Finite Group Databasebeta

← #13164639

Subgroups — #13164639

Order 1728.

1035 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 1035 conjugacy classes of subgroups — too many to draw. Only its 78 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722438C211yesmaximal, characteristic, complemented
3#11722437C211yesmaximal, characteristic, complemented
4#11722432C212yesmaximal, complemented
5C2×#88623C212yesmaximal, complemented
6C3⋊D3×2−1+4C311yesmaximal, characteristic, complemented
7D3×2−1+4⋊C3—31nomaximal
8D4×(C3⋊D3)⋊C3—41nomaximal
9#41⋊H(3,𝔽3)C2211yescharacteristic, complemented
10C2×Q8⋊H(3,𝔽3)C2212yescomplemented
11#88623C2211yescharacteristic
12#88623C2212yescomplemented
13#88622C2211yescharacteristic, complemented
14C32×2−1+4C2×C311yescharacteristic, complemented
15#64367—32no—
16#64367—31no—
17#64367C2×C311yescharacteristic, complemented
18#41×C3⋊D3—31no—
19#41×C3⋊D3—32no—
20#41×C3⋊D3C2×C311yescharacteristic, complemented
21#64364—31no—
22#64364C2×C312yescomplemented
23C2×Q8×C3⋊D3—31no—
24C2×Q8×C3⋊D3C2×C312yescomplemented
25C3×2−1+4⋊C3—21no—
26C3×2−1+4⋊C3D311yescharacteristic, complemented
27#1010⋊C3—31no—
28D3×#41⋊C3—31no—
29#1007⋊C3—32no—
30C2×D3×SL(2,𝔽3)—32no—
31C22×(C3⋊D3)⋊C3—42no—
32D4×H(3,𝔽3)—41no—
33#638⋊C3—42no—
34C4⋊((C3⋊D3)⋊C3)—41no—
35C4×(C3⋊D3)⋊C3—41no—
36Q8⋊H(3,𝔽3)C2311yescharacteristic
37D3×2−1+4—31no—
38D3×2−1+4—91no—
39C2×2−1+4⋊C3—91no—
40C32×#41—31no—
41C32×#41—32no—
42C32×#41C22×C311yescharacteristic, complemented
43C2×C32×Q8—31no—
44C2×C32×Q8C22×C312yescomplemented
45#3832—31no—
46#3832—32no—
47#3832C22×C311yescharacteristic, complemented
48Q8×C3⋊D3—32no—
49Q8×C3⋊D3—31no—
50Q8×C3⋊D3—32no—
51Q8×C3⋊D3—32no—
52Q8×C3⋊D3—32no—
53Q8×C3⋊D3C22×C311yescharacteristic
54Q8×C3⋊D3C22×C312yescomplemented
55#3830—34no—
56#3830—31no—
57#3830—32no—
58#3830—32no—
59#3830—31no—
60D4×C3⋊D3—32no—
61D4×C3⋊D3—31no—
62D4×C3⋊D3A411yescharacteristic
63#3828—34no—
64#3828—32no—
65#3828—32no—
66#3828—31no—
67#3828—31no—
68C2×C4×C3⋊D3—32no—
69C2×C4×C3⋊D3—32no—
70C2×C4×C3⋊D3—31no—
71C2×#634—32no—
72C2×#634—32no—
73C2×#634—31no—
74C3×D3×D4—121no—
75C3×#41⋊C3—21no—
76C3×#41⋊C3C2×D311yescharacteristic, complemented
77C2×C3×SL(2,𝔽3)—22no—
78C2×C3×SL(2,𝔽3)C2×D312yescomplemented
79D3×SL(2,𝔽3)—31no—
80D3×SL(2,𝔽3)—32no—
81C3⋊(#41⋊C3)—31no—
82C22×H(3,𝔽3)—42no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—42no—
85C2×(C3⋊D3)⋊C3—84no—
86C4×H(3,𝔽3)—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×2−1+4—31no—
89C3×2−1+4C3×D311yescharacteristic
90#1010—31no—
91#1010—91no—
92#1010—93no—
93#1010—96no—
94#1010—92no—
95#1010—91no—
96D3×#41—31no—
97D3×#41—91no—
98D3×#41—91no—
99D3×#41—93no—
100D3×#41—96no—
101D3×#41—92no—
