Finite Group Databasebeta

← #13177271

Subgroups — #13177271

Order 1728.

1628 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723965C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C32×#924C211yesmaximal, characteristic
4#11723641C212yesmaximal
5#11723595C212yesmaximal
6#11723535C211yesmaximal, characteristic, complemented
7#11723455C211yesmaximal, characteristic, complemented
8#11723445C212yesmaximal, complemented
9#11723406C211yesmaximal, characteristic, complemented
10Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
11#11723390C212yesmaximal, complemented
12#10596093—31nomaximal
13#10594808—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C2×#6293C2212yes—
20#88864C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857C2212yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30#88807C2212yes—
31#88807C2212yes—
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38Q8×C3⋊Dic3—31no—
39#64153—31no—
40#64133—31no—
41#64133—34no—
42#64133—64no—
43C2×C4×C3⋊Dic3D311yescharacteristic
44#64098—32no—
45#64097—31no—
46#64096—32no—
47#64096—31no—
48#64087—32no—
49#64086—32no—
50C3×#924D314yes—
51C2×C3×C4×Dic3—34no—
52#63964—38no—
53#63955—38no—
54#63918—34no—
55#63867—38no—
56#63866—34no—
57#63844—34no—
58Dic3×Dic6—34no—
59#63837—38no—
60C23×C33C2311yescharacteristic
61C2×C33×C4—22no—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3—22no—
68C2×C3×C3⋊Dic3Q814yescomplemented
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C32×Dic6—22no—
76#6293—24no—
77#5622—94no—
78#5470—91no—
79C2×C32×Q8—31no—
80C22×C32×C4—24no—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C2×D311yescharacteristic
83C22×C3⋊Dic3C2×D311yescharacteristic
84C22×C3⋊Dic3C2×D311yescharacteristic
85C2×#634—31no—
86C2×#634—34no—
87C2×#634—32no—
88C2×#634—64no—
89C22×C3×Dic3—34no—
90C22×C3×Dic3—34no—
91C2×C3×Dic6C2×D314yes—
92C2×#627—38no—
93C32×C4⋊C4—31no—
94C32×C4⋊C4—32no—
95C32×C22⋊C4—32no—
96#3757—32no—
97#3757—38no—
98#3757—68no—
99C4⋊(C3⋊Dic3)—31no—
100C4⋊(C3⋊Dic3)—34no—
101C4⋊(C3⋊Dic3)—32no—
102C4⋊(C3⋊Dic3)—64no—
103#3750—32no—
104#3750—38no—
105#3750—32no—
106#3750—32no—
107#3750—32no—
108#3750—32no—
109#3750—68no—
110C4×C3⋊Dic3—32no—
111C4×C3⋊Dic3—32no—
112C4×C3⋊Dic3C2×D311yescharacteristic
113C4×C3⋊Dic3C2×D311yescharacteristic
114C4×C3⋊Dic3C2×D311yescharacteristic
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×#217C2×D318yes—
117C3×C4⋊Dic3C2×D314yes—
118C3×Dic3⋊C4C2×D318yes—
119C3×C4×Dic3—34no—
120C3×C4×Dic3—34no—
121C3×C4×Dic3—34no—
122C3×C4×Dic3—34no—
123#3724—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—38no—
128Dic32—38no—
129Dic32—38no—
130C22×C33—21no—
131C22×C33—21no—
132C22×C33C2×Q812yes—
133C22×C33C2411yescharacteristic
134C33×C4—21no—
135C33×C4#4112yes—
136C3⋊(C3⋊Dic3)—22no—
137C3⋊(C3⋊Dic3)—22no—
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3C2×Q812yes—
140C3×C3⋊Dic3C2×Q812yes—
141C3×C3⋊Dic3#4112yes—
142C32×Dic3—22no—
143C32×Dic3—22no—
144C3×C22⋊Q8—34no—
145C2×C3×C42—91no—
146Q8×Dic3—94no—
147#944—94no—
148#926—92no—
149#924—94no—
150#924—91no—
151#924—92no—
152#924—184no—
153#924C3⋊D311yescharacteristic
154C2×C4×Dic3—34no—
155#889—98no—
156#888—94no—
157#887—98no—
158#887—94no—
159#878—98no—
160#877—98no—
