Finite Group Databasebeta

← #13177604

Subgroups — #13177604

Order 1728.

2128 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×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C32×#237C211yesmaximal, characteristic, complemented
3D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#11723637C212yesmaximal, complemented
5C2×#88806C212yesmaximal, complemented
6#11723605C212yesmaximal, complemented
7#11723602C212yesmaximal, complemented
8C4×#6290C211yesmaximal, characteristic, complemented
9#11723451C211yesmaximal, characteristic, complemented
10#11723435C211yesmaximal, characteristic, complemented
11#11723425C211yesmaximal, characteristic, complemented
12#10596295—31nomaximal
13#10595252—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C22×C3×C3⋊Dic3C2212yes—
18C3×#3830—24no—
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#634C2211yescharacteristic
21C2×C32×#67C2212yes—
22C22×C32×Dic3C2212yes—
23C32×#237—24no—
24C2×C32×C4×D3C2211yescharacteristic
25C2×C32×Dic6C2211yescharacteristic
26C2×#6290C2211yescharacteristic
27#88864C2212yes—
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30#88806C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33#88806C2211yescharacteristic
34#88806C2212yes—
35#88803C2211yescharacteristic
36#88803C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×C32×#41—31no—
40C2×#3830D311yescharacteristic, complemented
41C2×C3×#237—34no—
42C2×C3×#237D314yescomplemented
43#64155—31no—
44#64153—31no—
45#64148—32no—
46#64144—32no—
47D4×C3⋊Dic3—31no—
48D4×C3⋊Dic3—34no—
49D4×C3⋊Dic3—64no—
50#64138—32no—
51C4×#638—31no—
52#64133—31no—
53C2×#3750—32no—
54#63980—38no—
55C2×#3724—38no—
56#63961—38no—
57#63961—38no—
58C4×#625—34no—
59#63873—34no—
60#63856—34no—
61#63856—34no—
62C23×C33C2312yes—
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C33×D4D414yescomplemented
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)—24no—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C3×#638—24no—
72C2×C3×C3⋊Dic3—24no—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4×C3⋊D3—22no—
76C3×#634—22no—
77C32×#67—24no—
78C2×C32×Dic3—24no—
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81C32×C4×D3—22no—
82C32×Dic6—22no—
83#6290—22no—
84#5824—94no—
85#5824—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—
96C22×C3⋊Dic3—32no—
97C22×C3⋊Dic3—38no—
98C22×C3⋊Dic3—68no—
99C22×C3⋊Dic3C2×D312yes—
100#3830—24no—
101C2×C4×C3⋊D3C2×D311yescharacteristic
102C2×#634C2×D311yescharacteristic
103C2×C3×#67—38no—
104C2×C3×#67C2×D318yes—
105C22×C3×Dic3—38no—
106C22×C3×Dic3C2×D318yes—
107C3×#237—216no—
108C3×#237—616no—
109C2×C3×C4×D3—34no—
110C2×C3×C4×D3C2×D314yes—
111C2×C3×Dic6—34no—
112C2×C3×Dic6C2×D314yes—
113C2×#625—34no—
114#3757—32no—
115#3757—38no—
116#3757—32no—
117#3757—31no—
118#3757—68no—
119#3752—31no—
120#3752—32no—
121C4⋊(C3⋊Dic3)—31no—
122C4⋊(C3⋊Dic3)—34no—
123C4⋊(C3⋊Dic3)—64no—
124#3750—31no—
125#3750—32no—
126#3750—32no—
