Finite Group Databasebeta

← C2×#64315⋊C3 #13180894

Subgroups — C2×#64315⋊C3 #13180894

Order 1728.

959 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 959 conjugacy classes of subgroups — too many to draw. Only its 96 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
2C2×C3×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
4#64315⋊C3C214yesmaximal, direct product factor
5C2×#64315C311yesmaximal, characteristic, complemented
6C2×#1010⋊C3—31nomaximal
7C2×#1007⋊C3—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×#626—41nomaximal
10C3⋊D3×SL(2,𝔽3)C2212yescomplemented
11C3×D3×SL(2,𝔽3)C2212yescomplemented
12C3×C3⋊(#41⋊C3)C2212yescomplemented
13C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
14C2×C3×#239C2×C311yescharacteristic, complemented
15C2×C3×D3×Q8C2×C311yescharacteristic, complemented
16C2×C3×#41⋊C3—31no—
17C22×C3×SL(2,𝔽3)—31no—
18#64315—34no—
19#64315C2×C314yescomplemented
20C2×#3798—31no—
21C2×#3796—31no—
22C2×#3795—31no—
23C2×D3×Dic6—31no—
24#1010⋊C3—34no—
25#1007⋊C3—34no—
26C2×D3×SL(2,𝔽3)—21no—
27C2×D3×SL(2,𝔽3)—31no—
28C2×D3×SL(2,𝔽3)—31no—
29C2×D3×SL(2,𝔽3)—61no—
30C2×D3×SL(2,𝔽3)D311yescharacteristic, complemented
31C2×C3⋊(#41⋊C3)—21no—
32C2×C3⋊(#41⋊C3)D311yescharacteristic, complemented
33C22×C3×C3⋊D3—41no—
34C22×C32×D3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×#626—44no—
38C2×#1010—31no—
39C2×#1007—31no—
40C2×2−1+4⋊C3—91no—
41C2×C32×Q8C22×C311yescharacteristic
42Q8×C3⋊D3—32no—
43Q8×C3⋊D3C22×C312yescomplemented
44C2×C4×C3⋊D3—31no—
45C2×#634—31no—
46C2×C3×#67—121no—
47C3×#239—32no—
48C3×#239C22×C312yescomplemented
49C3×D3×Q8—32no—
50C3×D3×Q8C22×C312yescomplemented
51C2×C3×D12—31no—
52C2×C3×D12—121no—
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—31no—
55C2×C3×Dic6—31no—
56C3×#41⋊C3—32no—
57C2×C3×SL(2,𝔽3)—21no—
58C2×C3×SL(2,𝔽3)—21no—
59C2×C3×SL(2,𝔽3)—21no—
60C2×C3×SL(2,𝔽3)—32no—
61C2×C3×SL(2,𝔽3)—41no—
62C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
63C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
64C2×#627—31no—
65C2×#626—31no—
66C2×#626A411yescharacteristic
67C2×#625—31no—
68C2×D3×Dic3—32no—
69#3798—34no—
70#3798—34no—
71#3796—34no—
72#3796—34no—
73#3795—34no—
74#3795—34no—
75D3×Dic6—34no—
76D3×Dic6—34no—
77D3×SL(2,𝔽3)—22no—
78D3×SL(2,𝔽3)—32no—
79D3×SL(2,𝔽3)—32no—
80D3×SL(2,𝔽3)—62no—
81D3×SL(2,𝔽3)C2×D312yescomplemented
82C3⋊(#41⋊C3)—22no—
83C3⋊(#41⋊C3)C2×D312yescomplemented
84C22×C33—41no—
85C2×C3×C3⋊D3—42no—
86C2×C3×C3⋊D3—82no—
87C2×C32×D3—42no—
88C2×C32×D3—82no—
89C32×Dic3—42no—
90C2×C3×#41—31no—
91C22×C3×Q8—31no—
92#1010—34no—
93#1010—94no—
94#1007—34no—
95#1007—94no—
96C2×#239—91no—
97C2×#239C3×D311yescharacteristic, complemented
98C2×D3×Q8—31no—
99C2×D3×Q8—31no—
100C2×D3×Q8—61no—
101C2×D3×Q8—91no—
102C2×D3×Q8C3×D311yescharacteristic, complemented
103C2×#237—91no—
104C2×#237—91no—
