Finite Group Databasebeta

← #10596687⋊C3 #13180898

Subgroups — #10596687⋊C3 #13180898

Order 1728.

943 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 943 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#1007⋊C3C212yesmaximal, complemented
3D32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#64315⋊C3C212yesmaximal, complemented
5#64319⋊C3C211yesmaximal, characteristic, complemented
6#10596687C311yesmaximal, characteristic, complemented
7D3×2−1+4⋊C3—32nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C3×#3811—41nomaximal
10C3⋊D3×SL(2,𝔽3)C2211yescharacteristic
11C3⋊D3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3×D3×SL(2,𝔽3)C2212yescomplemented
13C3×C3⋊(#41⋊C3)C2212yescomplemented
14C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
15C3×#1007C2×C312yescomplemented
16C3×2−1+4⋊C3—32no—
17#64319—31no—
18#64319C2×C311yescharacteristic, complemented
19D32×Q8—31no—
20D32×Q8C2×C311yescharacteristic, complemented
21#64315—32no—
22#64315C2×C312yescomplemented
23#64306—31no—
24#64306—31no—
25#64298—32no—
26#1010⋊C3—32no—
27D3×#41⋊C3—32no—
28#1007⋊C3—22no—
29#1007⋊C3—32no—
30#1007⋊C3D312yescomplemented
31C2×D3×SL(2,𝔽3)—32no—
32C2×D3×SL(2,𝔽3)—32no—
33C2×D3×SL(2,𝔽3)—61no—
34C22×C3×C3⋊D3—41no—
35C2×C3×D32—41no—
36C32×#67—42no—
37C32×SL(2,𝔽3)C2311yescharacteristic
38C3×#626—42no—
39C3×#624—41no—
40D3×2−1+4—32no—
41C2×2−1+4⋊C3—91no—
42C2×C32×Q8C22×C311yescharacteristic, complemented
43Q8×C3⋊D3—31no—
44Q8×C3⋊D3—31no—
45Q8×C3⋊D3C22×C311yescharacteristic
46Q8×C3⋊D3C22×C311yescharacteristic, complemented
47C2×C4×C3⋊D3—31no—
48C2×#634—31no—
49C3×#239—32no—
50C3×#239C22×C312yescomplemented
51C3×D3×Q8—32no—
52C3×D3×Q8C22×C312yescomplemented
53C3×D3×D4—122no—
54C3×#235—32no—
55C3×#235—32no—
56C3×#41⋊C3—32no—
57C2×C3×SL(2,𝔽3)—22no—
58C2×C3×SL(2,𝔽3)—21no—
59C2×C3×SL(2,𝔽3)—32no—
60C2×C3×SL(2,𝔽3)—41no—
61C2×C3×SL(2,𝔽3)C2×D312yescomplemented
62#3811—31no—
63#3811A411yescharacteristic
64#3805—32no—
65#3805—32no—
66#3802—31no—
67#3802—31no—
68C4×D32—32no—
69C4×D32—31no—
70C4×D32—31no—
71#3798—32no—
72#3798—32no—
73#3797—31no—
74#3797—31no—
75#3796—32no—
76#3796—32no—
77#3795—34no—
78#3795—32no—
79#3795—32no—
80D3×Dic6—34no—
81D3×Dic6—32no—
82D3×Dic6—32no—
83D3×SL(2,𝔽3)—22no—
84D3×SL(2,𝔽3)—32no—
85D3×SL(2,𝔽3)—32no—
86D3×SL(2,𝔽3)—61no—
87D3×SL(2,𝔽3)—61no—
88D3×SL(2,𝔽3)C2×D312yescomplemented
89C3⋊(#41⋊C3)—22no—
90C3⋊(#41⋊C3)C2×D312yescomplemented
91C22×C33—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—82no—
95C2×C32×D3—42no—
96C3×D32—82no—
97C32×Dic3—42no—
98C3×2−1+4—32no—
99#1010—32no—
100#1010—92no—
101#1010—92no—
102#1010—92no—
103D3×#41—32no—
104D3×#41—92no—
105D3×#41—92no—
106D3×#41—92no—
107#1007—32no—
108#1007—92no—
109#1007C3×D312yescomplemented
110C2×D3×Q8—32no—
111C2×D3×Q8—32no—
