Finite Group Databasebeta

← D3×#64259 #13180794

Subgroups — D3×#64259 #13180794

Order 1728.

926 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 926 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C3×C4×D3×A4C211yesmaximal, characteristic
2#11724463C211yesmaximal, characteristic, complemented
3C3×#64259C211yesmaximal, characteristic, complemented
4D3×C3⋊SL(2,ℤ/4)C212yesmaximal
5#11724199C212yesmaximal, complemented
6C2×#64259—31nomaximal
7D3×A4⋊Q8—33nomaximal
8D3×#924—31nomaximal
9C2×C3×D3×A4C2211yescharacteristic
10C32×C4×A4C2211yescharacteristic
11D3×#634—41nomaximal
12C3×Dic3×A4C2211yescharacteristic
13C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
14C3×C3⋊SL(2,ℤ/4)C2212yes—
15C22×C3×C4×D3D311yescharacteristic
16C2×C3×C4×A4—31no—
17C2×D3×Dic6—31no—
18C4×D3×A4D313yes—
19C2×C3⋊SL(2,ℤ/4)—32no—
20#64259—31no—
21#64259—33no—
22#64259—32no—
23#64259—61no—
24#64259—63no—
25#64259D311yescharacteristic, direct product factor
26C3×A4⋊Q8—33no—
27D3×SL(2,ℤ/4)—36no—
28#64204—36no—
29#64133—31no—
30C3×#924—31no—
31D3×#217—32no—
32#63964—32no—
33D3×C4⋊Dic3—31no—
34D3×Dic3⋊C4—32no—
35#63857—31no—
36#63856—31no—
37#63851—32no—
38C2×C32×A4C2311yescharacteristic
39C3×D3×A4Q812yescomplemented
40#6308—41no—
41C3×#634—41no—
42C32×C4×D3—41no—
43#6293—42no—
44D3×C3⋊Dic3—42no—
45C2×A4⋊Q8—93no—
46C2×#924—91no—
47D3×C22⋊Q8—91no—
48C23×C3×D3C2×D311yescharacteristic
49C22×C3×A4—31no—
50C2×D3×A4C2×D313yes—
51C22×C32×C4C2×D311yescharacteristic
52C2×#634—31no—
53C2×#634—121no—
54C22×C3×Dic3C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—31no—
57C2×C3×C4×D3—31no—
58C2×C3×C4×D3—31no—
59C2×C3×Dic6—31no—
60C3×C4×A4—21no—
61C3×C4×A4—23no—
62C3×C4×A4—31no—
63C3×C4×A4C2×D313yes—
64C3×C4×A4C2×D311yescharacteristic
65C2×#627—32no—
66C2×D3×Dic3—32no—
67C2×D3×Dic3—32no—
68D3×Dic6—61no—
69D3×Dic6—61no—
70D3×Dic6—61no—
71D3×Dic6—61no—
72D3×Dic6—123no—
73Dic3×A4C2×D313yes—
74C3⋊SL(2,ℤ/4)—32no—
75C3⋊SL(2,ℤ/4)—36no—
76C3⋊SL(2,ℤ/4)—62no—
77C3⋊SL(2,ℤ/4)—66no—
78C3⋊SL(2,ℤ/4)C2×D312yes—
79C3×SL(2,ℤ/4)—36no—
80#3757—32no—
81C4⋊(C3⋊Dic3)—31no—
82#3750—32no—
83C3×#217—32no—
84C3×C4⋊Dic3—31no—
85C3×Dic3⋊C4—32no—
86#3724—32no—
87Dic3⋊Dic3—32no—
88Dic3⋊Dic3—32no—
89(C2×D3)⋊Dic3—32no—
90(C2×D3)⋊Dic3—32no—
91C2×C32×D3—41no—
92C32×A4C2×Q811yescharacteristic, complemented
93C33×C4—41no—
94C3⋊(C3⋊Dic3)—42no—
95C3×C3⋊Dic3—42no—
96C32×Dic3—41no—
97C23×C3×C4—31no—
98C2×D3×Q8—91no—
99C22×C4×D3C3⋊D311yescharacteristic
100C22×Dic6—91no—
101C2×C4×A4—33no—
102C2×SL(2,ℤ/4)—96no—
103A4⋊Q8—33no—
104A4⋊Q8—96no—
105A4⋊Q8—93no—
106A4⋊Q8—183no—
107C3×C22⋊Q8—91no—
108C2×#217—92no—
109#946—91no—
110C2×C4⋊Dic3—91no—
111#924—31no—
112#924—92no—
113#924—91no—
114#924—92no—
