no—
117#3752—31no—
118C4⋊(C3⋊Dic3)—31no—
119C4⋊(C3⋊Dic3)—34no—
120C4⋊(C3⋊Dic3)—32no—
121C4⋊(C3⋊Dic3)—31no—
122C4⋊(C3⋊Dic3)—64no—
123C4⋊(C3⋊Dic3)C2×D311yescharacteristic
124#3750—32no—
125#3750—32no—
126C3×#217—38no—
127C3×#217C2×D318yes—
128C3×C4⋊Dic3—34no—
129C3×C4⋊Dic3C2×D314yes—
130C3×C4×Dic3C2×D314yes—
131#3724—38no—
132Dic3⋊Dic3—38no—
133Dic3⋊Dic3—34no—
134Dic3⋊Dic3—38no—
135#3722—34no—
136(C2×D3)⋊Dic3—34no—
137C22×C33—22no—
138C22×C33—22no—
139C22×C33—22no—
140C22×C33C2411yescharacteristic
141C2×C3⋊(C3⋊D3)—21no—
142C2×C3⋊(C3⋊D3)—21no—
143C2×C3⋊(C3⋊D3)C2×D412yes—
144C2×C3×C3⋊D3—21no—
145C2×C3×C3⋊D3—21no—
146C2×C3×C3⋊D3C2×D412yes—
147C33×C4#4112yescomplemented
148C3⋊(C3⋊Dic3)—22no—
149C3⋊(C3⋊Dic3)—21no—
150C3×C3⋊Dic3—22no—
151C3×C3⋊Dic3—21no—
152C32×Dic3—22no—
153C32×Dic3—21no—
154C32×Dic3C2×D412yes—
155C2×#237—92no—
156C2×#235—98no—
157C3×#121—91no—
158C3×C4×D4—34no—
159#938—34no—
160#938—94no—
161#938—91no—
162#938—184no—
163#935—92no—
164D4×Dic3C3⋊D311yescharacteristic, complemented
165C4×#67—91no—
166C2×C4⋊Dic3—91no—
167#924—98no—
168D3×C4⋊C4—94no—
169D3×C4⋊C4—94no—
170D3×C4⋊C4—91no—
171#888—91no—
172#883—98no—
173#883—98no—
174#883—92no—
175#878—92no—
176C4×D12—94no—
177#869—94no—
178C23×C32—28no—
179C23×C32C22×D318yes—
180C23×C32C22×D312yes—
181C22×C3⋊D3—31no—
182C22×C3⋊D3—34no—
183C22×C3⋊D3—64no—
184C22×C3⋊D3C22×D311yescharacteristic
185C22×C3×D3—34no—
186C32×D4—22no—
187C32×D4—28no—
188C32×D4—48no—
189C2×C32×C4—24no—
190C2×C32×C4—31no—
191C2×C32×C4—32no—
192C2×C32×C4—31no—
193C2×C32×C4—64no—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C22×D311yescharacteristic
196#638—24no—
197#638—64no—
198#638—616no—
199#638—1216no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—32no—
202C2×C3⋊Dic3—34no—
203C2×C3⋊Dic3—38no—
204C2×C3⋊Dic3—68no—
205C2×C3⋊Dic3—64no—
206C2×C3⋊Dic3C22×D311yescharacteristic
207C2×C3⋊Dic3C22×D312yes—
208C4⋊(C3⋊D3)—34no—
209C4×C3⋊D3—22no—
210C4×C3⋊D3—64no—
211C4×C3⋊D3—64no—
212C4×C3⋊D3—62no—
213C4×C3⋊D3—68no—
214C4×C3⋊D3—128no—
215#634—64no—
216C3×#67—616no—
217C2×C3×Dic3—216no—
218C2×C3×Dic3—34no—
219C2×C3×Dic3—38no—
220C2×C3×Dic3C22×D314yes—
221C2×C3×Dic3C22×D318yes—
222C2×C3×Dic3C22×D314yes—
223C3×C4×D3—68no—
224#627—616no—
225#626—316no—
226#624—616no—
227D3×Dic3—616no—
228#547—271noSylow
229C2×C33—42no—
230C2×C33C22×D411yescharacteristic
231C2×C33C2×#4111yescharacteristic
232C2×C332−1+411yescharacteristic
233C3⋊(C3⋊D3)—22no—
234C3×C3⋊D3—22no—
235C2×C3×D4—24no—
236C2×C3×D4—92no—
237C2×C3×D4D3214yes—
238C2×C3×D4C2×C3⋊D311yescharacteristic
239C22×C3×C4—38no—
240C22×C3×C4—91no—
241C2×#67—38no—
242C2×#67—98no—
243C2×#67—92no—
244C2×#67—91no—
245C2×#67—188no—
246C22×Dic3—92no—
247C22×Dic3C2×C3⋊D312yes—
248#237—188no—
249#235—1832no—
250C2×D12—94no—
251C2×C4×D3—34no—
252C2×C4×D3—98no—
253C2×C4×D3—98no—
254C2×C4×D3—94no—
255C2×C4×D3—91no—
256C2×C4×D3—92no—
257C2×C4×D3—184no—
258C2×Dic6—98no—
259C2×Dic6—92no—
260C3×C4⋊C4—34no—
261C3×C4⋊C4—91no—
262C3×C22⋊C4—38no—
263C3×C22⋊C4—92no—
264C3×C42—34no—
265#217—38no—
266#217—98no—
267#217—92no—
268#217—91no—
269#217—188no—
