Finite Group Databasebeta

← #13178722

Subgroups — #13178722

Order 1728.

1132 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C212yesmaximal, complemented
2C3×#64155C211yesmaximal, characteristic, complemented
3#11723695C211yesmaximal, characteristic
4#11723694C212yesmaximal, complemented
5#11723690C212yesmaximal, complemented
6#11723684C212yesmaximal, complemented
7#11723682C211yesmaximal, characteristic, complemented
8#11723675C212yesmaximal, complemented
9#11723669C212yesmaximal, complemented
10#10595415—31nomaximal
11#10595378—32nomaximal
12C2×C33×Q8C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2212yes—
15C2×#6295C2212yes—
16C2×#6295C2211yescharacteristic
17C2×#6294C2211yescharacteristic
18C2×#6294C2211yescharacteristic
19C3×#3752C2212yes—
20C3×#3752C2212yes—
21C3×#3752C2212yes—
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×#3750C2212yes—
27#88815C2211yescharacteristic
28#88815C2212yes—
29#88815C2212yes—
30#88814C2212yes—
31#88814C2212yes—
32#88813C2212yes—
33#64155D312yescomplemented
34#64155D311yescharacteristic, complemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—32no—
38C3×#946—62no—
39C3×#946—61no—
40#63912—31no—
41#63906—32no—
42#63905—32no—
43#63904—32no—
44#63891—32no—
45#63890—32no—
46#63883—31no—
47#63881—32no—
48#63879—32no—
49D3×Dic3⋊C4—32no—
50#63861—32no—
51#63858—32no—
52#63854—32no—
53#63853—32no—
54#63850—32no—
55#63848—32no—
56#63847—32no—
57#63846—32no—
58#63845—32no—
59C22×C3×C3⋊D3C2312yes—
60C22×C3×C3⋊D3C2311yescharacteristic
61C33×Q8—41no—
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×C4×C3⋊D3—42no—
70C3×C4×C3⋊D3—41no—
71#6295—24no—
72#6295—41no—
73#6294—22no—
74#6294—41no—
75#5645—91no—
76#5638—92no—
77C2×C32×Q8—22no—
78C2×C32×Q8—21no—
79C2×C32×Q8—41no—
80C2×C32×Q8C2×D312yes—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D312yes—
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—62no—
89C2×#626—32no—
90C2×#626—32no—
91C2×#626—32no—
92C2×#625—32no—
93C2×#625—31no—
94C2×#624—31no—
95C2×#624—31no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98#3752C2×D312yes—
99#3752C2×D312yes—
100#3752C2×D312yes—
101C4⋊(C3⋊Dic3)C2×D312yes—
102C4⋊(C3⋊Dic3)C2×D311yescharacteristic
103#3750C2×D312yes—
104#3750C2×D312yes—
105#3750C2×D312yes—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—62no—
113C3×(C2×D3)⋊C4—62no—
114C3×(C2×D3)⋊C4—62no—
115C3×C4⋊Dic3—32no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—32no—
118C3×C4⋊Dic3—62no—
119C3×C4⋊Dic3—61no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—62no—
127C3×Dic3⋊C4—62no—
128C3×Dic3⋊C4—62no—
129#3724—32no—
130#3724—31no—
131#3724—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137#3722—32no—
138#3722—32no—
139(C2×D3)⋊Dic3—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142(C2×D3)⋊Dic3—32no—
143Dic32—32no—
144Dic32—32no—
145Dic32—32no—
146C22×C33C2411yescharacteristic
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3—21no—
