Finite Group Databasebeta

← #13147808

Subgroups — #13147808

Order 1728.

927 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×#272C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721159C211yesmaximal, characteristic
5#11721136C211yesmaximal, characteristic, complemented
6#11721124C212yesmaximal, complemented
7#11721121C212yesmaximal, complemented
8#11721112C212yesmaximal, complemented
9#11721104C212yesmaximal, complemented
10#11721094C211yesmaximal, characteristic, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10593976—31nomaximal
13#10593921—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#6200C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C42×H(3,𝔽3)C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#88451C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64084D311yescharacteristic, complemented
38#64079D311yescharacteristic, complemented
39C3×#875—31no—
40C3×C42×D3—31no—
41C3×C42×D3—31no—
42C3×C42×D3—61no—
43C3×#870—31no—
44#63909—31no—
45#63908—31no—
46#63905—32no—
47#63904—32no—
48#63894—31no—
49Dic3×D12—31no—
50#63879—32no—
51#63877—32no—
52#63868—32no—
53#63859—32no—
54#63857—31no—
55#63856—31no—
56#63850—32no—
57#63844—31no—
58Dic3×Dic6—31no—
59#63836—32no—
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—
69C4×#272—22no—
70C4×#272—22no—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C4⋊((C3⋊D3)⋊C3)—22no—
75#634⋊C3—22no—
76#6200—24no—
77((C3⋊D3)⋊C3)⋊C4—24no—
78#5691—91no—
79#5680—91no—
80#5479—91no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×#634C2×D311yescharacteristic
83C2×C3×D12—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—61no—
91C2×C3×C4×D3—61no—
92C2×C3×C4×D3—61no—
93C2×C3×Dic6—31no—
94C2×#626—32no—
95C2×#625—32no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98C32×C42—21no—
99C32×C42C2×D311yescharacteristic
100C32×C42C2×D311yescharacteristic
101#3752C2×D312yes—
102#3752C2×D312yes—
103C4⋊(C3⋊Dic3)C2×D311yescharacteristic
104C4⋊(C3⋊Dic3)C2×D311yescharacteristic
105#3750C2×D312yes—
106#3750C2×D312yes—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—31no—
119C3×C4×Dic3—61no—
120C3×C4×Dic3—61no—
121C3×C4×Dic3—61no—
122#3724—32no—
123#3724—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126#3722—32no—
127#3722—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—32no—
132Dic32—32no—
133Dic32—32no—
134C22×H(3,𝔽3)C2411yescharacteristic
135C2×#272—21no—
136C2×#272—21no—
137C2×#272#4112yes—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—22no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)—21no—
143C4×H(3,𝔽3)#4112yes—
144#1084—21no—
145#1084—21no—
146#1084#4112yes—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—22no—
149(C3⋊Dic3)⋊C3—22no—
150C3×#125—91no—
151C3×#124—91no—
152C2×C3×C42—91no—
153#946—91no—
154Q8×Dic3—91no—
155#938—91no—
156D4×Dic3—91no—
157#930—91no—
158#929—92no—
159C4×#67—92no—
160#924—91no—
161#898—92no—
162#897—91no—
163#895—91no—
164#894—92no—
165#893—91no—
166#893—92no—
167#892—92no—
168#887—91no—
169#883—92no—
170#67⋊C4—92no—
171#879—92no—
172#877—92no—
173#876—92no—
174#875—31no—
175#875—91no—
176C4×D12—91no—
177#872—92no—
178C42×D3—31no—
179C42×D3—31no—
180C42×D3—61no—
181#870—31no—
182#870—91no—
183C4×Dic6—91no—
184C22×C3⋊D3C22×D312yes—
185C22×C3×D3—31no—
186C22×C3×D3—31no—
187C22×C3×D3—32no—
188C22×C3×D3—61no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C4⋊(C3⋊D3)—22no—
