Finite Group Databasebeta

← #13177368

Subgroups — #13177368

Order 1728.

2292 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89028C212yesmaximal, complemented
2#11723981C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723622C212yesmaximal, complemented
6#11723616C212yesmaximal, complemented
7#11723592C212yesmaximal, complemented
8#11723536C211yesmaximal, characteristic, complemented
9#11723516C211yesmaximal, characteristic, complemented
10#11723494C211yesmaximal, characteristic, complemented
11C4×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596067—31nomaximal
13#10595131—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C22×C3×C3⋊Dic3C2212yes—
19C2×C32×#67C2212yes—
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6293C2212yes—
22C2×#6291C2212yes—
23C2×#6289C2212yes—
24#89028—28no—
25C2×D3×C3⋊Dic3C2211yescharacteristic
26C2×D3×C3⋊Dic3C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3757C2212yes—
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×C4×C3⋊Dic3C2211yescharacteristic
31C32×#217C2212yes—
32C32×C4⋊Dic3C2211yescharacteristic
33#88806C2212yes—
34#88804C2212yes—
35#88804C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×#3830—32no—
39C2×#3804—38no—
40C32×#121—31no—
41#64149—31no—
42#64149—34no—
43#64149—64no—
44#64148—32no—
45D4×C3⋊Dic3D311yescharacteristic, complemented
46#64139—31no—
47C2×C4×C3⋊Dic3—31no—
48#64102—31no—
49#64097—31no—
50#64094—32no—
51#64090—32no—
52C3×#938D314yescomplemented
53C3×D4×Dic3—34no—
54#63971—38no—
55#63953—38no—
56#63952—38no—
57#63919—34no—
58#63909—34no—
59#63894—34no—
60C4×D3×Dic3—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)C2312yes—
63C22×C32×D3C2311yescharacteristic
64C33×D4—22no—
65C2×C33×C4C2311yescharacteristic
66#6312—24no—
67#6312—24no—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C4⋊(C3⋊(C3⋊D3))—22no—
71C2×C3×C3⋊Dic3—24no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C32×#67—24no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78C32×C4×D3—22no—
79#6293—24no—
80#6291—24no—
81#6289—24no—
82D3×C3⋊Dic3—22no—
83D3×C3⋊Dic3D414yescomplemented
84#5596—94no—
85#5548—91no—
86C2×C32×D4—24no—
87C2×C32×D4—32no—
88C2×C32×D4C2×D314yes—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4—31no—
91C2×#638—32no—
92C2×#638—32no—
93C2×#638—38no—
94C2×#638—38no—
95C2×#638—32no—
96C2×#638—68no—
97C2×#638—68no—
98C22×C3⋊Dic3—31no—
99C22×C3⋊Dic3—31no—
100C22×C3⋊Dic3C2×D312yes—
101#3830—68no—
102C2×C4⋊(C3⋊D3)—31no—
103C2×C4⋊(C3⋊D3)—34no—
104C2×C4⋊(C3⋊D3)—64no—
105C2×C4×C3⋊D3—32no—
106C2×#634—32no—
107C2×C3×#67C2×D318yes—
108C22×C3×Dic3—38no—
109C2×C3×C4×D3C2×D314yes—
110C2×#627—38no—
111C2×#626—38no—
112C2×#624—38no—
113#3804—632no—
114C2×D3×Dic3—34no—
115C2×D3×Dic3—34no—
116C32×C4⋊C4—31no—
117C32×C22⋊C4—32no—
118#3757C2×D312yes—
119#3752—32no—
120#3752—32no—
121C4⋊(C3⋊Dic3)C2×D311yescharacteristic
122#3750—32no—
123C4×C3⋊Dic3—31no—
124C4×C3⋊Dic3—34no—
125C4×C3⋊Dic3—31no—
126C4×C3⋊Dic3—31no—
127C4×C3⋊Dic3—64no—
128C4×C3⋊Dic3C2×D311yescharacteristic
129C3×#217—38no—
130C3×#217C2×D318yes—
