Finite Group Databasebeta

← #13151378

Subgroups — #13151378

Order 1728.

1179 conjugacy classes of subgroups.

The subgroup lattice has 598 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
6#11721296C212yesmaximal, complemented
7C2×#88447C212yesmaximal, complemented
8C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
9#11721084C211yesmaximal, characteristic, complemented
10#11721082C211yesmaximal, characteristic, complemented
11#11721081C211yesmaximal, characteristic, complemented
12#10595190—31nomaximal
13Dic3×#237—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)⋊C4C2211yescharacteristic
23C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
24#88552C2212yes—
25C4⋊#1084C2211yescharacteristic
26C4×#1084C2211yescharacteristic
27#3757⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88450C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88447C2211yescharacteristic
34#88447C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37#88447C2211yescharacteristic
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—
47#63957—32no—
48#63955—32no—
49C2×Dic32—32no—
50C2×Dic32—32no—
51#63952—32no—
52#63952—32no—
53C4×#624—31no—
54C4×D3×Dic3—31no—
55#63849—31no—
56#63844—31no—
57#63844—31no—
58#63843—31no—
59Dic3×Dic6—31no—
60#63841—31no—
61C23×H(3,𝔽3)C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63D4×H(3,𝔽3)C2×C414yescomplemented
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084—24no—
66C2×#1084C2311yescharacteristic
67C2×#1084C2312yes—
68C2×#1084C2311yescharacteristic
69#638⋊C3C2×C418yescomplemented
70C2×(C3⋊Dic3)⋊C3—24no—
71C2×(C3⋊Dic3)⋊C3C2×C418yescomplemented
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2312yes—
75C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
76C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
77C4×(C3⋊D3)⋊C3C2×C414yescomplemented
78#634⋊C3C2×C414yescomplemented
79((C3⋊D3)⋊C3)⋊C4—22no—
80((C3⋊D3)⋊C3)⋊C4—22no—
81Dic3×#41—91no—
82C4×#237—91no—
83C4×#237—91no—
84C2×C32×D4—21no—
85C2×C32×D4C2×D311yescharacteristic
86C2×C32×D4C2×D311yescharacteristic
87C2×#638C2×D312yes—
88C22×C3⋊Dic3C2×D312yes—
89C22×C3⋊Dic3C2×D312yes—
90#3830Dic318yescomplemented
91C2×C4×C3⋊D3C2×D311yescharacteristic
92C2×#634C2×D311yescharacteristic
93C2×C3×#67—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3—32no—
97C22×C3×Dic3—32no—
98C22×C3×Dic3—62no—
99C3×#237—38no—
100C2×C3×C4×D3—31no—
101C2×C3×Dic6—31no—
102C2×#624—31no—
103C2×#624—31no—
104C2×D3×Dic3—31no—
105C2×D3×Dic3—31no—
106#3757C2×D312yes—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108C4×C3⋊Dic3C2×D311yescharacteristic
109C3×#217—32no—
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—62no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C3×C4×Dic3—31no—
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(C2×D3)⋊Dic3—32no—
127Dic32—32no—
128Dic32—32no—
129Dic32—32no—
130Dic32—32no—
131Dic32—32no—
132Dic32—32no—
133Dic32—31no—
134Dic32—31no—
135Dic32—31no—
136Dic32—31no—
137C22×H(3,𝔽3)—22no—
138C22×H(3,𝔽3)—22no—
139C22×H(3,𝔽3)C22×C414yes—
140C22×H(3,𝔽3)C2411yescharacteristic