202#634—22no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
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—62no—
217C3×C4×D3—62no—
218C3×C4×D3—62no—
219C3×C4×D3—62no—
220C3×C4×D3—64no—
221C3×C4×D3—64no—
222C3×C4×D3—64no—
223C3×C4×D3—62no—
224C3×C4×D3—62no—
225C3×Dic6—62no—
226#626—64no—
227#625—64no—
228#624—64no—
229D3×Dic3—64no—
230#529—271noSylow
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232C2×H(3,𝔽3)C2×#4111yescharacteristic
233C2×H(3,𝔽3)C2×#4111yescharacteristic
234#272—22no—
235(C3⋊D3)⋊C3—42no—
236C2×C3×Q8—91no—
237C2×C3×D4—91no—
238C22×C3×C4—91no—
239C22×C3×C4—91no—
240C22×C3×C4—91no—
241C2×#67—92no—
242C2×#67—92no—
243C22×Dic3—92no—
244C2×D12—31no—
245C2×D12—91no—
246C2×D12—92no—
247C2×C4×D3—31no—
248C2×C4×D3—31no—
249C2×C4×D3—31no—
250C2×C4×D3—31no—
251C2×C4×D3—31no—
252C2×C4×D3—31no—
253C2×C4×D3—61no—
254C2×C4×D3—61no—
255C2×C4×D3—61no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×Dic6—31no—
259C2×Dic6—91no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C42—31no—
267C3×C42—31no—
268C3×C42—61no—
269C3×C42—91no—
270C3×C42—91no—
271C3×C42—91no—
272C3×C42D3211yescharacteristic
273#217—92no—
274#217—92no—
275#217—92no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—32no—
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—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—31no—
287C4⋊Dic3—31no—
288C4⋊Dic3—91no—
289C4⋊Dic3—91no—
290C4⋊Dic3—92no—
291C4⋊Dic3—92no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300C4×Dic3—31no—
301C4×Dic3—31no—
302C4×Dic3—31no—
303C4×Dic3—31no—
304C4×Dic3—31no—
305C4×Dic3—31no—
306C4×Dic3—61no—
307C4×Dic3—61no—
308C4×Dic3—61no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C22×C32—21no—
313C22×C32C23×D311yescharacteristic
314C22×C32C23×D311yescharacteristic
315C2×C3⋊D3—22no—
316C2×C3⋊D3—22no—
317C2×C3⋊D3—22no—
318C2×C3×D3—32no—
319C2×C3×D3—32no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C2×C3×D3—61no—
326C2×C3×D3—61no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—62no—
330C32×C4—22no—
331C32×C4—22no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—22no—
337C32×C4#23712yes—
338C32×C4#23912yes—
339C3⋊Dic3—22no—
340C3⋊Dic3—22no—
341C3⋊Dic3—22no—
342C3×Dic3—32no—
343C3×Dic3—32no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—61no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—61no—
353C3×Dic3—61no—
354#126—272no—
355#125—91no—
356#124—91no—
357#123—272no—
358C22⋊Q8—271no—
359#121—271no—
360C4×Q8—271no—
361C4×D4—271no—
362C4×D4—272no—
363#117—272no—
364C2×C42—91no—
365H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
366C23×C3—92no—
367C23×C3—91no—
368C22×D3—31no—
369C22×D3—31no—
370C22×D3—32no—
371C22×D3—61no—
372C22×D3—92no—
373C3×Q8—182no—
374C3×D4—182no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—91no—
385C2×C3×C4—92no—
386C2×C3×C4—92no—
387C2×C3×C4—92no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—182no—
391C2×C3×C4—182no—
392C2×C3×C4—182no—
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C2×D3211yescharacteristic
396#67—184no—
397#67—184no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C2×Dic3—92no—
413C2×Dic3—184no—
414D12—62no—
415D12—182no—
416D12—184no—
417C4×D3—62no—
418C4×D3—62no—
419C4×D3—62no—
420C4×D3—62no—
421C4×D3—62no—
422C4×D3—62no—
423C4×D3—64no—
424C4×D3—64no—
425C4×D3—64no—
426C4×D3—184no—
427C4×D3—184no—
428Dic6—62no—
