Finite Group Databasebeta

← #10596695⋊C3 #13180903

Subgroups — #10596695⋊C3 #13180903

Order 1728.

891 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 88 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
3#64320⋊C3C211yesmaximal, characteristic, complemented
4#64315⋊C3C212yesmaximal, complemented
5#64316⋊C3C211yesmaximal, characteristic, complemented
6#10596695C311yesmaximal, characteristic, complemented
7#5964⋊C3—32nomaximal
8C32×#41⋊C3C2211yescharacteristic, complemented
9C3×#3798—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#64320C2×C311yescharacteristic, complemented
18#64316C2×C311yescharacteristic, complemented
19#64315C2×C312yescomplemented
20#64314—32no—
21#64313—31no—
22#64309—31no—
23#64306—31no—
24D3×#235—32no—
25C2×#3798—31no—
26#1010⋊C3—32no—
27D3×#41⋊C3—22no—
28D3×#41⋊C3—32no—
29D3×#41⋊C3—61no—
30D3×#41⋊C3D312yescomplemented
31C3⋊#994—32no—
32#1007⋊C3—32no—
33#1007⋊C3—32no—
34C3×C4×C3⋊D3—41no—
35C32×C4×D3—42no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×#627—41no—
38C3×#626—42no—
39C3×#625—41no—
40#5964—32no—
41#5945—91no—
42C32×#41C22×C311yescharacteristic
43#3832C22×C311yescharacteristic
44Q8×C3⋊D3C22×C311yescharacteristic
45#3830—31no—
46D4×C3⋊D3—31no—
47#3828—31no—
48C2×C4×C3⋊D3—31no—
49C3×#239C22×C312yes—
50C3×D3×Q8C22×C312yes—
51C3×#237—32no—
52C3×D3×D4—32no—
53C3×#235—32no—
54C3×#235—122no—
55C2×C3×C4×D3—32no—
56C3×#41⋊C3—22no—
57C3×#41⋊C3—21no—
58C3×#41⋊C3—32no—
59C3×#41⋊C3—41no—
60C3×#41⋊C3C2×D312yescomplemented
61C2×C3×SL(2,𝔽3)—32no—
62#3811—32no—
63D3×#67—34no—
64C2×#627—31no—
65C2×#626—32no—
66C2×#625—31no—
67#3805—32no—
68#3804—34no—
69#3802—31no—
70D3×D12—32no—
71C4×D32—32no—
72#3799—32no—
73#3798—31no—
74#3798—32no—
75#3798—31no—
76#3798—32no—
77#3798—31no—
78#3798A411yescharacteristic
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—
93C33×C4—41no—
94C3×C3⋊Dic3—41no—
95C32×Dic3—42no—
96C2×C3×#41—32no—
97#1010—32no—
98#1010—92no—
99#1009—32no—
100#1009—92no—
101D3×#41—32no—
102D3×#41—61no—
103D3×#41—92no—
104D3×#41—92no—
105D3×#41C3×D312yescomplemented
106#1007—32no—
107#1007—32no—
108#1004—92no—
109#1004—92no—
110C2×#235—92no—
111C2×#235—92no—
1122−1+4⋊C3—91no—
1132−1+4⋊C3—92no—
114#994—91no—
115C2×#41⋊C3—32no—
116C2×#41⋊C3—62no—
117C2×#41⋊C3—91no—
118C22×C3⋊D3—31no—
119C22×C3×D3—32no—
120C2×D32—32no—
121C32×Q8C23×C311yescharacteristic
122C32×D4—31no—
123C2×C32×C4—31no—
124C2×C32×C4—122no—
125#638—32no—
126C2×C3⋊Dic3—31no—
127C4⋊(C3⋊D3)—31no—
128C4×C3⋊D3—31no—
129C4×C3⋊D3—31no—
130C4×C3⋊D3—31no—
131C4×C3⋊D3C2×A411yescharacteristic
132#634—31no—
133C3×#67—34no—
134C3×#67—122no—
135C3×#67—122no—
136C2×C3×Dic3—32no—
137C3×D12—32no—
138C3×D12—122no—
139C3×C4×D3—32no—
140C3×C4×D3—32no—
141C3×C4×D3—32no—
142C3×C4×D3—42no—
143C3×C4×D3—82no—
144C3×C4×D3—122no—
145C3×C4×D3—241no—
146C3×C4×D3C2×A412yes—
147C3×Dic6—32no—
148C3×Dic6—122no—
149C3×SL(2,𝔽3)—22no—
150C3×SL(2,𝔽3)—21no—
151C3×SL(2,𝔽3)—41no—
152C3×SL(2,𝔽3)C22×D312yes—
153#627—31no—
154#627—32no—
155#627C2×A411yescharacteristic
156#626—32no—
157#626—32no—
158#626—32no—
159#626C2×A412yes—
160#625—32no—
161#625—31no—
162#625C2×A411yescharacteristic
