Finite Group Databasebeta

← #13175501

Subgroups — #13175501

Order 1728.

2500 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic, complemented
2C2×#89031C211yesmaximal, characteristic, complemented
3#11723923C211yesmaximal, characteristic, complemented
4C3×#64095C211yesmaximal, characteristic, complemented
5C32×#877C211yesmaximal, characteristic, complemented
6#11723639C211yesmaximal, characteristic, complemented
7#11723608C211yesmaximal, characteristic, complemented
8#11723486C212yesmaximal, complemented
9#11723485C212yesmaximal, complemented
10#11723484C212yesmaximal, complemented
11#11723471C212yesmaximal, complemented
12#10595995—31nomaximal
13#10594079—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
17C2×C3×#638C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×#6293C2211yescharacteristic, complemented
21C2×#6292C2212yes—
22C2×#6292C2211yescharacteristic, complemented
23C2×#6289—24no—
24C2×#6289C2212yes—
25C2×#6289C2212yes—
26C2×#6289C2211yescharacteristic, complemented
27#89031—24no—
28C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
29C33×C22⋊C4C2211yescharacteristic
30#88859C2212yes—
31C3×#3752C2212yescomplemented
32C3×C4⋊(C3⋊Dic3)C2212yescomplemented
33C32×#217C2211yescharacteristic
34C32×Dic3⋊C4C2212yescomplemented
35C32×C4×Dic3C2212yescomplemented
36Dic3⋊(C3⋊Dic3)C2212yescomplemented
37#88804C2212yes—
38#88804C2212yescomplemented
39C2×#3828—31no—
40C22×C4×C3⋊D3—31no—
41C22×#624—34no—
42C2×#3804—34no—
43C32×#117—31no—
44#64106—32no—
45#64104—32no—
46#64095—31no—
47#64095D311yescharacteristic, complemented
48#64091—31no—
49#64091—31no—
50#64091—34no—
51#64091—64no—
52C4×C4⋊(C3⋊D3)—32no—
53C4×C4⋊(C3⋊D3)—32no—
54C3×#886—34no—
55C3×#877D314yescomplemented
56#63981—34no—
57#63966—34no—
58#63910—38no—
59#63895—38no—
60#63894—38no—
61#63880—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)—22no—
64C22×C3⋊(C3⋊D3)—24no—
65C22×C3⋊(C3⋊D3)C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2312yes—
67C22×C3×C3⋊D3C2311yescharacteristic
68C2×C33×C4C2312yes—
69#6312—22no—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4⋊(C3⋊(C3⋊D3))—24no—
72C3×#638—22no—
73C2×C3×C3⋊Dic3—22no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C2×C32×Dic3D414yescomplemented
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2311yescharacteristic
80#6293—22no—
81#6292—24no—
82#6292—22no—
83#6289—22no—
84#6289—24no—
85#6289—24no—
86Dic3×C3⋊D3—22no—
87#5656—91no—
88#5524—94no—
89C23×C3⋊D3—31no—
90C23×C3⋊D3—34no—
91C23×C3⋊D3—64no—
92C22×C32×C4—31no—
93C2×#638—31no—
94C2×#638—34no—
95C2×#638—64no—
96C2×#638C2×D311yescharacteristic
97C22×C3⋊Dic3C2×D311yescharacteristic
98#3828—64no—
