Finite Group Databasebeta

← #13147888

Subgroups — #13147888

Order 1728.

934 conjugacy classes of subgroups.

The subgroup lattice has 612 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#64081⋊C3C211yesmaximal, characteristic, complemented
4#11721158C211yesmaximal, characteristic, complemented
5#11721156C211yesmaximal, characteristic, complemented
6#11721142C212yesmaximal, complemented
7#11721141C211yesmaximal, characteristic, complemented
8#11721140C212yesmaximal, complemented
9#11721104C212yesmaximal, complemented
10#11721100C212yesmaximal, complemented
11#11721093C211yesmaximal, characteristic, complemented
12#10593986—31nomaximal
13#10593984—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×#6200C2212yes—
22C2×#6198C2212yes—
23C42×H(3,𝔽3)C2211yescharacteristic
24#88542C2212yes—
25C4⋊#1084C2211yescharacteristic
26#3752⋊C3C2212yes—
27#3752⋊C3C2211yescharacteristic
28#3752⋊C3C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33#88451C2212yes—
34#88450C2212yes—
35#88449C2212yes—
36#88448C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
38#64081D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×C4×D12—31no—
42C3×C4×D12—61no—
43C3×#872—31no—
44#63916—31no—
45#63916—31no—
46#63914—31no—
47#63912—32no—
48#63911—31no—
49#63909—31no—
50#63908—31no—
51#63906—32no—
52#63906—32no—
53#63906—32no—
54#63905—32no—
55#63905—32no—
56#63872—31no—
57#63869—31no—
58#63853—32no—
59#63850—32no—
60C22×#272C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
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)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4⋊((C3⋊D3)⋊C3)—22no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—42no—
76#6200—42no—
77#6200—42no—
78#6200—42no—
79#6198—42no—
80#5562—91no—
81#5562—91no—
82#5535—91no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D312yes—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—31no—
89C2×C3×D12—61no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—32no—
94C2×C3×C4×D3—62no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#626—32no—
100C2×#625—32no—
101C2×#625—32no—
102C2×#625—32no—
103C32×C42—21no—
104C32×C42C2×D311yescharacteristic
105C32×C42C2×D311yescharacteristic
106#3752C2×D311yescharacteristic
107#3752C2×D311yescharacteristic
108#3752C2×D312yes—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750C2×D311yescharacteristic
111#3750C2×D311yescharacteristic
112C4×C3⋊Dic3C2×D311yescharacteristic
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—31no—
117C3×(C2×D3)⋊C4—31no—
118C3×(C2×D3)⋊C4—62no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—61no—
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—31no—
125C3×C4×Dic3—31no—
126#3724—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130#3722—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133(C2×D3)⋊Dic3—32no—
134C22×H(3,𝔽3)C2411yescharacteristic
135C2×#272—22no—
136C2×#272—22no—
137C2×#272—22no—
138C2×(C3⋊D3)⋊C3—21no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3—21no—
143C2×(C3⋊D3)⋊C3—22no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)—21no—
146C4×H(3,𝔽3)#4112yes—
147#1084—22no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—21no—
151(C3⋊Dic3)⋊C3—22no—
152C3×C4×D4—91no—
153C3×C4×D4—91no—
154C3×#117—91no—
155#940—91no—
156#940—91no—
157#938—91no—
158#938—91no—
159#930—91no—
160#930—92no—
161#930—92no—
162#930—91no—
163#929—92no—
164#929—92no—
165#895—91no—
166#895—91no—
167#894—92no—
168#894—92no—
169#890—91no—
170#890—91no—
171#884—92no—
172#884—92no—
173#884—92no—
174#884—92no—
175#883—92no—
176#883—92no—
177#874—91no—
178C4×D12—31no—
179C4×D12—31no—
180C4×D12—31no—
181C4×D12—61no—
182C4×D12—91no—
183#872—31no—
184#872—91no—
185#870—91no—
186C22×C3⋊D3C22×D312yes—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3—32no—
191C22×C3×D3—32no—