112C2×D3×Q8—61no—
113C2×D3×Q8—92no—
1142−1+4⋊C3—32no—
1152−1+4⋊C3—62no—
1162−1+4⋊C3—92no—
117C2×#41⋊C3—91no—
118C22×SL(2,𝔽3)—91no—
119C22×SL(2,𝔽3)—91no—
120C22×C3⋊D3C2×A411yescharacteristic
121C22×C3×D3—122no—
122C22×C3×D3—122no—
123C22×C3×D3—241no—
124C2×D32—31no—
125C2×D32C2×A411yescharacteristic
126C32×Q8—31no—
127C32×Q8C23×C311yescharacteristic
128C32×D4—122no—
129C2×C32×C4—31no—
130C2×C3⋊Dic3—31no—
131C4×C3⋊D3—31no—
132C4×C3⋊D3—31no—
133C4×C3⋊D3—31no—
134C4×C3⋊D3—31no—
135#634—32no—
136#634—31no—
137#634—31no—
138C3×#67—32no—
139C3×#67—42no—
140C3×#67—82no—
141C3×#67—122no—
142C3×#67C2×A412yes—
143C3×D12—32no—
144C3×D12—32no—
145C3×D12—122no—
146C3×C4×D3—34no—
147C3×C4×D3—32no—
148C3×C4×D3—32no—
149C3×C4×D3—122no—
150C3×Dic6—32no—
151C3×Dic6—32no—
152C3×SL(2,𝔽3)—22no—
153C3×SL(2,𝔽3)—21no—
154C3×SL(2,𝔽3)—41no—
155C3×SL(2,𝔽3)C22×D312yes—
156#627—32no—
157#627—32no—
158#626—32no—
159#626C2×A412yes—
160#625—32no—
161#625—32no—
162#624—31no—
163#624C2×A411yescharacteristic
164D3×Dic3—34no—
165D3×Dic3—32no—
166D3×Dic3—32no—
167C2×2−1+4—91noSylow
168C2×C33—41no—
169C2×C33—81no—
170C3×C3⋊D3—42no—
171C3×C3⋊D3—81no—
172C32×D3—82no—
173C3×#41—32no—
174C3×#41—92no—
175C3×#41—92no—
176C3×#41—92no—
177C2×C3×Q8—21no—
178C2×C3×Q8—32no—
179C2×C3×Q8—92no—
180C2×C3×Q8C2×C3×D312yescomplemented
181C2×C3×D4—361no—
182#239—32no—
183#239—92no—
184#239—92no—
185#239C2×C3×D312yescomplemented
186D3×Q8—32no—
187D3×Q8—32no—
188D3×Q8—32no—
189D3×Q8—61no—
190D3×Q8—61no—
191D3×Q8—92no—
192D3×Q8—92no—
193D3×Q8—94no—
194D3×Q8—92no—
195D3×Q8—92no—
196D3×Q8—92no—
197D3×Q8—92no—
198D3×Q8—181no—
199D3×Q8—181no—
200D3×Q8C2×C3×D312yescomplemented
201#237—92no—
202#237—92no—
203#237—94no—
204#237—92no—
205#237—92no—
206#237—92no—
207#237—94no—
208#237—92no—
209D3×D4—32no—
210D3×D4—92no—
211D3×D4—92no—
212D3×D4—92no—
213#235—32no—
214#235—32no—
215#235—92no—
216#235—92no—
217#235—94no—
218#235—94no—
219#235—92no—
220#235—92no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×C4×D3—94no—
224C2×C4×D3—92no—
225C2×C4×D3—181no—
226C2×Dic6—94no—
227C2×Dic6—92no—
228C2×Dic6—92no—
229C2×Dic6—92no—
230C2×Dic6—181no—
231#41⋊C3—32no—
232#41⋊C3—62no—
233C2×SL(2,𝔽3)—22no—
234C2×SL(2,𝔽3)—32no—
235C2×SL(2,𝔽3)—41no—
236C2×SL(2,𝔽3)—62no—
237C2×SL(2,𝔽3)—91no—
238C2×SL(2,𝔽3)—91no—
239C2×SL(2,𝔽3)D3211yescharacteristic, complemented
240C22×C32—42no—
241C22×C32—81no—
242C22×C32—82no—
243C22×C32—122no—
244C22×C32—161no—
245C22×C32C22×A411yescharacteristic
246C2×C3⋊D3—22no—
247C2×C3⋊D3C22×A411yescharacteristic
248C2×C3⋊D3C22×A411yescharacteristic
249C2×C3×D3—32no—
250C2×C3×D3—42no—
251C2×C3×D3—82no—
252C2×C3×D3—122no—
253C2×C3×D3—122no—
254C2×C3×D3—241no—
255C2×C3×D3—241no—
256C2×C3×D3—244no—
257C2×C3×D3—244no—
258C2×C3×D3—244no—
259C2×C3×D3C22×A412yes—