99C2×C4⋊(C3⋊D3)—32no—
100C2×C4⋊(C3⋊D3)—31no—
101C2×C4⋊(C3⋊D3)—32no—
102C2×C4⋊(C3⋊D3)—38no—
103C2×C4⋊(C3⋊D3)—68no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—32no—
108C2×C4×C3⋊D3—64no—
109C2×#634—31no—
110C2×C3×#67—34no—
111C22×C3×Dic3—34no—
112C22×C3×Dic3C2×D314yes—
113C2×#627—34no—
114C2×#626—34no—
115C2×#626—38no—
116C2×#624—34no—
117C2×#624—38no—
118C2×#624—38no—
119C2×#624—616no—
120#3804—616no—
121C2×D3×Dic3—34no—
122C32×C4⋊C4—32no—
123C32×C22⋊C4—24no—
124C32×C22⋊C4—31no—
125C32×C22⋊C4C2×D314yes—
126C32×C22⋊C4C2×D311yescharacteristic
127C32×C42—32no—
128#3752—32no—
129#3752—38no—
130#3752—32no—
131#3752—32no—
132#3752—68no—
133#3752C2×D312yescomplemented
134C4⋊(C3⋊Dic3)—32no—
135C4⋊(C3⋊Dic3)C2×D312yescomplemented
136C3×#217C2×D314yes—
137C3×(C2×D3)⋊C4—38no—
138C3×C4⋊Dic3—38no—
139C3×Dic3⋊C4C2×D318yescomplemented
140C3×C4×Dic3C2×D318yescomplemented
141Dic3⋊Dic3—38no—
142#3722—38no—
143#3722—38no—
144C22×C33—22no—
145C22×C33C2×D412yes—
146C22×C33C2411yescharacteristic
147C2×C3⋊(C3⋊D3)—28no—
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—22no—
152C2×C3⋊(C3⋊D3)#4114yescomplemented
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C33×C4—22no—
156C3⋊(C3⋊Dic3)—21no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3—21no—
159C32×Dic3—22no—
160C32×Dic3C2×D412yes—
161C32×Dic3C2×D412yes—
162C2×#237—91no—
163C2×#235—94no—
164C22×C4×D3—94no—
165C22×C4×D3—91no—
166C3×#123—91no—
167C3×#117—34no—
168#937—91no—
169#929—91no—
170#897—98no—
171#896—92no—
172#895—98no—
173#893—92no—
174#886—34no—
175#886—94no—
176#884—92no—
177#882—94no—
178#882—94no—
179#882—91no—
180#882—184no—
181#67⋊C4—92no—
182#877C3⋊D311yescharacteristic, complemented
183C4×D12—98no—
184C4×D12—98no—
185C23×C32—24no—
186C23×C32C22×D314yes—
187C23×C32C22×D311yescharacteristic
188C22×C3⋊D3—32no—
189C22×C3⋊D3—31no—
190C22×C3⋊D3—38no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—68no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—64no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3—616no—
197C22×C3⋊D3—68no—
198C22×C3⋊D3—1216no—
199C22×C3⋊D3—128no—
200C22×C3⋊D3C22×D311yescharacteristic
201C22×C3×D3—34no—
202C2×C32×C4—28no—
203C2×C32×C4—34no—
204C2×C32×C4—31no—
205C2×C32×C4—31no—
206C2×C32×C4—32no—
207C2×C32×C4C22×D318yes—
208C2×C32×C4C22×D312yes—
209#638—22no—
210#638—62no—
211#638—68no—
212#638—128no—
213C2×C3⋊Dic3—22no—
214C2×C3⋊Dic3—31no—
215C2×C3⋊Dic3—34no—
216C2×C3⋊Dic3—64no—
217C2×C3⋊Dic3C22×D312yes—
218C2×C3⋊Dic3C22×D311yescharacteristic
219C4⋊(C3⋊D3)—64no—
220C4⋊(C3⋊D3)—62no—
221C4⋊(C3⋊D3)—64no—
222C4⋊(C3⋊D3)—616no—
223C4⋊(C3⋊D3)—1216no—
224C4×C3⋊D3—64no—
225C4×C3⋊D3—64no—
226C4×C3⋊D3—62no—