102#1007—32no—
103#1007—92no—
104#1007—93no—
105#1007—91no—
106C2×D3×Q8—32no—
107C2×D3×Q8—92no—
108C2×D3×Q8—93no—
109C2×D3×Q8—91no—
1102−1+4⋊C3—31no—
1112−1+4⋊C3—61no—
1122−1+4⋊C3—92no—
1132−1+4⋊C3—91no—
114C2×#41⋊C3—91no—
115C22×SL(2,𝔽3)—92no—
116C22×C3⋊D3—31no—
117C22×C3⋊D3C2×A412yes—
118C22×C3×D3—122no—
119C32×Q8—32no—
120C32×Q8—31no—
121C32×Q8C23×C311yescharacteristic
122C32×D4—31no—
123C32×D4—32no—
124C32×D4—41no—
125C32×D4—81no—
126C32×D4C2×A411yescharacteristic
127C2×C32×C4—32no—
128C2×C32×C4—32no—
129C2×C32×C4—31no—
130#638—32no—
131#638—32no—
132#638—32no—
133#638C2×A412yes—
134C2×C3⋊Dic3—32no—
135C2×C3⋊Dic3—32no—
136C2×C3⋊Dic3—31no—
137C4⋊(C3⋊D3)—31no—
138C4⋊(C3⋊D3)—32no—
139C4⋊(C3⋊D3)C2×A411yescharacteristic
140C4×C3⋊D3—34no—
141C4×C3⋊D3—32no—
142C4×C3⋊D3—31no—
143C4×C3⋊D3—32no—
144C4×C3⋊D3—32no—
145C4×C3⋊D3—31no—
146C4×C3⋊D3—31no—
147C4×C3⋊D3C2×A411yescharacteristic
148#634—34no—
149#634—31no—
150#634—32no—
151#634—31no—
152#634—32no—
153C3×#67—122no—
154C3×D12—121no—
155C3×C4×D3—121no—
156C3×SL(2,𝔽3)—21no—
157C3×SL(2,𝔽3)C22×D311yescharacteristic
158C2×2−1+4—91noSylow
159C2×H(3,𝔽3)—41no—
160C2×H(3,𝔽3)—82no—
161(C3⋊D3)⋊C3—42no—
162(C3⋊D3)⋊C3—82no—
163C3×#41—33no—
164C3×#41—36no—
165C3×#41—31no—
166C3×#41—32no—
167C3×#41—31no—
168C3×#41C2×C3×D311yescharacteristic
169C2×C3×Q8—31no—
170C2×C3×Q8—33no—
171C2×C3×Q8—32no—
172C2×C3×Q8C2×C3×D312yes—
173C2×C3×D4—361no—
174#239—31no—
175#239—91no—
176#239—91no—
177#239—93no—
178#239—96no—
179#239—92no—
180D3×Q8—31no—
181D3×Q8—32no—
182D3×Q8—91no—
183D3×Q8—92no—
184D3×Q8—91no—
185D3×Q8—93no—
186D3×Q8—96no—
187D3×Q8—96no—
188D3×Q8—96no—
189D3×Q8—92no—
190D3×Q8—96no—
191D3×Q8—92no—
192D3×Q8—92no—
193D3×Q8—92no—
194#237—92no—
195#237—96no—
196#237—912no—
197#237—94no—
198#237—96no—
199#237—93no—
200#237—93no—
201#237—91no—
202#237—92no—
203#237—91no—
204D3×D4—31no—
205D3×D4—91no—
206D3×D4—92no—
207D3×D4—91no—
208D3×D4—96no—
209D3×D4—93no—
210#235—91no—
211#235—92no—
212#235—93no—
213#235—96no—
214#235—94no—
215#235—912no—
216#235—96no—
217#235—93no—
218#235—92no—
219#235—91no—
220C2×C4×D3—91no—
221C2×C4×D3—92no—
222C2×C4×D3—93no—
223C2×C4×D3—96no—
224C2×C4×D3—92no—
225C2×C4×D3—96no—
226C2×Dic6—93no—
227C2×Dic6—92no—
228C2×Dic6—96no—
229C2×Dic6—96no—
230C2×Dic6—92no—
231C2×Dic6—91no—
232#41⋊C3—31no—
233#41⋊C3—61no—
234#41⋊C3—91no—
235C2×SL(2,𝔽3)—32no—
236C2×SL(2,𝔽3)—62no—
237C2×SL(2,𝔽3)—91no—
238C2×SL(2,𝔽3)—92no—
239C22×C32—31no—
240C22×C32—42no—
241C22×C32—82no—
242C22×C32C22×A412yes—
243C2×C3⋊D3—24no—
244C2×C3⋊D3—31no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3C22×A412yes—
247C2×C3⋊D3C22×A411yescharacteristic
248C2×C3×D3—121no—
249C2×C3×D3—122no—
250C2×C3×D3—244no—
