Finite Group Databasebeta

← #13147816

Subgroups — #13147816

Order 1728.

832 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721718C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#11721159C211yesmaximal, characteristic
5#11721155C212yesmaximal, complemented
6#11721136C211yesmaximal, characteristic, complemented
7#11721130C212yesmaximal, complemented
8#11721128C212yesmaximal, complemented
9#11721108C212yesmaximal, complemented
10#11721094C211yesmaximal, characteristic, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10593961—31nomaximal
13#10593877—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×#634⋊C3C2212yes—
18C2×#6200C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21#88542C2211yescharacteristic
22#88542C2211yescharacteristic
23#88540C2211yescharacteristic
24#88540C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2212yes—
31#88451C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64084D311yescharacteristic, complemented
38#64078D311yescharacteristic, complemented
39C3×#875—31no—
40C3×#872—31no—
41C3×#872—31no—
42C3×#872—61no—
43C3×#869—31no—
44#63913—32no—
45#63909—31no—
46#63908—31no—
47#63907—32no—
48#63896—32no—
49#63894—31no—
50Dic3×D12—31no—
51#63891—32no—
52#63888—32no—
53#63861—32no—
54#63857—31no—
55#63856—31no—
56#63854—32no—
57#63848—32no—
58#63844—31no—
59Dic3×Dic6—31no—
60C22×#272C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C4×#272—22no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4⋊((C3⋊D3)⋊C3)—22no—
73#634⋊C3—22no—
74#634⋊C3—42no—
75#6200—42no—
76((C3⋊D3)⋊C3)⋊C4—42no—
77#5733—91no—
78#5682—91no—
79#5534—91no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×#634C2×D311yescharacteristic
82C2×#634C2×D312yes—
83C2×C3×D12—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—61no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—32no—
89C2×#626—32no—
90C2×#625—32no—
91C2×#624—32no—
92C2×D3×Dic3—32no—
93C32×C42—21no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96#3752C2×D312yes—
97#3752C2×D312yes—
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100C4⋊(C3⋊Dic3)C2×D312yes—
101C3×(C2×D3)⋊C4—31no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—31no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—31no—
106C3×(C2×D3)⋊C4—31no—
107C3×(C2×D3)⋊C4—61no—
108C3×(C2×D3)⋊C4—61no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—31no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—31no—
114C3×Dic3⋊C4—31no—
115C3×Dic3⋊C4—31no—
116C3×Dic3⋊C4—61no—
117C3×Dic3⋊C4—61no—
118C3×C4×Dic3—31no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—61no—
121#3724—32no—
122#3724—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126#3722—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—21no—
135C2×#272—21no—
136C2×#272—21no—
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)#4112yes—
143#1084—21no—
144#1084—21no—
145#1084—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—22no—
149C3×C4⋊Q8—91no—
150C3×#124—91no—
151C3×#117—91no—
152#947—92no—
153#946—92no—
154#946—91no—
155Q8×Dic3—91no—
156#938—91no—
157D4×Dic3—91no—
158#930—91no—
159#924—91no—
160#898—92no—
161#898—92no—
162#897—91no—
163#896—92no—
164#895—91no—
165#894—92no—
166#893—91no—
167#887—91no—
168#886—92no—
169#885—92no—
170#885—92no—
171#883—92no—
172#879—92no—
173#878—92no—
174#875—31no—
175#875—91no—
176C4×D12—91no—
177#872—31no—
178#872—31no—
179#872—61no—
180#869—31no—
181#869—91no—
182C4×Dic6—91no—
183C22×C3⋊D3C22×D312yes—
184C22×C3×D3—32no—
185C22×C3×D3—31no—
186C22×C3×D3—31no—
187C22×C3×D3—61no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C4⋊(C3⋊D3)—22no—
201#634—22no—
202#634—42no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—61no—
215C3×D12—62no—
216C3×C4×D3—64no—
217C3×C4×D3—62no—
218C3×C4×D3—62no—
219C3×Dic6—62no—
220C3×Dic6—122no—
221#626—122no—
222#625—122no—
223#624—122no—
224D3×Dic3—122no—
225#542—271noSylow
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
228C2×H(3,𝔽3)2−1+411yescharacteristic
229#272—41no—
230(C3⋊D3)⋊C3—42no—
231C2×C3×Q8—91no—
232C2×C3×Q8—92no—
233C2×C3×D4—91no—
234C22×C3×C4—91no—
235C2×#67—92no—
236C2×#67—92no—
237C22×Dic3—92no—
238C2×D12—31no—
239C2×D12—91no—
240C2×D12—92no—
241C2×C4×D3—31no—
242C2×C4×D3—31no—
243C2×C4×D3—61no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×Dic6—31no—
247C2×Dic6—32no—
248C2×Dic6—91no—
249C2×Dic6—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—91no—
258C3×C22⋊C4—91no—
259C3×C42—31no—
260C3×C42—31no—
261C3×C42—61no—
262C3×C42—91no—
263C3×C42D3211yescharacteristic
264#217—92no—
265#217—92no—
266#217—92no—
267(C2×D3)⋊C4—31no—
268(C2×D3)⋊C4—31no—
269(C2×D3)⋊C4—31no—
270(C2×D3)⋊C4—31no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—61no—
274(C2×D3)⋊C4—61no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283C4⋊Dic3—31no—
284C4⋊Dic3—31no—
285C4⋊Dic3—32no—
286C4⋊Dic3—92no—
287C4⋊Dic3—91no—
288C4⋊Dic3—91no—
289C4⋊Dic3—92no—
290C4⋊Dic3—92no—
291Dic3⋊C4—31no—
292Dic3⋊C4—31no—
293Dic3⋊C4—31no—
294Dic3⋊C4—31no—
295Dic3⋊C4—61no—
296Dic3⋊C4—61no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301C4×Dic3—31no—
