Finite Group Databasebeta

← D4×(C3⋊S4)⋊C3 #13164402

Subgroups — D4×(C3⋊S4)⋊C3 #13164402

Order 1728.

948 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C3⋊S4)⋊C3C212yesmaximal, complemented
2D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64269⋊C3C212yesmaximal, complemented
4C4⋊((C3⋊S4)⋊C3)C211yesmaximal, characteristic, complemented
5C4×(C3⋊S4)⋊C3C211yesmaximal, characteristic, complemented
6D4×C3⋊S4C311yesmaximal, characteristic, complemented
7C3×D4×#67—31nomaximal
8C22×C22⋊H(3,𝔽3)C2212yescomplemented
9C2×(C3⋊S4)⋊C3—24no—
10C2×(C3⋊S4)⋊C3C2211yescharacteristic
11C2×(C3⋊S4)⋊C3C2212yescomplemented
12D4×(C3⋊D3)⋊C3—41nomaximal
13C4×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
14(C3⋊SL(2,ℤ/4))⋊C3C2211yescharacteristic, complemented
15C22×C3⋊S4C2×C312yescomplemented
16C22×C32×D4D311yescharacteristic, complemented
17C22×C3×#67—32no—
18C2×C3×D3×D4—31no—
19C3×D4×A4—21no—
20C3×D4×A4C2×C311yescharacteristic, complemented
21#64269C2×C312yescomplemented
22C4⋊(C3⋊S4)C2×C311yescharacteristic, complemented
23C4×C3⋊S4C2×C311yescharacteristic, complemented
24C3×#953—32no—
25C3×#940—31no—
26C3×#939—32no—
27C3×#938—31no—
28C3×#937—32no—
29C3×D4×Dic3—31no—
30C3×#930—31no—
31C3×C4×#67—31no—
32C22×(C3⋊D3)⋊C3—42no—
33C2×C22⋊H(3,𝔽3)—22no—
34C2×C22⋊H(3,𝔽3)C2311yescharacteristic
35(C3⋊S4)⋊C3—22no—
36(C3⋊S4)⋊C3D412yesdirect product factor
37D4×H(3,𝔽3)—41no—
38#638⋊C3—42no—
39C4⋊((C3⋊D3)⋊C3)—41no—
40C4×(C3⋊D3)⋊C3—41no—
41D4×S4—91no—
42C3×D42—91no—
43D4×#67—31no—
44C24×C32C2×D312yescomplemented
45C23×C3×D3—32no—
46C22×C3×A4—22no—
47C22×C3×A4C22×C312yescomplemented
48C2×C3⋊S4—24no—
49C2×C3⋊S4C22×C311yescharacteristic
50C2×C3⋊S4C22×C312yescomplemented
51C2×C32×D4—31no—
52C2×C32×D4—32no—
53C2×C32×D4—31no—
54C22×C32×C4C2×D311yescharacteristic, complemented
55D4×C3⋊D3—41no—
56C2×C3×#67—31no—
57C2×C3×#67—32no—
58C2×C3×#67—32no—
59C2×C3×#67—32no—
60C2×C3×#67—32no—
61C2×C3×#67—32no—
62C2×C3×#67—64no—
63C2×C3×#67—64no—
64C22×C3×Dic3—32no—
65C3×D3×D4—62no—
66C3×D3×D4—61no—
67C3×D3×D4—61no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—31no—
70C3×C4×A4—21no—
71C3×C4×A4C22×C311yescharacteristic, complemented
72C3⋊SL(2,ℤ/4)C22×C311yescharacteristic, complemented
73C3×#217—32no—
74C3×#217—32no—
75C3×#217—31no—
76C3×(C2×D3)⋊C4—31no—
77C3×(C2×D3)⋊C4—32no—
78C3×C4⋊Dic3—31no—
79C3×Dic3⋊C4—31no—
80C3×C4×Dic3—31no—
81C22×H(3,𝔽3)—42no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—42no—
84C2×(C3⋊D3)⋊C3—84no—
85C22⋊H(3,𝔽3)C2×D411yescharacteristic, complemented
86C4×H(3,𝔽3)—41no—
87(C3⋊Dic3)⋊C3—41no—
88C22×S4—92no—
89C22×C3×D4—31no—
90C22×C3×D4—91no—
91C22×C3×D4—92no—
92C22×C3×D4C3×D311yescharacteristic
93C22×#67—32no—
94C2×D3×D4—31no—
95D4×A4—31no—
96D4×A4—61no—
97GL(2,ℤ/4)—92no—
98C4⋊S4—91no—
99C4×S4—91no—
100C3×#127—91no—
101C3×#121—91no—
102C3×#121—91no—
103C3×#121—92no—
104C3×#120—92no—
105C3×#120—92no—
106C3×C4×D4—91no—
107C3×C4×D4—91no—
108#953—32no—
109#940—31no—
110#939—32no—
111#938—31no—
112#937—32no—
113D4×Dic3—31no—
114#930—31no—
115C4×#67—31no—
116C23×C32—22no—
117C23×C32—32no—
