Finite Group Databasebeta

← #13177513

Subgroups — #13177513

Order 1728.

2128 conjugacy classes of subgroups.

The subgroup lattice has 544 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×#64147C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723642C212yesmaximal, complemented
6#11723621C212yesmaximal, complemented
7#11723614C212yesmaximal, complemented
8#11723536C211yesmaximal, characteristic, complemented
9C4×#6290C211yesmaximal, characteristic, complemented
10C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
11D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
12#10596082—31nomaximal
13#10595134—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×#67C2212yes—
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6293C2212yes—
21C2×#6290C2211yescharacteristic
22C2×Dic3×C3⋊D3C2212yes—
23#89029—28no—
24C2×D3×C3⋊Dic3C2212yes—
25#88864C2212yes—
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3757C2212yes—
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C32×#217C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32#88807C2212yes—
33#88806C2212yes—
34#88806C2211yescharacteristic
35Dic3⋊(C3⋊Dic3)C2212yes—
36#88803C2211yescharacteristic
37(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
38C2×#3830—32no—
39C2×#3805—38no—
40C32×#121—31no—
41#64147D311yescharacteristic, complemented
42#64144—32no—
43D4×C3⋊Dic3—31no—
44D4×C3⋊Dic3—34no—
45D4×C3⋊Dic3—64no—
46C4×#638—31no—
47C2×C4⋊(C3⋊Dic3)—31no—
48C4⋊C4×C3⋊D3—31no—
49#64097—31no—
50#64092—32no—
51#64087—32no—
52C3×#938—34no—
53C3×#938D314yescomplemented
54#63982—38no—
55#63969—38no—
56#63969—38no—
57#63919—34no—
58C4×#625—34no—
59D3×C4⋊Dic3—34no—
60D3×C4⋊Dic3—34no—
61C23×C33C2312yes—
62C22×C3×C3⋊D3C2311yescharacteristic
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—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—22no—
74C32×#67—24no—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2312yes—
77C32×C4×D3—22no—
78#6293—24no—
79#6290D414yescomplemented
80Dic3×C3⋊D3—24no—
81D3×C3⋊Dic3—24no—
82#5611—94no—
83#5611—91no—
84C2×C32×D4—24no—
85C2×C32×D4—32no—
86C2×C32×D4C2×D314yes—
87C2×C32×D4C2×D311yescharacteristic
88C22×C32×C4—31no—
89C2×#638—31no—
90C2×#638C2×D312yes—
91C22×C3⋊Dic3—32no—
92C22×C3⋊Dic3—38no—
93C22×C3⋊Dic3—32no—
94C22×C3⋊Dic3—68no—
95#3830—68no—
96C2×C4×C3⋊D3—32no—
97C2×C4×C3⋊D3C2×D311yescharacteristic
98C2×#634—32no—
99C2×C3×#67—38no—
100C2×C3×#67C2×D318yes—
101C2×C3×C4×D3—34no—
102C2×C3×C4×D3C2×D314yes—
103C2×#627—38no—
104C2×#625—34no—
105#3805—632no—
106C2×D3×Dic3—38no—
107C2×D3×Dic3—38no—
108C32×C4⋊C4—31no—
109C32×C22⋊C4—32no—
110#3757—32no—
111#3757—38no—
112#3757—31no—
113#3757—68no—
114#3757C2×D312yes—
115#3752—31no—
116C4⋊(C3⋊Dic3)—31no—
117C4⋊(C3⋊Dic3)—34no—
118C4⋊(C3⋊Dic3)—32no—
119C4⋊(C3⋊Dic3)—64no—
120C4⋊(C3⋊Dic3)C2×D311yescharacteristic
121#3750—31no—
122#3750—32no—
123#3750—32no—
124C4×C3⋊Dic3—31no—
125C4×C3⋊Dic3—34no—
126C4×C3⋊Dic3—64no—
127C3×#217—38no—