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3—21no—
143C2×(C3⋊D3)⋊C3C22×C412yes—
144C4×H(3,𝔽3)C22×C412yes—
145#1084—21no—
146#1084—22no—
147#1084#4112yescomplemented
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3C22×C414yes—
151(C3⋊Dic3)⋊C3C22×C412yes—
152(C3⋊Dic3)⋊C3#4112yescomplemented
153C2×C3×#41—91no—
154C2×#237—31no—
155C2×#237—91no—
156C3×C4×D4—91no—
157C3×C4×D4—91no—
158Q8×Dic3—91no—
159Q8×Dic3—91no—
160D4×Dic3—31no—
161D4×Dic3—31no—
162D4×Dic3—31no—
163D4×Dic3—61no—
164D4×Dic3—91no—
165D4×Dic3—92no—
166D4×Dic3—92no—
167C4×#67—92no—
168C4×#67—92no—
169#926—91no—
170#926—92no—
171C2×C4×Dic3—91no—
172C2×C4×Dic3—92no—
173C2×C4×Dic3—92no—
174C2×C4×Dic3—92no—
175#892—91no—
176#892—91no—
177#887—91no—
178#887—91no—
179#67⋊C4—92no—
180#67⋊C4—92no—
181#877—92no—
182#877—92no—
183C42×D3—91no—
184C42×D3—91no—
185C4×Dic6—91no—
186C4×Dic6—91no—
187C23×C32—22no—
188C23×C32C22×D312yes—
189C23×C32C22×D312yes—
190C22×C3⋊D3C22×D311yescharacteristic
191C22×C3×D3—31no—
192C32×D4—24no—
193C32×D4C4×D314yescomplemented
194C32×D4C2×Dic314yescomplemented
195C2×C32×C4—21no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198#638C2×Dic318yescomplemented
199C2×C3⋊Dic3—24no—
200C2×C3⋊Dic3C2×Dic318yescomplemented
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D312yes—
206C4×C3⋊D3C2×Dic314yescomplemented
207#634C2×Dic314yescomplemented
208C3×#67—38no—
209C2×C3×Dic3—38no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—62no—
224C2×C3×Dic3—64no—
225C2×C3×Dic3—64no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3—124no—
228C3×C4×D3—34no—
229C3×Dic6—34no—
230#624—62no—
231#624—62no—
232D3×Dic3—62no—
233D3×Dic3—62no—
234C4×#41—271noSylow
235C2×H(3,𝔽3)—24no—
236C2×H(3,𝔽3)C23×C411yescharacteristic
237C2×H(3,𝔽3)C2×#4111yescharacteristic
238C2×H(3,𝔽3)C2×#4111yescharacteristic
239(C3⋊D3)⋊C3—22no—
240C3×#41—98no—
241C2×C3×Q8—91no—
242C2×C3×D4—31no—
243C2×C3×D4—31no—
244C2×C3×D4—61no—
245C2×C3×D4—92no—
246C2×C3×D4D3211yescharacteristic
247C22×C3×C4—91no—
248C22×C3×C4—92no—
249C22×C3×C4—92no—
250C22×C3×C4—92no—
251C2×#67—32no—
252C2×#67—92no—
253C22×Dic3—32no—
254C22×Dic3—32no—
255C22×Dic3—32no—
256C22×Dic3—32no—
257C22×Dic3—62no—
258C22×Dic3—92no—
259C22×Dic3—92no—
260C22×Dic3—91no—
261C22×Dic3—91no—
262#237—38no—
263#237—98no—
264C2×C4×D3—31no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—91no—
268C2×C4×D3—91no—
269C2×C4×D3—91no—
270C2×Dic6—31no—
271C2×Dic6—91no—
272C3×C4⋊C4—91no—
273C3×C4⋊C4—91no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—92no—
276C3×C42—91no—
277C3×C42—91no—
278#217—32no—
279#217—32no—
280#217—32no—
281#217—62no—
282#217—92no—
283#217—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—31no—
287C4⋊Dic3—31no—
288C4⋊Dic3—31no—
289C4⋊Dic3—61no—
290C4⋊Dic3—91no—
291C4⋊Dic3—92no—
292C4⋊Dic3—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—31no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—61no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C4×Dic3—91no—
304C4×Dic3—91no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3—92no—