163#624—32no—
164D3×Dic3—34no—
165#585—91noSylow
166C2×C33—41no—
167C3×C3⋊D3—81no—
168C32×D3—82no—
169C3×#41—21no—
170C3×#41—32no—
171C3×#41—92no—
172C3×#41—92no—
173C3×#41—361no—
174C3×#41C2×C3×D312yes—
175C2×C3×Q8—32no—
176C2×C3×D4—92no—
177C22×C3×C4—92no—
178C2×#67—92no—
179C2×#67—92no—
180C2×#67—94no—
181#239—32no—
182#239—61no—
183#239—92no—
184#239—92no—
185#239C2×C3×D312yes—
186D3×Q8—32no—
187D3×Q8—61no—
188D3×Q8—92no—
189D3×Q8—92no—
190D3×Q8C2×C3×D312yes—
191#237—32no—
192#237—92no—
193#237—92no—
194#237—94no—
195#237—94no—
196#237—92no—
197#237—181no—
198D3×D4—32no—
199D3×D4—92no—
200D3×D4—92no—
201D3×D4—94no—
202D3×D4—92no—
203D3×D4—94no—
204D3×D4—181no—
205#235—32no—
206#235—32no—
207#235—92no—
208#235—92no—
209#235—92no—
210#235—92no—
211#235—92no—
212#235—92no—
213#235—92no—
214#235—92no—
215#235—94no—
216#235—92no—
217#235—92no—
218#235—181no—
219C2×D12—92no—
220C2×D12—92no—
221C2×C4×D3—32no—
222C2×C4×D3—92no—
223C2×C4×D3—94no—
224C2×C4×D3—181no—
225C2×Dic6—92no—
226C2×Dic6—92no—
227#41⋊C3—22no—
228#41⋊C3—32no—
229#41⋊C3—41no—
230#41⋊C3—62no—
231#41⋊C3—91no—
232#41⋊C3D3211yescharacteristic, complemented
233C2×SL(2,𝔽3)—32no—
234C2×SL(2,𝔽3)—62no—
235C2×SL(2,𝔽3)—91no—
236C22×C32—31no—
237C22×C32—122no—
238C2×C3⋊D3—31no—
239C2×C3⋊D3—121no—
240C2×C3⋊D3C22×A411yescharacteristic
241C2×C3×D3—32no—
242C2×C3×D3—42no—
243C2×C3×D3—82no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246C2×C3×D3—241no—
247C2×C3×D3C22×A412yes—
248D32—122no—
249C32×C4—31no—
250C32×C4—42no—
251C32×C4—81no—
252C32×C4—82no—
253C32×C4—122no—
254C32×C4—161no—
255C32×C4C22×A411yescharacteristic
256C3⋊Dic3—31no—
257C3⋊Dic3C22×A411yescharacteristic
258C3×Dic3—32no—
259C3×Dic3—42no—
260C3×Dic3—82no—
261C3×Dic3—122no—
262C3×Dic3—241no—
263C3×Dic3C22×A412yes—
2642−1+4—91no—
2652−1+4—92no—
2662−1+4—271no—
267H(5,𝔽2)—91no—
268H(5,𝔽2)—271no—
269H(5,𝔽2)—272no—
270C2×#41—32no—
271C2×#41—91no—
272C2×#41—271no—
273C2×#41—272no—
274C2×#41—271no—
275C33—41noSylow
276C23×C3—92no—
277C22×D3—32no—
278C22×D3—92no—
279C22×D3—94no—
280C22×D3—181no—
281C3×Q8—21no—
282C3×Q8—92no—
283C3×Q8—361no—
284C3×Q8C22×C3×D312yes—
285C3×D4—32no—
286C3×D4—61no—
287C3×D4—92no—
288C3×D4—94no—
289C3×D4—362no—
290C3×D4—361no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—61no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—122no—
299C2×C3×C4—242no—
300C2×C3×C4—361no—
301#67—34no—
302#67—32no—
303#67—32no—
304#67—92no—
305#67—92no—
306#67—92no—
307#67—92no—
308#67—94no—
309#67—92no—
310#67—92no—
311#67—182no—
312C2×Dic3—32no—
313C2×Dic3—92no—
314C2×Dic3—94no—
315C2×Dic3—181no—
316D12—32no—
317D12—32no—
318D12—92no—
319D12—92no—
320D12—92no—
321D12—92no—
322D12—181no—
323C4×D3—32no—
324C4×D3—32no—
325C4×D3—32no—
326C4×D3—32no—
327C4×D3—61no—
328C4×D3—92no—
329C4×D3—94no—
330C4×D3—92no—
331C4×D3—94no—
332C4×D3—92no—
333C4×D3—181no—
334C4×D3—181no—
335C4×D3—181no—
336C4×D3D3×A412yes—
337Dic6—32no—
338Dic6—32no—
339Dic6—92no—
340Dic6—92no—
341Dic6—92no—
342Dic6—92no—
343Dic6—181no—
344SL(2,𝔽3)—22no—
345SL(2,𝔽3)—41no—
346SL(2,𝔽3)C2×D3211yescharacteristic
347C2×C32—42no—
348C2×C32—61no—
349C2×C32—81no—