105C2×#235—91no—
106C2×#235—91no—
107C2×#235—91no—
108C22×Dic6—91no—
1092−1+4⋊C3—94no—
110C2×#41⋊C3—31no—
111C2×#41⋊C3—61no—
112C22×SL(2,𝔽3)—31no—
113C22×SL(2,𝔽3)—61no—
114C22×SL(2,𝔽3)—91no—
115C23×C32—121no—
116C22×C3⋊D3C2×A411yescharacteristic
117C22×C3×D3—31no—
118C22×C3×D3—41no—
119C22×C3×D3—81no—
120C22×C3×D3—121no—
121C22×C3×D3—121no—
122C22×C3×D3—241no—
123C22×C3×D3C2×A411yescharacteristic
124C32×Q8—31no—
125C32×Q8C23×C311yescharacteristic
126C2×C32×C4—31no—
127C2×C32×C4—121no—
128C2×C3⋊Dic3—31no—
129C4×C3⋊D3—32no—
130C4×C3⋊D3—32no—
131#634—32no—
132#634—32no—
133C3×#67—124no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—81no—
137C2×C3×Dic3C2×A411yescharacteristic
138C3×D12—32no—
139C3×D12—32no—
140C3×D12—124no—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×Dic6—32no—
146C3×Dic6—32no—
147C3×SL(2,𝔽3)—21no—
148C3×SL(2,𝔽3)—21no—
149C3×SL(2,𝔽3)—21no—
150C3×SL(2,𝔽3)—41no—
151C3×SL(2,𝔽3)C22×D311yescharacteristic
152C3×SL(2,𝔽3)C22×D311yescharacteristic
153#627—34no—
154#626—34no—
155#626C2×A414yes—
156#625—34no—
157D3×Dic3—38no—
158C2×2−1+4—91noSylow
159C2×C33—42no—
160C2×C33—41no—
161C3×C3⋊D3—82no—
162C32×D3—82no—
163C3×#41—32no—
164C3×#41—92no—
165C2×C3×Q8—21no—
166C2×C3×Q8—32no—
167C2×C3×Q8—91no—
168C2×C3×Q8—92no—
169C2×C3×Q8C2×C3×D311yescharacteristic
170C2×C3×Q8C2×C3×D311yescharacteristic
171C2×C3×D4—91no—
172C2×C3×D4—361no—
173C22×C3×C4—91no—
174C22×C3×C4—91no—
175C2×#67—31no—
176C2×#67—91no—
177C2×#67—91no—
178C2×#67—91no—
179C22×Dic3—92no—
180#239—32no—
181#239—94no—
182#239—94no—
183#239C2×C3×D312yescomplemented
184D3×Q8—32no—
185D3×Q8—32no—
186D3×Q8—32no—
187D3×Q8—62no—
188D3×Q8—94no—
189D3×Q8—92no—
190D3×Q8—92no—
191D3×Q8—94no—
192D3×Q8—182no—
193D3×Q8C2×C3×D312yescomplemented
194#237—94no—
195#237—94no—
196#237—94no—
197#237—94no—
198#235—94no—
199#235—94no—
200#235—94no—
201#235—94no—
202#235—94no—
203#235—94no—
204C2×D12—31no—
205C2×D12—31no—
206C2×D12—91no—
207C2×C4×D3—31no—
208C2×C4×D3—31no—
209C2×C4×D3—91no—
210C2×C4×D3—91no—
211C2×C4×D3—92no—
212C2×C4×D3—181no—
213C2×Dic6—31no—
214C2×Dic6—94no—
215C2×Dic6—94no—
216C2×Dic6—91no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C2×Dic6—181no—
221#41⋊C3—32no—
222#41⋊C3—62no—
223C2×SL(2,𝔽3)—21no—
224C2×SL(2,𝔽3)—21no—
225C2×SL(2,𝔽3)—32no—
226C2×SL(2,𝔽3)—41no—
227C2×SL(2,𝔽3)—62no—
228C2×SL(2,𝔽3)—92no—
229C2×SL(2,𝔽3)D3211yescharacteristic
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—81no—
233C22×C32—81no—
234C22×C32—81no—
235C22×C32—122no—
236C22×C32—161no—
237C22×C32—242no—
238C22×C32C22×A411yescharacteristic
239C2×C3⋊D3—22no—
240C2×C3⋊D3C22×A412yes—
241C2×C3×D3—22no—
242C2×C3×D3—32no—
243C2×C3×D3—42no—
244C2×C3×D3—62no—
245C2×C3×D3—82no—
246C2×C3×D3—84no—
247C2×C3×D3—82no—
248C2×C3×D3—122no—
249C2×C3×D3—122no—
250C2×C3×D3—242no—
251C2×C3×D3—242no—
252C2×C3×D3—242no—
