Finite Group Databasebeta

← #13178748

Subgroups — #13178748

Order 1728.

1142 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic, complemented
2C3×#64155C212yesmaximal, complemented
3#11723695C212yesmaximal, complemented
4#11723694C212yesmaximal, complemented
5#11723692C212yesmaximal, complemented
6#11723690C212yesmaximal, complemented
7#11723687C212yesmaximal, complemented
8#11723679C211yesmaximal, characteristic, complemented
9#11723672C211yesmaximal, characteristic, complemented
10#10595418—31nomaximal
11#10595417—32nomaximal
12C2×C33×Q8C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2212yes—
15C2×#6296C2211yescharacteristic
16C2×#6295C2211yescharacteristic
17C2×#6295C2212yes—
18C2×#6295C2212yes—
19C2×#6294C2212yes—
20C3×#3752C2212yes—
21C3×#3752C2212yes—
22C3×#3752C2212yes—
23C3×#3752C2212yes—
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×#3750C2212yes—
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88815C2211yescharacteristic
29#88815C2212yes—
30#88815C2211yescharacteristic
31#88814C2212yes—
32#88814C2212yes—
33#64156D311yescharacteristic, complemented
34#64155D312yescomplemented
35C3×#947—32no—
36C3×#947—61no—
37C3×#946—32no—
38C3×#946—32no—
39C3×#946—62no—
40#63915—31no—
41#63912—32no—
42#63910—32no—
43#63906—32no—
44#63906—32no—
45#63905—32no—
46#63899—32no—
47#63895—32no—
48#63893—32no—
49#63891—32no—
50#63890—32no—
51#63886—32no—
52#63885—32no—
53#63870—31no—
54#63867—32no—
55#63854—32no—
56#63853—32no—
57#63852—32no—
58#63850—32no—
59C22×C3×C3⋊D3C2312yes—
60C22×C3×C3⋊D3C2312yes—
61C33×Q8—41no—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2312yes—
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×C4⋊(C3⋊D3)—41no—
70C3×C4×C3⋊D3—42no—
71#6296—41no—
72#6295—42no—
73#6295—41no—
74#6295—42no—
75#6294—42no—
76#5684—91no—
77#5647—92no—
78C2×C32×Q8—22no—
79C2×C32×Q8—21no—
80C2×C32×Q8—41no—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C32×Q8C2×D312yes—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D312yes—
85C2×C3×D12—32no—
86C2×C3×D12—61no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—62no—
90C2×#627—32no—
91C2×#627—31no—
92C2×#626—32no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#625—32no—
98C2×#625—32no—
99C2×#625—31no—
100C2×#624—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103#3752C2×D312yes—
104#3752C2×D312yes—
105#3752C2×D312yes—
106#3752C2×D312yes—
107C4⋊(C3⋊Dic3)C2×D312yes—
108#3750C2×D312yes—
109#3750C2×D312yes—
110C4×C3⋊Dic3C2×D311yescharacteristic
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—62no—
120C3×(C2×D3)⋊C4—62no—
121C3×(C2×D3)⋊C4—62no—
122C3×(C2×D3)⋊C4—62no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—62no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—62no—
131C3×Dic3⋊C4—62no—
132C3×C4×Dic3—32no—
133C3×C4×Dic3—61no—
134#3724—31no—
135#3724—31no—
136#3724—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141#3722—32no—
142#3722—32no—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—32no—
