Finite Group Databasebeta

← #13176507

Subgroups — #13176507

Order 1728.

1648 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723934C211yesmaximal, characteristic
2C3×#64096C211yesmaximal, characteristic, complemented
3C32×#887C211yesmaximal, characteristic, complemented
4C4×#6293C211yesmaximal, characteristic, complemented
5#11723453C212yesmaximal, complemented
6#11723452C212yesmaximal, complemented
7#11723451C211yesmaximal, characteristic, complemented
8#11723446C212yesmaximal, complemented
9#11723442C212yesmaximal, complemented
10#11723441C211yesmaximal, characteristic
11#11723440C211yesmaximal, characteristic
12#10596039—31nomaximal
13#10594350—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6293C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20#88859C2212yes—
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750C2212yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24C3×C4×C3⋊Dic3C2212yes—
25C32×Dic3⋊C4C2212yes—
26C32×C4×Dic3C2211yescharacteristic
27C32×C4×Dic3C2212yes—
28#88807C2211yescharacteristic
29#88806C2211yescharacteristic
30#88806C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33#88804C2212yes—
34#88804C2212yes—
35#88804C2212yes—
36#88804C2212yes—
37C32×C4×Q8—31no—
38#64156—31no—
39#64153—31no—
40#64107—32no—
41#64105—32no—
42#64102—31no—
43#64102—34no—
44#64102—64no—
45#64096D311yescharacteristic, complemented
46#64085—32no—
47#64084—31no—
48#64081—32no—
49C4×#634—31no—
50C3×#887—34no—
51C3×#887D314yescomplemented
52C4×#627—34no—
53#63874—38no—
54#63873—34no—
55#63868—38no—
56#63868—38no—
57#63865—38no—
58#63864—34no—
59#63864—34no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C4⋊(C3⋊(C3⋊D3))—22no—
66C4×C3⋊(C3⋊D3)—24no—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—22no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C32×Dic6—22no—
75#6293—22no—
76#5568—94no—
77#5568—91no—
78C2×C32×Q8—31no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—64no—
82C2×C4×C3⋊D3—32no—
83C2×C4×C3⋊D3—38no—
84C2×C4×C3⋊D3—68no—
85C2×#634—31no—
86C2×#634C2×D311yescharacteristic
87C2×C3×Dic6—34no—
88C2×C3×Dic6C2×D314yes—
89C2×#627—34no—
90C32×C4⋊C4—24no—
91C32×C4⋊C4—32no—
92C32×C4⋊C4C2×D314yes—
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C42—31no—
95C32×C42—32no—
96#3752—32no—
97#3752—38no—
98#3752—32no—
99#3752—32no—
100#3752—32no—
101#3752—32no—
102#3752—68no—
103C4⋊(C3⋊Dic3)—31no—
104#3750—31no—
105#3750—31no—
106#3750—32no—
107#3750—32no—
108#3750C2×D312yes—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—34no—
111C4×C3⋊Dic3—64no—
112C4×C3⋊Dic3C2×D311yescharacteristic
113C4×C3⋊Dic3C2×D312yes—
114C3×Dic3⋊C4—38no—
115C3×Dic3⋊C4C2×D318yes—
116C3×C4×Dic3—34no—
117C3×C4×Dic3—38no—
118C3×C4×Dic3C2×D314yes—
119C3×C4×Dic3C2×D318yes—
120#3724—38no—
121#3724—38no—
122#3724—34no—
123Dic3⋊Dic3—34no—
124Dic3⋊Dic3—34no—
125#3722—38no—
126#3722—38no—
127#3722—38no—
128#3722—38no—
129C22×C33C2411yescharacteristic
130C2×C3⋊(C3⋊D3)—22no—
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)—22no—
133C33×C4—22no—
134C33×C4#4112yes—
135C3⋊(C3⋊Dic3)—21no—
136C3⋊(C3⋊Dic3)#4112yes—
137C3×C3⋊Dic3—21no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—21no—
140C32×Dic3—21no—
141C32×Dic3—22no—
142C32×Dic3—21no—
143C3×C4×Q8—34no—
144C3×C4×Q8—91no—
145#947—94no—
146#947—91no—
147#944—94no—
148#944—91no—
149#898—98no—
150#898—92no—
151#896—98no—
152#896—92no—
153#893—94no—
154#893—91no—
155#893—184no—
156#887—34no—
157#887C3⋊D311yescharacteristic, complemented
158#876—98no—