227C4×C3⋊D3—62no—
228#634—62no—
229C3×#67—68no—
230C2×C3×Dic3—34no—
231C2×C3×Dic3—38no—
232C2×C3×Dic3—68no—
233C2×C3×Dic3D12116yescomplemented
234C2×C3×Dic3C22×D318yes—
235C2×C3×Dic3C22×D314yes—
236C2×C3×Dic3C22×D314yes—
237#627—68no—
238#626—616no—
239#626—68no—
240#624—68no—
241#624—616no—
242#624—616no—
243D3×Dic3—68no—
244#525—271noSylow
245C2×C33—22no—
246C2×C33C22×D411yescharacteristic
247C2×C33C2×#4112yes—
248C3⋊(C3⋊D3)—24no—
249C3⋊(C3⋊D3)—41no—
250C3×C3⋊D3—41no—
251C23×D3—94no—
252C23×D3—91no—
253C23×D3—184no—
254C2×C3×D4—91no—
255C22×C3×C4—34no—
256C22×C3×C4—91no—
257C2×#67—34no—
258C2×#67—94no—
259C2×#67—91no—
260C2×#67—92no—
261C2×#67—91no—
262C2×#67—184no—
263C22×Dic3—34no—
264C22×Dic3—91no—
265C22×Dic3C2×C3⋊D311yescharacteristic
266#237—184no—
267#235—1816no—
268C2×D12—98no—
269C2×D12—94no—
270C2×D12—98no—
271C2×D12—92no—
272C2×D12—188no—
273C2×C4×D3—98no—
274C2×C4×D3—94no—
275C2×C4×D3—98no—
276C2×C4×D3—94no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—91no—
280C2×C4×D3—1816no—
281C2×C4×D3—184no—
282C2×Dic6—94no—
283C2×Dic6—91no—
284C3×C4⋊C4—38no—
285C3×C4⋊C4—92no—
286C3×C22⋊C4—24no—
287C3×C22⋊C4—34no—
288C3×C22⋊C4—92no—
289C3×C22⋊C4D3214yes—
290C3×C22⋊C4C2×C3⋊D311yescharacteristic
291C3×C42—38no—
292#217C2×C3⋊D311yescharacteristic
293(C2×D3)⋊C4—38no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—98no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—188no—
301C4⋊Dic3—38no—
302C4⋊Dic3—98no—
303Dic3⋊C4—92no—
304Dic3⋊C4C2×C3⋊D312yescomplemented
305C4×Dic3C2×C3⋊D312yescomplemented
306C22×C32—22no—
307C22×C32—24no—
308C22×C32—28no—
309C22×C32—28no—
310C22×C32—48no—
311C22×C32C2×D1218yes—
312C22×C32D3×D412yes—
313C22×C32C23×D314yes—
314C22×C32C23×D311yescharacteristic
315C2×C3⋊D3—21no—
316C2×C3⋊D3—22no—
317C2×C3⋊D3—34no—
318C2×C3⋊D3—316no—
319C2×C3⋊D3—616no—
320C2×C3⋊D3—68no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—62no—
323C2×C3⋊D3—61no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—632no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—68no—
328C2×C3⋊D3—64no—
329C2×C3⋊D3—68no—
330C2×C3⋊D3—128no—
331C2×C3⋊D3—128no—
332C2×C3⋊D3—1232no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—124no—
335C2×C3×D3—64no—
336C2×C3×D3—68no—
337C32×C4—22no—
338C32×C4—28no—
339C32×C4—32no—
340C32×C4—32no—
341C32×C4—48no—
342C32×C4—62no—
343C3⋊Dic3—21no—
344C3⋊Dic3—22no—
345C3⋊Dic3—61no—
346C3⋊Dic3—64no—
347C3⋊Dic3—124no—
348C3×Dic3—28no—
349C3×Dic3—68no—
350C3×Dic3—64no—
351C3×Dic3C2×D1218yes—
352C3×Dic3C2×D1218yes—
353C2×#41—271no—
354C23×C4—271no—
355#123—91no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—271no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—31no—