161#870—92no—
162#869—91no—
163#869—91no—
164C4×Dic6—91no—
165C4×Dic6—91no—
166C4×Dic6—92no—
167C23×C32—24no—
168C23×C32C22×D314yes—
169C23×C32C22×D311yescharacteristic
170C32×Q8—62no—
171C2×C32×C4—22no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—28no—
175C2×C32×C4—32no—
176C2×C32×C4—32no—
177C2×C32×C4—48no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C3⋊Dic3—22no—
183C2×C3⋊Dic3—32no—
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—38no—
186C2×C3⋊Dic3—38no—
187C2×C3⋊Dic3—68no—
188C2×C3⋊Dic3—68no—
189C2×C3⋊Dic3Dic614yescomplemented
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194#634—62no—
195#634—64no—
196#634—68no—
197#634—128no—
198C2×C3×Dic3—316no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—68no—
204C2×C3×Dic3C22×D318yes—
205C2×C3×Dic3C22×D318yes—
206C3×Dic6—28no—
207#627—616no—
208#533—271noSylow
209C2×C33—22no—
210C2×C33C22×Q811yescharacteristic
211C2×C33C2×#4111yescharacteristic
212C2×C33C2×#4111yescharacteristic
213C2×C3×Q8—34no—
214C22×C3×C4—24no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4D3214yes—
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C22×Dic3—34no—
220C22×Dic3—34no—
221C2×Dic6—98no—
222C2×Dic6—94no—
223C2×Dic6—91no—
224C2×Dic6—92no—
225C2×Dic6—184no—
226C2×Dic6C2×C3⋊D311yescharacteristic
227C3×C4⋊C4—34no—
228C3×C4⋊C4—38no—
229C3×C22⋊C4—38no—
230C3×C42—91no—
231C3×C42—91no—
232C3×C42—91no—
233C3×C42—91no—
234#217—98no—
235#217—92no—
236#217—188no—
237#217C2×C3⋊D312yes—
238C4⋊Dic3—98no—
239C4⋊Dic3—94no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—184no—
245C4⋊Dic3C2×C3⋊D311yescharacteristic
246Dic3⋊C4—98no—
247Dic3⋊C4—98no—
248Dic3⋊C4—98no—
249Dic3⋊C4—98no—
250Dic3⋊C4—98no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—188no—
255Dic3⋊C4C2×C3⋊D312yes—
256C4×Dic3—34no—
257C4×Dic3—34no—
258C4×Dic3—34no—
259C4×Dic3—34no—
260C4×Dic3—98no—
261C4×Dic3—98no—
262C4×Dic3—92no—
263C4×Dic3—92no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—24no—
267C22×C32—28no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—44no—
271C22×C32—44no—
272C22×C32C2×Dic612yes—
273C22×C32D3×Q818yes—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C32×C4—21no—
277C32×C4—28no—
278C32×C4—24no—
279C32×C4—44no—
280C32×C4—62no—
281C32×C4—62no—
282C32×C4#23512yes—
283C32×C4#23718yes—
284C3⋊Dic3—21no—
285C3⋊Dic3—62no—
286C3⋊Dic3—62no—
287C3⋊Dic3—68no—
288C3⋊Dic3—68no—
289C3⋊Dic3—128no—
290C3⋊Dic3—128no—
291C3⋊Dic3C2×Dic612yes—
292C3⋊Dic3C2×Dic612yes—
293C3⋊Dic3#23512yes—
294C3×Dic3—28no—
295C3×Dic3—28no—
296C3×Dic3—38no—
297C3×Dic3—38no—
298C3×Dic3—38no—
299C3×Dic3—64no—
300C4⋊Q8—271no—
301C4⋊Q8—271no—
302#125—272no—
303C22⋊Q8—31no—
304C22⋊Q8—272no—
305C22⋊Q8—271no—
306C4×Q8—272no—
307C4×Q8—271no—
308C4×Q8—271no—
309#117—272no—
310C2×C42—91no—
311C33#53311yescharacteristic, complemented, Sylow
312C23×C3—24no—
313C23×C3C2×D3214yes—
314C23×C3C22×C3⋊D311yescharacteristic
315C3×Q8—68no—
316C2×C3×C4—28no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—38no—
321C2×C3×C4—38no—
322C2×C3×C4—48no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—94no—
327C2×C3×C4—91no—
328C2×C3×C4—182no—
329C2×C3×C4C2×D3214yes—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C22×C3⋊D311yescharacteristic
