Finite Group Databasebeta

← #13177421

Subgroups — #13177421

Order 1728.

2396 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C32×#936C211yesmaximal, characteristic, complemented
5#11723642C212yesmaximal, complemented
6#11723626C212yesmaximal, complemented
7#11723624C212yesmaximal, complemented
8#11723592C212yesmaximal, complemented
9C4×#6292C211yesmaximal, characteristic, complemented
10#11723456C211yesmaximal, characteristic, complemented
11#11723399C211yesmaximal, characteristic, complemented
12#10596148—31nomaximal
13#10595161—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C22×C3⋊(C3⋊Dic3)C2212yes—
17#89078—24no—
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C2×C3×#638C2212yes—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C32×Dic6C2211yescharacteristic
23C2×#6293C2212yes—
24C2×#6292C2211yescharacteristic
25C2×Dic3×C3⋊D3C2212yes—
26C3⋊D3×Dic6—24no—
27C3×#3757C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C32×#217C2212yes—
30C32×#217C2212yes—
31C32×C4×Dic3C2211yescharacteristic
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2211yescharacteristic
35#88804C2211yescharacteristic
36#88804C2212yes—
37#88803C2211yescharacteristic
38Dic3×C3⋊Dic3C2212yes—
39C2×Q8×C3⋊D3—31no—
40C2×#3830—31no—
41C2×#3830—34no—
42C2×#3830—64no—
43C2×D3×Dic6—34no—
44C32×#124—31no—
45#64155—31no—
46#64147D311yescharacteristic, complemented
47#64094—32no—
48#64093—32no—
49#64090—32no—
50#64087—32no—
51C4×C4⋊(C3⋊D3)—31no—
52C4×#634—31no—
53C3×#938—34no—
54C3×#936D314yescomplemented
55#63982—38no—
56#67⋊Dic3—38no—
57#63971—38no—
58#63953—38no—
59C4×#626—34no—
60#63871—34no—
61#63844—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
65C33×D4—22no—
66C2×C33×C4C2311yescharacteristic
67#6312—24no—
68C2×C3⋊(C3⋊Dic3)—24no—
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C2×C3⋊(C3⋊Dic3)C2312yes—
71C4×C3⋊(C3⋊D3)—22no—
72#6308D414yescomplemented
73C3×#638—24no—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4×C3⋊D3—22no—
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2311yescharacteristic
80C32×Dic6—22no—
81#6293—24no—
82#6292—22no—
83Dic3×C3⋊D3—24no—
84#5677—94no—
85#5598—91no—
86C2×C32×Q8—31no—
87C2×C32×D4—24no—
88C2×C32×D4C2×D314yes—
89C2×C32×D4C2×D311yescharacteristic
90C2×#638—32no—
91C2×#638—38no—
92C2×#638—68no—
93C2×#638C2×D312yes—
94C22×C3⋊Dic3—32no—
95C22×C3⋊Dic3—38no—
96C22×C3⋊Dic3—68no—
97Q8×C3⋊D3—64no—
98#3830—64no—
99#3830—616no—
100#3830—632no—
101C2×C4⋊(C3⋊D3)—31no—
102C2×C4×C3⋊D3—32no—
103C2×C4×C3⋊D3—31no—
104C2×C4×C3⋊D3—34no—
105C2×C4×C3⋊D3—64no—
106C2×C4×C3⋊D3C2×D311yescharacteristic
107C2×#634—31no—
108C2×#634—34no—
109C2×#634—32no—
110C2×#634—64no—
111C2×C3×#67—38no—
112C2×C3×C4×D3—34no—
113C2×C3×Dic6C2×D314yes—
114C2×#627—38no—
115C2×#626—34no—
116C2×D3×Dic3—38no—
117D3×Dic6—616no—
118C32×C22⋊C4—32no—
119C32×C22⋊C4—32no—
120C32×C42—31no—
121#3757C2×D312yes—
122#3752—31no—
123#3752—31no—
124#3752—32no—
125C4⋊(C3⋊Dic3)—31no—
126C4⋊(C3⋊Dic3)C2×D311yescharacteristic
127#3750—32no—
128#3750—32no—
129C4×C3⋊Dic3—32no—