192C22×C3×D3—62no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D311yescharacteristic
206C4⋊(C3⋊D3)—22no—
207C4×C3⋊D3—22no—
208C4×C3⋊D3—42no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—62no—
216C3×D12—64no—
217C3×D12—62no—
218C3×D12—62no—
219C3×D12—62no—
220C3×C4×D3—62no—
221C3×C4×D3—122no—
222C3×C4×D3—122no—
223C3×C4×D3—122no—
224C3×C4×D3—124no—
225#626—122no—
226#626—122no—
227#626—122no—
228#626—122no—
229#626—122no—
230#625—122no—
231#625—122no—
232#625—122no—
233#540—271noSylow
234C2×H(3,𝔽3)C2×#4111yescharacteristic
235C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
236C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
237#272—42no—
238(C3⋊D3)⋊C3—42no—
239(C3⋊D3)⋊C3—41no—
240C2×C3×D4—91no—
241C2×C3×D4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—92no—
244C22×C3×C4—92no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—92no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—92no—
252C2×#67—92no—
253C2×D12—31no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—61no—
257C2×D12—91no—
258C2×D12—92no—
259C2×D12—92no—
260C2×D12—92no—
261C2×D12—92no—
262C2×C4×D3—31no—
263C2×C4×D3—32no—
264C2×C4×D3—32no—
265C2×C4×D3—32no—
266C2×C4×D3—62no—
267C2×C4×D3—91no—
268C2×C4×D3—92no—
269C3×C4⋊C4—91no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4—91no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—92no—
277C3×C42—31no—
278C3×C42—31no—
279C3×C42—61no—
280C3×C42—91no—
281C3×C42D3211yescharacteristic
282#217—92no—
283#217—92no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—32no—
286(C2×D3)⋊C4—31no—
287(C2×D3)⋊C4—31no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—62no—
290(C2×D3)⋊C4—91no—
291(C2×D3)⋊C4—91no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297C4⋊Dic3—31no—
298C4⋊Dic3—31no—
299C4⋊Dic3—31no—
300C4⋊Dic3—61no—
301C4⋊Dic3—91no—
302C4⋊Dic3—92no—
303C4⋊Dic3—92no—
304Dic3⋊C4—31no—
305Dic3⋊C4—31no—
306Dic3⋊C4—91no—
307Dic3⋊C4—91no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312C4×Dic3—31no—
313C4×Dic3—91no—
314C22×C32—21no—
315C22×C32C23×D311yescharacteristic
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—21no—
318C2×C3⋊D3—22no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—21no—
321C2×C3⋊D3—21no—
322C2×C3⋊D3—22no—
323C2×C3×D3—64no—
324C2×C3×D3—61no—
325C2×C3×D3—62no—
326C2×C3×D3—62no—
327C2×C3×D3—62no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—61no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—122no—
337C2×C3×D3—122no—
338C32×C4—22no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—22no—
344C32×C4—41no—
345C32×C4#23512yes—
346C32×C4#23512yes—
347C3⋊Dic3—21no—
348C3⋊Dic3—22no—
349C3⋊Dic3—21no—
350C3⋊Dic3—21no—
351C3×Dic3—61no—
352C3×Dic3—61no—
353C3×Dic3—61no—
354C3×Dic3—62no—
355C3×Dic3—62no—
356C3×Dic3—62no—
357C3×Dic3—122no—
358#127—271no—
359#125—271no—
360#123—272no—
361#123—272no—
362#121—272no—
363#121—272no—
364#121—271no—
365#121—271no—
366C4×D4—91no—
367C4×D4—91no—
368#117—91no—
369H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
370C23×C3—92no—
371C23×C3—91no—
372C23×C3—92no—
373C22×D3—32no—
374C22×D3—32no—
375C22×D3—31no—
376C22×D3—32no—
377C22×D3—62no—
378C22×D3—92no—
379C22×D3—91no—
380C3×D4—182no—
381C3×D4—182no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—92no—
394C2×C3×C4—91no—
395C2×C3×C4—92no—
396C2×C3×C4—182no—
397C2×C3×C4—362no—
398C2×C3×C4—362no—
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—362no—
403#67—362no—
404#67—362no—
405#67—362no—
406#67—362no—
407#67—362no—
408#67—362no—
409#67—362no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—31no—
415C2×Dic3—32no—
416C2×Dic3—62no—
417C2×Dic3—91no—
418C2×Dic3—91no—
419C2×Dic3—92no—
420C2×Dic3—91no—
421D12—62no—
422D12—62no—
423D12—64no—
424D12—62no—
425D12—182no—
426D12—362no—
427D12—362no—
428D12—362no—
429D12—362no—
430C4×D3—62no—