364C33#52511yescharacteristic, complemented, Sylow
365C23×C3—24no—
366C23×C3—91no—
367C23×C3C2×D3214yes—
368C23×C3C22×C3⋊D311yescharacteristic
369C22×D3—34no—
370C22×D3—98no—
371C22×D3—92no—
372C22×D3—94no—
373C22×D3—91no—
374C22×D3—188no—
375C22×D3—184no—
376C22×D3—184no—
377C22×D3—1816no—
378C22×D3—188no—
379C22×D3—182no—
380C22×D3—3616no—
381C22×D3—368no—
382C3×D4—182no—
383C2×C3×C4—28no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—316no—
387C2×C3×C4—38no—
388C2×C3×C4—92no—
389C2×C3×C4—91no—
390C2×C3×C4—182no—
391C2×C3×C4C2×D3218yes—
392C2×C3×C4C22×C3⋊D312yes—
393#67—68no—
394#67—184no—
395#67—188no—
396#67—182no—
397#67—182no—
398#67—368no—
399C2×Dic3—34no—
400C2×Dic3—38no—
401C2×Dic3—68no—
402C2×Dic3—94no—
403C2×Dic3—91no—
404C2×Dic3—184no—
405C2×Dic3—182no—
406C2×Dic3C4⋊(C3⋊D3)14yescomplemented
407C2×Dic3C22×C3⋊D312yes—
408C2×Dic3C22×C3⋊D311yescharacteristic
409C2×Dic3C22×C3⋊D311yescharacteristic
410D12—184no—
411D12—1816no—
412D12—188no—
413D12—1816no—
414D12—3616no—
415C4×D3—1816no—
416C4×D3—1816no—
417C4×D3—188no—
418C4×D3—188no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—182no—
422Dic6—188no—
423Dic6—182no—
424C2×C32—22no—
425C2×C32—24no—
426C2×C32—28no—
427C2×C32—28no—
428C2×C32—48no—
429C2×C32C22×D1214yes—
430C2×C32C2×D3×D411yescharacteristic
431C2×C32D3×#4118yes—
432C2×C32D3×#4112yes—
433C3⋊D3—41no—
434C3⋊D3—64no—
435C3⋊D3—616no—
436C3⋊D3—1216no—
437C3⋊D3—121no—
438C3⋊D3—124no—
439C3⋊D3—244no—
440C3×D3—124no—
441C24—271no—
442#41—544no—
443C2×Q8—271no—
444C2×D4—91no—
445C2×D4—272no—
446C2×D4—271no—
447C2×D4—271no—
448C2×D4—272no—
449C22×C4—31no—
450C22×C4—91no—
451C22×C4—272no—
452C22×C4—271no—
453C22×C4—272no—
454C22×C4—271no—
455C22×C4—544no—
456C4⋊C4—32no—
457C4⋊C4—92no—
458C4⋊C4—272no—
459C22⋊C4—31no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—272no—
464C22⋊C4D3×C3⋊D311yescharacteristic
465C42—32no—
466C22×C3—28no—
467C22×C3—22no—
468C22×C3—24no—
469C22×C3—28no—
470C22×C3—48no—
471C22×C3—181no—
472C22×C3—182no—
473C22×C3D3×D1218yes—
474C22×C3C2×C4⋊(C3⋊D3)12yes—
475C22×C3C22×D3214yes—
476C22×C3C23×C3⋊D311yescharacteristic
477C2×D3—64no—
478C2×D3—68no—
479C2×D3—916no—
480C2×D3—94no—
481C2×D3—1816no—
482C2×D3—188no—
483C2×D3—1832no—
484C2×D3—188no—
485C2×D3—188no—
486C2×D3—182no—
487C2×D3—182no—
488C2×D3—184no—
489C2×D3—188no—
490C2×D3—181no—
491C2×D3—182no—
492C2×D3—368no—
493C2×D3—368no—
494C2×D3—3632no—
495C2×D3—368no—
496C2×D3—364no—
497C3×C4—28no—
498C3×C4—22no—
499C3×C4—38no—
500C3×C4—38no—
501C3×C4—48no—
502C3×C4—68no—
503C3×C4—181no—
504C3×C4—182no—
505Dic3—22no—
506Dic3—64no—
507Dic3—68no—
508Dic3—184no—
509Dic3—181no—
510Dic3—364no—