131C3×C4⋊Dic3—34no—
132C3×C4⋊Dic3C2×D314yes—
133C3×C4×Dic3—34no—
134#3724—38no—
135#3722—38no—
136#3722—38no—
137Dic32—34no—
138Dic32—34no—
139C22×C33—22no—
140C22×C33—22no—
141C22×C33—22no—
142C22×C33C2411yescharacteristic
143C2×C3⋊(C3⋊D3)—22no—
144C2×C3⋊(C3⋊D3)—22no—
145C2×C3⋊(C3⋊D3)—22no—
146C2×C32×D3—21no—
147C2×C32×D3—21no—
148C2×C32×D3C2×D412yes—
149C33×C4#4112yescomplemented
150C3⋊(C3⋊Dic3)C2×D412yes—
151C3⋊(C3⋊Dic3)#4112yescomplemented
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3C2×D412yes—
155C32×Dic3—22no—
156C32×Dic3—21no—
157C2×#237—98no—
158C2×#235—92no—
159C3×#121—34no—
160C3×C4×D4—91no—
161#940—94no—
162#940—91no—
163#940—184no—
164#939—98no—
165#938C3⋊D311yescharacteristic, complemented
166D4×Dic3—34no—
167#930—94no—
168C4×#67—92no—
169C2×C4×Dic3—94no—
170#895—91no—
171#893—94no—
172#888—94no—
173#888—91no—
174#885—98no—
175#885—92no—
176#884—92no—
177#67⋊C4—98no—
178C4×D12—91no—
179C42×D3—91no—
180C23×C32—28no—
181C23×C32—31no—
182C23×C32C22×D318yes—
183C23×C32C22×D312yes—
184C22×C3⋊D3—32no—
185C22×C3⋊D3—38no—
186C22×C3⋊D3—68no—
187C22×C3×D3C22×D314yes—
188C32×D4—22no—
189C32×D4—28no—
190C32×D4—48no—
191C32×D4—64no—
192C2×C32×C4—24no—
193C2×C32×C4—31no—
194C2×C32×C4—32no—
195C2×C32×C4—62no—
196C2×C32×C4C22×D314yes—
197C2×C32×C4C22×D311yescharacteristic
198#638—64no—
199#638—64no—
200#638—64no—
201#638—616no—
202#638—616no—
203#638—1216no—
204#638—1216no—
205C2×C3⋊Dic3—24no—
206C2×C3⋊Dic3—31no—
207C2×C3⋊Dic3—31no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—34no—
210C2×C3⋊Dic3—34no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—64no—
213C2×C3⋊Dic3—62no—
214C2×C3⋊Dic3C22×D311yescharacteristic
215C2×C3⋊Dic3C22×D312yes—
216C2×C3⋊Dic3C22×D311yescharacteristic
217C4⋊(C3⋊D3)—62no—
218C4⋊(C3⋊D3)—68no—
219C4⋊(C3⋊D3)—128no—
220C4×C3⋊D3—64no—
221#634—64no—
222C3×#67—216no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3—38no—
226C2×C3×Dic3—616no—
227C2×C3×Dic3C22×D314yes—
228C2×C3×Dic3C22×D318yes—
229C3×C4×D3—28no—
230#627—616no—
231#626—616no—
232#624—616no—
233D3×Dic3—316no—
234D3×Dic3—68no—
235#532—271noSylow
236C2×C33—42no—
237C2×C33C22×D411yescharacteristic
238C2×C33C2×#4111yescharacteristic
239C2×C33C2×#4111yescharacteristic
240C3⋊(C3⋊D3)—42no—
241C32×D3—22no—
242C2×C3×D4—24no—
243C2×C3×D4—38no—
244C2×C3×D4D3214yes—
245C2×C3×D4C2×C3⋊D311yescharacteristic
246C22×C3×C4—34no—
247C22×C3×C4—92no—
248C2×#67—98no—
249C2×#67—98no—
250C2×#67—98no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—188no—
254C2×#67—188no—
255C2×#67C2×C3⋊D312yes—
256C22×Dic3—38no—
257C22×Dic3—94no—
258C22×Dic3—94no—
259#237—1832no—
260#235—188no—
261C2×D12—92no—
262C2×D12—94no—
263C2×D12—91no—
264C2×D12—184no—
265C2×C4×D3—91no—
266C2×C4×D3—98no—
267C2×C4×D3—91no—
268C2×C4×D3—92no—
269C2×C4×D3C2×C3⋊D311yescharacteristic
270C2×Dic6—98no—
271C2×Dic6—92no—
272C3×C4⋊C4—34no—
273C3×C4⋊C4—91no—
274C3×C22⋊C4—38no—
275C3×C22⋊C4—92no—
276C3×C42—91no—
277#217—38no—
278#217C2×C3⋊D312yes—
279(C2×D3)⋊C4—98no—
280(C2×D3)⋊C4—98no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283C4⋊Dic3—34no—
