Finite Group Databasebeta

← #13151383

Subgroups — #13151383

Order 1728.

1123 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×#3830⋊C3C211yesmaximal, characteristic
2D4×#1084C211yesmaximal, characteristic, complemented
3D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#11721315C212yesmaximal, complemented
5#11721314C212yesmaximal, complemented
6C2×#88448C212yesmaximal, complemented
7#11721296C212yesmaximal, complemented
8#11721129C211yesmaximal, characteristic, complemented
9#11721105C211yesmaximal, characteristic, complemented
10#11721084C211yesmaximal, characteristic, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10595222—31nomaximal
13#10595219—31nomaximal
14C2×D4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2212yes—
16C2×#638⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2212yes—
19#3830⋊C3C418yescomplemented
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C2×#634⋊C3C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23#88552C2212yes—
24C4⋊#1084C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26#3757⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29#88450C2211yescharacteristic
30#88450C2211yescharacteristic
31#88448C2211yescharacteristic
32#88448C2212yes—
33#88448C2211yescharacteristic
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3830D311yescharacteristic
39C2×C3×#237—31no—
40D4×C3⋊Dic3D311yescharacteristic, complemented
41C3×D4×Dic3—31no—
42C3×D4×Dic3—31no—
43C3×D4×Dic3—31no—
44C3×D4×Dic3—61no—
45#67⋊Dic3—32no—
46Dic3×#67—32no—
47C2×#3724—32no—
48C2×Dic3⋊Dic3—32no—
49#63957—32no—
50#63955—32no—
51#63952—32no—
52#63952—32no—
53#63900—31no—
54D3×C4⋊Dic3—31no—
55#63874—31no—
56#63855—31no—
57#63844—31no—
58#63844—31no—
59#63843—31no—
60Dic3×Dic6—31no—
61C23×H(3,𝔽3)C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63D4×H(3,𝔽3)C2×C414yescomplemented
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084—42no—
66C2×#1084C2312yes—
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69#638⋊C3C2×C418yescomplemented
70C2×(C3⋊Dic3)⋊C3—42no—
71C2×(C3⋊Dic3)⋊C3C2×C418yescomplemented
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2312yes—
76C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
77C4×(C3⋊D3)⋊C3C2×C414yescomplemented
78#634⋊C3C2×C414yescomplemented
79((C3⋊D3)⋊C3)⋊C4—42no—
80#5827—91no—
81#5546—91no—
82#5546—91no—
83C2×C32×D4—21no—
84C2×C32×D4C2×D311yescharacteristic
85C2×C32×D4C2×D311yescharacteristic
86C2×#638C2×D312yes—
87C22×C3⋊Dic3C2×D312yes—
88C22×C3⋊Dic3C2×D312yes—
89#3830Dic318yescomplemented
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×#634C2×D311yescharacteristic
92C2×C3×#67—32no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3—32no—
97C22×C3×Dic3—62no—
98C3×#237—38no—
99C2×C3×C4×D3—31no—
100C2×C3×Dic6—31no—
101C2×#624—32no—
102C2×D3×Dic3—32no—
103#3757C2×D312yes—
104C4⋊(C3⋊Dic3)C2×D311yescharacteristic
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×#217—32no—
107C3×#217—32no—
108C3×#217—32no—
109C3×#217—62no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—61no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—61no—
118#3724—32no—
119#3724—32no—
120#3724—31no—
121#3724—31no—
