Finite Group Databasebeta

← #10592440⋊C3 #13141653

Subgroups — #10592440⋊C3 #13141653

Order 1728.

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

normal subgroups only

This group has 745 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
2#64146⋊C3C211yesmaximal, characteristic, complemented
3#64091⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
5#63633⋊C3C212yesmaximal, complemented
6#10592440C311yesmaximal, characteristic, complemented
7C3×#4741—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—21no—
11C2×#638⋊C3—21no—
12C2×#638⋊C3C2212yes—
13#3830⋊C3—22no—
14D4×(C3⋊D3)⋊C3—22no—
15#3828⋊C3—21no—
16C2×C4⋊((C3⋊D3)⋊C3)—22no—
17C22⋊C4×H(3,𝔽3)C2212yescomplemented
18#3757⋊C3C2211yescharacteristic, complemented
19#3752⋊C3—21no—
20#3752⋊C3—22no—
21#64361C2×C311yescharacteristic, complemented
22C3×#1004—31no—
23#64146C2×C311yescharacteristic, complemented
24#64091C2×C312yescomplemented
25C3×#937—31no—
26C3×#882—32no—
27C32×#99—21no—
28C32×#99D311yescharacteristic, complemented
29C32×#99C2×C311yescharacteristic, complemented
30#63633C2×C312yescomplemented
31C3×#806—32no—
32C23×H(3,𝔽3)D411yescharacteristic, complemented
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—22no—
35C22×(C3⋊D3)⋊C3—21no—
36C22×(C3⋊D3)⋊C3—21no—
37C22×(C3⋊D3)⋊C3—42no—
38C22×(C3⋊D3)⋊C3D412yescomplemented
39D4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)—22no—
41C2×C4×H(3,𝔽3)D411yescharacteristic, complemented
42#638⋊C3—22no—
43#638⋊C3—22no—
44#638⋊C3—22no—
45#638⋊C3—42no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D412yescomplemented
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—44no—
51C4×(C3⋊D3)⋊C3—22no—
52#634⋊C3—21no—
53C3×B(4,𝔽2)—91no—
54#4741—31no—
55#4741—91no—
56C23×C3⋊D3C22×C311yescharacteristic, complemented
57C23×C3×D3—31no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C22×C311yescharacteristic
61C2×#638—21no—
62C2×#638—21no—
63C2×#638C22×C312yes—
64#3830—22no—
65D4×C3⋊D3—22no—
66#3828—21no—
67C2×C4⋊(C3⋊D3)—22no—
68C2×C3×#67—32no—
69C2×C3×#67—61no—
70C2×C3×#67—61no—
71C3×#237—62no—
72C3×D3×D4—62no—
73C3×#235—61no—
74C2×C3×D12—62no—
75C32×C22⋊C4—22no—
76C32×C22⋊C4C2×D312yescomplemented
77C32×C22⋊C4C22×C312yescomplemented
78#3757C22×C311yescharacteristic, complemented
79#3752—21no—
80#3752—22no—
81C3×#217—31no—
82C3×(C2×D3)⋊C4—62no—
83C3×(C2×D3)⋊C4—61no—
84C22×H(3,𝔽3)—21no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C22×H(3,𝔽3)C2×D412yes—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—42no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—42no—
96C2×(C3⋊D3)⋊C3—81no—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—42no—
99(C3⋊Dic3)⋊C3—22no—
100(C3⋊Dic3)⋊C3—41no—
101C3×H(5,𝔽2)—91no—
102#1004—31no—
103#1004—91no—
104C3×#120—92no—
105C3×#120—91no—
106#937—31no—
107#937—91no—
108#882—32no—
109#882—92no—
110C3×#99—31no—
111C3×#99—31no—
112C3×#99—61no—
113C3×#99—92no—
114C3×#99C3×D311yescharacteristic
115#806—32no—
116#806—92no—
117C23×C32—21no—
118C23×C32—21no—
119C23×C32D1211yescharacteristic, complemented
120C23×C32C3×D411yescharacteristic, complemented
121C23×C32C22×D311yescharacteristic
122C23×C32C23×C311yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—42no—
127C22×C3⋊D3C3×D412yescomplemented
128C22×C3×D3—32no—
129C22×C3×D3—61no—
130C22×C3×D3—62no—
131C22×C3×D3—61no—
132C22×C3×D3—122no—
133C32×D4—22no—
134C32×D4—22no—
135C32×D4—42no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—21no—
139C2×C32×C4—42no—
140C2×C32×C4D1211yescharacteristic, complemented
141C2×C32×C4C3×D411yescharacteristic, complemented
142#638—22no—
143#638—22no—
144#638—22no—
145#638—42no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3C3×D412yescomplemented