145(C2×D3)⋊Dic3—32no—
146(C2×D3)⋊Dic3—32no—
147C22×C33C2411yescharacteristic
148C2×C3×C3⋊D3—22no—
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—22no—
154C33×C4—22no—
155C33×C4—21no—
156C3×C3⋊Dic3—22no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3—21no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3—21no—
161C3×#124—91no—
162C3×C22⋊Q8—92no—
163#947—32no—
164#947—61no—
165#946—32no—
166#946—32no—
167#946—62no—
168#939—92no—
169#930—92no—
170#929—92no—
171#897—92no—
172#896—92no—
173#895—92no—
174#894—92no—
175#894—92no—
176#889—92no—
177#886—92no—
178#885—92no—
179#884—92no—
180#884—92no—
181#884—92no—
182#883—92no—
183#883—92no—
184#878—92no—
185#875—91no—
186#870—91no—
187C22×C3⋊D3C22×D312yes—
188C22×C3⋊D3C22×D312yes—
189C22×C3×D3—32no—
190C22×C3×D3—32no—
191C22×C3×D3—32no—
192C22×C3×D3—32no—
193C22×C3×D3—62no—
194C22×C3×D3—62no—
195C32×Q8—42no—
196C32×Q8—41no—
197C32×Q8—44no—
198C32×Q8—42no—
199C32×Q8—44no—
200C2×C32×C4—21no—
201C2×C32×C4—22no—
202C2×C32×C4—22no—
203C2×C32×C4—22no—
204C2×C32×C4—22no—
205C2×C32×C4—41no—
206C2×C32×C4—42no—
207C2×C32×C4C22×D312yes—
208C2×C32×C4C22×D311yescharacteristic
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D312yes—
211C2×C32×C4C22×D312yes—
212C2×C3⋊Dic3C22×D311yescharacteristic
213C2×C3⋊Dic3C22×D312yes—
214C2×C3⋊Dic3C22×D312yes—
215C2×C3⋊Dic3C22×D311yescharacteristic
216C2×C3⋊Dic3C22×D312yes—
217C4⋊(C3⋊D3)—41no—
218C4×C3⋊D3—42no—
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—32no—
227C2×C3×Dic3—61no—
228C2×C3×Dic3—61no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3—62no—
231C2×C3×Dic3—62no—
232C3×D12—122no—
233C3×D12—122no—
234C3×C4×D3—122no—
235C3×C4×D3—122no—
236C3×C4×D3—124no—
237#627—122no—
238#627—121no—
239#626—122no—
240#626—122no—
241#626—122no—
242#626—122no—
243#626—122no—
244#625—122no—
245#625—122no—
246#625—121no—
247#624—122no—
248D3×Dic3—122no—
249D3×Dic3—122no—
250#562—271noSylow
251C2×C33H(5,𝔽2)11yescharacteristic
252C2×C33H(5,𝔽2)11yescharacteristic
253C2×C332−1+411yescharacteristic
254C3×C3⋊D3—42no—
255C3×C3⋊D3—42no—
256C2×C3×Q8—22no—
257C2×C3×Q8—21no—
258C2×C3×Q8—41no—
259C2×C3×Q8D3211yescharacteristic
260C2×C3×Q8D3212yes—
261C2×C3×D4—91no—
262C22×C3×C4—92no—
263C2×#67—92no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C22×Dic3—92no—
269C2×D12—32no—
270C2×D12—61no—
271C2×D12—91no—
272C2×D12—92no—
273C2×D12—92no—
274C2×C4×D3—32no—
275C2×C4×D3—32no—
276C2×C4×D3—62no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×Dic6—91no—
280C2×Dic6—92no—
281C3×C4⋊C4—92no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—92no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—92no—
287C3×C22⋊C4—92no—
288C3×C42—91no—
289#217—92no—
290#217—92no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—32no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—62no—
300(C2×D3)⋊C4—62no—
301(C2×D3)⋊C4—62no—
302(C2×D3)⋊C4—62no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307C4⋊Dic3—32no—
