Finite Group Databasebeta

← #13150738

Subgroups — #13150738

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
1C2×C4×#1084C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721315C212yesmaximal, complemented
5#11721307C212yesmaximal, complemented
6#11721136C211yesmaximal, characteristic, complemented
7#11721121C212yesmaximal, complemented
8#11721113C211yesmaximal, characteristic, complemented
9#11721112C212yesmaximal, complemented
10#11721095C211yesmaximal, characteristic, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10594955—31nomaximal
13#10594949—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×#1084C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×#634⋊C3C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C4×#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25#3757⋊C3C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30#88450C2212yes—
31#88450C2212yes—
32#88450C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64139D311yescharacteristic
38#64133D311yescharacteristic, complemented
39C3×#930—31no—
40C3×#924—31no—
41C2×C3×C4×Dic3—31no—
42C2×C3×C4×Dic3—31no—
43C2×C3×C4×Dic3—61no—
44Dic3×#67—32no—
45#63966—32no—
46#63961—32no—
47#63952—32no—
48#63894—31no—
49Dic3×D12—31no—
50#63877—32no—
51#63869—31no—
52#63869—31no—
53#63868—32no—
54#63864—31no—
55#63860—31no—
56#63859—32no—
57#63844—31no—
58Dic3×Dic6—31no—
59#63836—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084—22no—
66C2×#1084—22no—
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69C2×#1084C2311yescharacteristic
70C2×#1084C2311yescharacteristic
71#638⋊C3—24no—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2312yes—
75C4⋊((C3⋊D3)⋊C3)—22no—
76#634⋊C3—22no—
77((C3⋊D3)⋊C3)⋊C4—24no—
78#5624—91no—
79#5600—91no—
80#5479—91no—
81C22×C32×C4—21no—
82C22×C32×C4C2×D311yescharacteristic
83C22×C32×C4C2×D311yescharacteristic
84C2×#638C2×D312yes—
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×#634C2×D311yescharacteristic
87C2×C3×#67—32no—
88C22×C3×Dic3—31no—
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—61no—
93C22×C3×Dic3—61no—
94C2×C3×D12—31no—
95C2×C3×Dic6—31no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98#3757C2×D312yes—
99#3752C2×D312yes—
100C4⋊(C3⋊Dic3)C2×D311yescharacteristic
101C4⋊(C3⋊Dic3)C2×D311yescharacteristic
102#3750C2×D312yes—
103C3×#217—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×Dic3⋊C4—32no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—31no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—61no—
117C3×C4×Dic3—61no—
118C3×C4×Dic3—61no—
119C3×C4×Dic3—61no—
120#3724—32no—
121#3724—32no—
122#3724—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126#3722—32no—
127#3722—32no—
128#3722—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—32no—
132Dic32—32no—
133Dic32—32no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)#4112yes—
137C22×H(3,𝔽3)C2411yescharacteristic
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—22no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)#4112yes—
143#1084—21no—
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×C22⋊Q8—91no—
151C3×#121—91no—
152C2×C3×C42—91no—
153#947—91no—
154Q8×Dic3—91no—
155#936—91no—
156#935—92no—
157D4×Dic3—91no—
158D4×Dic3—92no—
159#930—31no—
160#930—91no—
161#929—92no—
162C4×#67—92no—
163#924—31no—
164#924—91no—
165C2×C4×Dic3—31no—
166C2×C4×Dic3—31no—
167C2×C4×Dic3—61no—
168#898—92no—
169#893—91no—
170#892—92no—
171#890—91no—
172#890—91no—
173#887—91no—
174#886—92no—
175#67⋊C4—92no—
176#879—92no—
177#877—92no—
178#876—92no—
179#875—91no—
180C4×D12—91no—
181#872—92no—
182#870—91no—
183C4×Dic6—91no—
184C23×C32—21no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—32no—
189C2×C32×C4—24no—
190C2×C32×C4—22no—
191C2×C32×C4—22no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198#638—24no—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D312yes—
202C4⋊(C3⋊D3)—22no—
203#634—22no—
204C3×#67—64no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—62no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—64no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3—62no—
225C2×C3×Dic3—62no—
226C3×D12—62no—
227C3×Dic6—62no—
228#624—64no—
229D3×Dic3—64no—
230#529—271noSylow
231C2×H(3,𝔽3)—22no—
232C2×H(3,𝔽3)C2×#4111yescharacteristic
233C2×H(3,𝔽3)C2×#4111yescharacteristic
234C2×H(3,𝔽3)C2×#4111yescharacteristic
235(C3⋊D3)⋊C3—42no—
236C2×C3×Q8—91no—
237C2×C3×D4—91no—
238C2×C3×D4—92no—
239C22×C3×C4—31no—
240C22×C3×C4—31no—
241C22×C3×C4—61no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C22×C3×C4D3211yescharacteristic
245C2×#67—32no—
246C2×#67—92no—
247C22×Dic3—31no—
248C22×Dic3—31no—
249C22×Dic3—31no—
250C22×Dic3—31no—
251C22×Dic3—61no—
252C22×Dic3—61no—
253C22×Dic3—92no—
254C2×D12—31no—
255C2×D12—91no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×Dic6—31no—
259C2×Dic6—91no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—92no—
263C3×C22⋊C4—92no—
264C3×C22⋊C4—92no—
265C3×C42—91no—
266C3×C42—91no—
267C3×C42—91no—
268C3×C42—91no—
269#217—32no—
270#217—92no—
271#217—92no—
272#217—92no—
273#217—92no—
274(C2×D3)⋊C4—32no—
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—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289Dic3⋊C4—32no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—31no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—31no—
