Finite Group Databasebeta

← #13177373

Subgroups — #13177373

Order 1728.

2236 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×#89029C212yesmaximal, complemented
2D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723626C212yesmaximal, complemented
6#11723615C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8#11723535C211yesmaximal, characteristic, complemented
9#11723515C211yesmaximal, characteristic, complemented
10#11723501C211yesmaximal, characteristic, complemented
11C4×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596067—31nomaximal
13#10595133—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C32×#67C2212yes—
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6293C2212yes—
22C2×#6290C2212yes—
23C2×Dic3×C3⋊D3C2212yes—
24#89029—28no—
25C2×D3×C3⋊Dic3C2211yescharacteristic
26C2×D3×C3⋊Dic3C2211yescharacteristic
27#88864C2212yes—
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30C3×C4×C3⋊Dic3C2211yescharacteristic
31C32×#217C2212yes—
32C32×C4⋊Dic3C2211yescharacteristic
33#88807C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×#3830—32no—
39C2×#3805—38no—
40C32×#121—31no—
41#64149D311yescharacteristic, complemented
42#64148—32no—
43D4×C3⋊Dic3—31no—
44D4×C3⋊Dic3—34no—
45D4×C3⋊Dic3—64no—
46#64139—31no—
47C2×C4×C3⋊Dic3—31no—
48#64102—31no—
49#64097—31no—
50#64094—32no—
51#64090—32no—
52C3×#940—34no—
53C3×#938D314yescomplemented
54#67⋊Dic3—38no—
55#63970—38no—
56#63959—38no—
57#63918—34no—
58#63908—34no—
59#63898—34no—
60C4×D3×Dic3—34no—
61C23×C33C2312yes—
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2311yescharacteristic
64C33×D4—22no—
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)—24no—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C3×#638—24no—
71C3×#638—24no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4⋊(C3⋊D3)—22no—
75C32×#67—24no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78C32×C4×D3—22no—
79#6293—24no—
80#6290—24no—
81Dic3×C3⋊D3—24no—
82D3×C3⋊Dic3—22no—
83D3×C3⋊Dic3D414yescomplemented
84#5716—91no—
85#5596—94no—
86C2×C32×D4—24no—
87C2×C32×D4—32no—
88C2×C32×D4C2×D314yes—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4—31no—
91C2×#638—32no—
92C2×#638C2×D312yes—
93C2×#638C2×D312yes—
94C22×C3⋊Dic3—31no—
95C22×C3⋊Dic3—32no—
96C22×C3⋊Dic3—38no—
97C22×C3⋊Dic3—31no—
98C22×C3⋊Dic3—68no—
99#3830—68no—
100C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
101C2×C4×C3⋊D3—32no—
102C2×#634—32no—
103C2×C3×#67—38no—
104C2×C3×#67—38no—
105C2×C3×#67C2×D318yes—
106C2×C3×D12—34no—
107C2×C3×C4×D3C2×D314yes—
108C2×#627—38no—
109C2×#625—38no—
110#3805—632no—
111C2×D3×Dic3—34no—
112C2×D3×Dic3—34no—
113C2×D3×Dic3—38no—
114C32×C4⋊C4—31no—
115C32×C22⋊C4—32no—
116#3757—32no—
117#3757—38no—
118#3757—68no—
119#3752—32no—
120#3752—32no—
121C4⋊(C3⋊Dic3)—31no—
122C4⋊(C3⋊Dic3)—34no—
123C4⋊(C3⋊Dic3)—64no—
124#3750—32no—
125C4×C3⋊Dic3—31no—
126C4×C3⋊Dic3—34no—
127C4×C3⋊Dic3—31no—
128C4×C3⋊Dic3—31no—
129C4×C3⋊Dic3—64no—