122#3724—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—31no—
126Dic3⋊Dic3—31no—
127Dic3⋊Dic3—32no—
128#3722—31no—
129#3722—31no—
130(C2×D3)⋊Dic3—31no—
131(C2×D3)⋊Dic3—31no—
132Dic32—32no—
133Dic32—32no—
134Dic32—32no—
135Dic32—32no—
136C22×H(3,𝔽3)—22no—
137C22×H(3,𝔽3)—22no—
138C22×H(3,𝔽3)C22×C414yes—
139C22×H(3,𝔽3)C2411yescharacteristic
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3C22×C412yes—
143C4×H(3,𝔽3)C22×C412yes—
144#1084—21no—
145#1084—22no—
146#1084—21no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3C22×C414yes—
151(C3⋊Dic3)⋊C3C22×C412yes—
152C2×C3×#41—91no—
153C2×#237—31no—
154C2×#237—91no—
155C3×C4×D4—91no—
156C3×C4×D4—91no—
157Q8×Dic3—91no—
158Q8×Dic3—91no—
159D4×Dic3—31no—
160D4×Dic3—31no—
161D4×Dic3—31no—
162D4×Dic3—61no—
163D4×Dic3—91no—
164D4×Dic3—92no—
165D4×Dic3—92no—
166C4×#67—92no—
167C4×#67—92no—
168#926—91no—
169#926—92no—
170C2×C4⋊Dic3—91no—
171C2×C4⋊Dic3—92no—
172C2×Dic3⋊C4—92no—
173C2×Dic3⋊C4—92no—
174D3×C4⋊C4—91no—
175D3×C4⋊C4—91no—
176#887—91no—
177#887—91no—
178#67⋊C4—92no—
179#67⋊C4—92no—
180#877—92no—
181#877—92no—
182#872—91no—
183#872—91no—
184C4×Dic6—91no—
185C4×Dic6—91no—
186C23×C32—22no—
187C23×C32C22×D312yes—
188C23×C32C22×D312yes—
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3×D3—31no—
191C32×D4—24no—
192C32×D4C4×D314yescomplemented
193C32×D4C2×Dic314yescomplemented
194C2×C32×C4—21no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197#638C2×Dic318yescomplemented
198C2×C3⋊Dic3—42no—
199C2×C3⋊Dic3C2×Dic318yescomplemented
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3C2×Dic314yescomplemented
206#634C2×Dic314yescomplemented
207C3×#67—38no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—62no—
223C2×C3×Dic3—122no—
224C2×C3×Dic3—122no—
225C2×C3×Dic3—122no—
226C2×C3×Dic3—124no—
227C3×C4×D3—34no—
228C3×Dic6—34no—
229#624—122no—
230D3×Dic3—122no—
231#520—271noSylow
232C2×H(3,𝔽3)—24no—
233C2×H(3,𝔽3)C23×C411yescharacteristic
234C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
235C2×H(3,𝔽3)2−1+411yescharacteristic
236(C3⋊D3)⋊C3—22no—
237C3×#41—98no—
238C2×C3×Q8—91no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—92no—
243C2×C3×D4D3211yescharacteristic
244C22×C3×C4—91no—
245C22×C3×C4—92no—
246C22×C3×C4—92no—
247C22×C3×C4—92no—
248C2×#67—32no—
249C2×#67—92no—
250C22×Dic3—32no—
251C22×Dic3—32no—
252C22×Dic3—32no—
253C22×Dic3—32no—
254C22×Dic3—62no—
255C22×Dic3—92no—
256C22×Dic3—92no—
257C22×Dic3—92no—
258#237—38no—
259#237—98no—
260C2×C4×D3—31no—
261C2×C4×D3—91no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×Dic6—31no—
265C2×Dic6—91no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4—91no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—92no—
270C3×C42—91no—
271C3×C42—91no—
272#217—32no—
273#217—32no—
274#217—32no—
275#217—62no—
276#217—92no—
277#217—91no—
278#217—91no—
279(C2×D3)⋊C4—91no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4—91no—
282(C2×D3)⋊C4—91no—
283C4⋊Dic3—31no—
284C4⋊Dic3—31no—
285C4⋊Dic3—31no—
286C4⋊Dic3—61no—
287C4⋊Dic3—91no—
288C4⋊Dic3—91no—
289C4⋊Dic3—92no—
290C4⋊Dic3—92no—
291C4⋊Dic3—92no—
292C4⋊Dic3—91no—
293Dic3⋊C4—92no—
294Dic3⋊C4—91no—
