Finite Group Databasebeta

← #10592731⋊C3 #13142814

Subgroups — #10592731⋊C3 #13142814

Order 1728.

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

normal subgroups only

This group has 823 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#64361⋊C3C211yesmaximal, characteristic, complemented
2H(3,𝔽3)×#120C211yesmaximal, characteristic, complemented
3#11721702C212yesmaximal, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#63633⋊C3C212yesmaximal, complemented
6#10592731C311yesmaximal, characteristic, complemented
7C3×#5032—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic, complemented
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—22no—
13C2×#638⋊C3C2212yes—
14#3830⋊C3—22no—
15D4×(C3⋊D3)⋊C3—22no—
16#3828⋊C3—21no—
17C22⋊C4×H(3,𝔽3)—21no—
18C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
19#3757⋊C3—22no—
20#3757⋊C3C2212yescomplemented
21#64361C2×C311yescharacteristic, complemented
22C3×#1004—31no—
23C32×#120—21no—
24C32×#120D311yescharacteristic, complemented
25C32×#120C2×C311yescharacteristic, complemented
26#64162C2×C312yescomplemented
27C3×#953—32no—
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—31no—
31C3×#806—32no—
32C23×H(3,𝔽3)—21no—
33C23×H(3,𝔽3)—22no—
34C23×H(3,𝔽3)—42no—
35C23×H(3,𝔽3)D411yescharacteristic, complemented
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C22×(C3⋊D3)⋊C3D412yescomplemented
39D4×H(3,𝔽3)—22no—
40D4×H(3,𝔽3)—42no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic, complemented
43#638⋊C3—22no—
44#638⋊C3—22no—
45#638⋊C3—22no—
46#638⋊C3—44no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3D412yescomplemented
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4×(C3⋊D3)⋊C3—22no—
52#634⋊C3—21no—
53C3×B(4,𝔽2)—91no—
54#5032—31no—
55#5032—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic, complemented
58C24×C32C22×C311yescharacteristic, complemented
59C2×C32×D4—22no—
60C2×C32×D4—21no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C22×C311yescharacteristic
65C2×#638—21no—
66C2×#638—22no—
67C2×#638C22×C312yes—
68#3830—22no—
69D4×C3⋊D3—22no—
70#3828—21no—
71C2×C3×#67—32no—
72C2×C3×#67—61no—
73C2×C3×#67—62no—
74C3×#237—62no—
75C3×D3×D4—62no—
76C3×#235—61no—
77C32×C22⋊C4—22no—
78C32×C22⋊C4—21no—
79C32×C22⋊C4—21no—
80C32×C22⋊C4—21no—
81C32×C22⋊C4C2×D311yescharacteristic, complemented
82C32×C22⋊C4C22×C311yescharacteristic, complemented
83#3757—22no—
84#3757C22×C312yescomplemented
85C3×#217—32no—
86C3×#217—62no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)—24no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)—41no—
92C22×H(3,𝔽3)—42no—
93C22×H(3,𝔽3)—41no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)C2×D412yes—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—42no—
99C2×(C3⋊D3)⋊C3—81no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)—41no—
102(C3⋊Dic3)⋊C3—22no—
103(C3⋊Dic3)⋊C3—42no—
104C3×H(5,𝔽2)—91no—
105#1004—31no—
106#1004—91no—
107C3×#120—31no—
108C3×#120—31no—
109C3×#120—61no—
110C3×#120—92no—
111C3×#120C3×D311yescharacteristic
112#953—32no—
113#953—92no—
114C3×#99—92no—
115C3×#99—91no—
116#834—31no—
117#834—91no—
118#806—32no—
119#806—92no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—21no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—42no—
128C23×C32—42no—
129C23×C32—42no—
130C23×C32—44no—
131C23×C32#6711yescharacteristic, complemented
132C23×C32C3×D411yescharacteristic, complemented
133C23×C32C22×D311yescharacteristic
134C23×C32C23×C311yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3⋊D3C3×D412yescomplemented
137C22×C3×D3—32no—
138C22×C3×D3—61no—
139C32×D4—22no—
140C32×D4—22no—
141C32×D4—42no—
142C32×D4—42no—
143C32×D4—42no—
144C32×D4—44no—
145C2×C32×C4—22no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4#6711yescharacteristic, complemented
150C2×C32×C4C3×D411yescharacteristic, complemented
