Finite Group Databasebeta

← #13147879

Subgroups — #13147879

Order 1728.

927 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#272C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721156C211yesmaximal, characteristic, complemented
5#11721156C211yesmaximal, characteristic, complemented
6#11721129C211yesmaximal, characteristic, complemented
7#11721127C212yesmaximal, complemented
8#11721124C212yesmaximal, complemented
9#11721104C212yesmaximal, complemented
10#11721088C212yesmaximal, complemented
11#11721081C211yesmaximal, characteristic, complemented
12#10593982—31nomaximal
13#10593923—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#88542C2212yes—
24C4⋊#1084C2211yescharacteristic
25#3752⋊C3C2211yescharacteristic
26#3752⋊C3C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#88451C2212yes—
33#88451C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
38#64079D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×C4×D12—31no—
42C3×C4×D12—61no—
43C3×#870—31no—
44#63916—31no—
45#63916—31no—
46#63910—32no—
47#63908—31no—
48#63908—31no—
49#63905—32no—
50#63904—32no—
51#63900—31no—
52D3×C4⋊Dic3—31no—
53#63885—32no—
54#63879—32no—
55#63861—32no—
56#63858—32no—
57#63850—32no—
58#63849—31no—
59#63841—31no—
60C22×#272C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2312yes—
67C4⋊#272—22no—
68C4×#272—42no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4⋊((C3⋊D3)⋊C3)—22no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—22no—
76#6200—24no—
77#6200—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79#5695—91no—
80#5565—91no—
81#5558—91no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—61no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—62no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#625—32no—
97C2×#625—32no—
98C2×#624—32no—
99C2×D3×Dic3—32no—
100C32×C42—21no—
101C32×C42C2×D311yescharacteristic
102C32×C42C2×D311yescharacteristic
103#3752C2×D311yescharacteristic
104#3752C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108#3750C2×D312yes—
109#3750C2×D312yes—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—31no—
113C3×(C2×D3)⋊C4—31no—
114C3×(C2×D3)⋊C4—62no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—61no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123#3724—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131Dic32—32no—
132Dic32—32no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×#272—22no—
135C2×#272—22no—
136C2×#272—22no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3—21no—
139C2×(C3⋊D3)⋊C3—21no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3#4112yes—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)#4112yes—
146#1084—22no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—22no—
151C3×#125—91no—
152C3×C4×D4—91no—
153C3×C4×D4—91no—
154#939—92no—
155#938—91no—
156#938—91no—
157#930—91no—
158#930—91no—
159#929—92no—
160C4×#67—92no—
161#926—91no—
162C2×C4⋊Dic3—91no—
163#898—92no—
164#898—92no—
165#895—92no—
166#895—91no—
167#895—91no—
168#894—92no—
169#893—92no—
170#892—91no—
171#892—91no—
172D3×C4⋊C4—91no—
173D3×C4⋊C4—91no—
174#884—92no—
175#883—92no—
176#67⋊C4—92no—
177#879—92no—
178C4×D12—31no—
179C4×D12—31no—
180C4×D12—31no—
181C4×D12—61no—
182C4×D12—91no—
183#870—31no—
184#870—91no—
185C22×C3⋊D3C22×D311yescharacteristic
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3×D3—31no—
188C22×C3×D3—31no—
189C22×C3×D3—32no—
190C22×C3×D3—32no—
191C22×C3×D3—62no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D312yes—
205C4⋊(C3⋊D3)—22no—
206C4×C3⋊D3—22no—
207C4×C3⋊D3—22no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—62no—