118C23×C32—32no—
119C23×C32—62no—
120C23×C32C22×D311yescharacteristic
121C22×C3⋊D3—42no—
122C22×C3×D3—31no—
123C22×C3×D3—32no—
124C22×C3×D3—64no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—122no—
128C2×C3×A4—22no—
129C2×C3×A4—21no—
130C2×C3×A4—42no—
131C2×C3×A4C23×C311yescharacteristic
132C3⋊S4—22no—
133C3⋊S4C3×D412yescomplemented
134C32×D4—32no—
135C32×D4—31no—
136C32×D4—41no—
137C32×D4—61no—
138C32×D4—81no—
139C32×D4S411yescharacteristic, complemented
140C2×C32×C4—31no—
141C2×C32×C4—31no—
142#638—42no—
143C4⋊(C3⋊D3)—41no—
144C4×C3⋊D3—41no—
145C3×#67—32no—
146C3×#67—32no—
147C3×#67—64no—
148C3×#67—64no—
149C3×#67—62no—
150C3×#67—62no—
151C3×#67—62no—
152C3×#67—62no—
153C3×#67—122no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—64no—
159C3×D12—61no—
160C3×D12—61no—
161C3×C4×D3—61no—
162C3×C4×D3—61no—
163D42—91noSylow
164C2×H(3,𝔽3)—41no—
165C2×H(3,𝔽3)—82no—
166(C3⋊D3)⋊C3—42no—
167(C3⋊D3)⋊C3—82no—
168C24×C3—32no—
169C24×C3—92no—
170C24×C3C2×C3×D312yes—
171C23×D3—32no—
172C22×A4—32no—
173C22×A4—62no—
174C2×S4—91no—
175C2×S4—92no—
176C2×S4—184no—
177C2×C3×D4—31no—
178C2×C3×D4—31no—
179C2×C3×D4—32no—
180C2×C3×D4—31no—
181C2×C3×D4—32no—
182C2×C3×D4—31no—
183C2×C3×D4—61no—
184C2×C3×D4—62no—
185C2×C3×D4—61no—
186C2×C3×D4—91no—
187C2×C3×D4—91no—
188C2×C3×D4—92no—
189C2×C3×D4—92no—
190C2×C3×D4—92no—
191C2×C3×D4—92no—
192C2×C3×D4—92no—
193C2×C3×D4—182no—
194C2×C3×D4—181no—
195C2×C3×D4—181no—
196C2×C3×D4—184no—
197C2×C3×D4—184no—
198C22×C3×C4—31no—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C22×C3×C4C2×C3×D311yescharacteristic
202C2×#67—31no—
203C2×#67—32no—
204C2×#67—32no—
205C2×#67—32no—
206C2×#67—32no—
207C2×#67—32no—
208C2×#67—64no—
209C2×#67—64no—
210C22×Dic3—32no—
211D3×D4—62no—
212D3×D4—61no—
213D3×D4—61no—
214D3×D4—361no—
215C2×D12—31no—
216C2×C4×D3—31no—
217C4×A4—31no—
218C4×A4—61no—
219SL(2,ℤ/4)—91no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—91no—
222C3×C22⋊C4—91no—
223C3×C22⋊C4—92no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—91no—
226C3×C22⋊C4—92no—
227C3×C42—91no—
228#217—32no—
229#217—32no—
230#217—31no—
231(C2×D3)⋊C4—31no—
232(C2×D3)⋊C4—32no—
233C4⋊Dic3—31no—
234Dic3⋊C4—31no—
235C4×Dic3—31no—
236C22×C32—31no—
237C22×C32—31no—
238C22×C32—32no—
239C22×C32—42no—
240C22×C32—62no—
241C22×C32—62no—
242C22×C32—62no—
243C22×C32—62no—
244C22×C32—82no—
245C22×C32D3×D411yescharacteristic, complemented
246C22×C32C2×S412yescomplemented
247C2×C3⋊D3—41no—
248C2×C3⋊D3—42no—
249C2×C3⋊D3—84no—
250C2×C3×D3—32no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—124no—
257C2×C3×D3—124no—
258C3×A4—21no—
259C3×A4C2×C3×D411yescharacteristic, complemented
260C32×C4—31no—
261C32×C4—41no—
262C32×C4—81no—
263C32×C4C2×S411yescharacteristic, complemented
264C3⋊Dic3—41no—
265C3×Dic3—32no—
266C3×Dic3—62no—
267C3×Dic3—61no—
268C3×Dic3—61no—
269C22×D4—91no—
270C22×D4—92no—
271C22×D4(C3⋊D3)⋊C311yescharacteristic, complemented
272#127—91no—
273#121—91no—
274#121—92no—
275#121—91no—
276#120—92no—
277#120—92no—
278C4×D4—91no—
279C4×D4—91no—