295Dic3⋊C4—91no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—91no—
300Dic3⋊C4—91no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303C4×Dic3—31no—
304C4×Dic3—31no—
305C4×Dic3—31no—
306C4×Dic3—61no—
307C4×Dic3—91no—
308C4×Dic3—92no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C22×C32—22no—
315C22×C32—22no—
316C22×C32—22no—
317C22×C32—22no—
318C22×C32—21no—
319C22×C32—24no—
320C22×C32—42no—
321C22×C32—42no—
322C22×C32C2×C4×D314yes—
323C22×C32C22×Dic314yes—
324C22×C32C23×D311yescharacteristic
325C22×C32C23×D311yescharacteristic
326C2×C3⋊D3—21no—
327C2×C3⋊D3—21no—
328C2×C3⋊D3C22×Dic312yes—
329C2×C3×D3—32no—
330C2×C3×D3—61no—
331C2×C3×D3—61no—
332C32×C4—22no—
333C32×C4C2×C4×D312yes—
334C32×C4C22×Dic312yes—
335C3⋊Dic3—22no—
336C3⋊Dic3—21no—
337C3⋊Dic3—21no—
338C3⋊Dic3C22×Dic314yes—
339C3⋊Dic3C22×Dic312yes—
340C3×Dic3—34no—
341C3×Dic3—32no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—61no—
345C3×Dic3—62no—
346C3×Dic3—61no—
347C3×Dic3—62no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—61no—
352C3×Dic3—121no—
353C3×Dic3—122no—
354C2×#41—91no—
355C4×Q8—271no—
356C4×Q8—271no—
357C4×D4—91no—
358C4×D4—91no—
359C4×D4—272no—
360C4×D4—272no—
361#117—271no—
362#117—272no—
363C2×C4⋊C4—271no—
364C2×C4⋊C4—272no—
365H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
366C23×C3—32no—
367C23×C3—32no—
368C23×C3—62no—
369C23×C3—91no—
370C23×C3C2×D3212yes—
371C22×D3—31no—
372C22×D3—91no—
373C3×Q8—94no—
374C3×D4—34no—
375C3×D4—34no—
376C3×D4—64no—
377C3×D4—98no—
378C3×D4D3×Dic314yes—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—61no—
382C2×C3×C4—94no—
383C2×C3×C4—92no—
384C2×C3×C4—91no—
385C2×C3×C4—98no—
386C2×C3×C4—91no—
387C2×C3×C4—92no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—92no—
391C2×C3×C4—91no—
392C2×C3×C4—362no—
393C2×C3×C4—362no—
394C2×C3×C4C2×D3211yescharacteristic
395#67—38no—
396#67—98no—
397C2×Dic3—32no—
398C2×Dic3—31no—
399C2×Dic3—38no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—32no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—62no—
412C2×Dic3—98no—
413C2×Dic3—92no—
414C2×Dic3—91no—
415C2×Dic3—91no—
416C2×Dic3—91no—
417C2×Dic3—92no—
418C2×Dic3—122no—
419C2×Dic3—122no—
420C2×Dic3—122no—
421C2×Dic3—124no—
422C2×Dic3—362no—
423C2×Dic3—362no—
424C4×D3—34no—
425C4×D3—94no—
426C4×D3—362no—
427C4×D3—362no—
428Dic6—34no—
429Dic6—94no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—44no—
436C2×C32C22×C4×D311yescharacteristic
437C2×C32#100411yescharacteristic
438C2×C32#100911yescharacteristic
439C2×C32#101011yescharacteristic
440C2×C32#101011yescharacteristic
441C2×C32C23×Dic311yescharacteristic
442C3⋊D3—22no—
443C3×D3—62no—
444#41—98no—
445C2×Q8—91no—
446C2×D4—92no—
447C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
448C22×C4—92no—
449C22×C4—91no—
450C22×C4—92no—
451C22×C4—92no—
452C22×C4—272no—
453C4⋊C4—91no—
454C4⋊C4—91no—
455C4⋊C4—271no—
456C4⋊C4—272no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C4⋊C4—271no—
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—271no—
463C22⋊C4—271no—
464C42—91no—
465C42—91no—
466C42—272no—
467C42—272no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—31no—
471C22×C3—31no—
472C22×C3—34no—
