Finite Group Databasebeta

← #13177402

Subgroups — #13177402

Order 1728.

2044 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#11723977C211yesmaximal, characteristic, complemented
4C3×#64147C211yesmaximal, characteristic, complemented
5#11723640C212yesmaximal, complemented
6#11723631C212yesmaximal, complemented
7Dic3×#638C212yesmaximal, complemented
8#11723600C212yesmaximal, complemented
9C4×#6290C211yesmaximal, characteristic, complemented
10#11723426C211yesmaximal, characteristic, complemented
11#11723397C211yesmaximal, characteristic, complemented
12#10596302—31nomaximal
13#10595157—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×#638C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×#67C2212yes—
19C22×C32×Dic3C2212yes—
20C32×#237—24no—
21C2×C32×C4×D3C2211yescharacteristic
22C2×C32×Dic6C2211yescharacteristic
23C2×#6293C2212yes—
24C2×#6290C2211yescharacteristic
25C2×Dic3×C3⋊D3C2212yes—
26C3⋊D3×Dic6—24no—
27#88864C2212yes—
28#88864C2212yes—
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3757C2212yes—
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32#88807C2212yes—
33#88806C2211yescharacteristic
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88803C2211yescharacteristic
36(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38Dic3×C3⋊Dic3C2212yes—
39C2×C32×#41—31no—
40C2×Q8×C3⋊D3—31no—
41C2×C3×#237D314yescomplemented
42C2×D3×Dic6—34no—
43Q8×C3⋊Dic3—31no—
44#64147—32no—
45#64147D311yescharacteristic, complemented
46#64145—31no—
47#64145—32no—
48#64145—34no—
49#64145—64no—
50C4×#638—32no—
51C4×#638—31no—
52#64133—32no—
53#64133—31no—
54C3×#938—34no—
55Dic3×#67—38no—
56#63970—38no—
57#63964—38no—
58#63959—38no—
59C4×#625—34no—
60#63857—34no—
61#63843—34no—
62C23×C33C2312yes—
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C33×D4—22no—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2312yes—
70#6308—22no—
71C3×#638—24no—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4×C3⋊D3—22no—
75C32×#67—24no—
76C2×C32×Dic3—24no—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2312yes—
79C32×C4×D3—22no—
80C32×Dic6D414yescomplemented
81#6293—24no—
82#6290—22no—
83Dic3×C3⋊D3—24no—
84#5831—94no—
85#5598—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—31no—
95C2×#638C2×D312yes—
96Q8×C3⋊D3—64no—
97C2×C4×C3⋊D3—32no—
98C2×C4×C3⋊D3C2×D311yescharacteristic
99C2×#634—31no—
100C2×#634—34no—
101C2×#634—32no—
102C2×#634—64no—
103C2×C3×#67—38no—
104C2×C3×#67C2×D318yes—
105C22×C3×Dic3C2×D318yes—
106C3×#237—216no—
107C2×C3×C4×D3—34no—
108C2×C3×C4×D3C2×D314yes—
109C2×C3×Dic6C2×D314yes—
110C2×#627—38no—
111C2×#625—34no—
112C2×D3×Dic3—38no—
113D3×Dic6—616no—
114#3757—32no—
115#3757—32no—
116#3757—38no—
117#3757—38no—
118#3757—32no—
119#3757—31no—
120#3757—68no—
121#3757—68no—
122#3757C2×D312yes—
123#3752—31no—
124C4⋊(C3⋊Dic3)—32no—
125C4⋊(C3⋊Dic3)C2×D311yescharacteristic