251C32×C4—32no—
252C32×C4—31no—
253C32×C4—41no—
254C32×C4—81no—
255C32×C4C22×A411yescharacteristic
256C3⋊Dic3—31no—
257C3⋊Dic3—32no—
258C3⋊Dic3C22×A411yescharacteristic
259C3×Dic3—121no—
2602−1+4—91no—
2612−1+4—92no—
2622−1+4—271no—
2632−1+4—271no—
2642−1+4—272no—
2652−1+4(C3⋊D3)⋊C311yescharacteristic, complemented
266C2×#41—91no—
267C2×#41—271no—
268C2×#41—272no—
269C22×Q8—92no—
270C22×Q8—271no—
271H(3,𝔽3)—41noSylow
272C23×C3—362no—
273C22×D3—32no—
274C22×D3—92no—
275C22×D3—91no—
276C22×D3—93no—
277C3×Q8—32no—
278C3×Q8—36no—
279C3×Q8—33no—
280C3×Q8—31no—
281C3×Q8—31no—
282C3×Q8C22×C3×D311yescharacteristic
283C3×D4—31no—
284C3×D4—33no—
285C3×D4—36no—
286C3×D4—32no—
287C3×D4—31no—
288C3×D4—121no—
289C3×D4—241no—
290C3×D4—361no—
291C3×D4—362no—
292C3×D4D3×A411yescharacteristic
293C2×C3×C4—36no—
294C2×C3×C4—32no—
295C2×C3×C4—36no—
296C2×C3×C4—33no—
297C2×C3×C4—32no—
298C2×C3×C4—31no—
299C2×C3×C4—361no—
300#67—32no—
301#67—92no—
302#67—92no—
303#67—96no—
304#67—96no—
305#67—92no—
306#67—96no—
307#67—92no—
308C2×Dic3—91no—
309C2×Dic3—93no—
310C2×Dic3—96no—
311C2×Dic3—92no—
312C2×Dic3—96no—
313C2×Dic3—92no—
314D12—31no—
315D12—91no—
316D12—92no—
317D12—91no—
318D12—96no—
319D12—93no—
320C4×D3—31no—
321C4×D3—91no—
322C4×D3—91no—
323C4×D3—91no—
324C4×D3—92no—
325C4×D3—92no—
326C4×D3—92no—
327C4×D3—93no—
328C4×D3—96no—
329C4×D3—93no—
330C4×D3—96no—
331C4×D3—96no—
332C4×D3—912no—
333C4×D3—94no—
334C4×D3—93no—
335C4×D3—91no—
336Dic6—92no—
337Dic6—96no—
338Dic6—93no—
339Dic6—96no—
340Dic6—912no—
341Dic6—94no—
342Dic6—93no—
343Dic6—91no—
344Dic6—91no—
345Dic6—92no—
346SL(2,𝔽3)—31no—
347SL(2,𝔽3)—61no—
348C2×C32—22no—
349C2×C32—41no—
350C2×C32—61no—
351C2×C32—82no—
352C2×C32—81no—
353C2×C32—162no—
354C2×C32C23×A411yescharacteristic
355C3⋊D3—22no—
356C3⋊D3—61no—
357C3⋊D32−1+4⋊C312yescomplemented
358C3×D3—122no—
359C3×D3—242no—
360#41—31no—
361#41—32no—
362#41—91no—
363#41—272no—
364#41—271no—
365#41—274no—
366#41—272no—
367#41—271no—
368#41—272no—
369#41—274no—
370#41—272no—
371#41—271no—
372#41—271no—
373#41—272no—
374#41—271no—
375#41C2×(C3⋊D3)⋊C311yescharacteristic, complemented
376C2×Q8—31no—
377C2×Q8—91no—
378C2×Q8—92no—
379C2×Q8—271no—
380C2×Q8—272no—
381C2×Q8—272no—
382C2×Q8—271no—
383C2×Q8—272no—
384C2×Q8—272no—
385C2×Q8—272no—
386C2×Q8—272no—
387C2×Q8C2×(C3⋊D3)⋊C312yescomplemented
388C2×D4—91no—
389C2×D4—272no—
390C2×D4—271no—
391C22×C4—271no—
392C22×C4—272no—
393C22×C4—272no—
394C22×C3—33no—
395C22×C3—31no—
396C22×C3—32no—
397C22×C3—122no—
398C22×C3—242no—
399C22×C3—361no—
400C22×C3—362no—
401C22×C3—724no—
402C22×C3C2×D3×A412yes—
403C2×D3—31no—
404C2×D3—32no—
405C2×D3—64no—
406C2×D3—91no—
407C2×D3—92no—
408C2×D3—93no—