151#638—22no—
152#638—22no—
153#638—22no—
154#638—44no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3C3×D412yescomplemented
158C4⋊(C3⋊D3)—21no—
159C4×C3⋊D3—22no—
160#634—21no—
161C3×#67—62no—
162C3×#67—62no—
163C3×#67—62no—
164C3×#67—124no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—62no—
168C3×D12—61no—
169C3×C4×D3—62no—
170C3×Dic6—61no—
171B(4,𝔽2)—91noSylow
172C2×H(3,𝔽3)—21no—
173C2×H(3,𝔽3)—22no—
174C2×H(3,𝔽3)—42no—
175C2×H(3,𝔽3)—41no—
176C2×H(3,𝔽3)#12011yescharacteristic
177(C3⋊D3)⋊C3—42no—
178C24×C3—31no—
179C24×C3—31no—
180C24×C3—61no—
181C24×C3C2×C3×D311yescharacteristic
182C3×#41—181no—
183C3×#41—182no—
184C2×C3×D4—21no—
185C2×C3×D4—31no—
186C2×C3×D4—31no—
187C2×C3×D4—61no—
188C2×C3×D4—61no—
189C2×C3×D4—62no—
190C2×C3×D4—61no—
191C2×C3×D4—92no—
192C2×C3×D4—182no—
193C2×C3×D4—181no—
194C2×C3×D4—182no—
195C2×C3×D4C2×C3×D311yescharacteristic
196C2×#67—32no—
197C2×#67—61no—
198C2×#67—62no—
199C2×#67—92no—
200C2×#67—181no—
201C2×#67—182no—
202#237—62no—
203#237—182no—
204D3×D4—62no—
205D3×D4—182no—
206#235—61no—
207#235—181no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—31no—
210C3×C22⋊C4—31no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—61no—
213C3×C22⋊C4—62no—
214C3×C22⋊C4—61no—
215C3×C22⋊C4—92no—
216C3×C22⋊C4—182no—
217C3×C22⋊C4C2×C3×D311yescharacteristic
218#217—32no—
219#217—62no—
220#217—92no—
221#217—182no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—24no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—41no—
232C22×C32—44no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32—41no—
241C22×C32—42no—
242C22×C32—41no—
243C22×C32—42no—
244C22×C32—42no—
245C22×C32—44no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32D3×D412yes—
249C22×C32C2×#6711yescharacteristic
250C22×C32C2×C3×D412yes—
251C22×C32C2×C3×D411yescharacteristic
252C2×C3⋊D3—22no—
253C2×C3⋊D3—42no—
254C2×C3⋊D3—81no—
255C2×C3×D3—62no—
256C2×C3×D3—122no—
257C2×C3×D3—241no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—41no—
261C32×C4—42no—
262C32×C4—41no—
263C32×C4—41no—
264C3⋊Dic3—22no—
265C3⋊Dic3—42no—
266C3×Dic3—62no—
267C3×Dic3—122no—
268H(5,𝔽2)—91no—
269#120—92no—
270#120(C3⋊D3)⋊C311yescharacteristic, complemented
271#99—91no—
272#99—92no—
273H(3,𝔽3)B(4,𝔽2)11yescharacteristic, complemented, Sylow
274C23×C3—22no—
275C23×C3—21no—
276C23×C3—31no—
277C23×C3—31no—
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—42no—
281C23×C3—62no—
282C23×C3—61no—
283C23×C3—61no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—61no—
287C23×C3—61no—
288C23×C3—92no—
289C23×C3—122no—
290C23×C3—122no—
291C23×C3—122no—
292C23×C3—124no—
293C23×C3—181no—
294C23×C3C3×#6711yescharacteristic
295C23×C3C22×C3×D311yescharacteristic
296C22×D3—32no—
297C22×D3—61no—
298C22×D3—92no—
299C22×D3—181no—
300C3×Q8—181no—
301C3×D4—22no—
302C3×D4—42no—
303C3×D4—62no—
304C3×D4—62no—
305C3×D4—122no—
306C3×D4—122no—
307C3×D4—122no—
308C3×D4—124no—
309C3×D4—181no—
310C3×D4—182no—
311C3×D4—182no—
312C3×D4—182no—
313C3×D4—364no—
314C2×C3×C4—21no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—92no—
322C2×C3×C4—182no—
323C2×C3×C4—181no—
324C2×C3×C4—182no—
325C2×C3×C4C3×#6711yescharacteristic
326#67—62no—
327#67—62no—
328#67—62no—
329#67—124no—
330#67—182no—
331#67—182no—
332#67—182no—
333#67—364no—
334C2×Dic3—32no—
335C2×Dic3—61no—
336C2×Dic3—62no—
337C2×Dic3—92no—
338C2×Dic3—181no—
339C2×Dic3—182no—
340D12—61no—
341D12—181no—
342C4×D3—62no—
343C4×D3—182no—
344Dic6—61no—
345Dic6—181no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—21no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—42no—
353C2×C32—44no—
354C2×C32—41no—
355C2×C32—42no—
356C2×C32—41no—
357C2×C32—42no—
358C2×C32—81no—
359C2×C32#93711yescharacteristic
360C2×C32C3×#12011yescharacteristic
361C3⋊D3—42no—
362C3×D3—122no—
363C24C2×(C3⋊D3)⋊C311yescharacteristic, complemented
