Finite Group Databasebeta

← #10592901⋊C3 #13143502

Subgroups — #10592901⋊C3 #13143502

Order 1728.

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

normal subgroups only

This group has 592 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
1#11722638C211yesmaximal, characteristic
2H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
3#64156⋊C3C211yesmaximal, characteristic, complemented
4#64151⋊C3C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#63652⋊C3C211yesmaximal, characteristic, complemented
7#63650⋊C3C211yesmaximal, characteristic, complemented
8#10592901C311yesmaximal, characteristic, complemented
9C3×#5202—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
12#88722—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—21no—
15#3828⋊C3—21no—
16#3828⋊C3C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)—21no—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19Q16×H(3,𝔽3)—21no—
20SD16×H(3,𝔽3)—21no—
21M16×H(3,𝔽3)C2211yescharacteristic, complemented
22#3755⋊C3—21no—
23#3756⋊C3—21no—
24#3752⋊C3—22no—
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
26#3748⋊C3C2211yescharacteristic, complemented
27#64366C2×C311yescharacteristic
28C3×#1009—31no—
29C32×#137—21no—
30C32×#137D311yescharacteristic, complemented
31C32×#137C2×C311yescharacteristic, complemented
32#64156C2×C311yescharacteristic, complemented
33#64151C2×C311yescharacteristic, complemented
34C3×#947—31no—
35C3×#942—31no—
36#63664C2×C311yescharacteristic, complemented
37#63652C2×C311yescharacteristic, complemented
38#63650C2×C311yescharacteristic, complemented
39C3×#837—31no—
40C3×#825—31no—
41C3×#823—31no—
42C22×(C3⋊D3)⋊C3—22no—
43C22×(C3⋊D3)⋊C3D412yes—
44Q8×H(3,𝔽3)—21no—
45Q8×H(3,𝔽3)—21no—
46Q8×H(3,𝔽3)D411yescharacteristic
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)D411yescharacteristic
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)—21no—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#638⋊C3—22no—
53#638⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55C4⋊((C3⋊D3)⋊C3)—22no—
56C4⋊((C3⋊D3)⋊C3)—21no—
57C4⋊((C3⋊D3)⋊C3)—21no—
58C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
59C4×(C3⋊D3)⋊C3—22no—
60C4×(C3⋊D3)⋊C3—21no—
61#634⋊C3D411yescharacteristic
62C8×H(3,𝔽3)—21no—
63(C3⋊(C3⋊C8))⋊C3—21no—
64C3×#455—91no—
65#5202—31no—
66#5202—91no—
67C32×#41—21no—
68C32×#41C2×D311yescharacteristic
69C32×#41C22×C311yescharacteristic
70C2×C32×Q8—21no—
71C2×C32×Q8C2×D311yescharacteristic, complemented
72C2×C32×Q8C22×C311yescharacteristic, complemented
73#3832—21no—
74D4×C3⋊D3—22no—
75D4×C3⋊D3—21no—
76#3828—21no—
77#3828C22×C311yescharacteristic
78C2×C4⋊(C3⋊D3)—21no—
79C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
80C3×#239—61no—
81C3×D3×D4—62no—
82C3×D3×D4—61no—
83C3×#235—31no—
84C3×#235—61no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C32×Q16—22no—
88C32×Q16—21no—
89C32×Q16—21no—
90C32×SD16—22no—