270#217C2×C3⋊D312yes—
271(C2×D3)⋊C4—94no—
272(C2×D3)⋊C4—94no—
273(C2×D3)⋊C4—91no—
274C4⋊Dic3—34no—
275C4⋊Dic3—98no—
276C4⋊Dic3—94no—
277C4⋊Dic3—94no—
278C4⋊Dic3—91no—
279C4⋊Dic3—92no—
280C4⋊Dic3—184no—
281C4⋊Dic3C2×C3⋊D311yescharacteristic
282Dic3⋊C4—98no—
283Dic3⋊C4—98no—
284Dic3⋊C4—91no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3C2×C3⋊D311yescharacteristic
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—22no—
291C22×C32—24no—
292C22×C32—28no—
293C22×C32—28no—
294C22×C32—28no—
295C22×C32—48no—
296C22×C32—48no—
297C22×C32—48no—
298C22×C32C23×D314yes—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—21no—
301C2×C3⋊D3—21no—
302C2×C3⋊D3—32no—
303C2×C3⋊D3—38no—
304C2×C3⋊D3—68no—
305C2×C3⋊D3—61no—
306C2×C3⋊D3—61no—
307C2×C3⋊D3—64no—
308C2×C3⋊D3—64no—
309C2×C3⋊D3—124no—
310C2×C3⋊D3—124no—
311C2×C3⋊D3D3×D412yes—
312C2×C3×D3—38no—
313C2×C3×D3—64no—
314C2×C3×D3—64no—
315C32×C4—28no—
316C32×C4—32no—
317C32×C4—62no—
318C32×C4—61no—
319C32×C4#23518yescomplemented
320C32×C4#23712yescomplemented
321C3⋊Dic3—22no—
322C3⋊Dic3—21no—
323C3⋊Dic3—62no—
324C3⋊Dic3—61no—
325C3⋊Dic3—64no—
326C3⋊Dic3—68no—
327C3⋊Dic3—124no—
328C3⋊Dic3—128no—
329C3×Dic3—28no—
330C3×Dic3—24no—
331C3×Dic3—64no—
332C3×Dic3—68no—
333C3×Dic3D3×D418yes—
334C2×#41—272no—
335C4⋊Q8—271no—
336#123—272no—
337#123—272no—
338C22⋊Q8—272no—
339#121—91no—
340#121—271no—
341C4×D4—31no—
342C4×D4—271no—
343C2×C4⋊C4—271no—
344C2×C4⋊C4—271no—
345C33#54711yescharacteristic, complemented, Sylow
346C23×C3—28no—
347C23×C3—91no—
348C23×C3C2×D3218yes—
349C23×C3C22×C3⋊D312yes—
350C22×D3—34no—
351C22×D3—94no—
352C22×D3—91no—
353C22×D3—184no—
354C3×D4—28no—
355C3×D4—22no—
356C3×D4—48no—
357C3×D4—184no—
358C2×C3×C4—24no—
359C2×C3×C4—34no—
360C2×C3×C4—38no—
361C2×C3×C4—34no—
362C2×C3×C4—616no—
363C2×C3×C4—92no—
364C2×C3×C4—91no—
365C2×C3×C4—182no—
366C2×C3×C4C2×D3214yes—
367C2×C3×C4C22×C3⋊D311yescharacteristic
368#67—616no—
369#67—94no—
370#67—184no—
371#67—1816no—
372#67—3616no—
373C2×Dic3—24no—
374C2×Dic3—34no—
375C2×Dic3—38no—
376C2×Dic3—94no—
377C2×Dic3—98no—
378C2×Dic3—91no—
379C2×Dic3—92no—
380C2×Dic3—188no—
381C2×Dic3—184no—
382C2×Dic3—184no—
383C2×Dic3C22×C3⋊D311yescharacteristic
384C2×Dic3C22×C3⋊D312yes—
385C2×Dic3C22×C3⋊D311yescharacteristic
386D12—916no—
387C4×D3—68no—
388C4×D3—184no—
389C4×D3—1816no—
390C4×D3—1816no—
391C4×D3—188no—
392C4×D3—182no—
393C4×D3—368no—
394Dic6—1816no—
395Dic6—184no—
396C2×C32—24no—
397C2×C32—24no—
398C2×C32—24no—
399C2×C32—42no—
400C2×C32—48no—
401C2×C32—88no—
402C2×C32C2×#23514yes—
403C2×C32C2×D3×D414yes—
404C2×C32C2×D3×D411yescharacteristic
405C2×C32C2×#23711yescharacteristic
406C2×C32#101014yes—
407C2×C32#101011yescharacteristic
408C3⋊D3—22no—
409C3⋊D3—62no—
410C3⋊D3—68no—
411C3⋊D3—128no—
412C3×D3—68no—
413#41—548no—
414C2×Q8—272no—
415C2×D4—92no—
416C2×D4—271no—
417C2×D4—272no—
418C2×D4D3×C3⋊D311yescharacteristic
419C22×C4—32no—
420C22×C4—91no—
421C22×C4—272no—
422C22×C4—272no—
423C22×C4—271no—
424C4⋊C4—31no—
425C4⋊C4—91no—
426C4⋊C4—272no—
427C4⋊C4—272no—
428C4⋊C4—271no—
429C4⋊C4—272no—
430C4⋊C4—271no—
431C22⋊C4—32no—
