Finite Group Databasebeta

← D32×#41⋊C3 #13180905

Subgroups — D32×#41⋊C3 #13180905

Order 1728.

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

normal subgroups only

This group has 991 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⋊D3×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×D3×#41⋊C3C212yesmaximal, complemented
3D32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#64319⋊C3C211yesmaximal, characteristic, complemented
5D3×C3⋊(#41⋊C3)C212yesmaximal, complemented
6D32×#41C311yesmaximal, characteristic, complemented
7C2×D3×#41⋊C3—32nomaximal
8C32×#41⋊C3C2211yescharacteristic, complemented
9C3×C4×D32—41nomaximal
10C3⋊D3×SL(2,𝔽3)C2211yescharacteristic, complemented
11C3⋊(C3⋊(#41⋊C3))C2211yescharacteristic, complemented
12C3×D3×SL(2,𝔽3)C2212yescomplemented
13C3×C3⋊(#41⋊C3)C2212yescomplemented
14#41×C3⋊D3C2×C311yescharacteristic, complemented
15C3×D3×#41C2×C312yescomplemented
16C2×C3×#41⋊C3—32no—
17#64319C2×C311yescharacteristic, complemented
18D3×#239C2×C312yescomplemented
19D32×Q8C2×C311yescharacteristic
20#64312—31no—
21D3×#237—32no—
22D32×D4—31no—
23#64306—31no—
24D3×#235—32no—
25C2×C4×D32—31no—
26D3×#41⋊C3—22no—
27D3×#41⋊C3—32no—
28D3×#41⋊C3—32no—
29D3×#41⋊C3—32no—
30D3×#41⋊C3—61no—
31D3×#41⋊C3D312yesdirect product factor
32C2×D3×SL(2,𝔽3)—32no—
33C2×C3⋊(#41⋊C3)—32no—
34C2×C3×D32—41no—
35C3×C4×C3⋊D3—41no—
36C32×C4×D3—42no—
37C32×SL(2,𝔽3)C2311yescharacteristic, complemented
38C3×#624—41no—
39C3×D3×Dic3—42no—
40C2×D3×#41—32no—
41C22×#41⋊C3—91no—
42C22×D32—31no—
43C32×#41C22×C311yescharacteristic, complemented
44#3832C22×C311yescharacteristic, complemented
45Q8×C3⋊D3C22×C311yescharacteristic
46#3830—31no—
47D4×C3⋊D3—31no—
48#3828—31no—
49C2×C4×C3⋊D3—31no—
50C3×#239C22×C312yescomplemented
51C3×D3×Q8C22×C312yes—
52C3×#237—32no—
53C3×D3×D4—32no—
54C3×#235—32no—
55C2×C3×C4×D3—32no—
56C2×C3×C4×D3—122no—
57C3×#41⋊C3—22no—
58C3×#41⋊C3—21no—
59C3×#41⋊C3—32no—
60C3×#41⋊C3—41no—
61C3×#41⋊C3C2×D312yescomplemented
62C2×C3×SL(2,𝔽3)—32no—
63#3811—32no—
64D3×#67—34no—
65C2×#624—31no—
66#3805—32no—
67#3804—34no—
68C2×D3×Dic3—32no—
69#3802—31no—
70D3×D12—32no—
71C4×D32—31no—
72C4×D32—31no—
73C4×D32—32no—
74C4×D32—31no—
75C4×D32—32no—
76C4×D32A411yescharacteristic
77#3799—32no—
78#3798—32no—
79#3797—31no—
80#3796—32no—
81#3795—32no—
82D3×Dic6—32no—
83D3×SL(2,𝔽3)—22no—
84D3×SL(2,𝔽3)—32no—
85D3×SL(2,𝔽3)—61no—
86D3×SL(2,𝔽3)C2×D312yescomplemented
87C3⋊(#41⋊C3)—22no—
88C3⋊(#41⋊C3)—32no—
89C3⋊(#41⋊C3)—61no—
90C3⋊(#41⋊C3)C2×D312yescomplemented
91C2×C3×C3⋊D3—41no—
92C2×C32×D3—42no—
93C3×D32—44no—
94C33×C4—41no—
95C3×C3⋊Dic3—41no—
96C32×Dic3—42no—
97C2×C3×#41—32no—
98D3×#41—32no—
99D3×#41—32no—
100D3×#41—32no—
101D3×#41—61no—
102D3×#41—92no—
103D3×#41—92no—
104D3×#41—92no—
105D3×#41—92no—
106D3×#41C3×D312yescomplemented
107C2×#239—32no—
108C2×D3×Q8—32no—
109C2×#237—92no—