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3#4112yes—
153C33×C4—22no—
154C33×C4—21no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—22no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3—21no—
159C3×C3⋊Dic3—22no—
160C3×C22⋊Q8—92no—
161C3×C22⋊Q8—91no—
162#946—32no—
163#946—32no—
164#946—32no—
165#946—62no—
166#946—61no—
167#930—91no—
168#929—92no—
169C4×#67—92no—
170C2×Dic3⋊C4—91no—
171#898—92no—
172#898—92no—
173#894—92no—
174#894—92no—
175#893—92no—
176D3×C4⋊C4—92no—
177#890—92no—
178#889—92no—
179#889—92no—
180#886—92no—
181#884—92no—
182#883—92no—
183#883—92no—
184#67⋊C4—92no—
185#879—92no—
186C22×C3⋊D3C22×D312yes—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—32no—
189C22×C3×D3—32no—
190C22×C3×D3—32no—
191C22×C3×D3—61no—
192C22×C3×D3—62no—
193C32×Q8—42no—
194C32×Q8—41no—
195C32×Q8—44no—
196C32×Q8—42no—
197C32×Q8—44no—
198C2×C32×C4—22no—
199C2×C32×C4—21no—
200C2×C32×C4—22no—
201C2×C32×C4—22no—
202C2×C32×C4—22no—
203C2×C32×C4—42no—
204C2×C32×C4—41no—
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D312yes—
207C2×C32×C4C22×D312yes—
208C2×C32×C4C22×D312yes—
209C2×C32×C4C22×D312yes—
210C2×C3⋊Dic3C22×D312yes—
211C2×C3⋊Dic3C22×D312yes—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D312yes—
214C2×C3⋊Dic3C22×D311yescharacteristic
215C4×C3⋊D3—42no—
216C4×C3⋊D3—41no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—62no—
227C2×C3×Dic3—61no—
228C2×C3×Dic3—62no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3—62no—
231C3×C4×D3—122no—
232C3×C4×D3—122no—
233C3×C4×D3—122no—
234C3×C4×D3—124no—
235C3×C4×D3—122no—
236#626—64no—
237#626—64no—
238#626—122no—
239#625—64no—
240#625—121no—
241#624—62no—
242#624—121no—
243D3×Dic3—64no—
244D3×Dic3—122no—
245#539—271noSylow
246C2×C33C2×#4111yescharacteristic
247C2×C33H(5,𝔽2)11yescharacteristic
248C2×C332−1+411yescharacteristic
249C3×C3⋊D3—22no—
250C3×C3⋊D3—42no—
251C2×C3×Q8—22no—
252C2×C3×Q8—21no—
253C2×C3×Q8—41no—
254C2×C3×Q8D3212yes—
255C2×C3×Q8D3211yescharacteristic
256C22×C3×C4—91no—
257C22×C3×C4—92no—
258C2×#67—92no—
259C2×#67—92no—
260C2×#67—92no—
261C22×Dic3—91no—
262C22×Dic3—91no—
263C2×D12—92no—
264C2×D12—91no—
265C2×C4×D3—32no—
266C2×C4×D3—32no—
267C2×C4×D3—32no—
268C2×C4×D3—61no—
269C2×C4×D3—62no—
270C2×C4×D3—92no—
271C2×C4×D3—92no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4—92no—
276C3×C4⋊C4—91no—
277C3×C22⋊C4—92no—
278C3×C22⋊C4—92no—
279C3×C22⋊C4—92no—
280#217—92no—
281#217—92no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—32no—
286(C2×D3)⋊C4—32no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—62no—
289(C2×D3)⋊C4—62no—
290(C2×D3)⋊C4—62no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295C4⋊Dic3—32no—
296C4⋊Dic3—32no—
297C4⋊Dic3—32no—
298C4⋊Dic3—62no—
299C4⋊Dic3—61no—
300C4⋊Dic3—92no—
301C4⋊Dic3—92no—
302C4⋊Dic3—91no—
303Dic3⋊C4—32no—
304Dic3⋊C4—32no—
305Dic3⋊C4—32no—
306Dic3⋊C4—32no—