128C3×#217C2×D318yes—
129C3×C4⋊Dic3—34no—
130C3×C4⋊Dic3C2×D314yes—
131#3724—34no—
132#3724—38no—
133Dic3⋊Dic3—38no—
134Dic3⋊Dic3—38no—
135(C2×D3)⋊Dic3—34no—
136(C2×D3)⋊Dic3—34no—
137C22×C33—22no—
138C22×C33—22no—
139C22×C33—22no—
140C22×C33C2411yescharacteristic
141C2×C3×C3⋊D3—21no—
142C2×C3×C3⋊D3—21no—
143C2×C3×C3⋊D3C2×D412yes—
144C2×C32×D3—21no—
145C2×C32×D3—21no—
146C2×C32×D3C2×D412yes—
147C33×C4#4112yescomplemented
148C3⋊(C3⋊Dic3)—22no—
149C3⋊(C3⋊Dic3)—21no—
150C3⋊(C3⋊Dic3)C2×D412yes—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—21no—
153C32×Dic3—22no—
154C32×Dic3—21no—
155C2×#237—98no—
156C2×#237—92no—
157C3×#121—34no—
158C3×#121—91no—
159#938—34no—
160#938C3⋊D311yescharacteristic, complemented
161#935—98no—
162#935—92no—
163D4×Dic3—94no—
164D4×Dic3—91no—
165D4×Dic3—184no—
166C4×#67—94no—
167C4×#67—91no—
168C2×C4⋊Dic3—94no—
169C2×C4⋊Dic3—91no—
170D3×C4⋊C4—94no—
171D3×C4⋊C4—91no—
172#888—94no—
173#888—91no—
174#883—98no—
175#883—92no—
176#878—98no—
177#878—92no—
178C23×C32—28no—
179C23×C32—31no—
180C23×C32C22×D318yes—
181C23×C32C22×D312yes—
182C22×C3⋊D3C22×D311yescharacteristic
183C22×C3×D3—34no—
184C22×C3×D3C22×D314yes—
185C32×D4—216no—
186C32×D4—28no—
187C32×D4—22no—
188C32×D4—64no—
189C2×C32×C4—24no—
190C2×C32×C4—32no—
191C2×C32×C4—31no—
192C2×C32×C4—62no—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D311yescharacteristic
195#638—24no—
196#638—34no—
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—34no—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—38no—
203C2×C3⋊Dic3—64no—
204C2×C3⋊Dic3—64no—
205C2×C3⋊Dic3—68no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—616no—
209C2×C3⋊Dic3—1216no—
210C2×C3⋊Dic3C22×D311yescharacteristic
211C2×C3⋊Dic3C22×D312yes—
212C4×C3⋊D3—22no—
213C4×C3⋊D3—64no—
214#634—64no—
215C3×#67—216no—
216C3×#67—616no—
217C2×C3×Dic3—34no—
218C2×C3×Dic3—38no—
219C2×C3×Dic3C22×D314yes—
220C2×C3×Dic3C22×D318yes—
221C3×C4×D3—28no—
222C3×C4×D3—68no—
223#627—616no—
224#625—316no—
225D3×Dic3—616no—
226D3×Dic3—616no—
227#547—271noSylow
228C2×C33—42no—
229C2×C33C22×D411yescharacteristic
230C2×C33C2×#4111yescharacteristic
231C2×C332−1+411yescharacteristic
232C3×C3⋊D3—22no—
233C32×D3—22no—
234C2×C3×D4—24no—
235C2×C3×D4—38no—
236C2×C3×D4—92no—
237C2×C3×D4D3214yes—
238C2×C3×D4C2×C3⋊D311yescharacteristic
239C22×C3×C4—34no—
240C22×C3×C4—91no—
241C2×#67—38no—
242C2×#67—94no—
243C2×#67—91no—
244C2×#67C2×C3⋊D312yes—
245C22×Dic3—98no—
246C22×Dic3—98no—
247C22×Dic3—92no—
248C22×Dic3—92no—
249C22×Dic3—188no—
250#237—188no—
251#237—1832no—
252C2×C4×D3—34no—
253C2×C4×D3—98no—
254C2×C4×D3—92no—
255C2×C4×D3C2×C3⋊D311yescharacteristic
256C2×Dic6—98no—
257C2×Dic6—92no—
258C3×C4⋊C4—34no—
259C3×C4⋊C4—91no—
260C3×C22⋊C4—38no—
261C3×C22⋊C4—92no—
262#217—38no—
263#217—94no—
264#217—98no—
265#217—92no—
266#217—91no—
267#217—188no—
268#217C2×C3⋊D312yes—
269(C2×D3)⋊C4—94no—
270(C2×D3)⋊C4—91no—
271C4⋊Dic3—34no—
272C4⋊Dic3—98no—
273C4⋊Dic3—94no—