308C4⋊Dic3—32no—
309C4⋊Dic3—62no—
310C4⋊Dic3—91no—
311C4⋊Dic3—91no—
312C4⋊Dic3—92no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—62no—
318Dic3⋊C4—62no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323C4×Dic3—32no—
324C4×Dic3—61no—
325C22×C32—22no—
326C22×C32—21no—
327C22×C32—41no—
328C22×C32C23×D312yes—
329C22×C32C23×D311yescharacteristic
330C2×C3⋊D3—22no—
331C2×C3⋊D3—22no—
332C2×C3⋊D3—22no—
333C2×C3⋊D3—22no—
334C2×C3⋊D3—22no—
335C2×C3⋊D3—22no—
336C2×C3×D3—64no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—62no—
340C2×C3×D3—62no—
341C2×C3×D3—62no—
342C2×C3×D3—62no—
343C2×C3×D3—62no—
344C2×C3×D3—62no—
345C2×C3×D3—62no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—62no—
349C2×C3×D3—122no—
350C2×C3×D3—122no—
351C2×C3×D3—122no—
352C2×C3×D3—122no—
353C2×C3×D3—122no—
354C32×C4—22no—
355C32×C4—21no—
356C32×C4—24no—
357C32×C4—22no—
358C32×C4—22no—
359C32×C4—22no—
360C32×C4—22no—
361C32×C4—42no—
362C32×C4—42no—
363C32×C4—42no—
364C32×C4—44no—
365C32×C4—42no—
366C3⋊Dic3—22no—
367C3⋊Dic3—22no—
368C3⋊Dic3—22no—
369C3⋊Dic3—21no—
370C3⋊Dic3—21no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—62no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—122no—
380C3×Dic3—121no—
381C3×Dic3—122no—
382C3×Dic3—121no—
383C3×Dic3—122no—
384#125—271no—
385#124—91no—
386#124—271no—
387#123—272no—
388#123—272no—
389C22⋊Q8—92no—
390C22⋊Q8—272no—
391#121—272no—
392#121—272no—
393C33#56211yescharacteristic, complemented, Sylow
394C23×C3—92no—
395C23×C3—92no—
396C22×D3—32no—
397C22×D3—32no—
398C22×D3—32no—
399C22×D3—32no—
400C22×D3—62no—
401C22×D3—62no—
402C3×Q8—41no—
403C3×Q8—42no—
404C3×Q8—44no—
405C3×Q8—42no—
406C3×Q8—44no—
407C3×D4—361no—
408C2×C3×C4—21no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—41no—
414C2×C3×C4—42no—
415C2×C3×C4—91no—
416C2×C3×C4—92no—
417C2×C3×C4—92no—
418C2×C3×C4—91no—
419C2×C3×C4—92no—
420C2×C3×C4—362no—
421C2×C3×C4C2×D3212yes—
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C2×D3212yes—
424C2×C3×C4C2×D3212yes—
425C2×C3×C4C2×D3212yes—
426#67—362no—
427#67—362no—
428#67—362no—
429#67—362no—
430#67—362no—
431C2×Dic3—32no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—62no—
442C2×Dic3—62no—
443C2×Dic3—62no—
444C2×Dic3—362no—
445D12—122no—
446D12—122no—
447D12—361no—
448D12—362no—
449D12—362no—
450C4×D3—122no—
451C4×D3—122no—
452C4×D3—124no—
453C4×D3—362no—
454C4×D3—362no—
455Dic6—362no—
456Dic6—361no—
457C2×C32—22no—
458C2×C32—21no—
459C2×C32—22no—
460C2×C32—21no—
461C2×C32—22no—
462C2×C32—21no—
463C2×C32—41no—
464C2×C32—41no—
465C2×C32—41no—
466C2×C32#100412yes—
467C2×C32#100411yescharacteristic
468C2×C32#100712yes—
469C2×C32#100912yes—
470C2×C32#100911yescharacteristic
471C2×C32#101011yescharacteristic
472C3⋊D3—42no—
473C3⋊D3—42no—
474C3×D3—124no—
475C3×D3—122no—
476C3×D3—122no—
477C3×D3—122no—
478C3×D3—122no—
479C3×D3—242no—
480C2×Q8—271no—
481C2×Q8#111311yescharacteristic
482C2×D4—91no—
483C2×D4—272no—
484C2×D4—272no—