115#924—91no—
116#924—91no—
117#924—91no—
118#924—181no—
119C2×Dic3⋊C4—92no—
120#897—91no—
121#896—92no—
122D3×C4⋊C4—92no—
123D3×C4⋊C4—91no—
124D3×C22⋊C4—92no—
125#878—92no—
126C23×C32C22×D311yescharacteristic
127C22×C3×D3—31no—
128C22×C3×D3—32no—
129C22×C3×D3—31no—
130C22×C3×D3Dic612yescomplemented
131C2×C3×A4—21no—
132C2×C3×A4—23no—
133C2×C3×A4—32no—
134C2×C3×A4C22×D313yes—
135C2×C3×A4C22×D311yescharacteristic
136D3×A4Dic616yescomplemented
137C2×C32×C4—31no—
138C2×C32×C4—31no—
139C2×C32×C4—121no—
140C2×C3⋊Dic3—32no—
141C2×C3⋊Dic3—32no—
142C2×C3⋊Dic3—122no—
143#634—41no—
144#634—61no—
145#634—61no—
146#634—122no—
147#634—121no—
148#634—123no—
149#634—241no—
150#634—243no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C3×C4×D3—31no—
156C3×C4×D3—31no—
157C3×C4×D3—31no—
158C3×C4×D3—43no—
159C3×C4×D3—61no—
160C3×C4×D3S411yescharacteristic
161C3×Dic6—61no—
162C3×Dic6—61no—
163C3×Dic6—123no—
164#627—62no—
165#627—62no—
166#627—126no—
167D3×Dic3—62no—
168D3×Dic3—64no—
169D3×Dic3—62no—
170D3×Dic3—126no—
171C2×C22⋊Q8—271noSylow
172C2×C33—41no—
173C32×D3—42no—
174C24×C3—31no—
175C23×D3C2×C3⋊D311yescharacteristic
176C22×A4—33no—
177C2×C3×Q8—91no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C22×C3×C4—91no—
184C22×C3×C4D3211yescharacteristic
185C22×C3×C4C2×C3⋊D311yescharacteristic
186C22×Dic3—92no—
187C22×Dic3—92no—
188C22×Dic3C2×C3⋊D311yescharacteristic
189D3×Q8—181no—
190D3×Q8—181no—
191D3×Q8—181no—
192D3×Q8—181no—
193C2×C4×D3—31no—
194C2×C4×D3—31no—
195C2×C4×D3—31no—
196C2×C4×D3—31no—
197C2×C4×D3—92no—
198C2×C4×D3—92no—
199C2×Dic6—31no—
200C2×Dic6—92no—
201C2×Dic6—92no—
202C2×Dic6—91no—
203C2×Dic6—91no—
204C2×Dic6—181no—
205C2×Dic6—181no—
206C2×Dic6—181no—
207C2×Dic6—181no—
208C2×Dic6—181no—
209C2×Dic6—363no—
210C4×A4—23no—
211C4×A4—33no—
212C4×A4D3213yes—
213SL(2,ℤ/4)—36no—
214SL(2,ℤ/4)—96no—
215SL(2,ℤ/4)—186no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—91no—
218C3×C22⋊C4—92no—
219#217—32no—
220#217—92no—
221#217—92no—
222#217—92no—
223#217—92no—
224#217—182no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227C4⋊Dic3—31no—
228C4⋊Dic3—92no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—181no—
233Dic3⋊C4—32no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—182no—
241C22×C32—31no—
242C22×C32—31no—
243C22×C32—121no—
244C22×C32C2×Dic611yescharacteristic, complemented
245C2×C3×D3—31no—
246C2×C3×D3—32no—
247C2×C3×D3—32no—
248C2×C3×D3—32no—
249C2×C3×D3—43no—
250C2×C3×D3—61no—
251C2×C3×D3C2×S411yescharacteristic
252C3×A4—21no—
253C3×A4—23no—
254C3×A4C2×Dic613yescomplemented
255C3×A4D3×Q811yescharacteristic, complemented
256C32×C4—31no—
257C32×C4—41no—
258C32×C4—43no—
259C32×C4—83no—
260C32×C4—81no—