350C2×C32—82no—
351C2×C32—161no—
352C2×C32—242no—
353C2×C32C23×A411yescharacteristic
354C3⋊D3—21no—
355C3⋊D3—61no—
356C3×D3—22no—
357C3×D3—62no—
358C3×D3—82no—
359C3×D3—84no—
360C3×D3—242no—
361C3×D3—242no—
362#41—32no—
363#41—91no—
364#41—92no—
365#41—92no—
366#41—91no—
367#41—272no—
368#41—271no—
369#41—271no—
370#41—272no—
371#41—271no—
372#41—271no—
373#41—274no—
374#41—272no—
375#41—272no—
376#41—272no—
377#41—272no—
378#41—271no—
379#41C3×D3211yescharacteristic
380C2×Q8—32no—
381C2×Q8—91no—
382C2×Q8—271no—
383C2×Q8—272no—
384C2×Q8—271no—
385C2×D4—92no—
386C2×D4—272no—
387C2×D4—272no—
388C2×D4—271no—
389C2×D4—272no—
390C2×D4—271no—
391C2×D4—271no—
392C2×D4—274no—
393C22×C4—92no—
394C22×C4—271no—
395C22×C4—272no—
396C22×C3—32no—
397C22×C3—32no—
398C22×C3—61no—
399C22×C3—92no—
400C22×C3—122no—
401C22×C3—242no—
402C22×C3—361no—
403C22×C3—362no—
404C2×D3—32no—
405C2×D3—32no—
406C2×D3—61no—
407C2×D3—92no—
408C2×D3—122no—
409C2×D3—181no—
410C2×D3—362no—
411C2×D3—364no—
412C2×D3—362no—
413C2×D3C2×D3×A412yes—
414C3×C4—21no—
415C3×C4—32no—
416C3×C4—32no—
417C3×C4—41no—
418C3×C4—61no—
419C3×C4—82no—
420C3×C4—92no—
421C3×C4—122no—
422C3×C4—161no—
423C3×C4—242no—
424C3×C4—361no—
425C3×C4C2×D3×A412yes—
426Dic3—32no—
427Dic3—32no—
428Dic3—61no—
429Dic3—92no—
430Dic3—181no—
431Dic3C2×D3×A412yes—
432C32—42no—
433C32—81no—
434C32—82no—
435C32—161no—
436C32#594511yescharacteristic, complemented
437C23—92no—
438C23—271no—
439C23—272no—
440Q8—91no—
441Q8—92no—
442Q8—272no—
443Q8—271no—
444Q8—271no—
445Q8C2×C3×D3211yescharacteristic
446D4—31no—
447D4—92no—
448D4—92no—
449D4—91no—
450D4—94no—
451D4—272no—
452D4—271no—
453D4—272no—
454D4—272no—
455D4—271no—
456D4—272no—
457D4—272no—
458C2×C4—31no—
459C2×C4—32no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—91no—
465C2×C4—271no—
466C2×C4—271no—
467C2×C4—272no—
468C2×C4—271no—
469C2×C4—274no—
470C2×C4—271no—
471C2×C3—21no—
472C2×C3—41no—
473C2×C3—62no—
474C2×C3—62no—
475C2×C3—82no—
476C2×C3—121no—
477C2×C3—161no—
478C2×C3—182no—
479C2×C3—242no—
480C2×C3—244no—
481C2×C3—721no—
482C2×C3C22×D3×A412yes—
483D3—22no—
484D3—62no—
485D3—62no—
486D3—62no—
487D3—182no—
488D3—361no—
489C22—31no—
490C22—32no—
491C22—92no—
492C22—91no—
493C22—271no—
494C22—362no—
495C22—1081no—
496C22—1082no—
497C4—31no—
498C4—32no—
499C4—92no—
500C4—91no—
501C4—271no—
502C4D32×A411yescharacteristic
503C3—21no—
504C3—41no—
505C3—82no—
506C3—161no—
507C3#5964⋊C312yescomplemented
508C2—61no—
509C2—62no—
510C2—182no—
511C2—181no—
512C2—541no—
513C2C2×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. #64320⋊C3C2semidirect
4. #64315⋊C3C2semidirect
5. #64316⋊C3C2semidirect
6. #10596695C3semidirect
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. #64320C2×C3semidirect
18. #64316C2×C3semidirect
19. #64315C2×C3semidirect
30. D3×#41⋊C3D3semidirect
60. C3×#41⋊C3C2×D3semidirect
86. D3×SL(2,𝔽3)C2×D3semidirect
90. C3⋊(#41⋊C3)C2×D3semidirect
105. D3×#41C3×D3semidirect
232. #41⋊C3D32semidirect
436. C32#5945semidirect
507. C3#5964⋊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
#64320⋊C32
#64315⋊C32
#64316⋊C32
#105966953
#5964⋊C33
C3×#37984