91C32×SD16—21no—
92C32×SD16—21no—
93C32×M16—21no—
94C32×M16C2×D311yescharacteristic, complemented
95C32×M16C22×C311yescharacteristic, complemented
96#3756—21no—
97#3755—21no—
98#3752—22no—
99C4×C3⋊Dic3C22×C311yescharacteristic, complemented
100#3748C22×C311yescharacteristic, complemented
101C3×#216—61no—
102C3×#215—61no—
103C3×(C2×D3)⋊C4—62no—
104C3×C4×Dic3—31no—
105C3×#208—31no—
106C22×H(3,𝔽3)—21no—
107C22×H(3,𝔽3)C2×D411yescharacteristic
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—21no—
110C2×(C3⋊D3)⋊C3—42no—
111C2×(C3⋊D3)⋊C3—82no—
112C4×H(3,𝔽3)—21no—
113C4×H(3,𝔽3)—41no—
114C4×H(3,𝔽3)C2×D411yescharacteristic
115C4×H(3,𝔽3)C2×D411yescharacteristic
116(C3⋊Dic3)⋊C3—21no—
117(C3⋊Dic3)⋊C3—41no—
118C3×H(5,𝔽2)—91no—
119#1009—31no—
120#1009—91no—
121C3×#137—31no—
122C3×#137—31no—
123C3×#137—61no—
124C3×#137—91no—
125C3×#137C3×D311yescharacteristic
126C3×#124—91no—
127#947—31no—
128#947—91no—
129#942—31no—
130#942—91no—
131C3×#104—91no—
132C3×#104—91no—
133C3×#100—91no—
134#837—31no—
135#837—91no—
136#825—31no—
137#825—91no—
138#823—31no—
139#823—91no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3C3×D412yes—
142C22×C3×D3—32no—
143C22×C3×D3—62no—
144C32×Q8—22no—
145C32×Q8—21no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×Q8—21no—
150C32×Q8—41no—
151C32×Q8#6711yescharacteristic
152C32×Q8C3×D411yescharacteristic
153C32×D4—21no—
154C32×D4—21no—
155C32×D4—21no—
156C32×D4—41no—
157C32×D4#6711yescharacteristic
158C32×D4C3×D411yescharacteristic
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C23×C311yescharacteristic
168#638—22no—
169#638—21no—
170C2×C3⋊Dic3—21no—
171C4⋊(C3⋊D3)—22no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)C3×D411yescharacteristic
175C4×C3⋊D3—22no—
176C4×C3⋊D3—21no—
177#634C3×D411yescharacteristic
178C3×#67—62no—
179C3×#67—61no—
180C2×C3×Dic3—61no—
181C3×D12—31no—
182C3×D12—62no—
183C3×D12—61no—
184C3×D12—61no—
185C3×C4×D3—62no—
186C3×C4×D3—61no—
187C3×Dic6—31no—
188C32×C8—22no—
189C32×C8—21no—
190C32×C8—21no—
191C3⋊(C3⋊C8)—21no—
192C3×C3⋊C8—61no—
193#455—91noSylow
194C2×H(3,𝔽3)—21no—
195C2×H(3,𝔽3)—41no—
196C2×H(3,𝔽3)#12011yescharacteristic
197(C3⋊D3)⋊C3—42no—
198(C3⋊D3)⋊C3—41no—
199C3×#41—31no—
200C3×#41—31no—
201C3×#41—61no—
202C3×#41—91no—
203C3×#41—181no—
204C3×#41—181no—
205C3×#41C2×C3×D311yescharacteristic
206C2×C3×Q8—31no—
207C2×C3×Q8—31no—
208C2×C3×Q8—61no—
209C2×C3×Q8C2×C3×D311yescharacteristic
210C2×C3×D4—91no—
211C2×C3×D4—181no—
212C2×C3×D4—181no—
213C2×C3×D4—182no—
214#239—61no—
215#239—181no—
216D3×D4—62no—
217D3×D4—61no—
218D3×D4—181no—
219D3×D4—182no—
220#235—31no—
221#235—61no—
222#235—91no—
223#235—181no—
224C2×D12—31no—
225C2×D12—61no—
226C2×D12—91no—
227C2×D12—181no—
228C3×Q16—21no—
229C3×Q16—61no—
230C3×Q16—61no—
231C3×Q16—62no—
232C3×Q16—181no—
233C3×SD16—21no—
234C3×SD16—61no—
235C3×SD16—61no—
236C3×SD16—62no—
237C3×SD16—181no—
238C3×M16—31no—
239C3×M16—31no—
