Finite Group Databasebeta

← #13177419

Subgroups — #13177419

Order 1728.

2116 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×C32×#237C211yesmaximal, characteristic, complemented
2C2×#89019C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic, complemented
5#11723640C212yesmaximal, complemented
6#11723618C212yesmaximal, complemented
7#11723600C212yesmaximal, complemented
8#11723599C212yesmaximal, complemented
9C4×#6291C211yesmaximal, characteristic, complemented
10#11723425C211yesmaximal, characteristic, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596302—31nomaximal
13#10595237—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C32×#67C2212yes—
19C22×C32×Dic3C2212yes—
20C32×#237—24no—
21C2×C32×C4×D3C2211yescharacteristic
22C2×C32×Dic6C2211yescharacteristic
23C2×#6293C2212yes—
24C2×#6291C2211yescharacteristic
25C2×#6289C2212yes—
26#89019—24no—
27#88864C2212yes—
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29C3×#3757C2212yes—
30C3×#3757C2212yes—
31C3×C4×C3⋊Dic3C2211yescharacteristic
32#88807C2211yescharacteristic
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88804C2211yescharacteristic
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
38Dic3×C3⋊Dic3C2212yes—
39C2×C32×#41—31no—
40C2×Q8×C3⋊D3—31no—
41C2×C3×#237D314yescomplemented
42C2×#3797—34no—
43Q8×C3⋊Dic3—31no—
44#64147—31no—
45#64147—32no—
46#64147—34no—
47#64147—64no—
48#64145—32no—
49#64145D311yescharacteristic, complemented
50C4×#638—32no—
51C4×#638—31no—
52#64133—31no—
53#64133—32no—
54C3×#936—34no—
55#63964—38no—
56#63959—38no—
57#63958—38no—
58#63952—38no—
59C4×#626—34no—
60#63856—34no—
61Dic3×Dic6—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C33×D4—22no—
66C2×C33×C4C2311yescharacteristic
67#6312—24no—
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C4×C3⋊(C3⋊D3)—22no—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C3×#634—22no—
75C32×#67—24no—
76C2×C32×Dic3—24no—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2312yes—
79C32×C4×D3—22no—
80C32×Dic6D414yescomplemented
81#6293—24no—
82#6291—22no—
83#6289—24no—
84#5831—94no—
85#5677—91no—
86C32×#41—64no—
87C2×C32×Q8—31no—
88C2×C32×D4—24no—
89C2×C32×D4—32no—
90C2×C32×D4C2×D314yes—
91C2×C32×D4C2×D311yescharacteristic
92C22×C32×C4—32no—
93C22×C32×C4—31no—
94C2×#638—32no—
95C2×#638—38no—
96C2×#638—31no—
97C2×#638—68no—
98Q8×C3⋊D3—64no—
99C2×C4×C3⋊D3—31no—
100C2×C4×C3⋊D3—34no—
101C2×C4×C3⋊D3—32no—
102C2×C4×C3⋊D3—64no—
103C2×#634—32no—
104C2×#634C2×D311yescharacteristic
105C2×C3×#67C2×D318yes—
106C22×C3×Dic3C2×D318yes—
107C3×#237—216no—
108C2×C3×C4×D3C2×D314yes—
109C2×C3×Dic6—34no—
110C2×C3×Dic6C2×D314yes—
111C2×#627—38no—
112C2×#626—34no—
113C2×#624—38no—
114#3797—616no—
115#3757—32no—
116#3757—38no—
117#3757—32no—
118#3757—31no—
119#3757—68no—
120#3757C2×D312yes—
121#3757C2×D312yes—
122#3752—31no—
123C4⋊(C3⋊Dic3)—31no—
124C4⋊(C3⋊Dic3)—34no—
125C4⋊(C3⋊Dic3)—32no—
126C4⋊(C3⋊Dic3)—64no—
127#3750—32no—
128#3750—31no—
129C4×C3⋊Dic3—32no—