127#3750—32no—
128#3750—32no—
129C4×C3⋊Dic3—31no—
130C4×C3⋊Dic3—34no—
131C4×C3⋊Dic3—64no—
132#3724—38no—
133#3724—38no—
134#3724—38no—
135#3724—38no—
136#3724—34no—
137(C2×D3)⋊Dic3—38no—
138(C2×D3)⋊Dic3—34no—
139(C2×D3)⋊Dic3—38no—
140(C2×D3)⋊Dic3—34no—
141C22×C33—22no—
142C22×C33—22no—
143C22×C33C2×D414yes—
144C22×C33C2411yescharacteristic
145C2×C3×C3⋊D3—21no—
146C2×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3—21no—
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)#4112yescomplemented
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—21no—
157C32×Dic3—22no—
158C32×Dic3—21no—
159C2×C3×#41—34no—
160C2×C3×#41—91no—
161C2×#237—34no—
162C2×#237C3⋊D311yescharacteristic, complemented
163#946—94no—
164#946—91no—
165#944—94no—
166#944—91no—
167#939—98no—
168#939—92no—
169#935—98no—
170#935—92no—
171D4×Dic3—94no—
172D4×Dic3—91no—
173D4×Dic3—184no—
174#929—98no—
175#929—92no—
176C4×#67—94no—
177C4×#67—91no—
178#924—94no—
179#924—91no—
180C2×Dic3⋊C4—98no—
181C2×Dic3⋊C4—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—62no—
190C32×D4—216no—
191C32×D4—64no—
192C32×D4#67116yescomplemented
193C32×D4#6714yescomplemented
194C2×C32×C4—24no—
195C2×C32×C4—31no—
196C2×C32×C4—32no—
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—24no—
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—31no—
206C2×C3⋊Dic3—32no—
207C2×C3⋊Dic3—34no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—38no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—68no—
213C2×C3⋊Dic3—64no—
214C2×C3⋊Dic3—616no—
215C2×C3⋊Dic3—1216no—
216C2×C3⋊Dic3C22×D312yes—
217C2×C3⋊Dic3C22×D311yescharacteristic
218C4×C3⋊D3—22no—
219#634—22no—
220C3×#67—216no—
221C3×#67—616no—
222C2×C3×Dic3—216no—
223C2×C3×Dic3—38no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3—616no—
226C2×C3×Dic3C22×D318yes—
227C2×C3×Dic3C22×D314yes—
228C3×C4×D3—28no—
229C3×C4×D3—68no—
230C3×Dic6—28no—
231C3×Dic6—68no—
232#625—68no—
233#547—271noSylow
234C2×C33—24no—
235C2×C33C22×D411yescharacteristic
236C2×C33C2×#4111yescharacteristic
237C2×C332−1+411yescharacteristic
238C3×C3⋊D3—41no—
239C32×D3—41no—
240C3×#41—616no—
241C3×#41—184no—
242C2×C3×Q8—34no—
243C2×C3×Q8—91no—
244C2×C3×D4—24no—
245C2×C3×D4—38no—
246C2×C3×D4—92no—
247C2×C3×D4D3214yes—
248C2×C3×D4C2×C3⋊D311yescharacteristic
249C22×C3×C4—38no—
250C22×C3×C4—34no—
251C22×C3×C4—92no—
252C22×C3×C4—91no—
253C2×#67—38no—
254C2×#67—94no—
255C2×#67—91no—
256C2×#67C2×C3⋊D312yes—
257C22×Dic3—38no—
258C22×Dic3—98no—
259C22×Dic3—92no—
260C22×Dic3—188no—
261C22×Dic3C2×C3⋊D312yes—
262#237—24no—
263#237—616no—
264C2×C4×D3—34no—
265C2×C4×D3C2×C3⋊D311yescharacteristic
266C2×Dic6—34no—
267C2×Dic6C2×C3⋊D311yescharacteristic
268#217—98no—
269#217—94no—
270#217—98no—
271#217—92no—
272#217—92no—
273#217—91no—
274#217—188no—
275(C2×D3)⋊C4—94no—
276(C2×D3)⋊C4—98no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—92no—
279C4⋊Dic3—94no—
280C4⋊Dic3—91no—
281C4⋊Dic3—184no—