215C3×D12—62no—
216C3×D12—64no—
217C3×D12—62no—
218C3×D12—62no—
219C3×C4×D3—62no—
220C3×C4×D3—62no—
221C3×C4×D3—122no—
222C3×C4×D3—122no—
223C3×C4×D3—124no—
224#626—64no—
225#626—122no—
226#625—64no—
227#625—122no—
228#624—122no—
229D3×Dic3—122no—
230#553—271noSylow
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232C2×H(3,𝔽3)C2×#4111yescharacteristic
233C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
234#272—42no—
235(C3⋊D3)⋊C3—22no—
236(C3⋊D3)⋊C3—41no—
237C2×C3×D4—91no—
238C2×C3×D4—91no—
239C22×C3×C4—91no—
240C22×C3×C4—91no—
241C22×C3×C4—92no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—92no—
245C2×#67—92no—
246C22×Dic3—92no—
247C2×D12—31no—
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—61no—
251C2×D12—91no—
252C2×D12—92no—
253C2×D12—92no—
254C2×C4×D3—31no—
255C2×C4×D3—31no—
256C2×C4×D3—32no—
257C2×C4×D3—32no—
258C2×C4×D3—62no—
259C2×C4×D3—91no—
260C2×C4×D3—91no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C3×C4⋊C4—91no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4—91no—
268C3×C4⋊C4—91no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4—92no—
272C3×C42—31no—
273C3×C42—31no—
274C3×C42—61no—
275C3×C42D3211yescharacteristic
276#217—92no—
277#217—92no—
278#217—92no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—31no—
282(C2×D3)⋊C4—31no—
283(C2×D3)⋊C4—62no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—31no—
295C4⋊Dic3—31no—
296C4⋊Dic3—31no—
297C4⋊Dic3—61no—
298C4⋊Dic3—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302Dic3⋊C4—32no—
303Dic3⋊C4—32no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—92no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C22×C32—21no—
312C22×C32C23×D311yescharacteristic
313C22×C32C23×D311yescharacteristic
314C2×C3⋊D3—21no—
315C2×C3⋊D3—21no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3#23512yes—
320C2×C3×D3—32no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C2×C3×D3—64no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—61no—
327C2×C3×D3—62no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C2×C3×D3—61no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—122no—
334C2×C3×D3—122no—
335C32×C4—22no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—22no—
341C32×C4—41no—
342C32×C4#23712yes—
343C32×C4#23912yes—
344C3⋊Dic3—21no—
345C3⋊Dic3—21no—
346C3⋊Dic3—22no—
347C3⋊Dic3—22no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—122no—
355#126—272no—
356#125—91no—
357#123—272no—
358#121—272no—
359#121—271no—
360#121—271no—
361C4×D4—91no—
362C4×D4—91no—
363C4×D4—272no—
364#117—271no—
365C2×C4⋊C4—271no—
366H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
367C23×C3—91no—
368C23×C3—91no—
369C23×C3—92no—
370C22×D3—31no—
371C22×D3—32no—
372C22×D3—32no—
373C22×D3—31no—
374C22×D3—62no—
375C22×D3—91no—
376C22×D3—91no—
377C3×D4—182no—
378C3×D4—182no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—92no—
391C2×C3×C4—91no—
392C2×C3×C4—92no—
393C2×C3×C4—182no—
394C2×C3×C4—182no—
395C2×C3×C4—362no—
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399#67—184no—
400#67—184no—
401#67—362no—
402#67—362no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—31no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—31no—
409C2×Dic3—62no—
410C2×Dic3—91no—
411C2×Dic3—92no—
412C2×Dic3—92no—
413C2×Dic3—91no—
414C2×Dic3—362no—
415D12—62no—
416D12—62no—
417D12—62no—
418D12—64no—
419D12—182no—
420D12—184no—
421D12—362no—
422C4×D3—62no—
423C4×D3—62no—
424C4×D3—122no—
425C4×D3—122no—
426C4×D3—124no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3—362no—