511Dic3C2×C4⋊(C3⋊D3)12yes—
512Dic3C2×C4⋊(C3⋊D3)12yes—
513C32—24no—
514C32#552414yescomplemented
515C32#565611yescharacteristic, complemented
516C23—91no—
517C23—272no—
518C23—271no—
519C23—544no—
520C23—542no—
521C23C2×D3×C3⋊D311yescharacteristic
522Q8—542no—
523D4—182no—
524D4—544no—
525D4—542no—
526D4—544no—
527D4—542no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—34no—
531C2×C4—32no—
532C2×C4—92no—
533C2×C4—91no—
534C2×C4—182no—
535C2×C4—271no—
536C2×C4—544no—
537C2×C4—544no—
538C2×C4—542no—
539C2×C4—542no—
540C2×C4C2×D3×C3⋊D312yes—
541C2×C3—28no—
542C2×C3—22no—
543C2×C3—24no—
544C2×C3—28no—
545C2×C3—48no—
546C2×C3—361no—
547C2×C3C2×D3×D1214yes—
548C2×C3#6431218yes—
549C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
550C2×C3#41×C3⋊D312yes—
551D3—124no—
552D3—1816no—
553D3—184no—
554D3—3616no—
555D3—364no—
556D3—361no—
557D3—724no—
558C22—22no—
559C22—181no—
560C22—182no—
561C22—274no—
562C22—548no—
563C22—542no—
564C22—542no—
565C22—541no—
566C22—542no—
567C22D3×C4⋊(C3⋊D3)12yes—
568C22C22×D3×C3⋊D311yescharacteristic
569C4—22no—
570C4—32no—
571C4—32no—
572C4—62no—
573C4—182no—
574C4—181no—
575C4—541no—
576C3—24no—
577C3#1059407914yescomplemented
578C3#1059599511yescharacteristic, complemented
579C2—22no—
580C2—361no—
581C2—544no—
582C2—1081no—
583C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
584C2#1172458612yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#6289C2semidirect
2. C2×#89031C2semidirect
3. #11723923C2semidirect
4. C3×#64095C2semidirect
5. C32×#877C2semidirect
6. #11723639C2semidirect
7. #11723608C2semidirect
8. #11723486C2semidirect
9. #11723485C2semidirect
10. #11723484C2semidirect
11. #11723471C2semidirect
20. C2×#6293C22semidirect
22. C2×#6292C22semidirect
26. C2×#6289C22semidirect
28. C2×Dic3×C3⋊D3C22semidirect
31. C3×#3752C22semidirect
32. C3×C4⋊(C3⋊Dic3)C22semidirect
34. C32×Dic3⋊C4C22semidirect
35. C32×C4×Dic3C22semidirect
36. Dic3⋊(C3⋊Dic3)C22semidirect
38. #88804C22semidirect
47. #64095D3semidirect
55. C3×#877D3semidirect
76. C2×C32×Dic3D4semidirect
133. #3752C2×D3semidirect
135. C4⋊(C3⋊Dic3)C2×D3semidirect
139. C3×Dic3⋊C4C2×D3semidirect
140. C3×C4×Dic3C2×D3semidirect
152. C2×C3⋊(C3⋊D3)#41semidirect
182. #877C3⋊D3semidirect
233. C2×C3×Dic3D12semidirect
304. Dic3⋊C4C2×C3⋊D3semidirect
305. C4×Dic3C2×C3⋊D3semidirect
364. C33#525semidirect
406. C2×Dic3C4⋊(C3⋊D3)semidirect
514. C32#5524semidirect
515. C32#5656semidirect
577. C3#10594079semidirect
578. C3#10595995semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#890312
#117239232
C3×#640952
C32×#8772
#117236392
#117236082
#117234862
#117234852
#117234842
#117234712
#105959953
#105940793