282Dic3⋊C4—94no—
283Dic3⋊C4—98no—
284Dic3⋊C4—98no—
285Dic3⋊C4—98no—
286Dic3⋊C4—98no—
287Dic3⋊C4—91no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292C4×Dic3—94no—
293C4×Dic3—91no—
294C4×Dic3—184no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—24no—
298C22×C32—216no—
299C22×C32—28no—
300C22×C32—28no—
301C22×C32—48no—
302C22×C32—48no—
303C22×C32—61no—
304C22×C32—61no—
305C22×C32—61no—
306C22×C32C2×#67116yes—
307C22×C32C2×#6714yes—
308C22×C32C23×D314yes—
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—21no—
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3×D3—24no—
314C2×C3×D3—24no—
315C2×C3×D3—24no—
316C2×C3×D3—64no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C32×C4—28no—
320C32×C4—61no—
321C32×C4—62no—
322C32×C4C2×#6718yes—
323C32×C4C2×#6712yes—
324C3⋊Dic3—22no—
325C3⋊Dic3—21no—
326C3⋊Dic3—32no—
327C3⋊Dic3—38no—
328C3⋊Dic3—68no—
329C3⋊Dic3—62no—
330C3⋊Dic3—61no—
331C3⋊Dic3—68no—
332C3⋊Dic3—64no—
333C3⋊Dic3—124no—
334C3⋊Dic3—128no—
335C3×Dic3—28no—
336C3×Dic3—24no—
337C3×Dic3—68no—
338C3×Dic3—64no—
339C2×#41—31no—
340C2×#41—91no—
341C4⋊Q8—271no—
342#123—272no—
343#123—272no—
344C22⋊Q8—271no—
345C22⋊Q8—271no—
346#121—272no—
347C4×D4—271no—
348C4×D4—271no—
349C2×C4⋊C4—272no—
350C33#54711yescharacteristic, 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—68no—
359C3×Q8—182no—
360C3×D4—216no—
361C3×D4—616no—
362C3×D4—184no—
363C3×D4#625116yescomplemented
364C3×D4#63814yescomplemented
365C2×C3×C4—24no—
366C2×C3×C4—34no—
367C2×C3×C4—38no—
368C2×C3×C4—616no—
369C2×C3×C4—68no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—184no—
373C2×C3×C4—182no—
374C2×C3×C4C2×D3214yes—
375C2×C3×C4C22×C3⋊D311yescharacteristic
376#67—24no—
377#67—616no—
378#67—182no—
379#67—188no—
380C2×Dic3—24no—
381C2×Dic3—38no—
382C2×Dic3—34no—
383C2×Dic3—616no—
384C2×Dic3—94no—
385C2×Dic3—91no—
386C2×Dic3—94no—
387C2×Dic3—98no—
388C2×Dic3—91no—
389C2×Dic3—92no—
390C2×Dic3—184no—
391C2×Dic3—184no—
392C2×Dic3—188no—
393C2×Dic3—184no—
394C2×Dic3—1816no—
395C2×Dic3—3616no—
396C2×Dic3C22×C3⋊D312yes—
397C2×Dic3C22×C3⋊D311yescharacteristic
398C4×D3—22no—
399C4×D3—68no—
400Dic6—22no—
401Dic6—68no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—24no—
406C2×C32—216no—
407C2×C32—416no—
408C2×C32—121no—
409C2×C32D3×#4114yes—
410C2×C32D3×#4111yescharacteristic
411C2×C32#101014yes—
412C2×C32#101011yescharacteristic
413C2×C32C22×#6714yes—
414C2×C32C22×#6711yescharacteristic
415C3⋊D3—41no—
416C3×D3—44no—
417C3×D3—124no—
418#41—64no—
419#41—184no—
420C2×Q8—31no—
421C2×Q8—91no—
422C2×D4—32no—
423C2×D4—92no—
424C2×D4—271no—
425C2×D4D3×C3⋊D311yescharacteristic
426C22×C4—32no—
427C22×C4—31no—
428C22×C4—92no—
429C22×C4—91no—
430C22×C4—272no—
431C4⋊C4—271no—
432C4⋊C4—272no—
433C4⋊C4—272no—
434C4⋊C4—272no—
435C4⋊C4—272no—
436C4⋊C4—271no—
437C22⋊C4—271no—
438C22⋊C4—272no—
439C22⋊C4—271no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C42—271no—
443C22×C3—28no—
