Finite Group Databasebeta

← C4×((C3⋊D3)⋊C3)⋊Q8 #13147773

Subgroups — C4×((C3⋊D3)⋊C3)⋊Q8 #13147773

Order 1728.

1113 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic
2C4×#6229C211yesmaximal, characteristic, complemented
3C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721129C211yesmaximal, characteristic
6#11721119C212yesmaximal
7C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal
8C4×#6196C212yesmaximal, complemented
9#11721083C211yesmaximal, characteristic, complemented
10#11721082C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12C4×#3797—31nomaximal
13C4×D3×Dic6—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×#6196C2212yes—
22((C3⋊D3)⋊C3)⋊Q8C418yesdirect product factor
23C42×H(3,𝔽3)C2211yescharacteristic
24C4⋊#1084C2211yescharacteristic
25#88540C2212yes—
26C4×#1084C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2212yes—
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3797—31no—
39C2×D3×Dic6—31no—
40C42×C3⋊D3D311yescharacteristic
41C4×#634D311yescharacteristic, complemented
42C3×C42×D3—31no—
43C3×C4×Dic6—31no—
44C3×C4×Dic6—31no—
45C3×C4×Dic6—31no—
46C3×C4×Dic6—61no—
47C4×#627—32no—
48C4×#627—32no—
49#63900—31no—
50D3×C4⋊Dic3—31no—
51#63883—32no—
52C4×#624—32no—
53D3×Dic3⋊C4—32no—
54C4×D3×Dic3—32no—
55#63844—31no—
56#63843—31no—
57#63843—31no—
58Dic3×Dic6—31no—
59#63838—32no—
60#63837—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2312yes—
67#6229C2×C414yescomplemented
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3C2×C414yes—
75C4×(C3⋊D3)⋊C3Q814yescomplemented
76#634⋊C3C2×C414yescomplemented
77((C3⋊D3)⋊C3)⋊C4—24no—
78((C3⋊D3)⋊C3)⋊C4C2×C418yes—
79#6196C2×C418yescomplemented
80C4×D3×Q8—91no—
81C4×D3×Q8—91no—
82C2×C4×Dic6—91no—
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×#634C2×D311yescharacteristic
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—61no—
94C2×#627—32no—
95C2×#627—32no—
96C2×#624—32no—
97C2×#624—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100#3797—38no—
101D3×Dic6—38no—
102C32×C42—21no—
103C32×C42C2×D311yescharacteristic
104C32×C42C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750C2×D312yes—
107C4×C3⋊Dic3C2×D312yes—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D311yescharacteristic
110C4×C3⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—61no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—62no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—31no—
121C3×C4×Dic3—32no—
122C3×C4×Dic3—32no—
123C3×C4×Dic3—32no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—62no—
126#3724—32no—
127#3724—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic32—32no—
133Dic32—32no—
134Dic32—32no—
135Dic32—32no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3C22×C412yes—
138C2×(C3⋊D3)⋊C3C2×Q812yes—
139C2×(C3⋊D3)⋊C3#4112yes—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)C22×C412yes—
142C4×H(3,𝔽3)C2×Q812yescomplemented
143#1084—22no—
144#1084C22×C414yes—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3C22×C412yes—
148(C3⋊Dic3)⋊C3C22×C414yes—
149(C3⋊Dic3)⋊C3C2×Q812yescomplemented
150C2×D3×Q8—91no—
151C2×D3×Q8—91no—
152C22×Dic6—91no—
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C2×C3×C42—91no—
156Q8×Dic3—91no—
157Q8×Dic3—91no—
158C2×C4⋊Dic3—91no—
159C2×Dic3⋊C4—92no—
160C2×C4×Dic3—92no—
161D3×C4⋊C4—92no—
162D3×C4⋊C4—92no—
163D3×C4⋊C4—91no—
164D3×C4⋊C4—91no—
165#887—92no—
166#887—91no—
167#887—92no—
168#887—91no—
169C42×D3—31no—
170C42×D3—91no—
171C42×D3—92no—
172C42×D3—92no—
173C4×Dic6—31no—
174C4×Dic6—31no—
175C4×Dic6—31no—
176C4×Dic6—61no—
177C4×Dic6—91no—
178C4×Dic6—92no—
179C4×Dic6—92no—