253C2×C3×D3—244no—
254C2×C3×D3C22×A412yes—
255C32×C4—31no—
256C32×C4—31no—
257C32×C4—122no—
258C3⋊Dic3—32no—
259C3×Dic3—32no—
260C3×Dic3—42no—
261C3×Dic3—82no—
262C3×Dic3C22×A412yes—
2632−1+4—94no—
2642−1+4—274no—
265C2×#41—31no—
266C2×#41—271no—
267C2×#41—271no—
268C2×#41—271no—
269C22×Q8—31no—
270C22×Q8—91no—
271C22×Q8—271no—
272C33—41noSylow
273C23×C3—31no—
274C23×C3—91no—
275C23×C3—121no—
276C23×C3—241no—
277C23×C3—361no—
278C22×D3—31no—
279C22×D3—31no—
280C22×D3—31no—
281C22×D3—61no—
282C22×D3D3×A411yescharacteristic
283C3×Q8—21no—
284C3×Q8—31no—
285C3×Q8—31no—
286C3×Q8—61no—
287C3×Q8—92no—
288C3×Q8—92no—
289C3×Q8C22×C3×D311yescharacteristic
290C3×Q8C22×C3×D311yescharacteristic
291C3×D4—92no—
292C3×D4—92no—
293C3×D4—364no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—61no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—91no—
303C2×C3×C4—121no—
304C2×C3×C4—241no—
305#67—34no—
306#67—94no—
307#67—94no—
308#67—94no—
309C2×Dic3—31no—
310C2×Dic3—98no—
311C2×Dic3—91no—
312C2×Dic3—91no—
313C2×Dic3—181no—
314C2×Dic3D3×A411yescharacteristic
315D12—32no—
316D12—32no—
317D12—34no—
318D12—94no—
319C4×D3—32no—
320C4×D3—32no—
321C4×D3—32no—
322C4×D3—32no—
323C4×D3—92no—
324C4×D3—92no—
325C4×D3—92no—
326C4×D3—92no—
327C4×D3—98no—
328C4×D3—182no—
329C4×D3—182no—
330Dic6—32no—
331Dic6—32no—
332Dic6—94no—
333Dic6—92no—
334Dic6—92no—
335Dic6—94no—
336Dic6—92no—
337Dic6—92no—
338Dic6—182no—
339Dic6—182no—
340SL(2,𝔽3)—21no—
341SL(2,𝔽3)—21no—
342SL(2,𝔽3)—41no—
343SL(2,𝔽3)C2×D3211yescharacteristic
344C2×C32—42no—
345C2×C32—42no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32—81no—
349C2×C32—82no—
350C2×C32—82no—
351C2×C32—82no—
352C2×C32—81no—
353C2×C32—81no—
354C2×C32—161no—
355C2×C32—162no—
356C2×C32—242no—
357C2×C322−1+4⋊C312yescomplemented
358C2×C32C23×A411yescharacteristic
359C3⋊D3—22no—
360C3×D3—22no—
361C3×D3—62no—
362C3×D3—82no—
363C3×D3—84no—
364C3×D3—242no—
365C3×D3—242no—
366C3×D3—244no—
367#41—32no—
368#41—92no—
369#41—274no—
370#41—274no—
371#41—274no—
372#41—274no—
373#41—274no—
374#41—274no—
375C2×Q8—32no—
376C2×Q8—91no—
377C2×Q8—92no—
378C2×Q8—92no—
379C2×Q8—271no—
380C2×Q8—271no—
381C2×Q8—274no—
382C2×Q8—274no—
383C2×Q8—272no—
384C2×Q8C3×D3211yescharacteristic
385C2×D4—91no—
386C2×D4—91no—
387C2×D4—271no—
388C2×D4—271no—
389C22×C4—91no—
390C22×C4—91no—
391C22×C4—271no—
392C22×C4—272no—
393C22×C3—21no—
394C22×C3—32no—
395C22×C3—41no—
396C22×C3—62no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—92no—
400C22×C3—122no—
401C22×C3—161no—
402C22×C3—182no—
403C22×C3—242no—
404C22×C3—242no—
405C22×C3—244no—
406C22×C3—362no—
407C22×C3—722no—
408C22×C3C2×D3×A411yescharacteristic
409C22×C3C2×D3×A411yescharacteristic
410C2×D3—22no—
411C2×D3—32no—
412C2×D3—32no—
413C2×D3—32no—
414C2×D3—62no—
415C2×D3—64no—