261C32×C4—121no—
262C32×C4C2×S411yescharacteristic
263C3⋊Dic3—42no—
264C3⋊Dic3—62no—
265C3⋊Dic3—62no—
266C3⋊Dic3—122no—
267C3⋊Dic3—126no—
268C3⋊Dic3—242no—
269C3⋊Dic3—246no—
270C3×Dic3—31no—
271C3×Dic3—43no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—126no—
275C3×Dic3C2×S411yescharacteristic
276C22×Q8—271no—
277C23×C4—31no—
278C22⋊Q8—91no—
279C22⋊Q8—272no—
280C22⋊Q8—272no—
281C22⋊Q8—271no—
282C22⋊Q8—271no—
283C22⋊Q8—271no—
284C2×C4⋊C4—272no—
285C2×C4⋊C4—271no—
286C2×C22⋊C4—272no—
287C33—41noSylow
288C23×C3—21no—
289C23×C3—32no—
290C23×C3—91no—
291C23×C3—91no—
292C23×C3—92no—
293C23×C3C2×D3211yescharacteristic
294C23×C3C22×C3⋊D311yescharacteristic
295C22×D3—31no—
296C22×D3—31no—
297C22×D3—32no—
298C22×D3#63412yescomplemented
299C2×A4—23no—
300C2×A4—36no—
301C2×A4C2×D3213yes—
302C3×Q8—181no—
303C3×Q8—181no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—91no—
315C2×C3×C4—91no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
318C2×C3×C4—123no—
319C2×C3×C4—181no—
320C2×Dic3—31no—
321C2×Dic3—31no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—182no—
329C2×Dic3—182no—
330C2×Dic3—182no—
331C2×Dic3—182no—
332C2×Dic3—184no—
333C2×Dic3—366no—
334C4×D3—31no—
335C4×D3—31no—
336C4×D3—31no—
337C4×D3—61no—
338C4×D3—184no—
339C4×D3—182no—
340C4×D3—182no—
341C4×D3C3⋊S411yescharacteristic
342Dic6—61no—
343Dic6—61no—
344Dic6—123no—
345Dic6—182no—
346Dic6—182no—
347Dic6—182no—
348Dic6—181no—
349Dic6—181no—
350Dic6—181no—
351Dic6—181no—
352Dic6—182no—
353Dic6—366no—
354Dic6—361no—
355Dic6—363no—
356Dic6—361no—
357Dic6—723no—
358C2×C32—31no—
359C2×C32—31no—
360C2×C32—41no—
361C2×C32—43no—
362C2×C32—83no—
363C2×C32—81no—
364C2×C32—122no—
365C2×C32C22×S411yescharacteristic
366C3×D3—32no—
367C3×D3—46no—
368C3×D3A4⋊Q812yescomplemented
369C24—31no—
370C2×Q8—91no—
371C2×Q8—271no—
372C2×Q8—272no—
373C2×Q8—541no—
374C2×Q8—541no—
375C2×Q8—541no—
376C2×Q8—541no—
377C22×C4—31no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—272no—
383C22×C4—272no—
384C22×C4D3×C3⋊D311yescharacteristic
385C4⋊C4—92no—
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—271no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—92no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22×C3—21no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—31no—
400C22×C3—31no—
401C22×C3—31no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—92no—
405C22×C3—92no—
406C22×C3—92no—
407C22×C3—91no—
408C22×C3—123no—
409C22×C3—181no—
410C22×C3D3×Dic611yescharacteristic, complemented
411C22×C3C2×#63411yescharacteristic, complemented
412C2×D3—31no—
413C2×D3—32no—
414C2×D3—32no—
415C2×D3—32no—
416C2×D3—61no—
417C2×D3C2×C3⋊S411yescharacteristic
418A4—23no—