260D32—22no—
261D32—62no—
262C32×C4—31no—
263C32×C4—31no—
264C32×C4—122no—
265C3⋊Dic3—31no—
266C3⋊Dic3—31no—
267C3×Dic3—32no—
268C3×Dic3—42no—
269C3×Dic3—82no—
270C3×Dic3C22×A412yes—
2712−1+4—32no—
2722−1+4—92no—
2732−1+4—272no—
2742−1+4—272no—
275C2×#41—91no—
276C2×#41—271no—
277C2×#41—271no—
278C2×#41—271no—
279C22×Q8—91no—
280C22×Q8—91no—
281C22×Q8—271no—
282C33—41noSylow
283C23×C3—361no—
284C23×C3—361no—
285C22×D3—32no—
286C22×D3—32no—
287C22×D3—61no—
288C22×D3—92no—
289C3×Q8—21no—
290C3×Q8—32no—
291C3×Q8—61no—
292C3×Q8—92no—
293C3×Q8—92no—
294C3×Q8C22×C3×D312yes—
295C3×D4—32no—
296C3×D4—92no—
297C3×D4—92no—
298C3×D4—92no—
299C3×D4—122no—
300C3×D4—242no—
301C3×D4—362no—
302C2×C3×C4—32no—
303C2×C3×C4—61no—
304C2×C3×C4—92no—
305C2×C3×C4—92no—
306C2×C3×C4—94no—
307C2×C3×C4—361no—
308#67—32no—
309#67—32no—
310#67—92no—
311#67—92no—
312#67—94no—
313#67—92no—
314#67D3×A412yes—
315C2×Dic3—92no—
316C2×Dic3—94no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—181no—
320D12—32no—
321D12—32no—
322D12—32no—
323D12—92no—
324C4×D3—34no—
325C4×D3—32no—
326C4×D3—32no—
327C4×D3—32no—
328C4×D3—92no—
329C4×D3—92no—
330C4×D3—92no—
331C4×D3—92no—
332C4×D3—92no—
333C4×D3—92no—
334C4×D3—94no—
335C4×D3—92no—
336C4×D3—181no—
337C4×D3—181no—
338C4×D3—181no—
339C4×D3—181no—
340Dic6—32no—
341Dic6—32no—
342Dic6—92no—
343Dic6—94no—
344Dic6—94no—
345Dic6—92no—
346Dic6—92no—
347Dic6—92no—
348Dic6—182no—
349Dic6—181no—
350Dic6—181no—
351SL(2,𝔽3)—22no—
352SL(2,𝔽3)—41no—
353SL(2,𝔽3)C2×D3211yescharacteristic
354C2×C32—21no—
355C2×C32—42no—
356C2×C32—81no—
357C2×C32—82no—
358C2×C32—82no—
359C2×C32—82no—
360C2×C32—161no—
361C2×C32—162no—
362C2×C32—162no—
363C2×C32—242no—
364C2×C32C23×A411yescharacteristic
365C3⋊D3—21no—
366C3⋊D32−1+4⋊C312yescomplemented
367C3×D3—22no—
368C3×D3—62no—
369C3×D3—82no—
370C3×D3—124no—
371C3×D3—162no—
372C3×D3—242no—
373C3×D3—242no—
374C3×D3—242no—
375#41—32no—
376#41—92no—
377#41—92no—
378#41—92no—
379#41—272no—
380#41—272no—
381#41—274no—
382#41—272no—
383#41—272no—
384#41—272no—
385#41—272no—
386#41—272no—
387#41—272no—
388C2×Q8—32no—
389C2×Q8—92no—
390C2×Q8—91no—
391C2×Q8—91no—
392C2×Q8—271no—
393C2×Q8—271no—
394C2×Q8—272no—
395C2×Q8—274no—
396C2×Q8—272no—
397C2×Q8—271no—
398C2×Q8—271no—
399C2×Q8—271no—
400C2×Q8C3×D3211yescharacteristic, complemented
401C2×D4—91no—
402C2×D4—271no—
403C2×D4—271no—
404C2×D4—271no—
405C22×C4—271no—
406C22×C4—271no—
407C22×C4—271no—
408C22×C4—272no—
409C22×C3—21no—
410C22×C3—32no—
411C22×C3—41no—
412C22×C3—82no—
413C22×C3—92no—
414C22×C3—122no—
415C22×C3—161no—
416C22×C3—242no—
417C22×C3—361no—
418C22×C3—361no—
419C22×C3—722no—
420C22×C3—722no—
421C22×C3C2×D3×A412yes—
422C2×D3—32no—
423C2×D3—32no—
424C2×D3—32no—
425C2×D3—61no—
426C2×D3—61no—
427C2×D3—64no—