110C2×D3×D4—92no—
111C2×#235—92no—
112C22×C4×D3—92no—
113C2×#41⋊C3—32no—
114C2×#41⋊C3—62no—
115C2×#41⋊C3—92no—
116C2×#41⋊C3—91no—
117C2×#41⋊C3—91no—
118C22×SL(2,𝔽3)—91no—
119C22×C3⋊D3—31no—
120C22×C3×D3—32no—
121C22×C3×D3—122no—
122C2×D32—31no—
123C2×D32—32no—
124C2×D32—64no—
125C2×D32C2×A411yescharacteristic
126C32×Q8C23×C311yescharacteristic
127C32×D4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—122no—
130#638—32no—
131C2×C3⋊Dic3—31no—
132C4⋊(C3⋊D3)—31no—
133C4×C3⋊D3—31no—
134C4×C3⋊D3—31no—
135C4×C3⋊D3—31no—
136C4×C3⋊D3C2×A411yescharacteristic
137#634—31no—
138C3×#67—34no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—122no—
141C3×D12—32no—
142C3×C4×D3—32no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×C4×D3—42no—
146C3×C4×D3—82no—
147C3×C4×D3—122no—
148C3×C4×D3—122no—
149C3×C4×D3—122no—
150C3×C4×D3—241no—
151C3×C4×D3C2×A412yes—
152C3×Dic6—32no—
153C3×SL(2,𝔽3)—22no—
154C3×SL(2,𝔽3)—21no—
155C3×SL(2,𝔽3)—41no—
156C3×SL(2,𝔽3)C22×D312yescomplemented
157#627—32no—
158#626—34no—
159#625—32no—
160#624—31no—
161#624—32no—
162#624C2×A411yescharacteristic
163D3×Dic3—32no—
164D3×Dic3—32no—
165D3×Dic3—32no—
166D3×Dic3C2×A412yes—
167C22×#41—91noSylow
168C2×C33—41no—
169C3×C3⋊D3—42no—
170C32×D3—44no—
171C23×D3—92no—
172C3×#41—21no—
173C3×#41—32no—
174C3×#41—92no—
175C3×#41—92no—
176C3×#41C2×C3×D312yescomplemented
177C2×C3×Q8—32no—
178C2×C3×D4—92no—
179C22×C3×C4—92no—
180C22×C3×C4—361no—
181C2×#67—94no—
182C22×Dic3—92no—
183#239—32no—
184#239—61no—
185#239—92no—
186#239—92no—
187#239C2×C3×D312yescomplemented
188D3×Q8—32no—
189D3×Q8—61no—
190D3×Q8—92no—
191D3×Q8—92no—
192D3×Q8C2×C3×D312yes—
193#237—32no—
194#237—92no—
195#237—92no—
196#237—94no—
197#237—94no—
198#237—92no—
199#237—181no—
200D3×D4—32no—
201D3×D4—92no—
202D3×D4—92no—
203D3×D4—94no—
204D3×D4—92no—
205D3×D4—94no—
206D3×D4—181no—
207#235—32no—
208#235—92no—
209#235—94no—
210#235—94no—
211#235—92no—
212#235—92no—
213#235—181no—
214C2×D12—92no—
215C2×C4×D3—32no—
216C2×C4×D3—32no—
217C2×C4×D3—92no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×C4×D3—92no—
224C2×C4×D3—92no—
225C2×C4×D3—92no—
226C2×C4×D3—92no—
227C2×C4×D3—181no—
228C2×Dic6—92no—
229#41⋊C3—22no—
230#41⋊C3—32no—
231#41⋊C3—41no—
232#41⋊C3—62no—
233#41⋊C3—91no—
234#41⋊C3D3211yescharacteristic, direct product factor
235C2×SL(2,𝔽3)—32no—
236C2×SL(2,𝔽3)—62no—
237C2×SL(2,𝔽3)—91no—
238C22×C32—31no—
239C22×C32—122no—
240C2×C3⋊D3—31no—
241C2×C3⋊D3—62no—
242C2×C3⋊D3C22×A411yescharacteristic
243C2×C3×D3—32no—
244C2×C3×D3—42no—
245C2×C3×D3—64no—
246C2×C3×D3—82no—
247C2×C3×D3—128no—
248C2×C3×D3—122no—
249C2×C3×D3—241no—
250C2×C3×D3C22×A412yes—
251D32—62no—
252D32—64no—
253D32#41⋊C314yesdirect product factor
254C32×C4—31no—
255C32×C4—42no—
256C32×C4—81no—
257C32×C4—82no—
258C32×C4—122no—
259C32×C4—161no—
260C32×C4C22×A411yescharacteristic