307Dic3⋊C4—32no—
308Dic3⋊C4—32no—
309Dic3⋊C4—62no—
310Dic3⋊C4—62no—
311Dic3⋊C4—62no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C4×Dic3—92no—
320C22×C32—22no—
321C22×C32—21no—
322C22×C32—41no—
323C22×C32C23×D311yescharacteristic
324C22×C32C23×D312yes—
325C2×C3⋊D3—22no—
326C2×C3⋊D3—21no—
327C2×C3⋊D3—21no—
328C2×C3⋊D3—22no—
329C2×C3⋊D3—22no—
330C2×C3⋊D3#23912yes—
331C2×C3×D3—34no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—64no—
336C2×C3×D3—62no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—62no—
340C2×C3×D3—62no—
341C2×C3×D3—62no—
342C2×C3×D3—121no—
343C2×C3×D3—122no—
344C2×C3×D3—121no—
345C2×C3×D3—122no—
346C32×C4—22no—
347C32×C4—21no—
348C32×C4—24no—
349C32×C4—22no—
350C32×C4—22no—
351C32×C4—22no—
352C32×C4—22no—
353C32×C4—42no—
354C32×C4—42no—
355C32×C4—42no—
356C32×C4—44no—
357C32×C4—42no—
358C3⋊Dic3—22no—
359C3⋊Dic3—21no—
360C3⋊Dic3—22no—
361C3⋊Dic3—22no—
362C3⋊Dic3—22no—
363C3×Dic3—62no—
364C3×Dic3—62no—
365C3×Dic3—62no—
366C3×Dic3—62no—
367C3×Dic3—62no—
368C3×Dic3—62no—
369C3×Dic3—62no—
370C3×Dic3—62no—
371C3×Dic3—62no—
372C3×Dic3—122no—
373C3×Dic3—121no—
374C3×Dic3—122no—
375C3×Dic3—122no—
376C3×Dic3—122no—
377#126—272no—
378#125—272no—
379#123—272no—
380#123—272no—
381C22⋊Q8—91no—
382C22⋊Q8—92no—
383#121—271no—
384C4×D4—272no—
385C2×C4⋊C4—271no—
386C33#53911yescharacteristic, complemented, Sylow
387C23×C3—91no—
388C23×C3—92no—
389C22×D3—32no—
390C22×D3—32no—
391C22×D3—32no—
392C22×D3—61no—
393C22×D3—62no—
394C3×Q8—42no—
395C3×Q8—41no—
396C3×Q8—44no—
397C3×Q8—42no—
398C3×Q8—44no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—42no—
405C2×C3×C4—41no—
406C2×C3×C4—92no—
407C2×C3×C4—92no—
408C2×C3×C4—92no—
409C2×C3×C4—91no—
410C2×C3×C4—92no—
411C2×C3×C4—362no—
412C2×C3×C4—361no—
413C2×C3×C4C2×D3212yes—
414C2×C3×C4C2×D3211yescharacteristic
415C2×C3×C4C2×D3212yes—
416C2×C3×C4C2×D3212yes—
417C2×C3×C4C2×D3212yes—
418#67—184no—
419#67—184no—
420#67—362no—
421C2×Dic3—32no—
422C2×Dic3—32no—
423C2×Dic3—32no—
424C2×Dic3—32no—
425C2×Dic3—32no—
426C2×Dic3—32no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—62no—
431C2×Dic3—61no—
432C2×Dic3—62no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—182no—
436C2×Dic3—361no—
437D12—184no—
438D12—361no—
439C4×D3—122no—
440C4×D3—122no—
441C4×D3—122no—
442C4×D3—124no—
443C4×D3—122no—
444C4×D3—184no—
445C4×D3—362no—
446C2×C32—22no—
447C2×C32—21no—
448C2×C32—22no—
449C2×C32—21no—
450C2×C32—22no—
451C2×C32—21no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32#100412yes—
456C2×C32#100411yescharacteristic
457C2×C32C2×#23911yescharacteristic
458C2×C32#100712yes—
459C2×C32D3×#4112yes—
460C2×C32#101011yescharacteristic
461C3⋊D3—22no—
462C3⋊D3—42no—
463C3×D3—64no—
464C3×D3—122no—
465C3×D3—124no—
466C3×D3—122no—
467C3×D3—122no—
468C2×Q8#111311yescharacteristic
469C2×D4—272no—
470C2×D4—271no—
471C22×C4—91no—
472C22×C4—92no—
473C22×C4—271no—
474C22×C4—271no—
475C4⋊C4—92no—