308C4×Dic3—91no—
309C4×Dic3—91no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C4×Dic3—91no—
313C4×Dic3—91no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C22×C32—22no—
318C22×C32—22no—
319C22×C32—22no—
320C22×C32—22no—
321C22×C32—21no—
322C22×C32—24no—
323C22×C32—42no—
324C22×C32—42no—
325C22×C32C2×C4×D314yes—
326C22×C32C22×Dic314yes—
327C22×C32C23×D311yescharacteristic
328C22×C32C23×D311yescharacteristic
329C2×C3⋊D3—21no—
330C2×C3⋊D3—21no—
331C2×C3⋊D3C22×Dic312yes—
332C2×C3×D3—32no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C32×C4—22no—
336C32×C4C2×C4×D312yes—
337C32×C4C22×Dic312yes—
338C3⋊Dic3—22no—
339C3⋊Dic3—21no—
340C3⋊Dic3#23712yescomplemented
341C3⋊Dic3C22×Dic314yes—
342C3⋊Dic3C22×Dic312yes—
343C3×Dic3—34no—
344C3×Dic3—32no—
345C3×Dic3—32no—
346C3×Dic3—32no—
347C3×Dic3—32no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—62no—
352C3×Dic3—61no—
353C3×Dic3—62no—
354C3×Dic3—62no—
355C3×Dic3—121no—
356C3×Dic3—122no—
357C2×#41—91no—
358C4×Q8—271no—
359C4×Q8—271no—
360C4×D4—91no—
361C4×D4—91no—
362C4×D4—272no—
363C4×D4—272no—
364#117—272no—
365#117—271no—
366C2×C42—271no—
367C2×C42—272no—
368H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
369C23×C3—32no—
370C23×C3—32no—
371C23×C3—62no—
372C23×C3—91no—
373C23×C3C2×D3212yes—
374C22×D3—31no—
375C22×D3—91no—
376C3×Q8—94no—
377C3×D4—34no—
378C3×D4—34no—
379C3×D4—64no—
380C3×D4—98no—
381C3×D4D3×Dic314yes—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—61no—
385C2×C3×C4—94no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—92no—
389C2×C3×C4—98no—
390C2×C3×C4—91no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—92no—
394C2×C3×C4—91no—
395C2×C3×C4—184no—
396C2×C3×C4—184no—
397C2×C3×C4C2×D3211yescharacteristic
398#67—38no—
399#67—98no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—32no—
405C2×Dic3—38no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—31no—
411C2×Dic3—32no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—62no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3—64no—
418C2×Dic3—98no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—92no—
422C2×Dic3—91no—
423C2×Dic3—92no—
424C2×Dic3—124no—
425C2×Dic3—184no—
426C2×Dic3—182no—
427C2×Dic3—182no—
428C4×D3—34no—
429C4×D3—94no—
430C4×D3—182no—
431C4×D3—182no—
432C4×D3—182no—
433C4×D3—182no—
434Dic6—34no—
435Dic6—94no—
436C2×C32—24no—
437C2×C32—24no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—44no—
442C2×C32C22×C4×D311yescharacteristic
443C2×C32C2×#23711yescharacteristic
444C2×C32D3×#4111yescharacteristic
445C2×C32D3×#4111yescharacteristic
446C2×C32D3×#4111yescharacteristic
447C2×C32C23×Dic311yescharacteristic
448C3⋊D3—22no—
449C3×D3—62no—
450#41—98no—
451C2×Q8—91no—
452C2×D4—92no—
453C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
454C22×C4—92no—
455C22×C4—91no—
456C22×C4—92no—
457C22×C4—92no—
458C22×C4—271no—
459C22×C4—271no—
460C4⋊C4—91no—
461C4⋊C4—91no—
462C4⋊C4—272no—
463C4⋊C4—272no—
464C22⋊C4—92no—
465C22⋊C4—92no—
466C22⋊C4—272no—
467C42—91no—
468C42—91no—
469C42—271no—
470C42—271no—
471C42—272no—