364#41—182no—
365#41—181no—
366C2×D4—21no—
367C2×D4—92no—
368C2×D4—182no—
369C2×D4—181no—
370C2×D4—182no—
371C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
372C22⋊C4—21no—
373C22⋊C4—92no—
374C22⋊C4—182no—
375C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—21no—
379C22×C3—32no—
380C22×C3—32no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3—64no—
391C22×C3—62no—
392C22×C3—62no—
393C22×C3—64no—
394C22×C3—64no—
395C22×C3—62no—
396C22×C3—61no—
397C22×C3—121no—
398C22×C3—124no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3—122no—
406C22×C3—122no—
407C22×C3—121no—
408C22×C3—121no—
409C22×C3—122no—
410C22×C3—122no—
411C22×C3—122no—
412C22×C3—122no—
413C22×C3—124no—
414C22×C3—182no—
415C22×C3—362no—
416C22×C3—721no—
417C22×C3C3×D3×D412yes—
418C22×C3C2×C3×#6711yescharacteristic
419C2×D3—62no—
420C2×D3—122no—
421C2×D3—182no—
422C2×D3—241no—
423C2×D3—362no—
424C2×D3—721no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—61no—
428C3×C4—61no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4—122no—
433C3×C4—182no—
434C3×C4—362no—
435Dic3—62no—
436Dic3—122no—
437Dic3—182no—
438Dic3—362no—
439C32—21no—
440C32#503211yescharacteristic, complemented
441C32C3×B(4,𝔽2)11yescharacteristic, complemented
442C23—22no—
443C23—21no—
444C23—42no—
445C23—92no—
446C23—181no—
447C23#638⋊C311yescharacteristic, complemented
448C23C22×(C3⋊D3)⋊C311yescharacteristic
449Q8—181no—
450D4—22no—
451D4—42no—
452D4—181no—
453D4—182no—
454D4—182no—
455D4—182no—
456D4—364no—
457C2×C4—21no—
458C2×C4—92no—
459C2×C4—182no—
460C2×C4—181no—
461C2×C4—182no—
462C2×C4#638⋊C311yescharacteristic, complemented
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—41no—
468C2×C3—42no—
469C2×C3—62no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3—62no—
473C2×C3—62no—
474C2×C3—61no—
475C2×C3—122no—
476C2×C3—121no—
477C2×C3—122no—
478C2×C3—122no—
479C2×C3—121no—
480C2×C3—124no—
481C2×C3—241no—
482C2×C3—362no—
483C2×C3C3×#93711yescharacteristic
484D3—122no—
485D3—362no—
486C22—22no—
487C22—24no—
488C22—21no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—42no—
493C22—41no—
494C22—182no—
495C22—362no—
496C22—721no—
497C22D4×(C3⋊D3)⋊C312yes—
498C22C2×#638⋊C311yescharacteristic
499C4—21no—
500C4—41no—
501C4—182no—
502C4—362no—
503C3—31no—
504C3—31no—
505C3—61no—
506C3C3×#503211yescharacteristic
507C2—21no—
508C2—22no—
509C2—42no—
510C2—41no—
511C2—362no—
512C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC23×C32C23×C3
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×#120C2
socleC2×C3C3×#937
hypercenter#120(C3⋊D3)⋊C3
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #64361⋊C3C2semidirect
2. H(3,𝔽3)×#120C2semidirect
3. #11721702C2semidirect
4. #63661⋊C3C2semidirect
5. #63633⋊C3C2semidirect
6. #10592731C3semidirect
8. C24×H(3,𝔽3)C22semidirect
18. C22⋊C4×H(3,𝔽3)C22semidirect
20. #3757⋊C3C22semidirect
21. #64361C2×C3semidirect
24. C32×#120D3semidirect
25. C32×#120C2×C3semidirect
26. #64162C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
35. C23×H(3,𝔽3)D4semidirect
38. C22×(C3⋊D3)⋊C3D4semidirect
42. C2×C4×H(3,𝔽3)D4semidirect
49. C2×(C3⋊Dic3)⋊C3D4semidirect
57. C24×C32C2×D3semidirect
58. C24×C32C22×C3semidirect
81. C32×C22⋊C4C2×D3semidirect
82. C32×C22⋊C4C22×C3semidirect
84. #3757C22×C3semidirect
131. C23×C32#67semidirect
132. C23×C32C3×D4semidirect
136. C22×C3⋊D3C3×D4semidirect
149. C2×C32×C4#67semidirect
150. C2×C32×C4C3×D4semidirect
157. C2×C3⋊Dic3C3×D4semidirect
270. #120(C3⋊D3)⋊C3semidirect
273. H(3,𝔽3)B(4,𝔽2)semidirect
363. C24C2×(C3⋊D3)⋊C3semidirect
375. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
440. C32#5032semidirect
441. C32C3×B(4,𝔽2)semidirect
447. C23#638⋊C3semidirect
462. C2×C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
H(3,𝔽3)×#1202
#117217022
#63661⋊C32
#63633⋊C32
#105927313
C3×#50323