274C4⋊Dic3—91no—
275C4⋊Dic3—92no—
276C4⋊Dic3—184no—
277C4⋊Dic3C2×C3⋊D311yescharacteristic
278Dic3⋊C4—94no—
279Dic3⋊C4—98no—
280Dic3⋊C4—98no—
281Dic3⋊C4—91no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284C4×Dic3—94no—
285C4×Dic3—91no—
286C4×Dic3—184no—
287C22×C32—216no—
288C22×C32—22no—
289C22×C32—28no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—24no—
293C22×C32—28no—
294C22×C32—28no—
295C22×C32—32no—
296C22×C32—48no—
297C22×C32—48no—
298C22×C32—61no—
299C22×C32—61no—
300C22×C32C23×D314yes—
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—21no—
303C2×C3⋊D3—21no—
304C2×C3⋊D3D3×D412yes—
305C2×C3×D3—24no—
306C2×C3×D3—24no—
307C2×C3×D3—38no—
308C2×C3×D3—64no—
309C2×C3×D3—64no—
310C2×C3×D3D3×D418yes—
311C32×C4—28no—
312C32×C4—62no—
313C32×C4—61no—
314C32×C4#23718yescomplemented
315C32×C4#23712yescomplemented
316C3⋊Dic3—22no—
317C3⋊Dic3—21no—
318C3⋊Dic3—32no—
319C3⋊Dic3—38no—
320C3⋊Dic3—68no—
321C3⋊Dic3—62no—
322C3⋊Dic3—61no—
323C3⋊Dic3—68no—
324C3⋊Dic3—64no—
325C3⋊Dic3—124no—
326C3⋊Dic3—128no—
327C3×Dic3—28no—
328C3×Dic3—24no—
329C3×Dic3—68no—
330C3×Dic3—64no—
331C2×#41—272no—
332C4⋊Q8—271no—
333#123—272no—
334#123—272no—
335C22⋊Q8—272no—
336#121—31no—
337#121—91no—
338C4×D4—271no—
339C4×D4—271no—
340C2×C4⋊C4—271no—
341C2×C4⋊C4—271no—
342C33#54711yescharacteristic, complemented, Sylow
343C23×C3—28no—
344C23×C3—34no—
345C23×C3—91no—
346C23×C3C2×D3218yes—
347C23×C3C22×C3⋊D312yes—
348C22×D3—34no—
349C22×D3C22×C3⋊D311yescharacteristic
350C3×D4—216no—
351C3×D4—28no—
352C3×D4—22no—
353C3×D4—616no—
354C3×D4—184no—
355C2×C3×C4—24no—
356C2×C3×C4—38no—
357C2×C3×C4—34no—
358C2×C3×C4—68no—
359C2×C3×C4—92no—
360C2×C3×C4—91no—
361C2×C3×C4—182no—
362C2×C3×C4C2×D3214yes—
363C2×C3×C4C22×C3⋊D311yescharacteristic
364#67—24no—
365#67—616no—
366#67—916no—
367#67—94no—
368C2×Dic3—34no—
369C2×Dic3—38no—
370C2×Dic3—94no—
371C2×Dic3—91no—
372C2×Dic3—94no—
373C2×Dic3—98no—
374C2×Dic3—91no—
375C2×Dic3—92no—
376C2×Dic3—184no—
377C2×Dic3—184no—
378C2×Dic3—188no—
379C2×Dic3—184no—
380C2×Dic3—1816no—
381C2×Dic3—184no—
382C2×Dic3—1816no—
383C2×Dic3—3616no—
384C2×Dic3C22×C3⋊D311yescharacteristic
385C2×Dic3C22×C3⋊D312yes—
386C4×D3—22no—
387C4×D3—68no—
388C4×D3—1816no—
389C4×D3—184no—
390Dic6—184no—
391Dic6—1816no—
392C2×C32—24no—
393C2×C32—24no—
394C2×C32—24no—
395C2×C32—416no—
396C2×C32—48no—
397C2×C32—42no—
398C2×C32—62no—
399C2×C32C2×D3×D414yes—
400C2×C32C2×D3×D411yescharacteristic
401C2×C32C2×#23714yes—
402C2×C32C2×#23711yescharacteristic
403C2×C32#101014yes—
404C2×C32#101011yescharacteristic
405C3⋊D3—22no—
406C3×D3—28no—
407C3×D3—68no—
408#41—548no—
409C2×Q8—272no—
410C2×D4—32no—
411C2×D4—92no—
412C2×D4—271no—
413C2×D4D3×C3⋊D311yescharacteristic
414C22×C4—31no—
415C22×C4—91no—
416C22×C4—272no—
417C22×C4—272no—
418C22×C4—272no—
419C4⋊C4—31no—
420C4⋊C4—91no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—271no—
424C4⋊C4—272no—
425C4⋊C4—271no—
426C22⋊C4—32no—
427C22⋊C4—92no—
428C22⋊C4—271no—