444C22×C3—28no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—24no—
448C22×C3—216no—
449C22×C3—48no—
450C22×C3—48no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—64no—
454C22×C3—181no—
455C22×C3—181no—
456C22×C3—181no—
457C22×C3C2×#625116yes—
458C22×C3C2×#63814yes—
459C22×C3C22×D3214yes—
460C22×C3C23×C3⋊D311yescharacteristic
461C2×D3—21no—
462C2×D3—21no—
463C2×D3—21no—
464C2×D3—64no—
465C2×D3—64no—
466C2×D3—64no—
467C3×C4—28no—
468C3×C4—64no—
469C3×C4—68no—
470C3×C4—181no—
471C3×C4—182no—
472C3×C4C2×#62518yes—
473C3×C4C2×#63812yes—
474Dic3—22no—
475Dic3—21no—
476Dic3—68no—
477Dic3—64no—
478Dic3—98no—
479Dic3—92no—
480Dic3—188no—
481Dic3—184no—
482Dic3—188no—
483Dic3—182no—
484Dic3—181no—
485Dic3—368no—
486Dic3—364no—
487C32—24no—
488C32#582414yescomplemented
489C32#582411yescharacteristic, complemented
490C23—31no—
491C23—91no—
492C23C2×D3×C3⋊D312yes—
493Q8—62no—
494Q8—182no—
495D4—64no—
496D4—184no—
497D4—542no—
498D4#629014yescomplemented
499C2×C4—31no—
500C2×C4—32no—
501C2×C4—64no—
502C2×C4—62no—
503C2×C4—91no—
504C2×C4—92no—
505C2×C4—184no—
506C2×C4—182no—
507C2×C4—271no—
508C2×C4—271no—
509C2×C4—272no—
510C2×C4—544no—
511C2×C4C2×D3×C3⋊D311yescharacteristic
512C2×C3—216no—
513C2×C3—24no—
514C2×C3—24no—
515C2×C3—24no—
516C2×C3—24no—
517C2×C3—416no—
518C2×C3—124no—
519C2×C3—361no—
520C2×C3#6430914yes—
521C2×C3#6431214yes—
522C2×C3C22×#62514yes—
523C2×C3#41×C3⋊D311yescharacteristic
524C2×C3#6436711yescharacteristic
525C2×C3C22×#63811yescharacteristic
526D3—41no—
527D3—124no—
528C22—22no—
529C22—22no—
530C22—61no—
531C22—61no—
532C22—61no—
533C22—181no—
534C22—181no—
535C22—181no—
536C22C2×#629014yes—
537C22C22×D3×C3⋊D311yescharacteristic
538C4—61no—
539C4—62no—
540C4—181no—
541C4—182no—
542C4—272no—
543C4—541no—
544C4—542no—
545C4C2×#629012yes—
546C3—24no—
547C3#1059525214yescomplemented
548C3#1059629511yescharacteristic, complemented
549C2—24no—
550C2—121no—
551C2—361no—
552C2#1172458211yescharacteristic
553C2#1172458611yescharacteristic
554C2C22×#629011yescharacteristic
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×C3×#3830C2semidirect
2. C2×C32×#237C2semidirect
3. D4×C3⋊(C3⋊Dic3)C2semidirect
4. #11723637C2semidirect
5. C2×#88806C2semidirect
6. #11723605C2semidirect
7. #11723602C2semidirect
8. C4×#6290C2semidirect
9. #11723451C2semidirect
10. #11723435C2semidirect
11. #11723425C2semidirect
40. C2×#3830D3semidirect
42. C2×C3×#237D3semidirect
65. C33×D4D4semidirect
154. C3⋊(C3⋊Dic3)#41semidirect
162. C2×#237C3⋊D3semidirect
192. C32×D4#67semidirect
193. C32×D4#67semidirect
350. C33#547semidirect
363. C3×D4#625semidirect
364. C3×D4#638semidirect
488. C32#5824semidirect
489. C32#5824semidirect
498. D4#6290semidirect
547. C3#10595252semidirect
548. C3#10596295semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C32×#2372
D4×C3⋊(C3⋊Dic3)2
#117236372
C2×#888062
#117236052
#117236022
C4×#62902
#117234512
#117234352
#117234252
#105962953
#105952523