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—44no—
151C4×C3⋊D3—22no—
152#634—21no—
153C3×#67—62no—
154C3×#67—62no—
155C3×#67—62no—
156C3×#67—122no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C3×D12—61no—
161C3×D12—124no—
162C3×C4×D3—62no—
163C3×Dic6—61no—
164B(4,𝔽2)—91noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—22no—
167C2×H(3,𝔽3)—41no—
168C2×H(3,𝔽3)#12011yescharacteristic
169(C3⋊D3)⋊C3—42no—
170(C3⋊D3)⋊C3—42no—
171C24×C3—91no—
172C23×D3—31no—
173C23×D3—91no—
174C3×#41—181no—
175C3×#41—182no—
176C2×C3×D4—31no—
177C2×C3×D4—31no—
178C2×C3×D4—61no—
179C2×C3×D4—92no—
180C2×C3×D4—182no—
181C2×C3×D4—182no—
182C2×C3×D4—181no—
183C2×C3×D4—181no—
184C2×C3×D4C2×C3×D311yescharacteristic
185C2×#67—32no—
186C2×#67—61no—
187C2×#67—61no—
188C2×#67—92no—
189C2×#67—181no—
190C2×#67—181no—
191#237—62no—
192#237—182no—
193D3×D4—62no—
194D3×D4—182no—
195#235—61no—
196#235—181no—
197C2×D12—62no—
198C2×D12—182no—
199C3×C22⋊C4—32no—
200C3×C22⋊C4—32no—
201C3×C22⋊C4—62no—
202C3×C22⋊C4—91no—
203C3×C22⋊C4—181no—
204C3×C22⋊C4—182no—
205C3×C22⋊C4C2×C3×D312yes—
206#217—31no—
207#217—91no—
208(C2×D3)⋊C4—62no—
209(C2×D3)⋊C4—61no—
210(C2×D3)⋊C4—182no—
211(C2×D3)⋊C4—181no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—81no—
222C22×C32—81no—
223C22×C32C2×D1211yescharacteristic
224C22×C32D3×D412yes—
225C22×C32C2×C3×D411yescharacteristic
226C22×C32C2×C3×D412yes—
227C2×C3⋊D3—24no—
228C2×C3⋊D3—22no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—42no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—42no—
234C2×C3⋊D3—81no—
235C2×C3×D3—64no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C2×C3×D3—121no—
239C2×C3×D3—122no—
240C2×C3×D3—121no—
241C2×C3×D3—122no—
242C2×C3×D3—241no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—41no—
246C32×C4—42no—
247C32×C4—42no—
248C32×C4—82no—
249C3⋊Dic3—22no—
250C3⋊Dic3—41no—
251C3×Dic3—62no—
252C3×Dic3—121no—
253H(5,𝔽2)—91no—
254#120—91no—
255#120—92no—
256#99—92no—
257#99(C3⋊D3)⋊C311yescharacteristic, complemented
258H(3,𝔽3)B(4,𝔽2)11yescharacteristic, complemented, Sylow
259C23×C3—31no—
260C23×C3—31no—
261C23×C3—31no—
262C23×C3—31no—
263C23×C3—61no—
264C23×C3—61no—
265C23×C3—92no—
266C23×C3—181no—
267C23×C3—182no—
268C23×C3—181no—
269C23×C3—362no—
270C23×C3C3×D1211yescharacteristic
271C23×C3C22×C3×D311yescharacteristic
272C22×D3—32no—
273C22×D3—61no—
274C22×D3—62no—
275C22×D3—61no—
276C22×D3—92no—
277C22×D3—122no—
278C22×D3—182no—
279C22×D3—181no—
280C22×D3—181no—
281C22×D3—362no—
282C3×Q8—181no—
283C3×D4—22no—
284C3×D4—62no—
285C3×D4—62no—
286C3×D4—122no—
287C3×D4—181no—
288C3×D4—182no—
289C3×D4—182no—
290C3×D4—182no—
291C3×D4—364no—
292C3×D4—362no—
293C2×C3×C4—22no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—61no—
299C2×C3×C4—92no—
300C2×C3×C4—122no—
301C2×C3×C4—182no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4C3×D1211yescharacteristic
305#67—62no—
306#67—62no—
307#67—62no—
308#67—122no—
309#67—182no—
310#67—182no—
311#67—182no—
312#67—362no—
313C2×Dic3—32no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—92no—
317C2×Dic3—181no—
318C2×Dic3—181no—
319D12—61no—
320D12—124no—
321D12—181no—
322D12—364no—
323C4×D3—62no—
324C4×D3—182no—
325Dic6—61no—
326Dic6—181no—
327C2×C32—21no—
328C2×C32—22no—
329C2×C32—21no—
330C2×C32—22no—
331C2×C32—21no—
332C2×C32—42no—
333C2×C32—41no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—81no—
337C2×C32#88211yescharacteristic
338C2×C32C3×#12011yescharacteristic
339C3⋊D3—42no—
340C3⋊D3—42no—
341C3×D3—122no—
342C3×D3—122no—
343C24—91no—
344#41—182no—
345#41—181no—
346C2×D4—92no—
347C2×D4—182no—
348C2×D4—182no—
349C2×D4—181no—
350C2×D4—181no—
351C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