180C4×Dic6—92no—
181C4×Dic6—92no—
182C4×Dic6—91no—
183C4×Dic6—91no—
184C22×C3⋊D3C22×D311yescharacteristic
185C22×C3×D3—31no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C4×C3⋊D3—22no—
201C4×C3⋊D3Dic614yescomplemented
202C4×C3⋊D3C4×D314yes—
203#634C4×D314yescomplemented
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3—62no—
215C3×C4×D3—34no—
216C3×C4×D3—34no—
217C3×C4×D3—62no—
218C3×Dic6—34no—
219C3×Dic6—34no—
220C3×Dic6—34no—
221C3×Dic6—64no—
222#627—38no—
223#627—38no—
224#624—38no—
225#624—64no—
226D3×Dic3—38no—
227D3×Dic3—64no—
228C2×C4×Q8—271noSylow
229C2×H(3,𝔽3)C23×C411yescharacteristic
230C2×H(3,𝔽3)C22×Q811yescharacteristic
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232(C3⋊D3)⋊C3C4×Q814yescomplemented
233C2×C3×Q8—91no—
234C2×C3×Q8—91no—
235C22×C3×C4—91no—
236C22×C3×C4—91no—
237C22×C3×C4—91no—
238C22×Dic3—92no—
239C22×Dic3—92no—
240D3×Q8—98no—
241D3×Q8—98no—
242C2×C4×D3—31no—
243C2×C4×D3—31no—
244C2×C4×D3—31no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C2×C4×D3—92no—
251C2×C4×D3—92no—
252C2×Dic6—31no—
253C2×Dic6—31no—
254C2×Dic6—31no—
255C2×Dic6—61no—
256C2×Dic6—91no—
257C2×Dic6—98no—
258C2×Dic6—92no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—91no—
265C3×C42—31no—
266C3×C42—31no—
267C3×C42—61no—
268C3×C42—91no—
269C3×C42—91no—
270C3×C42—91no—
271C3×C42—92no—
272C3×C42—92no—
273C3×C42D3211yescharacteristic
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—61no—
278C4⋊Dic3—91no—
279C4⋊Dic3—92no—
280C4⋊Dic3—92no—
281C4⋊Dic3—92no—
282Dic3⋊C4—32no—
283Dic3⋊C4—32no—
284Dic3⋊C4—32no—
285Dic3⋊C4—62no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—32no—
294C4×Dic3—32no—
295C4×Dic3—32no—
296C4×Dic3—31no—
297C4×Dic3—31no—
298C4×Dic3—31no—
299C4×Dic3—62no—
300C4×Dic3—91no—
301C4×Dic3—92no—
302C4×Dic3—91no—
303C4×Dic3—91no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3—92no—
308C4×Dic3—92no—
309C4×Dic3—92no—
310C22×C32—21no—
311C22×C32C23×D311yescharacteristic
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3C2×Dic612yes—
314C2×C3⋊D3C2×C4×D312yes—
315C2×C3⋊D3#23512yes—
316C2×C3×D3—32no—
317C2×C3×D3—32no—
318C2×C3×D3—32no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—22no—
323C32×C4—41no—
324C32×C4C2×Dic612yescomplemented
325C32×C4C2×C4×D312yes—
326C32×C4C2×C4×D312yes—
327C32×C4D3×Q812yescomplemented
328C3⋊Dic3—21no—
329C3⋊Dic3—22no—
330C3⋊Dic3C2×Dic612yescomplemented
331C3⋊Dic3C2×C4×D314yes—
332C3⋊Dic3C2×C4×D312yes—
333C3×Dic3—34no—
334C3×Dic3—32no—
335C3×Dic3—34no—
336C3×Dic3—34no—
337C3×Dic3—32no—
338C3×Dic3—64no—
339C3×Dic3—62no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—62no—
343C3×Dic3—122no—
344C22×Q8—271no—
345C4×Q8—91no—
346C4×Q8—91no—
347C4×Q8—272no—
348C4×Q8—272no—
349C4×Q8—271no—
350C4×Q8—271no—
351C2×C4⋊C4—272no—
352C2×C4⋊C4—271no—
353C2×C42—91no—
354C2×C42—272no—
355H(3,𝔽3)C2×C4×Q811yescharacteristic, complemented, Sylow
356C23×C3—91no—
357C22×D3—31no—
358C22×D3—91no—
359C3×Q8—94no—
360C3×Q8—94no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—94no—
371C2×C3×C4—94no—
372C2×C3×C4—91no—
373C2×C3×C4—92no—
374C2×C3×C4—91no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—91no—
378C2×C3×C4—92no—
379C2×C3×C4—182no—
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—31no—
391C2×Dic3—32no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—92no—
397C2×Dic3—91no—
398C2×Dic3—92no—
399C2×Dic3—98no—
400C2×Dic3—184no—
401C4×D3—34no—
402C4×D3—34no—
403C4×D3—62no—
404C4×D3—94no—
405C4×D3—94no—
406C4×D3—98no—
407C4×D3—98no—
408C4×D3—182no—
409C4×D3—184no—
410C4×D3—184no—
411Dic6—34no—
412Dic6—34no—
413Dic6—34no—
414Dic6—64no—