472C42—272no—
473C42—272no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—31no—
477C22×C3—31no—
478C22×C3—34no—
479C22×C3—34no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—61no—
485C22×C3—64no—
486C22×C3—92no—
487C22×C3—122no—
488C22×C3—122no—
489C22×C3—181no—
490C22×C3—181no—
491C22×C3C2×D3×Dic314yes—
492C22×C3C22×D3211yescharacteristic
493C2×D3—32no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—92no—
497C2×D3—181no—
498C2×D3—181no—
499C3×C4—32no—
500C3×C4—32no—
501C3×C4—62no—
502C3×C4—92no—
503C3×C4—92no—
504C3×C4—94no—
505C3×C4—92no—
506C3×C4—181no—
507C3×C4—182no—
508C3×C4—181no—
509C3×C4—182no—
510C3×C4C2×D3×Dic312yes—
511Dic3—32no—
512Dic3—32no—
513Dic3—32no—
514Dic3—34no—
515Dic3—32no—
516Dic3—62no—
517Dic3—61no—
518Dic3—62no—
519Dic3—62no—
520Dic3—61no—
521Dic3—62no—
522Dic3—61no—
523Dic3—94no—
524Dic3—92no—
525Dic3—92no—
526Dic3—122no—
527Dic3—121no—
528Dic3—182no—
529Dic3—181no—
530C32—21no—
531C32C4×#23711yescharacteristic, complemented
532C32Dic3×#4111yescharacteristic, complemented
533C23—91no—
534C23C2×((C3⋊D3)⋊C3)⋊C212yes—
535Q8—94no—
536D4—98no—
537D4((C3⋊D3)⋊C3)⋊C414yescomplemented
538C2×C4—91no—
539C2×C4—91no—
540C2×C4—91no—
541C2×C4—92no—
542C2×C4—98no—
543C2×C4—94no—
544C2×C4—91no—
545C2×C4—91no—
546C2×C4—92no—
547C2×C4—92no—
548C2×C4—184no—
549C2×C4—184no—
550C2×C4—542no—
551C2×C4—542no—
552C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
553C2×C3—24no—
554C2×C3—31no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—64no—
561C2×C3—64no—
562C2×C3—61no—
563C2×C3—61no—
564C2×C3—61no—
565C2×C3—124no—
566C2×C3—182no—
567C2×C3D3×#23711yescharacteristic
568C2×C3#6431211yescharacteristic
569C2×C3C22×D3×Dic311yescharacteristic
570D3—62no—
571D3—182no—
572C22—22no—
573C22—22no—
574C22—92no—
575C22—181no—
576C22—181no—
577C22C2×((C3⋊D3)⋊C3)⋊C414yes—
578C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
579C4—92no—
580C4—92no—
581C4—92no—
582C4—94no—
583C4—182no—
584C4—181no—
585C4—182no—
586C4—181no—
587C4C2×((C3⋊D3)⋊C3)⋊C412yes—
588C3—31no—
589C3—31no—
590C3—61no—
591C3Dic3×#23711yescharacteristic
592C2—24no—
593C2—182no—
594C2#1172250411yescharacteristic
595C2#1172250511yescharacteristic
596C2C22×((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)C4×#41
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. #11721296C2semidirect
7. C2×#88447C2semidirect
8. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
9. #11721084C2semidirect
10. #11721082C2semidirect
11. #11721081C2semidirect
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
90. #3830Dic3semidirect
147. #1084#41semidirect
152. (C3⋊Dic3)⋊C3#41semidirect
193. C32×D4C4×D3semidirect
194. C32×D4C2×Dic3semidirect
198. #638C2×Dic3semidirect
200. C2×C3⋊Dic3C2×Dic3semidirect
206. C4×C3⋊D3C2×Dic3semidirect
207. #634C2×Dic3semidirect
340. C3⋊Dic3#237semidirect
368. H(3,𝔽3)C4×#41semidirect
531. C32C4×#237semidirect
532. C32Dic3×#41semidirect
537. 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
#117212962
C2×#884472
C4×((C3⋊D3)⋊C3)⋊C42
#117210842
#117210822
#117210812
#105951903
Dic3×#2373