261C3⋊Dic3—31no—
262C3⋊Dic3C22×A411yescharacteristic
263C3×Dic3—32no—
264C3×Dic3—42no—
265C3×Dic3—82no—
266C3×Dic3—122no—
267C3×Dic3—241no—
268C3×Dic3C22×A412yes—
269C2×#41—32no—
270C2×#41—92no—
271C2×#41—91no—
272C2×#41—91no—
273C2×#41—272no—
274C2×#41—272no—
275C2×#41—271no—
276C2×#41—271no—
277C22×Q8—91no—
278C22×D4—271no—
279C23×C4—271no—
280C33—41noSylow
281C23×C3—92no—
282C23×C3—361no—
283C22×D3—32no—
284C22×D3—32no—
285C22×D3—92no—
286C22×D3—92no—
287C22×D3—92no—
288C22×D3—92no—
289C22×D3—181no—
290C22×D3—188no—
291C3×Q8—21no—
292C3×Q8—92no—
293C3×Q8C22×C3×D312yes—
294C3×D4—32no—
295C3×D4—61no—
296C3×D4—92no—
297C3×D4—94no—
298C2×C3×C4—32no—
299C2×C3×C4—32no—
300C2×C3×C4—61no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—122no—
306C2×C3×C4—242no—
307C2×C3×C4—361no—
308C2×C3×C4—361no—
309C2×C3×C4—362no—
310#67—34no—
311#67—94no—
312#67—94no—
313#67—94no—
314#67—182no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—181no—
322D12—32no—
323D12—92no—
324D12—94no—
325D12—181no—
326C4×D3—32no—
327C4×D3—32no—
328C4×D3—32no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—32no—
332C4×D3—61no—
333C4×D3—92no—
334C4×D3—92no—
335C4×D3—92no—
336C4×D3—92no—
337C4×D3—92no—
338C4×D3—92no—
339C4×D3—92no—
340C4×D3—92no—
341C4×D3—92no—
342C4×D3—181no—
343C4×D3—181no—
344C4×D3—181no—
345C4×D3D3×A412yes—
346Dic6—32no—
347Dic6—94no—
348Dic6—92no—
349Dic6—181no—
350SL(2,𝔽3)—22no—
351SL(2,𝔽3)—41no—
352SL(2,𝔽3)C2×D3211yescharacteristic, complemented
353C2×C32—42no—
354C2×C32—61no—
355C2×C32—81no—
356C2×C32—82no—
357C2×C32—124no—
358C2×C32—161no—
359C2×C32C23×A411yescharacteristic
360C3⋊D3—61no—
361C3⋊D3C2×#41⋊C312yescomplemented
362C3×D3—44no—
363C3×D3—62no—
364C3×D3—84no—
365C3×D3—124no—
366C3×D3—242no—
367C3×D3C2×#41⋊C314yescomplemented
368C24—271no—
369#41—32no—
370#41—92no—
371#41—92no—
372#41—91no—
373#41—271no—
374#41—272no—
375#41—274no—
376#41—272no—
377#41—272no—
378#41—272no—
379#41—271no—
380#41—272no—
381#41C3×D3211yescharacteristic, complemented
382C2×Q8—32no—
383C2×Q8—91no—
384C2×Q8—272no—
385C2×Q8—271no—
386C2×D4—92no—
387C2×D4—272no—
388C2×D4—271no—
389C2×D4—271no—
390C2×D4—272no—
391C2×D4—274no—
392C22×C4—92no—
393C22×C4—91no—
394C22×C4—271no—
395C22×C4—272no—
396C22×C4—271no—
397C22×C4—272no—
398C22×C4—271no—
399C22×C4—271no—
400C22×C4—271no—
401C22×C4—272no—
402C22×C3—32no—
403C22×C3—32no—
404C22×C3—61no—
405C22×C3—92no—
406C22×C3—122no—
407C22×C3—184no—
408C22×C3—242no—
409C22×C3—361no—
410C22×C3—364no—
411C2×D3—32no—
412C2×D3—38no—
413C2×D3—32no—
414C2×D3—61no—
415C2×D3—64no—
416C2×D3—92no—
417C2×D3—181no—
418C2×D3—184no—
419C2×D3—184no—
420C2×D3—184no—
421C2×D3—184no—
422C2×D3—362no—
423C2×D3C2×D3×A412yes—
424C3×C4—21no—
425C3×C4—32no—
426C3×C4—32no—
427C3×C4—41no—
428C3×C4—61no—
429C3×C4—82no—
430C3×C4—92no—
431C3×C4—122no—
432C3×C4—161no—