130C4×C3⋊Dic3C2×D311yescharacteristic
131C3×#217C2×D318yes—
132C3×C4⋊Dic3C2×D314yes—
133C3×C4×Dic3—34no—
134Dic3⋊Dic3—38no—
135(C2×D3)⋊Dic3—38no—
136(C2×D3)⋊Dic3—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)—22no—
151C3⋊(C3⋊Dic3)—21no—
152C3⋊(C3⋊Dic3)C2×D412yes—
153C3×C3⋊Dic3C2×D412yes—
154C3×C3⋊Dic3#4112yescomplemented
155C32×Dic3—22no—
156C32×Dic3—21no—
157C2×#237—98no—
158C2×#237—92no—
159C3×#127—91no—
160C3×#121—34no—
161#940—34no—
162#939—98no—
163#938—92no—
164#938—91no—
165#938C3⋊D311yescharacteristic, complemented
166#936—92no—
167D4×Dic3—94no—
168D4×Dic3—91no—
169D4×Dic3—92no—
170D4×Dic3—91no—
171D4×Dic3—184no—
172#930—94no—
173C2×C4×Dic3—94no—
174#893—94no—
175#888—94no—
176#885—98no—
177#67⋊C4—98no—
178C42×D3—91no—
179#869—91no—
180C23×C32—28no—
181C23×C32—31no—
182C23×C32C22×D318yes—
183C23×C32C22×D312yes—
184C22×C3⋊D3C22×D312yes—
185C22×C3×D3—38no—
186C22×C3×D3C22×D314yes—
187C32×D4—216no—
188C32×D4—28no—
189C32×D4—22no—
190C32×D4—64no—
191C2×C32×C4—24no—
192C2×C32×C4—31no—
193C2×C32×C4—32no—
194C2×C32×C4—62no—
195C2×C32×C4C22×D314yes—
196C2×C32×C4C22×D311yescharacteristic
197#638—24no—
198#638—24no—
199#638—64no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—32no—
204C2×C3⋊Dic3—34no—
205C2×C3⋊Dic3—34no—
206C2×C3⋊Dic3—38no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—62no—
212C2×C3⋊Dic3—616no—
213C2×C3⋊Dic3—1216no—
214C2×C3⋊Dic3C22×D311yescharacteristic
215C2×C3⋊Dic3C22×D311yescharacteristic
216C4⋊(C3⋊D3)—22no—
217C4×C3⋊D3—64no—
218#634—64no—
219C3×#67—216no—
220C3×#67—616no—
221C3×#67—616no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3C22×D314yes—
225C2×C3×Dic3C22×D318yes—
226C3×D12—68no—
227C3×C4×D3—28no—
228#627—616no—
229#625—616no—
230D3×Dic3—316no—
231D3×Dic3—616no—
232D3×Dic3—68no—
233#532—271noSylow
234C2×C33—42no—
235C2×C33C22×D411yescharacteristic
236C2×C33C2×#4111yescharacteristic
237C2×C33C2×#4111yescharacteristic
238C3×C3⋊D3—42no—
239C32×D3—22no—
240C2×C3×D4—24no—
241C2×C3×D4—38no—
242C2×C3×D4—92no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C2×C3×D4D3214yes—
246C2×C3×D4C2×C3⋊D311yescharacteristic
247C22×C3×C4—34no—
248C2×#67—38no—
249C2×#67—38no—
250C2×#67—98no—
251C2×#67—92no—
252C2×#67C2×C3⋊D312yes—
253C22×Dic3—94no—
254C22×Dic3—94no—
255C22×Dic3—98no—
256C22×Dic3—92no—
257C22×Dic3—92no—
258C22×Dic3—188no—
259#237—188no—
260#237—1832no—
261C2×D12—34no—
262C2×C4×D3—91no—
263C2×C4×D3—98no—
264C2×C4×D3—91no—
265C2×C4×D3C2×C3⋊D311yescharacteristic
266C2×Dic6—98no—
267C2×Dic6—92no—
268C3×C4⋊C4—34no—
269C3×C22⋊C4—38no—
270C3×C42—91no—
271#217—98no—
272#217—92no—
273#217—92no—
274#217—92no—
275#217—188no—
276#217C2×C3⋊D312yes—
277(C2×D3)⋊C4—98no—
278(C2×D3)⋊C4—98no—
279C4⋊Dic3—94no—
280C4⋊Dic3—91no—
281C4⋊Dic3—92no—
282C4⋊Dic3—184no—
283C4⋊Dic3C2×C3⋊D311yescharacteristic
284Dic3⋊C4—98no—
285C4×Dic3—34no—
286C4×Dic3—94no—
287C4×Dic3—94no—
288C4×Dic3—94no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—184no—