429C22⋊C4—271no—
430C22⋊C4—272no—
431C42—271no—
432C22×C3—216no—
433C22×C3—28no—
434C22×C3—28no—
435C22×C3—28no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—24no—
440C22×C3—38no—
441C22×C3—48no—
442C22×C3—48no—
443C22×C3—64no—
444C22×C3—64no—
445C22×C3—92no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3C22×D3214yes—
449C22×C3C23×C3⋊D311yescharacteristic
450C2×D3—21no—
451C2×D3—21no—
452C2×D3—38no—
453C2×D3—64no—
454C2×D3—64no—
455C2×D3D4×C3⋊D312yes—
456C3×C4—28no—
457C3×C4—68no—
458C3×C4—64no—
459C3×C4—182no—
460C3×C4—181no—
461C3×C4#380518yescomplemented
462C3×C4#383012yescomplemented
463Dic3—22no—
464Dic3—21no—
465Dic3—68no—
466Dic3—64no—
467Dic3—98no—
468Dic3—92no—
469Dic3—188no—
470Dic3—184no—
471Dic3—188no—
472Dic3—182no—
473Dic3—181no—
474Dic3—368no—
475Dic3—364no—
476C32—24no—
477C32#561114yescomplemented
478C32#561111yescharacteristic, complemented
479C23—31no—
480C23—91no—
481C23C2×D3×C3⋊D312yes—
482Q8—544no—
483D4—22no—
484D4—64no—
485D4—184no—
486D4—274no—
487C2×C4—31no—
488C2×C4—32no—
489C2×C4—62no—
490C2×C4—91no—
491C2×C4—92no—
492C2×C4—182no—
493C2×C4—271no—
494C2×C4—271no—
495C2×C4—272no—
496C2×C4—544no—
497C2×C4—544no—
498C2×C4—544no—
499C2×C4C2×D3×C3⋊D311yescharacteristic
500C2×C3—24no—
501C2×C3—24no—
502C2×C3—24no—
503C2×C3—416no—
504C2×C3—48no—
505C2×C3—42no—
506C2×C3—68no—
507C2×C3—182no—
508C2×C3#6430914yes—
509C2×C3D32×D414yes—
510C2×C3C2×#380514yes—
511C2×C3C2×D4×C3⋊D311yescharacteristic
512C2×C3C2×#383011yescharacteristic
513C2×C3#6436711yescharacteristic
514D3—22no—
515D3—68no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—32no—
520C22—61no—
521C22—61no—
522C22—92no—
523C22—181no—
524C22—181no—
525C22C22×D3×C3⋊D311yescharacteristic
526C4—62no—
527C4—61no—
528C4—182no—
529C4—181no—
530C4—272no—
531C4—541no—
532C4—542no—
533C4#8902912yescomplemented
534C3—24no—
535C3#1059513414yescomplemented
536C3#1059608211yescharacteristic, complemented
537C2—42no—
538C2—62no—
539C2—182no—
540C2#1172458211yescharacteristic
541C2D3×D4×C3⋊D311yescharacteristic
542C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#547
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×#64147C2semidirect
4. C32×#938C2semidirect
5. #11723642C2semidirect
6. #11723621C2semidirect
7. #11723614C2semidirect
8. #11723536C2semidirect
9. C4×#6290C2semidirect
10. C3⋊D3×C4⋊Dic3C2semidirect
11. D3×C4⋊(C3⋊Dic3)C2semidirect
41. #64147D3semidirect
53. C3×#938D3semidirect
79. #6290D4semidirect
147. C33×C4#41semidirect
160. #938C3⋊D3semidirect
314. C32×C4#237semidirect
315. C32×C4#237semidirect
342. C33#547semidirect
461. C3×C4#3805semidirect
462. C3×C4#3830semidirect
477. C32#5611semidirect
478. C32#5611semidirect
533. C4#89029semidirect
535. C3#10595134semidirect
536. C3#10596082semidirect

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×#641472
C32×#9382
#117236422
#117236212
#117236142
#117235362
C4×#62902
C3⋊D3×C4⋊Dic32
D3×C4⋊(C3⋊Dic3)2
#105960823
#105951343