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3C2×D3×A412yes—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—32no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—92no—
428C3×C4—122no—
429C3×C4—242no—
430Dic3—32no—
431Dic3—92no—
432Dic3—92no—
433Dic3—182no—
434Dic3C2×D3×A412yes—
435C32—41no—
436C32—41no—
437C32—81no—
438C32—81no—
439C32—81no—
440C32—161no—
441C32C2×2−1+4⋊C311yescharacteristic, complemented
442C23—31no—
443C23—91no—
444C23—91no—
445Q8—31no—
446Q8—92no—
447Q8—92no—
448Q8—274no—
449Q8—272no—
450Q8—272no—
451Q8C2×C3×D3211yescharacteristic
452D4—94no—
453D4—92no—
454D4—92no—
455D4—274no—
456D4—274no—
457C2×C4—31no—
458C2×C4—31no—
459C2×C4—92no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4—92no—
463C2×C4—91no—
464C2×C4—272no—
465C2×C4—272no—
466C2×C4—278no—
467C2×C4—271no—
468C2×C3—21no—
469C2×C3—22no—
470C2×C3—41no—
471C2×C3—42no—
472C2×C3—62no—
473C2×C3—81no—
474C2×C3—81no—
475C2×C3—82no—
476C2×C3—82no—
477C2×C3—161no—
478C2×C3—162no—
479C2×C3—182no—
480C2×C3—242no—
481C2×C3—244no—
482C2×C3—722no—
483C2×C3#1007⋊C312yescomplemented
484C2×C3#1010⋊C312yescomplemented
485C2×C3C22×D3×A411yescharacteristic
486C2×C3C22×D3×A411yescharacteristic
487D3—22no—
488D3—64no—
489D3—62no—
490D3—62no—
491D3—62no—
492C22—32no—
493C22—62no—
494C22—92no—
495C22—92no—
496C22—182no—
497C22—182no—
498C22D32×A411yescharacteristic
499C4—31no—
500C4—31no—
501C4—32no—
502C4—92no—
503C4—272no—
504C3—21no—
505C3—41no—
506C3—81no—
507C3—81no—
508C3—161no—
509C3C2×#1007⋊C311yescharacteristic, complemented
510C3C2×#1010⋊C311yescharacteristic, complemented
511C2—62no—
512C2—182no—
513C2—182no—
514C2#64315⋊C312yesdirect product factor
515C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C2×D32×A4
FittingC2×C32×Q8C22×C3
socleC22×C32C22×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C3×D3×SL(2,𝔽3)C2semidirect
3. C2×C3×C3⋊(#41⋊C3)C2semidirect
4. #64315⋊C3C2direct (N × H)
5. C2×#64315C3semidirect
10. C3⋊D3×SL(2,𝔽3)C22semidirect
11. C3×D3×SL(2,𝔽3)C22semidirect
12. C3×C3⋊(#41⋊C3)C22semidirect
13. C2×Q8×C3⋊D3C2×C3semidirect
14. C2×C3×#239C2×C3semidirect
15. C2×C3×D3×Q8C2×C3semidirect
19. #64315C2×C3semidirect
30. C2×D3×SL(2,𝔽3)D3semidirect
32. C2×C3⋊(#41⋊C3)D3semidirect
43. Q8×C3⋊D3C22×C3semidirect
48. C3×#239C22×C3semidirect
50. C3×D3×Q8C22×C3semidirect
81. D3×SL(2,𝔽3)C2×D3semidirect
83. C3⋊(#41⋊C3)C2×D3semidirect
97. C2×#239C3×D3semidirect
102. C2×D3×Q8C3×D3semidirect
183. #239C2×C3×D3semidirect
193. D3×Q8C2×C3×D3semidirect
357. C2×C322−1+4⋊C3semidirect
441. C32C2×2−1+4⋊C3semidirect
483. C2×C3#1007⋊C3semidirect
484. C2×C3#1010⋊C3semidirect
509. C3C2×#1007⋊C3semidirect
510. C3C2×#1010⋊C3semidirect
514. C2#64315⋊C3direct (N × H)

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
C2×C3×D3×SL(2,𝔽3)2
C2×C3×C3⋊(#41⋊C3)2
#64315⋊C32
C2×#643153
C2×#1010⋊C33
C2×#1007⋊C33
C2×C3×#6264