293C22×C32—216no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—28no—
297C22×C32—22no—
298C22×C32—24no—
299C22×C32—28no—
300C22×C32—28no—
301C22×C32—32no—
302C22×C32—48no—
303C22×C32—48no—
304C22×C32—61no—
305C22×C32—61no—
306C22×C32C23×D314yes—
307C22×C32C23×D311yescharacteristic
308C2×C3⋊D3—22no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—22no—
311C2×C3×D3—24no—
312C2×C3×D3—24no—
313C2×C3×D3—68no—
314C2×C3×D3—68no—
315C2×C3×D3—68no—
316C2×C3×D3D3×D418yes—
317C32×C4—28no—
318C32×C4—62no—
319C32×C4—61no—
320C32×C4#23718yescomplemented
321C32×C4#23712yescomplemented
322C3⋊Dic3—32no—
323C3⋊Dic3—38no—
324C3⋊Dic3—68no—
325C3⋊Dic3—62no—
326C3⋊Dic3—61no—
327C3⋊Dic3—68no—
328C3⋊Dic3—64no—
329C3⋊Dic3—124no—
330C3⋊Dic3—128no—
331C3⋊Dic3D3×D412yes—
332C3⋊Dic3#23712yescomplemented
333C3×Dic3—28no—
334C3×Dic3—24no—
335C3×Dic3—38no—
336C3×Dic3—38no—
337C2×#41—272no—
338C4⋊Q8—271no—
339#127—91no—
340#124—272no—
341#121—31no—
342#121—271no—
343#121—272no—
344C4×D4—272no—
345C4×D4—271no—
346C4×D4—271no—
347C2×C42—271no—
348C33#53211yescharacteristic, complemented, Sylow
349C23×C3—28no—
350C23×C3—34no—
351C23×C3—92no—
352C23×C3C2×D3218yes—
353C23×C3C22×C3⋊D312yes—
354C22×D3—38no—
355C22×D3C22×C3⋊D311yescharacteristic
356C3×D4—216no—
357C3×D4—28no—
358C3×D4—22no—
359C3×D4—616no—
360C3×D4—184no—
361C3×D4—184no—
362C3×D4—182no—
363C2×C3×C4—24no—
364C2×C3×C4—34no—
365C2×C3×C4—38no—
366C2×C3×C4—68no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4C2×D3214yes—
370C2×C3×C4C22×C3⋊D311yescharacteristic
371#67—24no—
372#67—616no—
373#67—616no—
374#67—1816no—
375#67—184no—
376C2×Dic3—34no—
377C2×Dic3—34no—
378C2×Dic3—916no—
379C2×Dic3—94no—
380C2×Dic3—91no—
381C2×Dic3—94no—
382C2×Dic3—98no—
383C2×Dic3—91no—
384C2×Dic3—92no—
385C2×Dic3—184no—
386C2×Dic3—184no—
387C2×Dic3—188no—
388C2×Dic3—184no—
389C2×Dic3—188no—
390C2×Dic3—184no—
391C2×Dic3—1816no—
392C2×Dic3—3616no—
393C2×Dic3C22×C3⋊D311yescharacteristic
394C2×Dic3C22×C3⋊D312yes—
395D12—68no—
396C4×D3—22no—
397C4×D3—94no—
398C4×D3—1816no—
399C4×D3—182no—
400Dic6—184no—
401Dic6—1816no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—416no—
406C2×C32—42no—
407C2×C32—48no—
408C2×C32—62no—
409C2×C32C2×D3×D414yes—
410C2×C32C2×D3×D411yescharacteristic
411C2×C32C2×#23714yes—
412C2×C32C2×#23711yescharacteristic
413C2×C32C2×#23711yescharacteristic
414C2×C32D3×#4114yes—
415C3⋊D3—42no—
416C3×D3—28no—
417C3×D3—128no—
418#41—548no—
419C2×Q8—272no—
420C2×D4—32no—
421C2×D4—92no—
422C2×D4—92no—
423C2×D4—91no—
424C2×D4—272no—
425C2×D4D3×C3⋊D311yescharacteristic
426C22×C4—31no—
427C22×C4—272no—
428C22×C4—271no—
429C22×C4—271no—
430C22×C4—272no—
431C4⋊C4—31no—
432C4⋊C4—272no—
433C4⋊C4—271no—
434C22⋊C4—32no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C22⋊C4—272no—
438C42—91no—
439C42—271no—
440C42—271no—
441C42—271no—
442C22×C3—216no—
443C22×C3—28no—
444C22×C3—28no—
445C22×C3—28no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—24no—