431C4×D3—122no—
432C4×D3—122no—
433C4×D3—122no—
434C4×D3—124no—
435C4×D3—182no—
436C4×D3—362no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32C2×#23511yescharacteristic
441C2×C32C2×#23511yescharacteristic
442C2×C32#100411yescharacteristic
443C2×C32#100911yescharacteristic
444C2×C32#100911yescharacteristic
445C2×C32#100911yescharacteristic
446C3⋊D3—41no—
447C3⋊D3—42no—
448C3×D3—124no—
449C3×D3—122no—
450C3×D3—122no—
451C3×D3—121no—
452C3×D3—122no—
453C2×D4—91no—
454C2×D4—91no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4—272no—
459C22×C4—91no—
460C22×C4—92no—
461C22×C4—92no—
462C4⋊C4—91no—
463C4⋊C4—91no—
464C4⋊C4—91no—
465C4⋊C4—91no—
466C4⋊C4—272no—
467C4⋊C4—272no—
468C22⋊C4—91no—
469C22⋊C4—91no—
470C22⋊C4—92no—
471C22⋊C4—92no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—91no—
475C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
476C22×C3—31no—
477C22×C3—31no—
478C22×C3—61no—
479C22×C3—181no—
480C22×C3—182no—
481C22×C3—181no—
482C22×C3—182no—
483C22×C3—181no—
484C22×C3—182no—
485C22×C3—182no—
486C22×C3—182no—
487C22×C3—182no—
488C22×C3C22×D3211yescharacteristic
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—64no—
496C2×D3—61no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—62no—
502C2×D3—122no—
503C2×D3—122no—
504C2×D3—181no—
505C2×D3—182no—
506C2×D3—182no—
507C2×D3—181no—
508C2×D3—182no—
509C2×D3—181no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—32no—
513C3×C4—32no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—61no—
517C3×C4—61no—
518C3×C4—62no—
519C3×C4—62no—
520C3×C4—121no—
521C3×C4—181no—
522C3×C4—181no—
523C3×C4—182no—
524C3×C4—181no—
525C3×C4—182no—
526C3×C4#379812yes—
527Dic3—62no—
528Dic3—62no—
529Dic3—61no—
530Dic3—62no—
531Dic3—61no—
532Dic3—61no—
533Dic3—122no—
534Dic3—181no—
535Dic3—182no—
536Dic3—181no—
537Dic3—181no—
538C32—21no—
539C32#553511yescharacteristic, complemented
540C32#556211yescharacteristic, complemented
541C23—91no—
542C23—92no—
543C23—92no—
544D4—182no—
545D4—182no—
546D4—1082no—
547D4—1082no—
548D4—1082no—
549D4—1082no—
550C2×C4—91no—
551C2×C4—91no—
552C2×C4—92no—
553C2×C4—92no—
554C2×C4—91no—
555C2×C4—182no—
556C2×C4—362no—
557C2×C4—362no—
558C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
559C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
560C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
561C2×C3—31no—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—61no—
568C2×C3—61no—
569C2×C3—61no—
570C2×C3—361no—
571C2×C3—362no—
572C2×C3—362no—
573C2×C3C2×#379811yescharacteristic
574C2×C3#6430711yescharacteristic
575C2×C3#6430811yescharacteristic
576D3—122no—
577D3—122no—
578D3—124no—
579D3—121no—
580D3—122no—
581D3—362no—
582D3—361no—
583C22—181no—
584C22—182no—
585C22—182no—
586C22—182no—
587C22—182no—
588C22—181no—
589C22—182no—
590C22—181no—
591C22—182no—
592C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
593C4—21no—
594C4—21no—
595C4—181no—
596C4—182no—
597C4—181no—
598C4—182no—
599C4—181no—
600C4#8865012yes—
601C3—31no—
602C3—31no—
603C3—61no—
604C3#1059398611yescharacteristic
605C2—362no—
606C2—361no—
607C2—362no—
608C2C2×#8865011yescharacteristic
609C2#1172249211yescharacteristic
610C2#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)#540
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. #64081⋊C3C2semidirect
4. #11721158C2semidirect
5. #11721156C2semidirect
6. #11721142C2semidirect
7. #11721141C2semidirect
8. #11721140C2semidirect
9. #11721104C2semidirect
10. #11721100C2semidirect
11. #11721093C2semidirect
37. C4×C4⋊(C3⋊D3)D3semidirect
38. #64081D3semidirect
369. H(3,𝔽3)#540semidirect
539. C32#5535semidirect
540. C32#5562semidirect

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
#64081⋊C32
#117211582
#117211562
#117211422
#117211412
#117211402
#117211042
#117211002
#117210932
#105939863
#105939843