130C3×#217—38no—
131C3×#217C2×D318yes—
132C3×#217C2×D318yes—
133C3×C4⋊Dic3—34no—
134C3×C4×Dic3C2×D314yes—
135#3724—38no—
136Dic3⋊Dic3—38no—
137Dic3⋊Dic3—34no—
138#3722—38no—
139#3722—34no—
140(C2×D3)⋊Dic3—34no—
141Dic32—38no—
142C22×C33—22no—
143C22×C33—22no—
144C22×C33—22no—
145C22×C33C2411yescharacteristic
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)—21no—
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3#4112yescomplemented
152C33×C4C2×D412yes—
153C3⋊(C3⋊Dic3)—21no—
154C3⋊(C3⋊Dic3)C2×D414yes—
155C3×C3⋊Dic3—21no—
156C3×C3⋊Dic3—22no—
157C32×Dic3—22no—
158C32×Dic3—21no—
159C32×Dic3—21no—
160C2×D3×Q8—94no—
161C2×#237—94no—
162C2×#237—91no—
163C2×#237—184no—
164C22×Dic6—91no—
165C3×#124—34no—
166C3×#121—91no—
167#946—94no—
168#939—92no—
169#938—34no—
170#936C3⋊D311yescharacteristic, complemented
171D4×Dic3—92no—
172C4×#67—91no—
173#897—91no—
174#887—91no—
175#885—98no—
176#885—92no—
177#884—98no—
178#67⋊C4—98no—
179#878—98no—
180#878—92no—
181C4×D12—94no—
182C4×Dic6—94no—
183C23×C32—28no—
184C23×C32C22×D318yes—
185C23×C32C22×D312yes—
186C22×C3⋊D3—31no—
187C22×C3⋊D3—34no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3×D3—34no—
191C32×Q8—62no—
192C32×D4—216no—
193C32×D4—28no—
194C32×D4—22no—
195C2×C32×C4—24no—
196C2×C32×C4—31no—
197C2×C32×C4—31no—
198C2×C32×C4—32no—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D311yescharacteristic
201#638—24no—
202#638—64no—
203#638—616no—
204#638—632no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—31no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—616no—
213C2×C3⋊Dic3—632no—
214C2×C3⋊Dic3C22×D312yes—
215C2×C3⋊Dic3C22×D311yescharacteristic
216C4⋊(C3⋊D3)—62no—
217C4×C3⋊D3—22no—
218C4×C3⋊D3—64no—
219C4×C3⋊D3—616no—
220C4×C3⋊D3—68no—
221C4×C3⋊D3—62no—
222#634—34no—
223#634—316no—
224#634—616no—
225#634—64no—
226C3×#67—616no—
227C2×C3×Dic3—38no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3C22×D314yes—
230C2×C3×Dic3C22×D318yes—
231C2×C3×Dic3C22×D314yes—
232C3×C4×D3—68no—
233C3×Dic6—28no—
234#627—616no—
235#626—68no—
236D3×Dic3—616no—
237#548—271noSylow
238C2×C33—42no—
239C2×C33C22×D411yescharacteristic
240C2×C33C2×#4111yescharacteristic
241C2×C332−1+411yescharacteristic
242C3⋊(C3⋊D3)—41no—
243C3×C3⋊D3—22no—
244C2×C3×Q8—34no—
245C2×C3×D4—24no—
246C2×C3×D4—92no—
247C2×C3×D4D3214yes—
248C2×C3×D4C2×C3⋊D311yescharacteristic
249C22×C3×C4—91no—
250C2×#67—38no—
251C2×#67—98no—
252C2×#67—92no—
253C2×#67—91no—
254C2×#67—188no—
255C22×Dic3—98no—
256C22×Dic3—92no—
257C22×Dic3—92no—
258C22×Dic3—188no—
259D3×Q8—1816no—
260#237—1816no—
261#237—184no—
262#237—1832no—
263C2×D12—94no—
264C2×C4×D3—34no—
265C2×C4×D3—98no—
266C2×C4×D3—94no—
267C2×C4×D3—91no—
268C2×C4×D3—184no—
269C2×Dic6—94no—
270C2×Dic6—91no—
271C2×Dic6—98no—
272C2×Dic6—92no—
273C2×Dic6—184no—
274C2×Dic6—184no—
275C2×Dic6C2×C3⋊D311yescharacteristic
276C3×C4⋊C4—91no—
277C3×C22⋊C4—38no—
278C3×C22⋊C4—38no—
279C3×C22⋊C4—92no—
280C3×C42—34no—
281#217—38no—
282#217—91no—
283#217C2×C3⋊D312yes—
284#217C2×C3⋊D312yes—