450C22×C3—38no—
451C22×C3—48no—
452C22×C3—48no—
453C22×C3—64no—
454C22×C3—64no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3C22×D3214yes—
459C22×C3C23×C3⋊D311yescharacteristic
460C2×D3—21no—
461C2×D3—21no—
462C2×D3—68no—
463C2×D3—68no—
464C2×D3—68no—
465C2×D3D4×C3⋊D312yes—
466C3×C4—28no—
467C3×C4—68no—
468C3×C4—64no—
469C3×C4—92no—
470C3×C4—92no—
471C3×C4#380518yescomplemented
472C3×C4#383012yescomplemented
473Dic3—22no—
474Dic3—21no—
475Dic3—38no—
476Dic3—38no—
477Dic3—98no—
478Dic3—92no—
479Dic3—188no—
480Dic3—188no—
481Dic3—184no—
482Dic3—182no—
483Dic3—181no—
484Dic3—368no—
485Dic3—364no—
486C32—24no—
487C32#559614yescomplemented
488C32#571611yescharacteristic, complemented
489C23—31no—
490C23—92no—
491C23C2×D3×C3⋊D312yes—
492Q8—544no—
493D4—22no—
494D4—64no—
495D4—184no—
496D4—184no—
497D4—182no—
498D4—544no—
499C2×C4—31no—
500C2×C4—32no—
501C2×C4—62no—
502C2×C4—91no—
503C2×C4—91no—
504C2×C4—274no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4—272no—
508C2×C4—544no—
509C2×C4—542no—
510C2×C4—544no—
511C2×C4C2×D3×C3⋊D311yescharacteristic
512C2×C3—24no—
513C2×C3—24no—
514C2×C3—24no—
515C2×C3—416no—
516C2×C3—48no—
517C2×C3—42no—
518C2×C3—68no—
519C2×C3—362no—
520C2×C3D32×D414yes—
521C2×C3D3×#23714yes—
522C2×C3C2×#380514yes—
523C2×C3C2×D4×C3⋊D311yescharacteristic
524C2×C3C2×#383011yescharacteristic
525C2×C3#41×C3⋊D311yescharacteristic
526D3—22no—
527D3—128no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—32no—
532C22—61no—
533C22—61no—
534C22—182no—
535C22—182no—
536C22—182no—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—62no—
539C4—61no—
540C4—92no—
541C4—92no—
542C4—272no—
543C4—542no—
544C4—541no—
545C4#8902912yescomplemented
546C3—24no—
547C3#1059513314yescomplemented
548C3#1059606711yescharacteristic, complemented
549C2—42no—
550C2—62no—
551C2—362no—
552C2D3×D4×C3⋊D311yescharacteristic
553C2C3⋊D3×#23711yescharacteristic
554C2C2×#8902911yescharacteristic
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×#89029C2semidirect
2. D4×C3⋊(C3⋊Dic3)C2semidirect
3. C3×#64149C2semidirect
4. C32×#938C2semidirect
5. #11723626C2semidirect
6. #11723615C2semidirect
7. #11723591C2semidirect
8. #11723535C2semidirect
9. #11723515C2semidirect
10. #11723501C2semidirect
11. C4×D3×C3⋊Dic3C2semidirect
41. #64149D3semidirect
53. C3×#938D3semidirect
83. D3×C3⋊Dic3D4semidirect
149. C33×C4#41semidirect
154. C3×C3⋊Dic3#41semidirect
165. #938C3⋊D3semidirect
320. C32×C4#237semidirect
321. C32×C4#237semidirect
332. C3⋊Dic3#237semidirect
348. C33#532semidirect
471. C3×C4#3805semidirect
472. C3×C4#3830semidirect
487. C32#5596semidirect
488. C32#5716semidirect
545. C4#89029semidirect
547. C3#10595133semidirect
548. C3#10596067semidirect

Maximal subgroups

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

subgroupindex
C2×#890292
D4×C3⋊(C3⋊Dic3)2
C3×#641492
C32×#9382
#117236262
#117236152
#117235912
#117235352
#117235152
#117235012
C4×D3×C3⋊Dic32
#105960673
#105951333