130C4×C3⋊Dic3C2×D311yescharacteristic
131C3×#217—38no—
132C3×#217—38no—
133C3×C4×Dic3—34no—
134#3724—38no—
135Dic3⋊Dic3—38no—
136Dic3⋊Dic3—34no—
137#3722—34no—
138(C2×D3)⋊Dic3—38no—
139(C2×D3)⋊Dic3—34no—
140Dic32—38no—
141C22×C33—22no—
142C22×C33—22no—
143C22×C33—22no—
144C22×C33C2411yescharacteristic
145C2×C3⋊(C3⋊D3)—21no—
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)#4112yescomplemented
148C2×C32×D3—21no—
149C2×C32×D3—21no—
150C2×C32×D3—21no—
151C33×C4C2×D412yes—
152C3⋊(C3⋊Dic3)—21no—
153C3⋊(C3⋊Dic3)—22no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—21no—
156C3×C3⋊Dic3—21no—
157C32×Dic3—21no—
158C32×Dic3C2×D414yes—
159C2×C3×#41—34no—
160C2×D3×Q8—94no—
161C2×D3×Q8—91no—
162C2×#237C3⋊D311yescharacteristic, complemented
163C3×#124—91no—
164#946—91no—
165Q8×Dic3—94no—
166#938—98no—
167#938—94no—
168#938—91no—
169#938—184no—
170#936—34no—
171#936—98no—
172C4×#67—94no—
173C4×#67—98no—
174#924—98no—
175#924—94no—
176#885—92no—
177#884—92no—
178#67⋊C4—92no—
179#878—92no—
180C4×D12—91no—
181C4×Dic6—91no—
182C23×C32—28no—
183C23×C32—31no—
184C23×C32C22×D318yes—
185C23×C32C22×D312yes—
186C22×C3⋊D3—31no—
187C22×C3⋊D3—34no—
188C22×C3⋊D3—64no—
189C22×C3×D3C22×D314yes—
190C32×Q8—34no—
191C32×D4—22no—
192C32×D4—28no—
193C32×D4—48no—
194C32×D4—64no—
195C2×C32×C4—24no—
196C2×C32×C4—32no—
197C2×C32×C4—31no—
198C2×C32×C4—64no—
199C2×C32×C4—62no—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D311yescharacteristic
202#638—62no—
203#638—64no—
204#638—616no—
205#638—1216no—
206C2×C3⋊Dic3—32no—
207C2×C3⋊Dic3—31no—
208C2×C3⋊Dic3—38no—
209C2×C3⋊Dic3—34no—
210C2×C3⋊Dic3—68no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3C22×D311yescharacteristic
213C2×C3⋊Dic3C22×D311yescharacteristic
214C2×C3⋊Dic3C22×D312yes—
215C4×C3⋊D3—64no—
216C4×C3⋊D3—62no—
217C4×C3⋊D3—68no—
218C4×C3⋊D3—128no—
219#634—22no—
220#634—64no—
221C3×#67—216no—
222C2×C3×Dic3—216no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—38no—
225C2×C3×Dic3—34no—
226C2×C3×Dic3C22×D314yes—
227C2×C3×Dic3C22×D318yes—
228C3×C4×D3—28no—
229C3×Dic6—68no—
230C3×Dic6#67116yescomplemented
231#627—616no—
232#626—68no—
233#624—616no—
234#548—271noSylow
235C2×C33—42no—
236C2×C33C22×D411yescharacteristic
237C2×C33C2×#4111yescharacteristic
238C2×C332−1+411yescharacteristic
239C3⋊(C3⋊D3)—22no—
240C32×D3—41no—
241C3×#41—616no—
242C2×C3×Q8—34no—
243C2×C3×Q8—91no—
244C2×C3×D4—24no—
245C2×C3×D4—38no—
246C2×C3×D4D3214yes—
247C2×C3×D4C2×C3⋊D311yescharacteristic
248C22×C3×C4—38no—
249C22×C3×C4—34no—
250C2×#67—94no—
251C2×#67—98no—
252C2×#67—92no—
253C2×#67—188no—
254C2×#67C2×C3⋊D312yes—
255C22×Dic3C2×C3⋊D312yes—
256D3×Q8—1816no—
257D3×Q8—184no—
258#237—24no—
259C2×D12—91no—
260C2×C4×D3—98no—
261C2×C4×D3—94no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—184no—
265C2×C4×D3C2×C3⋊D311yescharacteristic
266C2×Dic6—34no—
267C2×Dic6—98no—
268C2×Dic6—92no—
269C2×Dic6C2×C3⋊D311yescharacteristic
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C42—91no—
273#217—38no—
274#217—38no—