433C3×C4—242no—
434C3×C4—361no—
435C3×C4C2×D3×A412yes—
436Dic3—32no—
437Dic3—32no—
438Dic3—61no—
439Dic3—92no—
440Dic3—181no—
441Dic3C2×D3×A412yes—
442C32—42no—
443C32—81no—
444C32—82no—
445C32—161no—
446C32C22×#41⋊C311yescharacteristic, complemented
447C23—91no—
448C23—92no—
449C23—272no—
450C23—271no—
451C23—271no—
452C23—544no—
453Q8—92no—
454Q8—271no—
455Q8—272no—
456Q8C2×C3×D3211yescharacteristic
457D4—31no—
458D4—92no—
459D4—94no—
460D4—271no—
461D4—274no—
462D4—272no—
463D4—272no—
464C2×C4—31no—
465C2×C4—32no—
466C2×C4—92no—
467C2×C4—91no—
468C2×C4—92no—
469C2×C4—92no—
470C2×C4—92no—
471C2×C4—92no—
472C2×C4—91no—
473C2×C4—271no—
474C2×C4—272no—
475C2×C4—271no—
476C2×C4—272no—
477C2×C4—271no—
478C2×C4—272no—
479C2×C4—272no—
480C2×C4—271no—
481C2×C4—271no—
482C2×C3—21no—
483C2×C3—34no—
484C2×C3—41no—
485C2×C3—62no—
486C2×C3—82no—
487C2×C3—124no—
488C2×C3—121no—
489C2×C3—161no—
490C2×C3—182no—
491C2×C3—244no—
492C2×C3—362no—
493C2×C3C22×D3×A412yes—
494D3—34no—
495D3—62no—
496D3—62no—
497D3—182no—
498D3—361no—
499D3D3×#41⋊C314yesdirect product factor
500C22—32no—
501C22—31no—
502C22—94no—
503C22—91no—
504C22—92no—
505C22—184no—
506C22—271no—
507C22—544no—
508C22—542no—
509C22—542no—
510C4—31no—
511C4—32no—
512C4—92no—
513C4—91no—
514C4—271no—
515C4D32×A411yescharacteristic
516C3—21no—
517C3—41no—
518C3—82no—
519C3—161no—
520C3C2×D3×#41⋊C312yescomplemented
521C2—34no—
522C2—61no—
523C2—92no—
524C2—182no—
525C2—541no—
526C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC4D32×A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C2×D32×A4
FittingC32×#41C22×C3
socleC2×C32C23×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×#41⋊C3C2semidirect
2. C3×D3×#41⋊C3C2semidirect
3. D32×SL(2,𝔽3)C2semidirect
4. #64319⋊C3C2semidirect
5. D3×C3⋊(#41⋊C3)C2semidirect
6. D32×#41C3semidirect
8. C32×#41⋊C3C22semidirect
10. C3⋊D3×SL(2,𝔽3)C22semidirect
11. C3⋊(C3⋊(#41⋊C3))C22semidirect
12. C3×D3×SL(2,𝔽3)C22semidirect
13. C3×C3⋊(#41⋊C3)C22semidirect
14. #41×C3⋊D3C2×C3semidirect
15. C3×D3×#41C2×C3semidirect
17. #64319C2×C3semidirect
18. D3×#239C2×C3semidirect
31. D3×#41⋊C3D3direct (N × H)
37. C32×SL(2,𝔽3)C23semidirect
43. C32×#41C22×C3semidirect
44. #3832C22×C3semidirect
50. C3×#239C22×C3semidirect
61. C3×#41⋊C3C2×D3semidirect
86. D3×SL(2,𝔽3)C2×D3semidirect
90. C3⋊(#41⋊C3)C2×D3semidirect
106. D3×#41C3×D3semidirect
156. C3×SL(2,𝔽3)C22×D3semidirect
176. C3×#41C2×C3×D3semidirect
187. #239C2×C3×D3semidirect
234. #41⋊C3D32direct (N × H)
253. D32#41⋊C3direct (N × H)
352. SL(2,𝔽3)C2×D32semidirect
361. C3⋊D3C2×#41⋊C3semidirect
367. C3×D3C2×#41⋊C3semidirect
381. #41C3×D32semidirect
446. C32C22×#41⋊C3semidirect
499. D3D3×#41⋊C3direct (N × H)
520. C3C2×D3×#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
C3×D3×#41⋊C32
D32×SL(2,𝔽3)2
#64319⋊C32
D3×C3⋊(#41⋊C3)2
D32×#413
C2×D3×#41⋊C33
C3×C4×D324