126#3750—31no—
127#3750—32no—
128C4×C3⋊Dic3—31no—
129C4×C3⋊Dic3—34no—
130C4×C3⋊Dic3—32no—
131C4×C3⋊Dic3—64no—
132C3×#217—38no—
133C3×C4⋊Dic3—34no—
134#3724—34no—
135Dic3⋊Dic3—38no—
136Dic3⋊Dic3—38no—
137(C2×D3)⋊Dic3—38no—
138(C2×D3)⋊Dic3—34no—
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)—22no—
153C3⋊(C3⋊Dic3)—21no—
154C3⋊(C3⋊Dic3)—21no—
155C3×C3⋊Dic3—21no—
156C3×C3⋊Dic3—22no—
157C32×Dic3—21no—
158C32×Dic3C2×D414yes—
159C2×C3×#41—34no—
160C2×D3×Q8—94no—
161C2×#237C3⋊D311yescharacteristic, complemented
162C22×Dic6—91no—
163C3×#121—91no—
164Q8×Dic3—94no—
165#939—92no—
166#938—34no—
167#938—98no—
168#936—98no—
169#936—94no—
170#936—91no—
171#936—184no—
172D4×Dic3—92no—
173C4×#67—94no—
174C4×#67—98no—
175C4×#67—91no—
176#924—94no—
177#924—98no—
178#897—91no—
179#887—91no—
180#885—92no—
181#878—92no—
182C23×C32—28no—
183C23×C32—31no—
184C23×C32C22×D318yes—
185C23×C32C22×D312yes—
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3×D3—34no—
188C22×C3×D3C22×D314yes—
189C32×Q8—34no—
190C32×D4—22no—
191C32×D4—28no—
192C32×D4—48no—
193C32×D4—64no—
194C2×C32×C4—24no—
195C2×C32×C4—32no—
196C2×C32×C4—31no—
197C2×C32×C4—64no—
198C2×C32×C4—62no—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D311yescharacteristic
201#638—24no—
202#638—62no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—34no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—68no—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D311yescharacteristic
214C4×C3⋊D3—22no—
215C4×C3⋊D3—64no—
216#634—64no—
217#634—62no—
218#634—68no—
219#634—128no—
220C3×#67—216no—
221C3×#67—616no—
222C2×C3×Dic3—216no—
223C2×C3×Dic3—38no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3C22×D314yes—
226C2×C3×Dic3C22×D318yes—
227C3×C4×D3—28no—
228C3×C4×D3—68no—
229C3×Dic6#67116yescomplemented
230#627—616no—
231#625—68no—
232D3×Dic3—616no—
233#548—271noSylow
234C2×C33—42no—
235C2×C33C22×D411yescharacteristic
236C2×C33C2×#4111yescharacteristic
237C2×C332−1+411yescharacteristic
238C3×C3⋊D3—22no—
239C32×D3—41no—
240C3×#41—616no—
241C2×C3×Q8—34no—
242C2×C3×D4—24no—
243C2×C3×D4—38no—
244C2×C3×D4—92no—
245C2×C3×D4D3214yes—
246C2×C3×D4C2×C3⋊D311yescharacteristic
247C22×C3×C4—38no—
248C22×C3×C4—34no—
249C22×C3×C4—91no—
250C2×#67—38no—
251C2×#67—94no—
252C2×#67—91no—
253C2×#67C2×C3⋊D312yes—
254C22×Dic3—92no—
255C22×Dic3C2×C3⋊D312yes—
256D3×Q8—1816no—
257#237—24no—
258C2×C4×D3—34no—
259C2×C4×D3—98no—
260C2×C4×D3C2×C3⋊D311yescharacteristic
261C2×Dic6—98no—
262C2×Dic6—94no—
263C2×Dic6—91no—
264C2×Dic6—92no—
265C2×Dic6—184no—
266C2×Dic6—184no—
267C2×Dic6C2×C3⋊D311yescharacteristic
268C3×C4⋊C4—91no—
269C3×C22⋊C4—92no—
270#217—38no—
271#217—98no—
272#217—94no—
273#217—98no—
274#217—98no—
275#217—92no—
276#217—92no—
277#217—91no—
278#217—188no—
279#217—188no—
280(C2×D3)⋊C4—94no—
281(C2×D3)⋊C4—91no—
282(C2×D3)⋊C4—92no—
283C4⋊Dic3—34no—
284C4⋊Dic3—98no—