284C4⋊Dic3C2×C3⋊D311yescharacteristic
285Dic3⋊C4—98no—
286Dic3⋊C4—92no—
287C4×Dic3—34no—
288C4×Dic3—94no—
289C4×Dic3—94no—
290C4×Dic3—94no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—91no—
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—32no—
303C22×C32—48no—
304C22×C32—48no—
305C22×C32—48no—
306C22×C32—61no—
307C22×C32—61no—
308C22×C32C23×D314yes—
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—62no—
311C2×C3⋊D3—62no—
312C2×C3⋊D3—62no—
313C2×C3⋊D3—68no—
314C2×C3⋊D3—68no—
315C2×C3⋊D3—68no—
316C2×C3⋊D3—128no—
317C2×C3⋊D3—128no—
318C2×C3⋊D3—128no—
319C2×C3×D3—24no—
320C2×C3×D3—24no—
321C2×C3×D3D3×D418yes—
322C32×C4—28no—
323C32×C4—61no—
324C32×C4—62no—
325C32×C4#23512yescomplemented
326C32×C4#23718yescomplemented
327C3⋊Dic3—22no—
328C3⋊Dic3—21no—
329C3⋊Dic3—32no—
330C3⋊Dic3—32no—
331C3⋊Dic3—38no—
332C3⋊Dic3—38no—
333C3⋊Dic3—68no—
334C3⋊Dic3—68no—
335C3⋊Dic3D3×D412yes—
336C3×Dic3—28no—
337C3×Dic3—24no—
338C3×Dic3—38no—
339C3×Dic3—68no—
340C3×Dic3—64no—
341C2×#41—272no—
342C4⋊Q8—271no—
343#127—271no—
344#124—272no—
345#121—31no—
346#121—272no—
347#121—271no—
348C4×D4—91no—
349C4×D4—271no—
350C4×D4—272no—
351C2×C42—271no—
352C33#53211yescharacteristic, complemented, Sylow
353C23×C3—28no—
354C23×C3—34no—
355C23×C3C2×D3218yes—
356C23×C3C22×C3⋊D312yes—
357C22×D3—98no—
358C22×D3—92no—
359C22×D3—188no—
360C22×D3C22×C3⋊D311yescharacteristic
361C3×D4—28no—
362C3×D4—22no—
363C3×D4—48no—
364C3×D4—616no—
365C2×C3×C4—24no—
366C2×C3×C4—34no—
367C2×C3×C4—38no—
368C2×C3×C4—68no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—184no—
373C2×C3×C4C2×D3214yes—
374C2×C3×C4C22×C3⋊D311yescharacteristic
375#67—24no—
376#67—1816no—
377#67—184no—
378#67—184no—
379#67—1816no—
380#67—1816no—
381#67—3616no—
382#67—3616no—
383C2×Dic3—34no—
384C2×Dic3—34no—
385C2×Dic3—38no—
386C2×Dic3—616no—
387C2×Dic3—916no—
388C2×Dic3—94no—
389C2×Dic3—94no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—184no—
393C2×Dic3—184no—
394C2×Dic3—188no—
395C2×Dic3C22×C3⋊D311yescharacteristic
396C2×Dic3C22×C3⋊D312yes—
397D12—184no—
398D12—188no—
399D12—182no—
400D12—368no—
401C4×D3—22no—
402C4×D3—94no—
403C4×D3—184no—
404C4×D3—1816no—
405C4×D3—182no—
406Dic6—184no—
407Dic6—1816no—
408C2×C32—24no—
409C2×C32—24no—
410C2×C32—24no—
411C2×C32—42no—
412C2×C32—48no—
413C2×C32—62no—
414C2×C32—88no—
415C2×C32C2×#23511yescharacteristic
416C2×C32C2×D3×D414yes—
417C2×C32C2×D3×D411yescharacteristic
418C2×C32C2×#23714yes—
419C2×C32D3×#4114yes—
420C2×C32D3×#4111yescharacteristic
421C3⋊D3—122no—
422C3⋊D3—128no—
423C3⋊D3—248no—
424C3×D3—28no—
425#41—548no—
426C2×Q8—272no—
427C2×D4—32no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—271no—
432C2×D4D3×C3⋊D311yescharacteristic
433C22×C4—31no—
434C22×C4—92no—
435C22×C4—272no—
436C22×C4—271no—
437C22×C4—271no—
438C4⋊C4—31no—
439C4⋊C4—91no—
440C4⋊C4—272no—
441C22⋊C4—32no—
442C22⋊C4—92no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C42—91no—
446C42—271no—
447C42—271no—
448C42—271no—
449C22×C3—28no—
450C22×C3—28no—