301C4×Dic3—31no—
302C4×Dic3—31no—
303C4×Dic3—31no—
304C4×Dic3—31no—
305C4×Dic3—61no—
306C4×Dic3—61no—
307C4×Dic3—61no—
308C4×Dic3—61no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—21no—
319C22×C32—22no—
320C22×C32#23712yes—
321C22×C32#23912yes—
322C22×C32C23×D311yescharacteristic
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—22no—
325C2×C3⋊D3—22no—
326C2×C3⋊D3—22no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—62no—
330C32×C4—22no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—22no—
334C32×C4#23712yes—
335C32×C4#23712yes—
336C3⋊Dic3—22no—
337C3⋊Dic3—22no—
338C3⋊Dic3—22no—
339C3×Dic3—32no—
340C3×Dic3—32no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—62no—
352C3×Dic3—61no—
353C3×Dic3—61no—
354#126—272no—
355#125—271no—
356#124—271no—
357#123—272no—
358C22⋊Q8—91no—
359#121—91no—
360C4×Q8—271no—
361C4×D4—271no—
362C4×D4—272no—
363#117—272no—
364C2×C42—91no—
365H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
366C23×C3—31no—
367C23×C3—31no—
368C23×C3—61no—
369C23×C3—92no—
370C23×C3C2×D3211yescharacteristic
371C22×D3—32no—
372C22×D3—92no—
373C3×Q8—182no—
374C3×D4—182no—
375C3×D4—184no—
376C2×C3×C4—22no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—64no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—91no—
387C2×C3×C4—92no—
388C2×C3×C4—92no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—182no—
394C2×C3×C4—182no—
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397#67—64no—
398#67—184no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—62no—
415C2×Dic3—62no—
416C2×Dic3—62no—
417C2×Dic3—62no—
418C2×Dic3—64no—
419C2×Dic3—64no—
420C2×Dic3—92no—
421C2×Dic3—92no—
422C2×Dic3—92no—
423C2×Dic3—184no—
424D12—62no—
425D12—182no—
426C4×D3—184no—
427C4×D3—184no—
428Dic6—62no—
429Dic6—182no—
430C2×C32—24no—
431C2×C32—22no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32C2×#23711yescharacteristic
437C2×C32C2×#23711yescharacteristic
438C2×C32C2×#23711yescharacteristic
439C2×C32C2×#23911yescharacteristic
440C2×C32D3×#4111yescharacteristic
441C2×C32D3×#4111yescharacteristic
442C3⋊D3—42no—
443C3×D3—122no—
444C2×Q8—91no—
445C2×D4—92no—
446C2×D4—91no—
447C22×C4—91no—
448C22×C4—91no—
449C22×C4—272no—
450C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
451C4⋊C4—92no—
452C4⋊C4—91no—
453C4⋊C4—91no—
454C4⋊C4—272no—
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—91no—
466C42—272no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—31no—
470C22×C3—31no—
471C22×C3—32no—
472C22×C3—32no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—61no—
480C22×C3—62no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3—182no—
484C22×C3#379612yes—
485C22×C3C22×D3211yescharacteristic
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—182no—
490C2×D3—182no—
491C2×D3—182no—
492C3×C4—21no—
493C3×C4—32no—
494C3×C4—32no—
495C3×C4—61no—
496C3×C4—61no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—92no—
500C3×C4—182no—
501C3×C4—182no—
502C3×C4—182no—
503C3×C4—181no—
504C3×C4—181no—
505C3×C4—181no—
506C3×C4#380512yes—
507Dic3—32no—
508Dic3—32no—
509Dic3—62no—
510Dic3—61no—
511Dic3—61no—
512Dic3—61no—
513Dic3—61no—
514Dic3—61no—
515Dic3—61no—
516Dic3—62no—
517Dic3—62no—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—182no—
523Dic3—182no—
524Dic3—182no—
525C32—21no—
526C32#560011yescharacteristic, complemented
527C32#562411yescharacteristic, complemented
528C23—92no—
529C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530Q8—182no—
531D4—184no—
532D4—182no—
533C2×C4—22no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—91no—
538C2×C4—92no—
539C2×C4—92no—
540C2×C4—92no—
541C2×C4—182no—
542C2×C4—182no—
543C2×C4—544no—
544C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
545C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
546C2×C3—22no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—31no—
553C2×C3—62no—
554C2×C3—62no—
555C2×C3—64no—
556C2×C3—61no—
557C2×C3—61no—
558C2×C3—61no—
559C2×C3—362no—
560C2×C3C2×#379611yescharacteristic
561C2×C3#6430611yescharacteristic
562C2×C3C2×#380511yescharacteristic
563D3—122no—
564D3—362no—
565C22—21no—
566C22—21no—
567C22—182no—
568C22—182no—
569C22—182no—
570C22#8864912yes—
571C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
572C4—21no—
573C4—92no—
574C4—182no—
575C4—181no—
576C4—181no—
577C4—182no—
578C4—181no—
579C4—182no—
580C4#8867212yes—
581C3—31no—
582C3—31no—
583C3—61no—
584C3#1059496511yescharacteristic
585C2—22no—
586C2—362no—
587C2C2×#8864911yescharacteristic
588C2#1172249111yescharacteristic
589C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#529
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×#1084C2semidirect
3. #64133⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721307C2semidirect
6. #11721136C2semidirect
7. #11721121C2semidirect
8. #11721113C2semidirect
9. #11721112C2semidirect
10. #11721095C2semidirect
11. #11721084C2semidirect
38. #64133D3semidirect
365. H(3,𝔽3)#529semidirect
526. C32#5600semidirect
527. C32#5624semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
#64139⋊C32
#64133⋊C32
#117213152
#117213072
#117211362
#117211212
#117211132
#117211122
#117210952
#117210842
#105949553
#105949493