285C4⋊Dic3—92no—
286Dic3⋊C4—98no—
287Dic3⋊C4—94no—
288Dic3⋊C4—91no—
289Dic3⋊C4—92no—
290C4×Dic3—98no—
291C4×Dic3—94no—
292C4×Dic3—91no—
293C4×Dic3—92no—
294C4×Dic3—184no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—24no—
299C22×C32—28no—
300C22×C32—28no—
301C22×C32—28no—
302C22×C32—48no—
303C22×C32—48no—
304C22×C32—48no—
305C22×C32—61no—
306C22×C32—61no—
307C22×C32—61no—
308C22×C32C23×D314yes—
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—21no—
311C2×C3⋊D3—21no—
312C2×C3⋊D3#23712yescomplemented
313C2×C3×D3—24no—
314C2×C3×D3—24no—
315C2×C3×D3—24no—
316C2×C3×D3—38no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C32×C4—28no—
320C32×C4—34no—
321C32×C4—61no—
322C32×C4D3×D412yes—
323C32×C4C2×#6718yes—
324C3⋊Dic3—21no—
325C3⋊Dic3—22no—
326C3⋊Dic3—62no—
327C3⋊Dic3—61no—
328C3⋊Dic3—61no—
329C3⋊Dic3—64no—
330C3⋊Dic3—64no—
331C3⋊Dic3—68no—
332C3⋊Dic3—128no—
333C3⋊Dic3—124no—
334C3⋊Dic3—124no—
335C3×Dic3—24no—
336C3×Dic3—68no—
337C3×Dic3—64no—
338C3×Dic3C2×#67116yes—
339C2×#41—31no—
340C22×Q8—271no—
341#124—272no—
342#124—271no—
343C22⋊Q8—272no—
344C22⋊Q8—271no—
345#121—91no—
346#121—272no—
347C4×Q8—271no—
348C4×D4—271no—
349C4×D4—272no—
350C33#54811yescharacteristic, complemented, Sylow
351C23×C3—28no—
352C23×C3—34no—
353C23×C3—91no—
354C23×C3C2×D3218yes—
355C23×C3C22×C3⋊D312yes—
356C22×D3—34no—
357C22×D3C22×C3⋊D311yescharacteristic
358C3×Q8—316no—
359C3×D4—28no—
360C3×D4—22no—
361C3×D4—48no—
362C3×D4—616no—
363C3×D4—184no—
364C2×C3×C4—24no—
365C2×C3×C4—38no—
366C2×C3×C4—34no—
367C2×C3×C4—616no—
368C2×C3×C4—68no—
369C2×C3×C4—92no—
370C2×C3×C4—91no—
371C2×C3×C4—182no—
372C2×C3×C4C2×D3214yes—
373C2×C3×C4C22×C3⋊D311yescharacteristic
374#67—24no—
375#67—616no—
376#67—182no—
377#67—188no—
378C2×Dic3—24no—
379C2×Dic3—38no—
380C2×Dic3—34no—
381C2×Dic3—94no—
382C2×Dic3—94no—
383C2×Dic3—98no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—92no—
387C2×Dic3—184no—
388C2×Dic3—184no—
389C2×Dic3—188no—
390C2×Dic3—184no—
391C2×Dic3C22×C3⋊D311yescharacteristic
392C2×Dic3C22×C3⋊D312yes—
393C4×D3—22no—
394C4×D3—68no—
395C4×D3—1816no—
396Dic6—1816no—
397Dic6—188no—
398Dic6—182no—
399Dic6—184no—
400Dic6—368no—
401Dic6#63814yescomplemented
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—42no—
406C2×C32—48no—
407C2×C32—88no—
408C2×C32—121no—
409C2×C32C2×D3×D411yescharacteristic
410C2×C32C2×#23711yescharacteristic
411C2×C32D3×#4114yes—
412C2×C32#101014yes—
413C2×C32#101011yescharacteristic
414C2×C32C22×#6714yes—
415C3⋊D3—22no—
416C3×D3—44no—
417C3×D3—68no—
418#41—64no—
419C2×Q8—31no—
420C2×Q8—272no—
421C2×Q8—271no—
422C2×Q8—544no—
423C2×D4—32no—
424C2×D4—92no—
425C2×D4—271no—
426C2×D4D3×C3⋊D311yescharacteristic
427C22×C4—32no—
428C22×C4—31no—
429C22×C4—91no—
430C22×C4—272no—
431C4⋊C4—91no—
432C4⋊C4—271no—
433C4⋊C4—272no—
434C4⋊C4—272no—
435C22⋊C4—92no—
436C22⋊C4—271no—