280H(3,𝔽3)—41noSylow
281C23×C3—22no—
282C23×C3—32no—
283C23×C3—32no—
284C23×C3—32no—
285C23×C3—32no—
286C23×C3—31no—
287C23×C3—62no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—91no—
293C23×C3—92no—
294C23×C3—122no—
295C23×C3—182no—
296C23×C3—182no—
297C23×C3—184no—
298C23×C3—362no—
299C23×C3C22×C3×D311yescharacteristic
300C22×D3—31no—
301C22×D3—32no—
302C22×D3—64no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—122no—
306C22×D3—362no—
307C2×A4—31no—
308C2×A4—62no—
309C2×A4—61no—
310C2×A4—122no—
311S4—92no—
312S4—182no—
313C3×D4—32no—
314C3×D4—31no—
315C3×D4—32no—
316C3×D4—31no—
317C3×D4—31no—
318C3×D4—61no—
319C3×D4—62no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—62no—
323C3×D4—92no—
324C3×D4—92no—
325C3×D4—121no—
326C3×D4—181no—
327C3×D4—181no—
328C3×D4—184no—
329C3×D4—182no—
330C3×D4—182no—
331C3×D4—182no—
332C3×D4—182no—
333C3×D4—184no—
334C3×D4—241no—
335C3×D4—362no—
336C3×D4C3×S411yescharacteristic
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—91no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—92no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—184no—
350#67—32no—
351#67—32no—
352#67—64no—
353#67—64no—
354#67—62no—
355#67—62no—
356#67—62no—
357#67—62no—
358#67—122no—
359#67—362no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—32no—
364C2×Dic3—64no—
365D12—61no—
366D12—61no—
367D12—361no—
368C4×D3—61no—
369C4×D3—61no—
370C4×D3—361no—
371C2×C32—22no—
372C2×C32—31no—
373C2×C32—31no—
374C2×C32—41no—
375C2×C32—62no—
376C2×C32—82no—
377C2×C32—81no—
378C2×C32—162no—
379C2×C32C22×S411yescharacteristic
380C3⋊D3—42no—
381C3⋊D3—82no—
382C3×D3—62no—
383C3×D3—122no—
384C24—92no—
385C24C2×(C3⋊D3)⋊C312yescomplemented
386C2×D4—31no—
387C2×D4—32no—
388C2×D4—31no—
389C2×D4—91no—
390C2×D4—91no—
391C2×D4—92no—
392C2×D4—92no—
393C2×D4—92no—
394C2×D4—92no—
395C2×D4—92no—
396C2×D4—182no—
397C2×D4—181no—
398C2×D4—184no—
399C2×D4—184no—
400C2×D4—181no—
401C22×C4—91no—
402C22×C4—92no—
403C22×C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
404C4⋊C4—91no—
405C4⋊C4—91no—
406C22⋊C4—91no—
407C22⋊C4—92no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4—91no—
411C42—91no—
412C22×C3—31no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—32no—
416C22×C3—31no—
417C22×C3—32no—
418C22×C3—32no—
419C22×C3—31no—
420C22×C3—61no—
421C22×C3—61no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—92no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—122no—
437C22×C3—181no—
438C22×C3—181no—
439C22×C3—182no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—242no—
443C22×C3—364no—
444C22×C3—364no—
445C22×C3C3×D3×D411yescharacteristic
446C22×C3C2×C3×S412yes—
447C2×D3—32no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—124no—
454C2×D3—124no—
455C2×D3—361no—
456C2×D3—362no—
457C2×D3—724no—
458A4—31no—
459A4—61no—
460C3×C4—31no—
461C3×C4—31no—
462C3×C4—31no—
463C3×C4—61no—
464C3×C4—92no—
465C3×C4—121no—
466C3×C4—182no—
467C3×C4—181no—
468C3×C4—181no—
469C3×C4—241no—
470C3×C4C2×C3×S411yescharacteristic
471Dic3—32no—
472Dic3—62no—
473Dic3—61no—