415Dic6—94no—
416Dic6—98no—
417Dic6—98no—
418Dic6—98no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32C22×Dic611yescharacteristic
423C2×C32C22×C4×D311yescharacteristic
424C2×C32C22×C4×D311yescharacteristic
425C2×C32C2×#23511yescharacteristic
426C2×C32C2×D3×Q811yescharacteristic
427C2×C32D3×#4111yescharacteristic
428C3⋊D3C4×Dic614yescomplemented
429C3×D3—34no—
430C2×Q8—91no—
431C2×Q8—91no—
432C2×Q8—272no—
433C2×Q8—278no—
434C22×C4—91no—
435C22×C4—91no—
436C22×C4—91no—
437C22×C4—272no—
438C22×C4—272no—
439C4⋊C4—92no—
440C4⋊C4—92no—
441C4⋊C4—91no—
442C4⋊C4—91no—
443C4⋊C4—272no—
444C4⋊C4—272no—
445C4⋊C4—272no—
446C42—91no—
447C42—91no—
448C42—91no—
449C42—92no—
450C42—92no—
451C42—272no—
452C42—272no—
453C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—61no—
457C22×C3—92no—
458C22×C3—92no—
459C22×C3—92no—
460C22×C3C22×D3211yescharacteristic
461C2×D3—32no—
462C2×D3—32no—
463C2×D3—32no—
464C2×D3—92no—
465C2×D3—92no—
466C2×D3—92no—
467C3×C4—21no—
468C3×C4—32no—
469C3×C4—32no—
470C3×C4—32no—
471C3×C4—32no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—62no—
475C3×C4—62no—
476C3×C4—92no—
477C3×C4—94no—
478C3×C4—92no—
479C3×C4—94no—
480C3×C4—121no—
481C3×C4—182no—
482C3×C4—181no—
483C3×C4—182no—
484C3×C4D3×Dic612yes—
485C3×C4C4×D3212yes—
486Dic3—34no—
487Dic3—34no—
488Dic3—32no—
489Dic3—34no—
490Dic3—32no—
491Dic3—64no—
492Dic3—62no—
493Dic3—62no—
494Dic3—61no—
495Dic3—62no—
496Dic3—92no—
497Dic3—94no—
498Dic3—92no—
499Dic3—122no—
500Dic3—181no—
501Dic3—182no—
502C32—21no—
503C32C2×C4×Dic611yescharacteristic, complemented
504C32C4×D3×Q811yescharacteristic, complemented
505C23—91no—
506Q8—94no—
507Q8—94no—
508Q8—278no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—92no—
512C2×C4—92no—
513C2×C4—94no—
514C2×C4—94no—
515C2×C4—91no—
516C2×C4—92no—
517C2×C4—92no—
518C2×C4—182no—
519C2×C4—278no—
520C2×C4—544no—
521C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—61no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—94no—
534C2×C3C2×D3×Dic611yescharacteristic
535C2×C3C2×C4×D3211yescharacteristic
536C2×C3D3×#23511yescharacteristic
537D3—34no—
538D3—94no—
539C22—92no—
540C22—92no—
541C22—92no—
542C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
543C4—21no—
544C4—92no—
545C4—92no—
546C4—94no—
547C4—94no—
548C4—181no—
549C4—182no—
550C4—182no—
551C4((C3⋊D3)⋊C3)⋊Q812yesdirect product factor
552C4C4×((C3⋊D3)⋊C3)⋊C212yes—
553C3—31no—
554C3—31no—
555C3—61no—
556C3C4×D3×Dic611yescharacteristic
557C2—94no—
558C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
559C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
560C2#1172249011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×Q8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6229C2semidirect
4. C4×#634⋊C3C2semidirect
8. C4×#6196C2semidirect
9. #11721083C2semidirect
10. #11721082C2semidirect
11. #11721077C2semidirect
22. ((C3⋊D3)⋊C3)⋊Q8C4direct (N × H)
41. C4×#634D3semidirect
67. #6229C2×C4semidirect
75. C4×(C3⋊D3)⋊C3Q8semidirect
76. #634⋊C3C2×C4semidirect
79. #6196C2×C4semidirect
142. C4×H(3,𝔽3)C2×Q8semidirect
149. (C3⋊Dic3)⋊C3C2×Q8semidirect
201. C4×C3⋊D3Dic6semidirect
203. #634C4×D3semidirect
232. (C3⋊D3)⋊C3C4×Q8semidirect
324. C32×C4C2×Dic6semidirect
327. C32×C4D3×Q8semidirect
330. C3⋊Dic3C2×Dic6semidirect
355. H(3,𝔽3)C2×C4×Q8semidirect
428. C3⋊D3C4×Dic6semidirect
503. C32C2×C4×Dic6semidirect
504. C32C4×D3×Q8semidirect
551. C4((C3⋊D3)⋊C3)⋊Q8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
C4×#62292
C42×(C3⋊D3)⋊C32
C4×#634⋊C32
#117211292
#117211192
C4×((C3⋊D3)⋊C3)⋊C42
C4×#61962
#117210832
#117210822
#117210772
C4×#37973
C4×D3×Dic63