240C3×M16—61no—
241C3×M16—91no—
242C3×M16C2×C3×D311yescharacteristic
243C3×C22⋊C4—182no—
244C3×C42—91no—
245#216—61no—
246#216—181no—
247#215—61no—
248#215—181no—
249(C2×D3)⋊C4—62no—
250(C2×D3)⋊C4—182no—
251C4×Dic3—31no—
252C4×Dic3—91no—
253#208—31no—
254#208—91no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32D3×D411yescharacteristic
260C22×C32C2×C3×D411yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—21no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3—82no—
265C2×C3×D3—61no—
266C2×C3×D3—62no—
267C2×C3×D3—122no—
268C2×C3×D3—242no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—41no—
274C32×C4—42no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4D3×D411yescharacteristic
278C32×C4C2×#6711yescharacteristic
279C32×C4C2×C3×D411yescharacteristic
280C32×C4C2×C3×D411yescharacteristic
281C3⋊Dic3—21no—
282C3⋊Dic3—41no—
283C3×Dic3—61no—
284C3×Dic3—121no—
285H(5,𝔽2)—91no—
286#137—91no—
287#137(C3⋊D3)⋊C311yescharacteristic, complemented
288#124—91no—
289#104—91no—
290#104—91no—
291#100—91no—
292H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
293C23×C3—92no—
294C23×C3—182no—
295C22×D3—32no—
296C22×D3—62no—
297C22×D3—92no—
298C22×D3—182no—
299C3×Q8—21no—
300C3×Q8—21no—
301C3×Q8—31no—
302C3×Q8—31no—
303C3×Q8—61no—
304C3×Q8—61no—
305C3×Q8—61no—
306C3×Q8—61no—
307C3×Q8—62no—
308C3×Q8—61no—
309C3×Q8—91no—
310C3×Q8—121no—
311C3×Q8C3×#6711yescharacteristic
312C3×D4—21no—
313C3×D4—31no—
314C3×D4—31no—
315C3×D4—61no—
316C3×D4—61no—
317C3×D4—61no—
318C3×D4—91no—
319C3×D4—121no—
320C3×D4—181no—
321C3×D4—181no—
322C3×D4—182no—
323C3×D4—181no—
324C3×D4—182no—
325C3×D4C3×#6711yescharacteristic
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—121no—
336C2×C3×C4—121no—
337C2×C3×C4—181no—
338C2×C3×C4—182no—
339C2×C3×C4—181no—
340C2×C3×C4C22×C3×D311yescharacteristic
341#67—62no—
342#67—61no—
343#67—181no—
344#67—182no—
345C2×Dic3—61no—
346C2×Dic3—181no—
347D12—31no—
348D12—62no—
349D12—61no—
350D12—61no—
351D12—91no—
352D12—181no—
353D12—181no—
354D12—182no—
355C4×D3—62no—
356C4×D3—61no—
357C4×D3—181no—
358C4×D3—182no—
359Dic6—31no—
360Dic6—91no—
361C3×C8—21no—
362C3×C8—61no—
363C3×C8—61no—
364C3×C8—62no—
365C3×C8—181no—
366C3⋊C8—61no—
367C3⋊C8—181no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—81no—
375C2×C32#93711yescharacteristic
376C2×C32C3×#12011yescharacteristic
377C3⋊D3—42no—
378C3⋊D3—41no—
379C3×D3—122no—
380C3×D3—121no—
381#41—91no—
382#41—181no—
383#41—181no—
384#41C2×(C3⋊D3)⋊C311yescharacteristic
385C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
386C2×D4—91no—
387C2×D4—181no—
388C2×D4—182no—
389C2×D4—181no—
390Q16—21no—
391Q16—181no—
392SD16—21no—
393SD16—181no—
394M16—91no—
395M16C2×(C3⋊D3)⋊C311yescharacteristic, complemented
396C22⋊C4—182no—
397C42—91no—
398C22×C3—21no—
399C22×C3—31no—
400C22×C3—31no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—121no—
405C22×C3—181no—
406C22×C3—182no—