428C2×D3—64no—
429C2×D3—64no—
430C2×D3—184no—
431C2×D3C2×D3×A412yes—
432C3×C4—32no—
433C3×C4—32no—
434C3×C4—32no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—92no—
438C3×C4—122no—
439C3×C4—242no—
440Dic3—32no—
441Dic3—92no—
442Dic3—92no—
443Dic3—181no—
444Dic3—181no—
445Dic3C2×D3×A412yes—
446C32—42no—
447C32—81no—
448C32—82no—
449C32—161no—
450C32C2×2−1+4⋊C311yescharacteristic, complemented
451C23—91no—
452C23—91no—
453C23—271no—
454Q8—31no—
455Q8—92no—
456Q8—92no—
457Q8—272no—
458Q8—271no—
459Q8—271no—
460Q8—272no—
461Q8—272no—
462Q8C2×C3×D3211yescharacteristic
463D4—32no—
464D4—92no—
465D4—92no—
466D4—92no—
467D4—92no—
468D4—272no—
469D4—272no—
470D4—272no—
471C2×C4—31no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—94no—
476C2×C4—271no—
477C2×C4—271no—
478C2×C4—271no—
479C2×C4—272no—
480C2×C4—271no—
481C2×C4—271no—
482C2×C4—272no—
483C2×C4—274no—
484C2×C4—271no—
485C2×C3—22no—
486C2×C3—22no—
487C2×C3—21no—
488C2×C3—41no—
489C2×C3—62no—
490C2×C3—81no—
491C2×C3—82no—
492C2×C3—162no—
493C2×C3—162no—
494C2×C3—161no—
495C2×C3—182no—
496C2×C3—242no—
497C2×C3—362no—
498C2×C3—482no—
499C2×C3—721no—
500C2×C3C22×D3×A412yes—
501D3—22no—
502D3—34no—
503D3—62no—
504D3—62no—
505D3—62no—
506D3—62no—
507C22—32no—
508C22—91no—
509C22—92no—
510C22—91no—
511C22—182no—
512C22—182no—
513C22—542no—
514C22D32×A411yescharacteristic
515C4—31no—
516C4—31no—
517C4—32no—
518C4—92no—
519C4—271no—
520C4—271no—
521C3—21no—
522C3—41no—
523C3—82no—
524C3—161no—
525C3D3×2−1+4⋊C312yescomplemented
526C2—21no—
527C2—62no—
528C2—92no—
529C2—181no—
530C2—182no—
531C2C2×D32×A411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FittingC2×C32×Q8C22×C3
socleC2×C32C23×A4
hypercenterC22D32×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C3×#1007⋊C3C2semidirect
3. D32×SL(2,𝔽3)C2semidirect
4. #64315⋊C3C2semidirect
5. #64319⋊C3C2semidirect
6. #10596687C3semidirect
8. C2×C32×SL(2,𝔽3)C22semidirect
11. C3⋊D3×SL(2,𝔽3)C22semidirect
12. C3×D3×SL(2,𝔽3)C22semidirect
13. C3×C3⋊(#41⋊C3)C22semidirect
14. C2×Q8×C3⋊D3C2×C3semidirect
15. C3×#1007C2×C3semidirect
18. #64319C2×C3semidirect
20. D32×Q8C2×C3semidirect
22. #64315C2×C3semidirect
30. #1007⋊C3D3semidirect
42. C2×C32×Q8C22×C3semidirect
46. Q8×C3⋊D3C22×C3semidirect
50. C3×#239C22×C3semidirect
52. C3×D3×Q8C22×C3semidirect
61. C2×C3×SL(2,𝔽3)C2×D3semidirect
88. D3×SL(2,𝔽3)C2×D3semidirect
90. C3⋊(#41⋊C3)C2×D3semidirect
109. #1007C3×D3semidirect
180. C2×C3×Q8C2×C3×D3semidirect
185. #239C2×C3×D3semidirect
200. D3×Q8C2×C3×D3semidirect
239. C2×SL(2,𝔽3)D32semidirect
366. C3⋊D32−1+4⋊C3semidirect
400. C2×Q8C3×D32semidirect
450. C32C2×2−1+4⋊C3semidirect
525. C3D3×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C3×#1007⋊C32
D32×SL(2,𝔽3)2
#64315⋊C32
#64319⋊C32
#105966873
D3×2−1+4⋊C33
C3×#38114