302C4×Dic3—31no—
303C4×Dic3—61no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C22×C32—21no—
308C22×C32C23×D311yescharacteristic
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—22no—
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3×D3—62no—
314C2×C3×D3—61no—
315C2×C3×D3—62no—
316C2×C3×D3—61no—
317C2×C3×D3—62no—
318C2×C3×D3—61no—
319C2×C3×D3—62no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C2×C3×D3—121no—
324C2×C3×D3—121no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—22no—
330C32×C4—41no—
331C32×C4—41no—
332C32×C4#23712yes—
333C32×C4#23912yes—
334C3⋊Dic3—22no—
335C3⋊Dic3—22no—
336C3⋊Dic3—22no—
337C3×Dic3—62no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345C3×Dic3—61no—
346C3×Dic3—61no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349C4⋊Q8—91no—
350#126—272no—
351#124—91no—
352#124—272no—
353#123—272no—
354C22⋊Q8—272no—
355C22⋊Q8—271no—
356#121—271no—
357C4×Q8—271no—
358C4×D4—271no—
359#117—91no—
360H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
361C23×C3—92no—
362C23×C3—91no—
363C22×D3—31no—
364C22×D3—31no—
365C22×D3—32no—
366C22×D3—61no—
367C22×D3—92no—
368C3×Q8—182no—
369C3×Q8—362no—
370C3×D4—182no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—182no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—362no—
391#67—362no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—92no—
405C2×Dic3—92no—
406C2×Dic3—92no—
407C2×Dic3—362no—
408D12—62no—
409D12—182no—
410D12—362no—
411C4×D3—62no—
412C4×D3—62no—
413C4×D3—64no—
414C4×D3—362no—
415C4×D3—362no—
416Dic6—62no—
417Dic6—122no—
418Dic6—182no—
419Dic6—362no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32C2×#23711yescharacteristic
424C2×C32#100411yescharacteristic
425C2×C32C2×#23911yescharacteristic
426C2×C32#100711yescharacteristic
427C2×C32#100911yescharacteristic
428C2×C32#101011yescharacteristic
429C3⋊D3—42no—
430C3×D3—122no—
431C3×D3—121no—
432C3×D3—122no—
433C3×D3—121no—
434C2×Q8—91no—
435C2×Q8—92no—
436C2×D4—91no—
437C2×D4—272no—
438C22×C4—91no—
439C22×C4—272no—
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—91no—
443C4⋊C4—91no—
444C4⋊C4—92no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C22⋊C4—91no—
448C22⋊C4—91no—
449C22⋊C4—92no—
450C22⋊C4—92no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C42—91no—
455C42—272no—
456C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—61no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3C22×D3211yescharacteristic
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—61no—
471C2×D3—61no—
472C2×D3—61no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—182no—
480C2×D3—182no—
481C2×D3—182no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—32no—
485C3×C4—32no—
486C3×C4—61no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—61no—
490C3×C4—62no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—182no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4—181no—
497C3×C4—181no—
498C3×C4—181no—
499C3×C4#379612yes—
500Dic3—61no—
501Dic3—61no—
502Dic3—61no—
503Dic3—61no—
504Dic3—61no—
505Dic3—61no—
506Dic3—62no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—121no—
511Dic3—121no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515C32—21no—
516C32#568211yescharacteristic, complemented
517C32#573311yescharacteristic, complemented
518C23—91no—
519C23—92no—
520Q8—182no—
521Q8—362no—
522D4—182no—
523D4—1082no—
524C2×C4—91no—
525C2×C4—91no—
526C2×C4—92no—
527C2×C4—92no—
528C2×C4—91no—
529C2×C4—92no—
530C2×C4—182no—
531C2×C4—1082no—
532C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
533C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—61no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—362no—
545C2×C3—361no—
546C2×C3C2×#379611yescharacteristic
547C2×C3#6429711yescharacteristic
548C2×C3#6430711yescharacteristic
549D3—121no—
550D3—121no—
551D3—122no—
552D3—122no—
553D3—362no—
554C22—181no—
555C22—182no—
556C22—181no—
557C22—182no—
558C22—182no—
559C22—181no—
560C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
561C4—21no—
562C4—21no—
563C4—181no—
564C4—182no—
565C4—181no—
566C4—181no—
567C4—182no—
568C4—182no—
569C4#8864912yes—
570C3—31no—
571C3—31no—
572C3—61no—
573C3#1059395511yescharacteristic
574C2—361no—
575C2—362no—
576C2C2×#8864911yescharacteristic
577C2#1172248211yescharacteristic
578C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721159C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#542
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. #11721718C2semidirect
2. #64084⋊C3C2semidirect
3. #64078⋊C3C2semidirect
5. #11721155C2semidirect
6. #11721136C2semidirect
7. #11721130C2semidirect
8. #11721128C2semidirect
9. #11721108C2semidirect
10. #11721094C2semidirect
11. #11721084C2semidirect
37. #64084D3semidirect
38. #64078D3semidirect
360. H(3,𝔽3)#542semidirect
516. C32#5682semidirect
517. C32#5733semidirect

Maximal subgroups

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

subgroupindex
#117217182
#64084⋊C32
#64078⋊C32
#117211592
#117211552
#117211362
#117211302
#117211282
#117211082
#117210942
#117210842
#105939613
#105938773