285(C2×D3)⋊C4—94no—
286(C2×D3)⋊C4—98no—
287(C2×D3)⋊C4—94no—
288(C2×D3)⋊C4—91no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—34no—
291C4⋊Dic3—94no—
292C4⋊Dic3—92no—
293Dic3⋊C4—98no—
294Dic3⋊C4—98no—
295Dic3⋊C4—91no—
296Dic3⋊C4—92no—
297C4×Dic3—98no—
298C4×Dic3—92no—
299C4×Dic3C2×C3⋊D311yescharacteristic
300C22×C32—216no—
301C22×C32—28no—
302C22×C32—22no—
303C22×C32—22no—
304C22×C32—22no—
305C22×C32—24no—
306C22×C32—28no—
307C22×C32—28no—
308C22×C32—48no—
309C22×C32—48no—
310C22×C32C23×D314yes—
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—21no—
313C2×C3⋊D3—21no—
314C2×C3⋊D3—68no—
315C2×C3⋊D3—64no—
316C2×C3⋊D3—61no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—64no—
320C2×C3⋊D3—64no—
321C2×C3⋊D3—124no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3#23712yescomplemented
324C2×C3×D3—38no—
325C2×C3×D3—64no—
326C2×C3×D3—64no—
327C32×C4—28no—
328C32×C4—62no—
329C32×C4—61no—
330C32×C4—61no—
331C32×C4D3×D418yes—
332C32×C4D3×D412yes—
333C3⋊Dic3—21no—
334C3⋊Dic3—22no—
335C3⋊Dic3—34no—
336C3⋊Dic3—316no—
337C3⋊Dic3—616no—
338C3⋊Dic3—68no—
339C3⋊Dic3—64no—
340C3⋊Dic3—61no—
341C3×Dic3—28no—
342C3×Dic3—24no—
343C3×Dic3—24no—
344C3×Dic3—64no—
345C3×Dic3—68no—
346C2×#41—271no—
347C22×Q8—271no—
348#124—31no—
349#124—272no—
350C22⋊Q8—272no—
351C22⋊Q8—271no—
352#121—91no—
353#121—272no—
354C4×Q8—271no—
355C4×D4—271no—
356C4×D4—272no—
357C33#54811yescharacteristic, complemented, Sylow
358C23×C3—28no—
359C23×C3—91no—
360C23×C3C2×D3218yes—
361C23×C3C22×C3⋊D312yes—
362C22×D3—34no—
363C22×D3—94no—
364C22×D3—91no—
365C22×D3—184no—
366C3×Q8—68no—
367C3×D4—216no—
368C3×D4—28no—
369C3×D4—22no—
370C3×D4—184no—
371C2×C3×C4—24no—
372C2×C3×C4—34no—
373C2×C3×C4—34no—
374C2×C3×C4—38no—
375C2×C3×C4—92no—
376C2×C3×C4—91no—
377C2×C3×C4—182no—
378C2×C3×C4C2×D3214yes—
379C2×C3×C4C22×C3⋊D311yescharacteristic
380#67—616no—
381#67—182no—
382#67—1816no—
383#67—184no—
384#67—1832no—
385C2×Dic3—38no—
386C2×Dic3—34no—
387C2×Dic3—98no—
388C2×Dic3—92no—
389C2×Dic3—94no—
390C2×Dic3—91no—
391C2×Dic3—188no—
392C2×Dic3—184no—
393C2×Dic3—184no—
394C2×Dic3—1816no—
395C2×Dic3—184no—
396C2×Dic3—1832no—
397C2×Dic3C22×C3⋊D311yescharacteristic
398C2×Dic3C22×C3⋊D311yescharacteristic
399C2×Dic3C22×C3⋊D312yes—
400D12—188no—
401C4×D3—68no—
402C4×D3—1816no—
403C4×D3—1816no—
404C4×D3—188no—
405C4×D3—182no—
406Dic6—22no—
407Dic6—916no—
408Dic6—94no—
409Dic6—1816no—
410Dic6—184no—
411Dic6—1816no—
412C2×C32—24no—
413C2×C32—24no—
414C2×C32—24no—
415C2×C32—416no—
416C2×C32—48no—
417C2×C32—42no—
418C2×C32C2×D3×D414yes—
419C2×C32C2×D3×D411yescharacteristic
420C2×C32C2×#23711yescharacteristic
421C2×C32D3×#4114yes—
422C2×C32#101014yes—
423C2×C32#101011yescharacteristic
424C3⋊D3—22no—
425C3⋊D3—128no—
426C3⋊D3—124no—
427C3⋊D3—121no—
428C3×D3—68no—
429#41—544no—
430C2×Q8—31no—
431C2×Q8—272no—
432C2×Q8—271no—
433C2×Q8—544no—
434C2×D4—92no—
435C2×D4—271no—
436C2×D4—272no—
437C2×D4D3×C3⋊D311yescharacteristic
438C22×C4—91no—
439C22×C4—272no—
440C22×C4—272no—
441C22×C4—271no—
442C4⋊C4—91no—
443C4⋊C4—272no—