275#217—98no—
276#217—94no—
277#217—98no—
278#217—92no—
279#217—188no—
280(C2×D3)⋊C4—94no—
281(C2×D3)⋊C4—91no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—91no—
284C4⋊Dic3—98no—
285C4⋊Dic3—94no—
286C4⋊Dic3—91no—
287C4⋊Dic3—91no—
288C4⋊Dic3—184no—
289Dic3⋊C4—94no—
290Dic3⋊C4—98no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—34no—
294C4×Dic3—98no—
295C4×Dic3—92no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32—24no—
300C22×C32—28no—
301C22×C32—28no—
302C22×C32—28no—
303C22×C32—48no—
304C22×C32—48no—
305C22×C32—48no—
306C22×C32—61no—
307C22×C32—61no—
308C22×C32—61no—
309C22×C32C23×D314yes—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—32no—
312C2×C3⋊D3—38no—
313C2×C3⋊D3—68no—
314C2×C3⋊D3—61no—
315C2×C3⋊D3—61no—
316C2×C3⋊D3—64no—
317C2×C3⋊D3—64no—
318C2×C3⋊D3—124no—
319C2×C3⋊D3—124no—
320C2×C3×D3—24no—
321C2×C3×D3—24no—
322C2×C3×D3—24no—
323C32×C4—28no—
324C32×C4—34no—
325C32×C4—61no—
326C32×C4D3×D412yes—
327C32×C4C2×#6718yes—
328C3⋊Dic3—22no—
329C3⋊Dic3—21no—
330C3⋊Dic3—21no—
331C3⋊Dic3—61no—
332C3⋊Dic3—62no—
333C3⋊Dic3—68no—
334C3⋊Dic3—64no—
335C3⋊Dic3—124no—
336C3⋊Dic3—128no—
337C3×Dic3—24no—
338C3×Dic3—64no—
339C3×Dic3—64no—
340C3×Dic3—68no—
341C3×Dic3C2×#67116yes—
342C2×#41—31no—
343C22×Q8—271no—
344#124—91no—
345#124—272no—
346C22⋊Q8—272no—
347C22⋊Q8—271no—
348#121—272no—
349#121—271no—
350C4×Q8—271no—
351C4×D4—271no—
352C4×D4—272no—
353C33#54811yescharacteristic, complemented, Sylow
354C23×C3—28no—
355C23×C3—34no—
356C23×C3C2×D3218yes—
357C23×C3C22×C3⋊D312yes—
358C22×D3—94no—
359C22×D3—91no—
360C22×D3—184no—
361C22×D3C22×C3⋊D311yescharacteristic
362C3×Q8—316no—
363C3×Q8—182no—
364C3×D4—28no—
365C3×D4—22no—
366C3×D4—48no—
367C3×D4—616no—
368C2×C3×C4—24no—
369C2×C3×C4—38no—
370C2×C3×C4—34no—
371C2×C3×C4—616no—
372C2×C3×C4—68no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—92no—
376C2×C3×C4C2×D3214yes—
377C2×C3×C4C22×C3⋊D311yescharacteristic
378#67—24no—
379#67—188no—
380#67—184no—
381#67—1816no—
382#67—3616no—
383C2×Dic3—24no—
384C2×Dic3—34no—
385C2×Dic3—34no—
386C2×Dic3—38no—
387C2×Dic3—98no—
388C2×Dic3—94no—
389C2×Dic3—92no—
390C2×Dic3—91no—
391C2×Dic3—188no—
392C2×Dic3—184no—
393C2×Dic3C22×C3⋊D311yescharacteristic
394C2×Dic3C22×C3⋊D312yes—
395D12—182no—
396C4×D3—22no—
397C4×D3—1816no—
398C4×D3—188no—
399C4×D3—182no—
400C4×D3—184no—
401C4×D3—368no—
402Dic6—68no—
403Dic6—1816no—
404Dic6—184no—
405Dic6#63814yescomplemented
406C2×C32—24no—
407C2×C32—24no—
408C2×C32—24no—
409C2×C32—42no—
410C2×C32—48no—
411C2×C32—88no—
412C2×C32—121no—
413C2×C32C2×D3×D411yescharacteristic
414C2×C32D3×#4114yes—
415C2×C32D3×#4111yescharacteristic
416C2×C32#101014yes—
417C2×C32#101011yescharacteristic
418C2×C32C22×#6714yes—
419C3⋊D3—62no—
420C3⋊D3—68no—
421C3⋊D3—128no—
422C3×D3—44no—
423#41—64no—
424C2×Q8—31no—
425C2×Q8—91no—
426C2×Q8—272no—
427C2×Q8—544no—
428C2×D4—32no—
429C2×D4—271no—
430C2×D4—272no—
431C2×D4D3×C3⋊D311yescharacteristic
432C22×C4—32no—
433C22×C4—31no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—272no—