437C22⋊C4—271no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C42—272no—
442C42—271no—
443C22×C3—28no—
444C22×C3—28no—
445C22×C3—28no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—24no—
450C22×C3—48no—
451C22×C3—48no—
452C22×C3—48no—
453C22×C3—64no—
454C22×C3—64no—
455C22×C3—64no—
456C22×C3—92no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C22×D3214yes—
460C22×C3C23×C3⋊D311yescharacteristic
461C2×D3—21no—
462C2×D3—21no—
463C2×D3—21no—
464C2×D3—38no—
465C2×D3—64no—
466C2×D3—64no—
467C3×C4—28no—
468C3×C4—316no—
469C3×C4—64no—
470C3×C4—182no—
471C3×C4—181no—
472C3×C4D3×#6718yes—
473C3×C4C2×#63812yes—
474Dic3—21no—
475Dic3—68no—
476Dic3—64no—
477Dic3—184no—
478Dic3—184no—
479Dic3—188no—
480Dic3—181no—
481Dic3—181no—
482Dic3—182no—
483Dic3—368no—
484Dic3—364no—
485Dic3—364no—
486Dic3C2×#63814yes—
487C32—24no—
488C32#559811yescharacteristic, complemented
489C32#583114yescomplemented
490C23—31no—
491C23—91no—
492C23C2×D3×C3⋊D312yes—
493Q8—34no—
494Q8—544no—
495Q8—542no—
496D4—22no—
497D4—64no—
498D4—184no—
499D4—542no—
500C2×C4—32no—
501C2×C4—31no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—92no—
505C2×C4—91no—
506C2×C4—182no—
507C2×C4—271no—
508C2×C4—271no—
509C2×C4—272no—
510C2×C4—544no—
511C2×C4C2×D3×C3⋊D311yescharacteristic
512C2×C3—24no—
513C2×C3—24no—
514C2×C3—24no—
515C2×C3—48no—
516C2×C3—42no—
517C2×C3—88no—
518C2×C3—124no—
519C2×C3—182no—
520C2×C3#6430914yes—
521C2×C3D3×#23714yes—
522C2×C3C2×D3×#6714yes—
523C2×C3#41×C3⋊D311yescharacteristic
524C2×C3#6436711yescharacteristic
525C2×C3C22×#63811yescharacteristic
526D3—41no—
527D3—68no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—61no—
532C22—61no—
533C22—61no—
534C22—92no—
535C22—181no—
536C22—181no—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—34no—
539C4—61no—
540C4—182no—
541C4—181no—
542C4—541no—
543C4—541no—
544C4—542no—
545C4D3×#63812yes—
546C3—24no—
547C3#1059515714yescomplemented
548C3#1059630211yescharacteristic, complemented
549C2—42no—
550C2—121no—
551C2—182no—
552C2#1172458211yescharacteristic
553C2C3⋊D3×#23711yescharacteristic
554C2C2×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×C3⋊D3×Dic6C2semidirect
3. #11723977C2semidirect
4. C3×#64147C2semidirect
5. #11723640C2semidirect
6. #11723631C2semidirect
7. Dic3×#638C2semidirect
8. #11723600C2semidirect
9. C4×#6290C2semidirect
10. #11723426C2semidirect
11. #11723397C2semidirect
41. C2×C3×#237D3semidirect
45. #64147D3semidirect
80. C32×Dic6D4semidirect
147. C2×C3×C3⋊D3#41semidirect
161. C2×#237C3⋊D3semidirect
229. C3×Dic6#67semidirect
312. C2×C3⋊D3#237semidirect
350. C33#548semidirect
401. Dic6#638semidirect
488. C32#5598semidirect
489. C32#5831semidirect
547. C3#10595157semidirect
548. 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×C3⋊D3×Dic62
#117239772
C3×#641472
#117236402
#117236312
Dic3×#6382
#117236002
C4×#62902
#117234262
#117233972
#105963023
#105951573