444C4⋊C4—271no—
445C4⋊C4—272no—
446C22⋊C4—32no—
447C22⋊C4—32no—
448C22⋊C4—92no—
449C22⋊C4—271no—
450C22⋊C4—272no—
451C22⋊C4—271no—
452C42—31no—
453C42—272no—
454C22×C3—216no—
455C22×C3—28no—
456C22×C3—28no—
457C22×C3—28no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—24no—
462C22×C3—48no—
463C22×C3—48no—
464C22×C3—92no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C22×D3214yes—
468C22×C3C23×C3⋊D311yescharacteristic
469C2×D3—38no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—188no—
473C2×D3—184no—
474C2×D3—184no—
475C2×D3—184no—
476C2×D3—181no—
477C2×D3—181no—
478C2×D3—181no—
479C2×D3—364no—
480C2×D3—364no—
481C3×C4—28no—
482C3×C4—64no—
483C3×C4—64no—
484C3×C4—68no—
485C3×C4—181no—
486C3×C4—182no—
487C3×C4#381118yes—
488C3×C4D4×C3⋊D312yes—
489Dic3—22no—
490Dic3—21no—
491Dic3—21no—
492Dic3—68no—
493Dic3—64no—
494Dic3—916no—
495Dic3—94no—
496Dic3—1816no—
497Dic3—188no—
498Dic3—184no—
499Dic3—181no—
500C32—24no—
501C32#559811yescharacteristic, complemented
502C32#567714yescomplemented
503C23—91no—
504C23—271no—
505C23C2×D3×C3⋊D312yes—
506Q8—62no—
507Q8—274no—
508Q8—544no—
509D4—22no—
510D4—184no—
511D4—542no—
512D4—544no—
513C2×C4—31no—
514C2×C4—31no—
515C2×C4—32no—
516C2×C4—92no—
517C2×C4—91no—
518C2×C4—182no—
519C2×C4—272no—
520C2×C4—271no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4—542no—
524C2×C4C2×D3×C3⋊D311yescharacteristic
525C2×C3—24no—
526C2×C3—24no—
527C2×C3—24no—
528C2×C3—416no—
529C2×C3—42no—
530C2×C3—48no—
531C2×C3—182no—
532C2×C3#6430914yes—
533C2×C3D3×#23714yes—
534C2×C3C2×#381114yes—
535C2×C3C2×D4×C3⋊D311yescharacteristic
536C2×C3#41×C3⋊D311yescharacteristic
537C2×C3#6436711yescharacteristic
538D3—68no—
539D3—368no—
540D3—361no—
541D3—364no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—92no—
546C22—181no—
547C22—181no—
548C22—541no—
549C22—541no—
550C22—541no—
551C22C22×D3×C3⋊D311yescharacteristic
552C4—61no—
553C4—61no—
554C4—62no—
555C4—182no—
556C4—181no—
557C4—274no—
558C4—541no—
559C4#8903912yes—
560C3—24no—
561C3#1059516114yescomplemented
562C3#1059614811yescharacteristic, complemented
563C2—42no—
564C2—182no—
565C2—1081no—
566C2#1172458211yescharacteristic
567C2C3⋊D3×#23711yescharacteristic
568C2C2×#8903911yescharacteristic
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×#89078C2semidirect
2. C2×C3⋊D3×Dic6C2semidirect
3. C3×#64147C2semidirect
4. C32×#936C2semidirect
5. #11723642C2semidirect
6. #11723626C2semidirect
7. #11723624C2semidirect
8. #11723592C2semidirect
9. C4×#6292C2semidirect
10. #11723456C2semidirect
11. #11723399C2semidirect
46. #64147D3semidirect
54. C3×#936D3semidirect
72. #6308D4semidirect
151. C2×C3×C3⋊D3#41semidirect
170. #936C3⋊D3semidirect
323. C2×C3⋊D3#237semidirect
357. C33#548semidirect
501. C32#5598semidirect
502. C32#5677semidirect
561. C3#10595161semidirect
562. C3#10596148semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×C3⋊D3×Dic62
C3×#641472
C32×#9362
#117236422
#117236262
#117236242
#117235922
C4×#62922
#117234562
#117233992
#105961483
#105951613