432C22⋊C4—92no—
433C22⋊C4—271no—
434C22⋊C4—271no—
435C22⋊C4—272no—
436C42—31no—
437C22×C3—28no—
438C22×C3—28no—
439C22×C3—28no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—24no—
444C22×C3—48no—
445C22×C3—48no—
446C22×C3—48no—
447C22×C3—92no—
448C22×C3—181no—
449C22×C3—181no—
450C22×C3C22×D3214yes—
451C22×C3C23×C3⋊D311yescharacteristic
452C2×D3—38no—
453C2×D3—64no—
454C2×D3—64no—
455C2×D3—98no—
456C2×D3—92no—
457C2×D3—188no—
458C2×D3—184no—
459C2×D3—184no—
460C2×D3—181no—
461C2×D3—181no—
462C2×D3—364no—
463C2×D3—364no—
464C3×C4—28no—
465C3×C4—38no—
466C3×C4—64no—
467C3×C4—68no—
468C3×C4—181no—
469C3×C4—182no—
470C3×C4#380418yescomplemented
471C3×C4#382812yescomplemented
472Dic3—22no—
473Dic3—21no—
474Dic3—68no—
475Dic3—64no—
476Dic3—188no—
477Dic3—184no—
478Dic3—181no—
479Dic3—182no—
480Dic3—364no—
481Dic3—368no—
482Dic3D4×C3⋊D312yes—
483C32—24no—
484C32#555114yescomplemented
485C32#561111yescharacteristic, complemented
486C23—91no—
487C23—271no—
488C23C2×D3×C3⋊D312yes—
489Q8—544no—
490D4—22no—
491D4—184no—
492D4—274no—
493D4—544no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—32no—
497C2×C4—64no—
498C2×C4—91no—
499C2×C4—92no—
500C2×C4—182no—
501C2×C4—271no—
502C2×C4—272no—
503C2×C4—544no—
504C2×C4—544no—
505C2×C4—542no—
506C2×C4C2×D3×C3⋊D311yescharacteristic
507C2×C3—24no—
508C2×C3—24no—
509C2×C3—24no—
510C2×C3—48no—
511C2×C3—42no—
512C2×C3—88no—
513C2×C3—182no—
514C2×C3#6430914yes—
515C2×C3D32×D414yes—
516C2×C3C2×#380414yes—
517C2×C3C2×#382811yescharacteristic
518C2×C3C2×D4×C3⋊D311yescharacteristic
519C2×C3#6436711yescharacteristic
520D3—68no—
521D3—188no—
522D3—182no—
523D3—368no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—92no—
528C22—181no—
529C22—181no—
530C22—272no—
531C22—541no—
532C22—541no—
533C22C22×D3×C3⋊D311yescharacteristic
534C4—32no—
535C4—61no—
536C4—62no—
537C4—182no—
538C4—181no—
539C4—542no—
540C4—541no—
541C4#8903112yescomplemented
542C3—24no—
543C3#1059513614yescomplemented
544C3#1059602211yescharacteristic, complemented
545C2—42no—
546C2—182no—
547C2—542no—
548C2#1172458211yescharacteristic
549C2D3×D4×C3⋊D311yescharacteristic
550C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#547
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×#64147C2semidirect
4. C32×D4×Dic3C2semidirect
5. #11723640C2semidirect
6. #11723628C2semidirect
7. #11723621C2semidirect
8. #11723534C2semidirect
9. C4×#6292C2semidirect
10. #11723503C2semidirect
11. C3⋊D3×C4⋊Dic3C2semidirect
44. #64147D3semidirect
53. C3×D4×Dic3D3semidirect
79. #6292D4semidirect
147. C33×C4#41semidirect
164. D4×Dic3C3⋊D3semidirect
319. C32×C4#235semidirect
320. C32×C4#237semidirect
345. C33#547semidirect
470. C3×C4#3804semidirect
471. C3×C4#3828semidirect
484. C32#5551semidirect
485. C32#5611semidirect
541. C4#89031semidirect
543. C3#10595136semidirect
544. C3#10596022semidirect

Maximal subgroups

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

subgroupindex
C2×#890312
#117239792
C3×#641472
C32×D4×Dic32
#117236402
#117236282
#117236212
#117235342
C4×#62922
#117235032
C3⋊D3×C4⋊Dic32
#105960223
#105951363