451C22×C3—28no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—38no—
457C22×C3—48no—
458C22×C3—48no—
459C22×C3—48no—
460C22×C3—64no—
461C22×C3—64no—
462C22×C3C22×D3214yes—
463C22×C3C23×C3⋊D311yescharacteristic
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—188no—
467C2×D3—188no—
468C2×D3—188no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—368no—
473C2×D3—368no—
474C2×D3—368no—
475C2×D3D4×C3⋊D312yes—
476C3×C4—28no—
477C3×C4—68no—
478C3×C4—64no—
479C3×C4—92no—
480C3×C4—182no—
481C3×C4—181no—
482C3×C4#380418yescomplemented
483C3×C4#383012yescomplemented
484Dic3—22no—
485Dic3—21no—
486Dic3—38no—
487Dic3—68no—
488Dic3—64no—
489Dic3—98no—
490Dic3—98no—
491Dic3—92no—
492Dic3—92no—
493Dic3—188no—
494Dic3—188no—
495C32—24no—
496C32#554811yescharacteristic, complemented
497C32#559614yescomplemented
498C23—31no—
499C23—272no—
500C23C2×D3×C3⋊D312yes—
501Q8—544no—
502D4—22no—
503D4—64no—
504D4—544no—
505D4—544no—
506D4—544no—
507D4—542no—
508C2×C4—31no—
509C2×C4—32no—
510C2×C4—62no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—184no—
515C2×C4—274no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—544no—
519C2×C4—542no—
520C2×C4C2×D3×C3⋊D311yescharacteristic
521C2×C3—24no—
522C2×C3—24no—
523C2×C3—24no—
524C2×C3—48no—
525C2×C3—42no—
526C2×C3—68no—
527C2×C3—88no—
528C2×C3D32×D414yes—
529C2×C3#6431214yes—
530C2×C3C2×#380414yes—
531C2×C3C2×D4×C3⋊D311yescharacteristic
532C2×C3C2×#383011yescharacteristic
533C2×C3#41×C3⋊D311yescharacteristic
534D3—22no—
535D3—368no—
536D3—362no—
537D3—728no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—32no—
542C22—61no—
543C22—61no—
544C22—542no—
545C22—542no—
546C22—542no—
547C22C22×D3×C3⋊D311yescharacteristic
548C4—62no—
549C4—61no—
550C4—92no—
551C4—182no—
552C4—181no—
553C4—272no—
554C4—272no—
555C4#8902812yescomplemented
556C3—24no—
557C3#1059513114yescomplemented
558C3#1059606711yescharacteristic, complemented
559C2—42no—
560C2—62no—
561C2—1082no—
562C2D3×D4×C3⋊D311yescharacteristic
563C2#1172458611yescharacteristic
564C2C2×#8902811yescharacteristic
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×#89028C2semidirect
2. #11723981C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. C32×#938C2semidirect
5. #11723622C2semidirect
6. #11723616C2semidirect
7. #11723592C2semidirect
8. #11723536C2semidirect
9. #11723516C2semidirect
10. #11723494C2semidirect
11. C4×D3×C3⋊Dic3C2semidirect
45. D4×C3⋊Dic3D3semidirect
52. C3×#938D3semidirect
83. D3×C3⋊Dic3D4semidirect
149. C33×C4#41semidirect
151. C3⋊(C3⋊Dic3)#41semidirect
165. #938C3⋊D3semidirect
325. C32×C4#235semidirect
326. C32×C4#237semidirect
352. C33#532semidirect
482. C3×C4#3804semidirect
483. C3×C4#3830semidirect
496. C32#5548semidirect
497. C32#5596semidirect
555. C4#89028semidirect
557. C3#10595131semidirect
558. C3#10596067semidirect

Maximal subgroups

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

subgroupindex
C2×#890282
#117239812
C3×D4×C3⋊Dic32
C32×#9382
#117236222
#117236162
#117235922
#117235362
#117235162
#117234942
C4×D3×C3⋊Dic32
#105960673
#105951313