352C22⋊C4—91no—
353C22⋊C4—182no—
354C22⋊C4—181no—
355C22⋊C4C2×(C3⋊D3)⋊C312yescomplemented
356C22×C3—21no—
357C22×C3—31no—
358C22×C3—31no—
359C22×C3—32no—
360C22×C3—32no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—62no—
367C22×C3—121no—
368C22×C3—121no—
369C22×C3—121no—
370C22×C3—121no—
371C22×C3—121no—
372C22×C3—182no—
373C22×C3—184no—
374C22×C3—182no—
375C22×C3—241no—
376C22×C3—241no—
377C22×C3—361no—
378C22×C3—362no—
379C22×C3—362no—
380C22×C3—361no—
381C22×C3—721no—
382C22×C3C2×C3×D1211yescharacteristic
383C22×C3C3×D3×D412yes—
384C2×D3—64no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—121no—
388C2×D3—122no—
389C2×D3—121no—
390C2×D3—122no—
391C2×D3—182no—
392C2×D3—184no—
393C2×D3—182no—
394C2×D3—241no—
395C2×D3—361no—
396C2×D3—361no—
397C2×D3—362no—
398C2×D3—362no—
399C2×D3—721no—
400C3×C4—21no—
401C3×C4—42no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—121no—
405C3×C4—122no—
406C3×C4—122no—
407C3×C4—182no—
408C3×C4—242no—
409C3×C4—361no—
410Dic3—62no—
411Dic3—121no—
412Dic3—182no—
413Dic3—361no—
414C32—21no—
415C32#474111yescharacteristic, complemented
416C32C3×B(4,𝔽2)11yescharacteristic, complemented
417C23—92no—
418C23—181no—
419C23—181no—
420C23—182no—
421C23—362no—
422C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
423C23C22×(C3⋊D3)⋊C311yescharacteristic
424Q8—181no—
425D4—22no—
426D4—181no—
427D4—182no—
428D4—182no—
429D4—182no—
430D4—364no—
431D4—362no—
432C2×C4—22no—
433C2×C4—92no—
434C2×C4—182no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
438C2×C3—21no—
439C2×C3—22no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—41no—
443C2×C3—61no—
444C2×C3—62no—
445C2×C3—62no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—122no—
449C2×C3—121no—
450C2×C3—121no—
451C2×C3—121no—
452C2×C3—241no—
453C2×C3—362no—
454C2×C3—362no—
455C2×C3C3×#88211yescharacteristic
456D3—122no—
457D3—122no—
458D3—362no—
459D3—362no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—184no—
464C22—182no—
465C22—182no—
466C22—362no—
467C22—362no—
468C22—361no—
469C22—361no—
470C22—721no—
471C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
472C22D4×(C3⋊D3)⋊C312yes—
473C4—21no—
474C4—42no—
475C4—182no—
476C4—361no—
477C3—31no—
478C3—31no—
479C3—61no—
480C3C3×#474111yescharacteristic
481C2—22no—
482C2—21no—
483C2—41no—
484C2—362no—
485C2—362no—
486C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC23×C32C23×C3
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×#99C2
socleC2×C3C3×#882
hypercenter#99(C3⋊D3)⋊C3
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #64361⋊C3C2semidirect
2. #64146⋊C3C2semidirect
3. #64091⋊C3C2semidirect
4. H(3,𝔽3)×#99C2semidirect
5. #63633⋊C3C2semidirect
6. #10592440C3semidirect
8. C23×(C3⋊D3)⋊C3C22semidirect
17. C22⋊C4×H(3,𝔽3)C22semidirect
18. #3757⋊C3C22semidirect
21. #64361C2×C3semidirect
23. #64146C2×C3semidirect
24. #64091C2×C3semidirect
28. C32×#99D3semidirect
29. C32×#99C2×C3semidirect
30. #63633C2×C3semidirect
32. C23×H(3,𝔽3)D4semidirect
38. C22×(C3⋊D3)⋊C3D4semidirect
41. C2×C4×H(3,𝔽3)D4semidirect
48. C2×(C3⋊Dic3)⋊C3D4semidirect
56. C23×C3⋊D3C22×C3semidirect
76. C32×C22⋊C4C2×D3semidirect
77. C32×C22⋊C4C22×C3semidirect
78. #3757C22×C3semidirect
119. C23×C32D12semidirect
120. C23×C32C3×D4semidirect
127. C22×C3⋊D3C3×D4semidirect
140. C2×C32×C4D12semidirect
141. C2×C32×C4C3×D4semidirect
148. C2×C3⋊Dic3C3×D4semidirect
257. #99(C3⋊D3)⋊C3semidirect
258. H(3,𝔽3)B(4,𝔽2)semidirect
355. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
415. C32#4741semidirect
416. C32C3×B(4,𝔽2)semidirect
422. C23C4⋊((C3⋊D3)⋊C3)semidirect
437. C2×C4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
#64146⋊C32
#64091⋊C32
H(3,𝔽3)×#992
#63633⋊C32
#105924403
C3×#47413