332C2×C3×C4C22×C3⋊D311yescharacteristic
333C2×Dic3—34no—
334C2×Dic3—34no—
335C2×Dic3—34no—
336C2×Dic3—316no—
337C2×Dic3—34no—
338C2×Dic3—68no—
339C2×Dic3—98no—
340C2×Dic3—98no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—188no—
344C2×Dic3—188no—
345C2×Dic3C22×C3⋊D312yes—
346C2×Dic3C22×C3⋊D312yes—
347Dic6—22no—
348Dic6—184no—
349Dic6—1816no—
350Dic6—188no—
351Dic6—182no—
352Dic6—368no—
353C2×C32—22no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—28no—
358C2×C32—48no—
359C2×C32C22×Dic611yescharacteristic
360C2×C32C2×#23511yescharacteristic
361C2×C32C2×#23511yescharacteristic
362C2×C32C2×#23714yes—
363C2×C32C2×D3×Q814yes—
364C2×C32D3×#4114yes—
365C2×Q8—31no—
366C2×Q8—272no—
367C2×Q8—271no—
368C22×C4—91no—
369C22×C4—91no—
370C22×C4D3×C3⋊D311yescharacteristic
371C4⋊C4—32no—
372C4⋊C4—31no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C22⋊C4—32no—
381C22⋊C4—272no—
382C42—91no—
383C42—91no—
384C42—91no—
385C42—91no—
386C42—272no—
387C42—272no—
388C22×C3—24no—
389C22×C3—24no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3D3×Dic618yes—
397C22×C3Q8×C3⋊D312yes—
398C22×C3C22×D3214yes—
399C22×C3C23×C3⋊D311yescharacteristic
400C3×C4—24no—
401C3×C4—21no—
402C3×C4—28no—
403C3×C4—44no—
404C3×C4—68no—
405C3×C4—68no—
406C3×C4—92no—
407C3×C4—92no—
408C3×C4—92no—
409C3×C4—181no—
410C3×C4#380418yes—
411C3×C4#383012yes—
412Dic3—22no—
413Dic3—22no—
414Dic3—38no—
415Dic3—38no—
416Dic3—38no—
417Dic3—64no—
418Dic3—188no—
419Dic3—188no—
420Dic3—182no—
421Dic3—182no—
422Dic3—368no—
423Dic3—368no—
424C32—24no—
425C32#547011yescharacteristic, complemented
426C32#562214yescomplemented
427C23C2×D3×C3⋊D311yescharacteristic
428Q8—62no—
429Q8—544no—
430Q8—542no—
431C2×C4—22no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—91no—
437C2×C4—94no—
438C2×C4—91no—
439C2×C4—182no—
440C2×C4—272no—
441C2×C4—272no—
442C2×C4C2×D3×C3⋊D311yescharacteristic
443C2×C4C2×D3×C3⋊D311yescharacteristic
444C2×C3—28no—
445C2×C3—22no—
446C2×C3—24no—
447C2×C3—24no—
448C2×C3—24no—
449C2×C3—48no—
450C2×C3C2×D3×Dic614yes—
451C2×C3D3×#23514yes—
452C2×C3C2×#380414yes—
453C2×C3C2×#383011yescharacteristic
454C2×C3C2×Q8×C3⋊D311yescharacteristic
455C2×C3#41×C3⋊D311yescharacteristic
456C22—21no—
457C22—21no—
458C22C3⋊D3×Dic612yes—
459C22C22×D3×C3⋊D311yescharacteristic
460C4—21no—
461C4—62no—
462C4—62no—
463C4—92no—
464C4—92no—
465C4—92no—
466C4—181no—
467C4—542no—
468C4—542no—
469C4#8902812yes—
470C3—24no—
471C3#1059480814yescomplemented
472C3#1059609311yescharacteristic, complemented
473C2—22no—
474C2C2×C3⋊D3×Dic611yescharacteristic
475C2C3⋊D3×#23511yescharacteristic
476C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11723535C2semidirect
7. #11723455C2semidirect
8. #11723445C2semidirect
9. #11723406C2semidirect
10. Dic6×C3⋊Dic3C2semidirect
11. #11723390C2semidirect
68. C2×C3×C3⋊Dic3Q8semidirect
189. C2×C3⋊Dic3Dic6semidirect
311. C33#533semidirect
425. C32#5470semidirect
426. C32#5622semidirect
471. C3#10594808semidirect
472. C3#10596093semidirect

Maximal subgroups

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

subgroupindex
#117239652
C2×C3×C4×C3⋊Dic32
C32×#9242
#117236412
#117235952
#117235352
#117234552
#117234452
#117234062
Dic6×C3⋊Dic32
#117233902
#105960933
#105948083