437C4⋊C4—271no—
438C4⋊C4—272no—
439C4⋊C4—271no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4—271no—
443C22⋊C4—271no—
444C22⋊C4—272no—
445C22⋊C4—272no—
446C42—91no—
447C42—272no—
448C22×C3—28no—
449C22×C3—28no—
450C22×C3—28no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—24no—
455C22×C3—48no—
456C22×C3—48no—
457C22×C3—48no—
458C22×C3—64no—
459C22×C3—64no—
460C22×C3—64no—
461C22×C3C22×D3214yes—
462C22×C3C23×C3⋊D311yescharacteristic
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—98no—
467C2×D3—92no—
468C2×D3—188no—
469C2×D3—184no—
470C2×D3—184no—
471C2×D3—181no—
472C2×D3—181no—
473C2×D3—364no—
474C2×D3—364no—
475C3×C4—28no—
476C3×C4—316no—
477C3×C4—64no—
478C3×C4—181no—
479C3×C4—181no—
480C3×C4—182no—
481C3×C4D3×#6718yes—
482C3×C4C2×#63812yes—
483Dic3—21no—
484Dic3—64no—
485Dic3—64no—
486Dic3—68no—
487Dic3—188no—
488Dic3—184no—
489Dic3—182no—
490Dic3—181no—
491Dic3—364no—
492Dic3—368no—
493Dic3C2×#63814yes—
494C32—24no—
495C32#567711yescharacteristic, complemented
496C32#583114yescomplemented
497C23—31no—
498C23—271no—
499C23C2×D3×C3⋊D312yes—
500Q8—34no—
501Q8—182no—
502Q8—544no—
503D4—22no—
504D4—64no—
505D4—542no—
506D4—544no—
507C2×C4—32no—
508C2×C4—31no—
509C2×C4—64no—
510C2×C4—62no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—272no—
515C2×C4—271no—
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—124no—
526C2×C3#6430914yes—
527C2×C3#6431214yes—
528C2×C3C2×D3×#6714yes—
529C2×C3#41×C3⋊D311yescharacteristic
530C2×C3#6436711yescharacteristic
531C2×C3C22×#63811yescharacteristic
532D3—41no—
533D3—188no—
534D3—182no—
535D3—368no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—61no—
540C22—61no—
541C22—61no—
542C22—272no—
543C22—541no—
544C22—541no—
545C22C22×D3×C3⋊D311yescharacteristic
546C4—34no—
547C4—61no—
548C4—181no—
549C4—181no—
550C4—182no—
551C4—542no—
552C4—541no—
553C4D3×#63812yes—
554C3—24no—
555C3#1059523714yescomplemented
556C3#1059630211yescharacteristic, complemented
557C2—42no—
558C2—121no—
559C2—542no—
560C2#1172458211yescharacteristic
561C2#1172458611yescharacteristic
562C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#548
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#237C2semidirect
2. C2×#89019C2semidirect
3. #11723979C2semidirect
4. C3×#64145C2semidirect
5. #11723640C2semidirect
6. #11723618C2semidirect
7. #11723600C2semidirect
8. #11723599C2semidirect
9. C4×#6291C2semidirect
10. #11723425C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
41. C2×C3×#237D3semidirect
49. #64145D3semidirect
80. C32×Dic6D4semidirect
147. C2×C3⋊(C3⋊D3)#41semidirect
162. C2×#237C3⋊D3semidirect
230. C3×Dic6#67semidirect
353. C33#548semidirect
405. Dic6#638semidirect
495. C32#5677semidirect
496. C32#5831semidirect
555. C3#10595237semidirect
556. C3#10596302semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×#890192
#117239792
C3×#641452
#117236402
#117236182
#117236002
#117235992
C4×#62912
#117234252
Dic6×C3⋊Dic32
#105963023
#105952373