419A4D3×Dic613yescomplemented
420C3×C4—21no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—43no—
425C3×C4—61no—
426C3×C4—83no—
427C3×C4—91no—
428C3×C4—123no—
429C3×C4—182no—
430C3×C4—182no—
431C3×C4D3×S411yescharacteristic
432C3×C4C2×C3⋊S411yescharacteristic
433Dic3—31no—
434Dic3—62no—
435Dic3—62no—
436Dic3—126no—
437Dic3—182no—
438Dic3—182no—
439Dic3—182no—
440Dic3—182no—
441Dic3—362no—
442Dic3—366no—
443Dic3—362no—
444Dic3—726no—
445Dic3C2×C3⋊S411yescharacteristic
446C32—41no—
447C32—43no—
448C32—83no—
449C32—81no—
450C32C2×A4⋊Q811yescharacteristic, complemented
451C23—32no—
452C23—91no—
453C23—91no—
454C23—92no—
455C23C2×D3×C3⋊D311yescharacteristic
456Q8—181no—
457Q8—181no—
458Q8—542no—
459Q8—541no—
460Q8—542no—
461Q8—541no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—91no—
468C2×C4—91no—
469C2×C4—91no—
470C2×C4—92no—
471C2×C4—92no—
472C2×C4—181no—
473C2×C4—272no—
474C2×C4—272no—
475C2×C4—542no—
476C2×C4—542no—
477C2×C4—544no—
478C2×C3—21no—
479C2×C3—31no—
480C2×C3—31no—
481C2×C3—32no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—43no—
485C2×C3—61no—
486C2×C3—61no—
487C2×C3—83no—
488C2×C3—92no—
489C2×C3—126no—
490C2×C3C2×D3×S411yescharacteristic
491C2×C3C22×C3⋊S411yescharacteristic
492D3—32no—
493D3#6425912yesdirect product factor
494C22—31no—
495C22—31no—
496C22—31no—
497C22—92no—
498C22—92no—
499C22—92no—
500C22—91no—
501C22—181no—
502C22D3×#63411yescharacteristic, complemented
503C4—31no—
504C4—31no—
505C4—91no—
506C4—182no—
507C4—182no—
508C4—542no—
509C4—542no—
510C4D3×C3⋊S411yescharacteristic
511C3—21no—
512C3—43no—
513C3—83no—
514C3D3×A4⋊Q811yescharacteristic, complemented
515C3C2×#6425911yescharacteristic, complemented
516C2—31no—
517C2—31no—
518C2—32no—
519C2—92no—
520C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊S4
commutator, cosocleC2×C32×A4C23
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
prime radicalC2×C3×D3×A4C22
hypercenterC4D3×C3⋊S4
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11724463C2semidirect
3. C3×#64259C2semidirect
5. #11724199C2semidirect
25. #64259D3direct (N × H)
39. C3×D3×A4Q8semidirect
92. C32×A4C2×Q8semidirect
130. C22×C3×D3Dic6semidirect
136. D3×A4Dic6semidirect
244. C22×C32C2×Dic6semidirect
254. C3×A4C2×Dic6semidirect
255. C3×A4D3×Q8semidirect
298. C22×D3#634semidirect
368. C3×D3A4⋊Q8semidirect
410. C22×C3D3×Dic6semidirect
411. C22×C3C2×#634semidirect
419. A4D3×Dic6semidirect
450. C32C2×A4⋊Q8semidirect
493. D3#64259direct (N × H)
502. C22D3×#634semidirect
514. C3D3×A4⋊Q8semidirect
515. C3C2×#64259semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×A42
#117244632
C3×#642592
D3×C3⋊SL(2,ℤ/4)2
#117241992
C2×#642593
D3×A4⋊Q83
D3×#9243
D3×#6344