473C22×C3—34no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—61no—
479C22×C3—64no—
480C22×C3—92no—
481C22×C3—122no—
482C22×C3—122no—
483C22×C3—181no—
484C22×C3—181no—
485C22×C3C2×D3×Dic314yes—
486C22×C3C22×D3211yescharacteristic
487C2×D3—32no—
488C2×D3—61no—
489C2×D3—61no—
490C2×D3—92no—
491C2×D3—181no—
492C2×D3—181no—
493C3×C4—32no—
494C3×C4—32no—
495C3×C4—62no—
496C3×C4—92no—
497C3×C4—94no—
498C3×C4—182no—
499C3×C4—181no—
500C3×C4—181no—
501C3×C4—181no—
502C3×C4—182no—
503C3×C4—181no—
504C3×C4C2×D3×Dic312yes—
505Dic3—34no—
506Dic3—32no—
507Dic3—62no—
508Dic3—61no—
509Dic3—61no—
510Dic3—62no—
511Dic3—61no—
512Dic3—61no—
513Dic3—62no—
514Dic3—61no—
515Dic3—62no—
516Dic3—61no—
517Dic3—94no—
518Dic3—92no—
519Dic3—122no—
520Dic3—121no—
521Dic3—181no—
522Dic3—182no—
523Dic3—181no—
524C32—21no—
525C32#554611yescharacteristic, complemented
526C32#582711yescharacteristic, complemented
527C23—91no—
528C23C2×((C3⋊D3)⋊C3)⋊C212yes—
529Q8—94no—
530D4—98no—
531D4((C3⋊D3)⋊C3)⋊C414yescomplemented
532C2×C4—92no—
533C2×C4—91no—
534C2×C4—98no—
535C2×C4—94no—
536C2×C4—91no—
537C2×C4—91no—
538C2×C4—91no—
539C2×C4—91no—
540C2×C4—92no—
541C2×C4—92no—
542C2×C4—362no—
543C2×C4—362no—
544C2×C4—1082no—
545C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
546C2×C3—24no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—31no—
553C2×C3—64no—
554C2×C3—64no—
555C2×C3—61no—
556C2×C3—61no—
557C2×C3—61no—
558C2×C3—124no—
559C2×C3—182no—
560C2×C3#6430911yescharacteristic
561C2×C3#6431411yescharacteristic
562C2×C3C22×D3×Dic311yescharacteristic
563D3—62no—
564D3—182no—
565C22—22no—
566C22—22no—
567C22—92no—
568C22—181no—
569C22—181no—
570C22C2×((C3⋊D3)⋊C3)⋊C414yes—
571C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
572C4—92no—
573C4—94no—
574C4—181no—
575C4—181no—
576C4—181no—
577C4—181no—
578C4—182no—
579C4—182no—
580C4C2×((C3⋊D3)⋊C3)⋊C412yes—
581C3—31no—
582C3—31no—
583C3—61no—
584C3#1059522211yescharacteristic
585C2—24no—
586C2—182no—
587C2#1172250211yescharacteristic
588C2#1172250711yescharacteristic
589C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radicalC2×#3830⋊C3C2
cosocleC22×H(3,𝔽3)C24
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×#1084C2semidirect
3. D4×(C3⋊Dic3)⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721314C2semidirect
6. C2×#88448C2semidirect
7. #11721296C2semidirect
8. #11721129C2semidirect
9. #11721105C2semidirect
10. #11721084C2semidirect
11. #11721082C2semidirect
19. #3830⋊C3C4semidirect
40. D4×C3⋊Dic3D3semidirect
63. D4×H(3,𝔽3)C2×C4semidirect
69. #638⋊C3C2×C4semidirect
71. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
77. C4×(C3⋊D3)⋊C3C2×C4semidirect
78. #634⋊C3C2×C4semidirect
89. #3830Dic3semidirect
192. C32×D4C4×D3semidirect
193. C32×D4C2×Dic3semidirect
197. #638C2×Dic3semidirect
199. C2×C3⋊Dic3C2×Dic3semidirect
205. C4×C3⋊D3C2×Dic3semidirect
206. #634C2×Dic3semidirect
365. H(3,𝔽3)#520semidirect
525. C32#5546semidirect
526. C32#5827semidirect
531. D4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
D4×#10842
D4×(C3⋊Dic3)⋊C32
#117213152
#117213142
C2×#884482
#117212962
#117211292
#117211052
#117210842
#117210822
#105952223
#105952193