429Dic6—182no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32C2×#23711yescharacteristic
434C2×C32C2×#23911yescharacteristic
435C2×C32D3×#4111yescharacteristic
436C2×C32D3×#4111yescharacteristic
437C2×C32D3×#4111yescharacteristic
438C2×C32D3×#4111yescharacteristic
439C3⋊D3—42no—
440C3×D3—64no—
441C3×D3—62no—
442C3×D3—62no—
443C3×D3—122no—
444C2×Q8—91no—
445C2×D4—91no—
446C2×D4—272no—
447C22×C4—91no—
448C22×C4—91no—
449C22×C4—91no—
450C22×C4—272no—
451C4⋊C4—92no—
452C4⋊C4—91no—
453C4⋊C4—92no—
454C4⋊C4—91no—
455C4⋊C4—272no—
456C4⋊C4—272no—
457C22⋊C4—92no—
458C22⋊C4—92no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22⋊C4—272no—
462C42—91no—
463C42—91no—
464C42—91no—
465C42—272no—
466C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
467C22×C3—31no—
468C22×C3—31no—
469C22×C3—61no—
470C22×C3—92no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—181no—
475C22×C3—181no—
476C22×C3C22×D3211yescharacteristic
477C2×D3—32no—
478C2×D3—32no—
479C2×D3—62no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—182no—
490C2×D3—182no—
491C2×D3—182no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—32no—
495C3×C4—32no—
496C3×C4—61no—
497C3×C4—61no—
498C3×C4—61no—
499C3×C4—61no—
500C3×C4—62no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—92no—
504C3×C4—182no—
505C3×C4—182no—
506C3×C4—182no—
507C3×C4—181no—
508C3×C4—181no—
509C3×C4#379612yes—
510Dic3—32no—
511Dic3—32no—
512Dic3—62no—
513Dic3—61no—
514Dic3—61no—
515Dic3—61no—
516Dic3—61no—
517Dic3—62no—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—182no—
523Dic3—182no—
524Dic3—182no—
525C32—21no—
526C32#568011yescharacteristic, complemented
527C32#569111yescharacteristic, complemented
528C23—91no—
529C23—92no—
530Q8—182no—
531D4—182no—
532D4—544no—
533C2×C4—91no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—92no—
537C2×C4—92no—
538C2×C4—92no—
539C2×C4—182no—
540C2×C4—182no—
541C2×C4—182no—
542C2×C4—544no—
543C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
544C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
545C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—61no—
555C2×C3—182no—
556C2×C3—362no—
557C2×C3C2×#379611yescharacteristic
558C2×C3#6430611yescharacteristic
559C2×C3#6430611yescharacteristic
560D3—62no—
561D3—62no—
562D3—64no—
563D3—122no—
564D3—362no—
565C22—92no—
566C22—181no—
567C22—181no—
568C22—182no—
569C22—182no—
570C22—182no—
571C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
572C4—21no—
573C4—21no—
574C4—92no—
575C4—181no—
576C4—182no—
577C4—181no—
578C4—182no—
579C4—182no—
580C4#8864912yes—
581C3—31no—
582C3—31no—
583C3—61no—
584C3#1059394311yescharacteristic
585C2—182no—
586C2—362no—
587C2C2×#8864911yescharacteristic
588C2#1172249111yescharacteristic
589C2#1172249111yescharacteristic
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)#529
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. C42×#272C2semidirect
2. #64084⋊C3C2semidirect
3. #64079⋊C3C2semidirect
5. #11721136C2semidirect
6. #11721124C2semidirect
7. #11721121C2semidirect
8. #11721112C2semidirect
9. #11721104C2semidirect
10. #11721094C2semidirect
11. #11721084C2semidirect
37. #64084D3semidirect
38. #64079D3semidirect
365. H(3,𝔽3)#529semidirect
526. C32#5680semidirect
527. C32#5691semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#64084⋊C32
#64079⋊C32
#117211592
#117211362
#117211242
#117211212
#117211122
#117211042
#117210942
#117210842
#105939763
#105939213