430C4×D3—362no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32C2×#23511yescharacteristic
435C2×C32C2×#23711yescharacteristic
436C2×C32C2×#23911yescharacteristic
437C2×C32D3×#4111yescharacteristic
438C2×C32#100911yescharacteristic
439C2×C32#100911yescharacteristic
440C3⋊D3—22no—
441C3⋊D3—41no—
442C3×D3—62no—
443C3×D3—124no—
444C3×D3—121no—
445C3×D3—122no—
446C3×D3—122no—
447C2×D4—91no—
448C2×D4—91no—
449C2×D4—272no—
450C2×D4—272no—
451C22×C4—91no—
452C22×C4—91no—
453C22×C4—92no—
454C22×C4—272no—
455C4⋊C4—91no—
456C4⋊C4—91no—
457C4⋊C4—91no—
458C4⋊C4—91no—
459C4⋊C4—92no—
460C4⋊C4—92no—
461C4⋊C4—272no—
462C22⋊C4—91no—
463C22⋊C4—92no—
464C22⋊C4—91no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C42—272no—
469C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C22×C3—31no—
471C22×C3—31no—
472C22×C3—61no—
473C22×C3—92no—
474C22×C3—181no—
475C22×C3—181no—
476C22×C3—181no—
477C22×C3—181no—
478C22×C3—181no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3C22×D3211yescharacteristic
483C2×D3—32no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—64no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—92no—
497C2×D3—122no—
498C2×D3—122no—
499C2×D3—181no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—181no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—32no—
507C3×C4—32no—
508C3×C4—61no—
509C3×C4—61no—
510C3×C4—61no—
511C3×C4—61no—
512C3×C4—62no—
513C3×C4—62no—
514C3×C4—121no—
515C3×C4—182no—
516C3×C4—182no—
517C3×C4—181no—
518C3×C4—181no—
519C3×C4—182no—
520C3×C4#379612yes—
521Dic3—62no—
522Dic3—62no—
523Dic3—61no—
524Dic3—61no—
525Dic3—62no—
526Dic3—62no—
527Dic3—122no—
528Dic3—181no—
529Dic3—181no—
530Dic3—182no—
531Dic3—182no—
532C32—21no—
533C32#556511yescharacteristic, complemented
534C32#569511yescharacteristic, complemented
535C23—91no—
536C23—92no—
537C23—91no—
538D4—182no—
539D4—182no—
540D4—544no—
541D4—1082no—
542C2×C4—91no—
543C2×C4—92no—
544C2×C4—92no—
545C2×C4—91no—
546C2×C4—92no—
547C2×C4—182no—
548C2×C4—182no—
549C2×C4—362no—
550C2×C4—1082no—
551C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
552C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
553C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
554C2×C3—31no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—61no—
561C2×C3—61no—
562C2×C3—61no—
563C2×C3—182no—
564C2×C3—361no—
565C2×C3—362no—
566C2×C3C2×#379611yescharacteristic
567C2×C3D3×#23511yescharacteristic
568C2×C3#6430811yescharacteristic
569D3—62no—
570D3—122no—
571D3—122no—
572D3—124no—
573D3—121no—
574D3—182no—
575D3—361no—
576C22—92no—
577C22—181no—
578C22—181no—
579C22—181no—
580C22—182no—
581C22—181no—
582C22—182no—
583C22—182no—
584C22—181no—
585C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
586C4—21no—
587C4—21no—
588C4—181no—
589C4—181no—
590C4—182no—
591C4—182no—
592C4—182no—
593C4#8864912yes—
594C3—31no—
595C3—31no—
596C3—61no—
597C3#1059392511yescharacteristic
598C2—182no—
599C2—362no—
600C2—361no—
601C2C2×#8864911yescharacteristic
602C2#1172249011yescharacteristic
603C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊#272C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #64079⋊C3C2semidirect
4. #11721156C2semidirect
5. #11721156C2semidirect
6. #11721129C2semidirect
7. #11721127C2semidirect
8. #11721124C2semidirect
9. #11721104C2semidirect
10. #11721088C2semidirect
11. #11721081C2semidirect
37. C4×C4⋊(C3⋊D3)D3semidirect
38. #64079D3semidirect
366. H(3,𝔽3)#553semidirect
533. C32#5565semidirect
534. C32#5695semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
#117211562
#117211292
#117211272
#117211242
#117211042
#117210882
#117210812
#105939823
#105939233