409C2×D3—91no—
410C2×D3—184no—
411C2×D3—182no—
412C2×D3—186no—
413C3×C4—33no—
414C3×C4—36no—
415C3×C4—31no—
416C3×C4—32no—
417C3×C4—31no—
418C3×C4—121no—
419C3×C4—241no—
420C3×C4—361no—
421C3×C4C2×D3×A411yescharacteristic
422Dic3—31no—
423Dic3—91no—
424Dic3—93no—
425Dic3—96no—
426Dic3—92no—
427Dic3—91no—
428C32—41no—
429C32—81no—
430C32C2×2−1+4⋊C311yescharacteristic, complemented
431C23—92no—
432C23—271no—
433Q8—32no—
434Q8—31no—
435Q8—272no—
436Q8—271no—
437Q8—272no—
438Q8—274no—
439Q8—271no—
440Q8C22×(C3⋊D3)⋊C311yescharacteristic
441D4—31no—
442D4—32no—
443D4—92no—
444D4—91no—
445D4—272no—
446D4—271no—
447D4—272no—
448D4—272no—
449D4—272no—
450D4(C3⋊D3)⋊A411yescharacteristic
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—31no—
454C2×C4—91no—
455C2×C4—271no—
456C2×C4—271no—
457C2×C4—272no—
458C2×C4—272no—
459C2×C4—272no—
460C2×C4—274no—
461C2×C4—271no—
462C2×C4—272no—
463C2×C4—272no—
464C2×C4—271no—
465C2×C3—22no—
466C2×C3—31no—
467C2×C3—62no—
468C2×C3—63no—
469C2×C3—61no—
470C2×C3—121no—
471C2×C3—242no—
472C2×C3—241no—
473C2×C3—362no—
474C2×C3—482no—
475C2×C3—722no—
476C2×C3C22×D3×A411yescharacteristic
477D3—32no—
478D3—62no—
479D3—92no—
480D3—182no—
481D3—183no—
482D3—181no—
483C22—31no—
484C22—91no—
485C22—92no—
486C22—184no—
487C22—271no—
488C22—542no—
489C22C2×(C3⋊D3)⋊A412yes—
490C4—31no—
491C4—32no—
492C4—91no—
493C4—271no—
494C4—272no—
495C4C2×(C3⋊D3)⋊A411yescharacteristic
496C3—31no—
497C3—121no—
498C3—241no—
499C3D3×2−1+4⋊C311yescharacteristic
500C2—22no—
501C2—61no—
502C2—92no—
503C2—182no—
504C2—541no—
505C2C22×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
centerC2C22×(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC32×Q8C23×C3
Frattini, socleC2×C3C22×D3×A4
FittingC32×2−1+4C2×C3
hypercenterD4(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2semidirect
2. #11722438C2semidirect
3. #11722437C2semidirect
4. #11722432C2semidirect
5. C2×#88623C2semidirect
6. C3⋊D3×2−1+4C3semidirect
9. #41⋊H(3,𝔽3)C22semidirect
10. C2×Q8⋊H(3,𝔽3)C22semidirect
12. #88623C22semidirect
13. #88622C22semidirect
14. C32×2−1+4C2×C3semidirect
17. #64367C2×C3semidirect
20. #41×C3⋊D3C2×C3semidirect
22. #64364C2×C3semidirect
24. C2×Q8×C3⋊D3C2×C3semidirect
26. C3×2−1+4⋊C3D3semidirect
42. C32×#41C22×C3semidirect
44. C2×C32×Q8C22×C3semidirect
47. #3832C22×C3semidirect
54. Q8×C3⋊D3C22×C3semidirect
76. C3×#41⋊C3C2×D3semidirect
78. C2×C3×SL(2,𝔽3)C2×D3semidirect
265. 2−1+4(C3⋊D3)⋊C3semidirect
357. C3⋊D32−1+4⋊C3semidirect
375. #41C2×(C3⋊D3)⋊C3semidirect
387. C2×Q8C2×(C3⋊D3)⋊C3semidirect
430. C32C2×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#117224382
#117224372
#117224322
C2×#886232
C3⋊D3×2−1+43
D3×2−1+4⋊C33
D4×(C3⋊D3)⋊C34