485C2×D4—271no—
486C22×C4—92no—
487C22×C4—272no—
488C4⋊C4—92no—
489C4⋊C4—92no—
490C4⋊C4—92no—
491C4⋊C4—272no—
492C4⋊C4—271no—
493C4⋊C4—271no—
494C22⋊C4—92no—
495C22⋊C4—92no—
496C22⋊C4—92no—
497C22⋊C4—92no—
498C22⋊C4—272no—
499C22⋊C4—272no—
500C42—91no—
501C22×C3—22no—
502C22×C3—21no—
503C22×C3—41no—
504C22×C3—182no—
505C22×C3—182no—
506C22×C3—182no—
507C22×C3—182no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3C22×D3212yes—
511C22×C3C22×D3211yescharacteristic
512C2×D3—64no—
513C2×D3—62no—
514C2×D3—62no—
515C2×D3—62no—
516C2×D3—62no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—122no—
526C2×D3—122no—
527C2×D3—122no—
528C2×D3—122no—
529C2×D3—122no—
530C3×C4—24no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—21no—
537C3×C4—42no—
538C3×C4—42no—
539C3×C4—42no—
540C3×C4—44no—
541C3×C4—42no—
542C3×C4—181no—
543C3×C4—182no—
544C3×C4—181no—
545C3×C4—182no—
546C3×C4—182no—
547Dic3—62no—
548Dic3—62no—
549Dic3—62no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—122no—
556Dic3—121no—
557Dic3—122no—
558Dic3—122no—
559Dic3—121no—
560C32—22no—
561C32—21no—
562C32—41no—
563C32#564712yescomplemented
564C32#568411yescharacteristic, complemented
565C23—92no—
566C23—92no—
567Q8—41no—
568Q8—1081no—
569D4—361no—
570D4—1082no—
571D4—1081no—
572D4—1082no—
573C2×C4—91no—
574C2×C4—91no—
575C2×C4—92no—
576C2×C4—92no—
577C2×C4—92no—
578C2×C4—362no—
579C2×C4—1082no—
580C2×C4C2×#111311yescharacteristic
581C2×C4C2×#111312yes—
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—22no—
585C2×C3—21no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—41no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—362no—
592C2×C3—362no—
593C2×C3#6431512yes—
594C2×C3#6431611yescharacteristic
595C2×C3#6432012yes—
596C2×C3#6432011yescharacteristic
597C2×C3#6433012yes—
598C2×C3#6433011yescharacteristic
599D3—124no—
600D3—122no—
601D3—122no—
602D3—122no—
603D3—122no—
604D3—242no—
605C22—182no—
606C22—182no—
607C22—182no—
608C22—182no—
609C22—182no—
610C22—182no—
611C22C22×#111311yescharacteristic
612C4—22no—
613C4—21no—
614C4—181no—
615C4—182no—
616C4—182no—
617C4—181no—
618C4—182no—
619C3—22no—
620C3—21no—
621C3—41no—
622C3#1059541712yescomplemented
623C3#1059541811yescharacteristic, complemented
624C2—362no—
625C2—362no—
626C2#1172462411yescharacteristic
627C2#1172462711yescharacteristic
628C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#562
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C3×#64155C2semidirect
3. #11723695C2semidirect
4. #11723694C2semidirect
5. #11723692C2semidirect
6. #11723690C2semidirect
7. #11723687C2semidirect
8. #11723679C2semidirect
9. #11723672C2semidirect
33. #64156D3semidirect
34. #64155D3semidirect
393. C33#562semidirect
563. C32#5647semidirect
564. C32#5684semidirect
622. C3#10595417semidirect
623. C3#10595418semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#641552
#117236952
#117236942
#117236922
#117236902
#117236872
#117236792
#117236722
#105954183
#105954173