476C4⋊C4—92no—
477C4⋊C4—92no—
478C4⋊C4—92no—
479C4⋊C4—91no—
480C4⋊C4—272no—
481C4⋊C4—271no—
482C4⋊C4—272no—
483C22⋊C4—92no—
484C22⋊C4—92no—
485C22⋊C4—92no—
486C22⋊C4—272no—
487C22⋊C4—272no—
488C42—272no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—92no—
493C22×C3—181no—
494C22×C3—181no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3—182no—
498C22×C3C22×D3212yes—
499C22×C3C22×D3211yescharacteristic
500C2×D3—34no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—64no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—121no—
512C2×D3—122no—
513C2×D3—121no—
514C2×D3—122no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—21no—
522C3×C4—42no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—44no—
526C3×C4—42no—
527C3×C4—182no—
528C3×C4—182no—
529C3×C4—182no—
530C3×C4—182no—
531C3×C4—181no—
532Dic3—62no—
533Dic3—62no—
534Dic3—62no—
535Dic3—62no—
536Dic3—62no—
537Dic3—62no—
538Dic3—62no—
539Dic3—62no—
540Dic3—62no—
541Dic3—122no—
542Dic3—121no—
543Dic3—122no—
544Dic3—122no—
545Dic3—122no—
546C32—22no—
547C32—21no—
548C32—41no—
549C32#563812yescomplemented
550C32#564511yescharacteristic, complemented
551C23—91no—
552C23—92no—
553Q8—41no—
554D4—544no—
555D4—1081no—
556C2×C4—92no—
557C2×C4—92no—
558C2×C4—92no—
559C2×C4—92no—
560C2×C4—91no—
561C2×C4—362no—
562C2×C4—361no—
563C2×C4—542no—
564C2×C4—1081no—
565C2×C4C2×#111312yes—
566C2×C4C2×#111311yescharacteristic
567C2×C3—21no—
568C2×C3—22no—
569C2×C3—22no—
570C2×C3—21no—
571C2×C3—22no—
572C2×C3—21no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—182no—
577C2×C3—362no—
578C2×C3#6431512yes—
579C2×C3#6431611yescharacteristic
580C2×C3D3×#23912yes—
581C2×C3#6431911yescharacteristic
582C2×C3#6433012yes—
583C2×C3#6433011yescharacteristic
584D3—64no—
585D3—122no—
586D3—124no—
587D3—122no—
588D3—122no—
589C22—92no—
590C22—181no—
591C22—181no—
592C22—182no—
593C22—182no—
594C22—182no—
595C22C22×#111311yescharacteristic
596C4—22no—
597C4—21no—
598C4—182no—
599C4—182no—
600C4—182no—
601C4—182no—
602C4—181no—
603C3—22no—
604C3—21no—
605C3—41no—
606C3#1059537812yescomplemented
607C3#1059541511yescharacteristic, complemented
608C2—182no—
609C2—362no—
610C2#1172462411yescharacteristic
611C2#1172462611yescharacteristic
612C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radical#11723695C2
hypercenterC2×Q8#1113
hypocenterC33#539
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64155C2semidirect
2. C3×#64155C2semidirect
4. #11723694C2semidirect
5. #11723690C2semidirect
6. #11723684C2semidirect
7. #11723682C2semidirect
8. #11723675C2semidirect
9. #11723669C2semidirect
33. #64155D3semidirect
34. #64155D3semidirect
386. C33#539semidirect
549. C32#5638semidirect
550. C32#5645semidirect
606. C3#10595378semidirect
607. C3#10595415semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
#117236952
#117236942
#117236902
#117236842
#117236822
#117236752
#117236692
#105954153
#105953783