474Dic3—61no—
475Dic3—361no—
476C32—41no—
477C32—81no—
478C32D4×S411yescharacteristic, complemented
479C23—22no—
480C23—32no—
481C23—32no—
482C23—62no—
483C23—91no—
484C23—92no—
485C23—182no—
486C23—184no—
487C23—182no—
488C23—362no—
489C23C22×(C3⋊D3)⋊C311yescharacteristic
490D4—32no—
491D4—31no—
492D4—61no—
493D4—92no—
494D4—92no—
495D4—181no—
496D4—184no—
497D4—182no—
498D4—182no—
499D4—184no—
500D4—182no—
501D4—182no—
502D4—181no—
503D4—362no—
504D4(C3⋊S4)⋊C311yescharacteristic, direct product factor
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—92no—
510C2×C4—91no—
511C2×C4—181no—
512C2×C4—184no—
513C2×C4—181no—
514C2×C3—22no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—62no—
521C2×C3—61no—
522C2×C3—61no—
523C2×C3—62no—
524C2×C3—62no—
525C2×C3—122no—
526C2×C3—121no—
527C2×C3—182no—
528C2×C3—242no—
529C2×C3—241no—
530C2×C3—362no—
531C2×C3—482no—
532C2×C3C22×C3×S411yescharacteristic
533D3—62no—
534D3—122no—
535D3—362no—
536D3—722no—
537C22—31no—
538C22—31no—
539C22—32no—
540C22—62no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—92no—
545C22—181no—
546C22—182no—
547C22—182no—
548C22—181no—
549C22—182no—
550C22—364no—
551C22—364no—
552C22D4×(C3⋊D3)⋊C311yescharacteristic, complemented
553C22C2×(C3⋊S4)⋊C312yescomplemented
554C4—31no—
555C4—92no—
556C4—182no—
557C4—181no—
558C4—181no—
559C4C2×(C3⋊S4)⋊C311yescharacteristic, complemented
560C3—31no—
561C3—121no—
562C3—241no—
563C3C3×D4×S411yescharacteristic
564C2—22no—
565C2—31no—
566C2—31no—
567C2—62no—
568C2—182no—
569C2—362no—
570C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC2C22×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×D4D3
socleC23×C3C22×C3×D3
hypercenterD4(C3⋊S4)⋊C3
hypocenterC3×A4C2×C3×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C22×(C3⋊S4)⋊C3C2semidirect
2. D4×C22⋊H(3,𝔽3)C2semidirect
3. #64269⋊C3C2semidirect
4. C4⋊((C3⋊S4)⋊C3)C2semidirect
5. C4×(C3⋊S4)⋊C3C2semidirect
6. D4×C3⋊S4C3semidirect
8. C22×C22⋊H(3,𝔽3)C22semidirect
11. C2×(C3⋊S4)⋊C3C22semidirect
13. C4×C22⋊H(3,𝔽3)C22semidirect
14. (C3⋊SL(2,ℤ/4))⋊C3C22semidirect
15. C22×C3⋊S4C2×C3semidirect
16. C22×C32×D4D3semidirect
20. C3×D4×A4C2×C3semidirect
21. #64269C2×C3semidirect
22. C4⋊(C3⋊S4)C2×C3semidirect
23. C4×C3⋊S4C2×C3semidirect
36. (C3⋊S4)⋊C3D4direct (N × H)
44. C24×C32C2×D3semidirect
47. C22×C3×A4C22×C3semidirect
50. C2×C3⋊S4C22×C3semidirect
54. C22×C32×C4C2×D3semidirect
71. C3×C4×A4C22×C3semidirect
72. C3⋊SL(2,ℤ/4)C22×C3semidirect
85. C22⋊H(3,𝔽3)C2×D4semidirect
133. C3⋊S4C3×D4semidirect
139. C32×D4S4semidirect
245. C22×C32D3×D4semidirect
246. C22×C32C2×S4semidirect
259. C3×A4C2×C3×D4semidirect
263. C32×C4C2×S4semidirect
271. C22×D4(C3⋊D3)⋊C3semidirect
385. C24C2×(C3⋊D3)⋊C3semidirect
403. C22×C4C2×(C3⋊D3)⋊C3semidirect
478. C32D4×S4semidirect
504. D4(C3⋊S4)⋊C3direct (N × H)
552. C22D4×(C3⋊D3)⋊C3semidirect
553. C22C2×(C3⋊S4)⋊C3semidirect
559. C4C2×(C3⋊S4)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×(C3⋊S4)⋊C32
D4×C22⋊H(3,𝔽3)2
#64269⋊C32
C4⋊((C3⋊S4)⋊C3)2
C4×(C3⋊S4)⋊C32
D4×C3⋊S43
C3×D4×#673
D4×(C3⋊D3)⋊C34