407C22×C3—362no—
408C22×C3—722no—
409C22×C3C3×D3×D411yescharacteristic
410C2×D3—61no—
411C2×D3—62no—
412C2×D3—122no—
413C2×D3—181no—
414C2×D3—182no—
415C2×D3—242no—
416C2×D3—362no—
417C2×D3—722no—
418C3×C4—21no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—41no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—61no—
428C3×C4—121no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—122no—
432C3×C4—181no—
433C3×C4—361no—
434C3×C4C3×D3×D411yescharacteristic
435C3×C4C2×C3×#6711yescharacteristic
436Dic3—61no—
437Dic3—121no—
438Dic3—181no—
439Dic3—361no—
440C32—21no—
441C32#520211yescharacteristic, complemented
442C32C3×#45511yescharacteristic, complemented
443C23—92no—
444C23—182no—
445Q8—21no—
446Q8—21no—
447Q8—91no—
448Q8#638⋊C311yescharacteristic
449D4—21no—
450D4—91no—
451D4—182no—
452D4—182no—
453D4—181no—
454D4—181no—
455D4—181no—
456D4#638⋊C311yescharacteristic
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—182no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
463C8—21no—
464C8—181no—
465C2×C3—21no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—41no—
469C2×C3—61no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3—121no—
473C2×C3—121no—
474C2×C3—121no—
475C2×C3—241no—
476C2×C3—362no—
477C2×C3—361no—
478C2×C3C3×#93711yescharacteristic
479D3—122no—
480D3—121no—
481D3—362no—
482D3—361no—
483C22—21no—
484C22—181no—
485C22—182no—
486C22—362no—
487C22—722no—
488C22D4×(C3⋊D3)⋊C311yescharacteristic
489C4—21no—
490C4—41no—
491C4—181no—
492C4—361no—
493C4D4×(C3⋊D3)⋊C311yescharacteristic
494C4C2×#638⋊C311yescharacteristic
495C3—31no—
496C3—31no—
497C3—61no—
498C3C3×#520211yescharacteristic
499C2—21no—
500C2—41no—
501C2—362no—
502C2—361no—
503C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#137C2
socleC2×C3C3×#937
prime radical#11722638C2
hypercenter#137(C3⋊D3)⋊C3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#137C2semidirect
3. #64156⋊C3C2semidirect
4. #64151⋊C3C2semidirect
5. #63664⋊C3C2semidirect
6. #63652⋊C3C2semidirect
7. #63650⋊C3C2semidirect
8. #10592901C3semidirect
11. C2×Q8×H(3,𝔽3)C22semidirect
21. M16×H(3,𝔽3)C22semidirect
25. C4×(C3⋊Dic3)⋊C3C22semidirect
26. #3748⋊C3C22semidirect
30. C32×#137D3semidirect
31. C32×#137C2×C3semidirect
32. #64156C2×C3semidirect
33. #64151C2×C3semidirect
36. #63664C2×C3semidirect
37. #63652C2×C3semidirect
38. #63650C2×C3semidirect
71. C2×C32×Q8C2×D3semidirect
72. C2×C32×Q8C22×C3semidirect
94. C32×M16C2×D3semidirect
95. C32×M16C22×C3semidirect
99. C4×C3⋊Dic3C22×C3semidirect
100. #3748C22×C3semidirect
287. #137(C3⋊D3)⋊C3semidirect
292. H(3,𝔽3)#455semidirect
385. C2×Q8C2×(C3⋊D3)⋊C3semidirect
395. M16C2×(C3⋊D3)⋊C3semidirect
441. C32#5202semidirect
442. C32C3×#455semidirect

Maximal subgroups

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

subgroupindex
#117226382
H(3,𝔽3)×#1372
#64156⋊C32
#64151⋊C32
#63664⋊C32
#63652⋊C32
#63650⋊C32
#105929013
C3×#52023