159#876—92no—
160#875—94no—
161#875—91no—
162#872—98no—
163#872—92no—
164C4×Dic6—94no—
165C4×Dic6—91no—
166C22×C3⋊D3—32no—
167C22×C3⋊D3—38no—
168C22×C3⋊D3—68no—
169C32×Q8—62no—
170C2×C32×C4—24no—
171C2×C32×C4—28no—
172C2×C32×C4—31no—
173C2×C32×C4—32no—
174C2×C32×C4—31no—
175C2×C32×C4C22×D318yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C4⋊(C3⋊D3)—616no—
189C4⋊(C3⋊D3)—68no—
190C4⋊(C3⋊D3)—62no—
191C4×C3⋊D3—64no—
192C4×C3⋊D3—616no—
193C4×C3⋊D3—1216no—
194#634—22no—
195#634—62no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—38no—
199C2×C3×Dic3C22×D314yes—
200C2×C3×Dic3C22×D314yes—
201C2×C3×Dic3C22×D318yes—
202C3×Dic6—28no—
203C3×Dic6—68no—
204#627—68no—
205#556—271noSylow
206C2×C33C2×#4111yescharacteristic
207C2×C33C2×#4111yescharacteristic
208C2×C332−1+411yescharacteristic
209C3⋊(C3⋊D3)—42no—
210C2×C3×Q8—34no—
211C2×C3×Q8—91no—
212C2×D12—94no—
213C2×D12—91no—
214C2×D12—184no—
215C2×C4×D3—98no—
216C2×C4×D3—92no—
217C2×C4×D3—188no—
218C2×Dic6—34no—
219C2×Dic6—94no—
220C2×Dic6—91no—
221C2×Dic6C2×C3⋊D311yescharacteristic
222C3×C4⋊C4—24no—
223C3×C4⋊C4—38no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4D3214yes—
226C3×C4⋊C4C2×C3⋊D311yescharacteristic
227C3×C42—34no—
228C3×C42—38no—
229C3×C42—91no—
230C3×C42—92no—
231(C2×D3)⋊C4—98no—
232(C2×D3)⋊C4—98no—
233(C2×D3)⋊C4—98no—
234(C2×D3)⋊C4—98no—
235(C2×D3)⋊C4—98no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—188no—
242C4⋊Dic3—94no—
243C4⋊Dic3—91no—
244Dic3⋊C4—38no—
245Dic3⋊C4—94no—
246Dic3⋊C4—94no—
247Dic3⋊C4—98no—
248Dic3⋊C4—98no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4C2×C3⋊D312yes—
254C4×Dic3—34no—
255C4×Dic3—38no—
256C4×Dic3—94no—
257C4×Dic3—91no—
258C4×Dic3—184no—
259C4×Dic3C2×C3⋊D311yescharacteristic
260C4×Dic3C2×C3⋊D312yes—
261C22×C32—24no—
262C22×C32C23×D314yes—
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—616no—
265C2×C3⋊D3—68no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—68no—
270C2×C3⋊D3—68no—
271C2×C3⋊D3—128no—
272C2×C3⋊D3—128no—
273C32×C4—22no—
274C32×C4—28no—
275C32×C4—28no—
276C32×C4—48no—
277C32×C4—62no—
278C32×C4—61no—
279C32×C4—61no—
280C32×C4#23518yes—
281C32×C4#23512yes—
282C3⋊Dic3—22no—
283C3⋊Dic3—21no—
284C3⋊Dic3—21no—
285C3⋊Dic3—32no—
286C3⋊Dic3—38no—
287C3⋊Dic3—68no—
288C3⋊Dic3—61no—
289C3⋊Dic3—64no—
290C3⋊Dic3—124no—
291C3×Dic3—28no—
292C3×Dic3—24no—
293C3×Dic3—24no—
294C3×Dic3—68no—
295C3×Dic3—64no—
296C3×Dic3—64no—
297C4⋊Q8—271no—
298#126—272no—
299#126—272no—
300#124—271no—
301#124—271no—
302C22⋊Q8—272no—
303C4×Q8—31no—
304C4×Q8—91no—
305C4×Q8—271no—
306C4×D4—271no—
307#117—272no—
308C33#55611yescharacteristic, complemented, Sylow
309C22×D3—98no—
310C22×D3—92no—
311C22×D3—188no—
312C3×Q8—68no—
313C3×Q8—182no—
314C2×C3×C4—24no—
315C2×C3×C4—28no—
316C2×C3×C4—34no—
317C2×C3×C4—38no—
318C2×C3×C4—34no—
319C2×C3×C4—91no—
320C2×C3×C4—92no—
321C2×C3×C4—91no—
322C2×C3×C4C2×D3218yes—
323C2×C3×C4C2×D3214yes—
324C2×C3×C4C22×C3⋊D312yes—
325C2×C3×C4C22×C3⋊D311yescharacteristic
326C2×Dic3—34no—
327C2×Dic3—34no—
328C2×Dic3—38no—
329C2×Dic3—94no—
330C2×Dic3—91no—
331C2×Dic3—94no—
332C2×Dic3—91no—
333C2×Dic3—184no—
334C2×Dic3—184no—
335C2×Dic3C22×C3⋊D311yescharacteristic
336C2×Dic3C22×C3⋊D312yes—
337C2×Dic3C22×C3⋊D311yescharacteristic
338D12—1816no—
339D12—188no—
340D12—182no—
341C4×D3—184no—
342C4×D3—1816no—
343C4×D3—3616no—
344Dic6—22no—
345Dic6—68no—
346Dic6—182no—
347Dic6—188no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32C2×#23514yes—
352C2×C32C2×#23511yescharacteristic
353C2×C32#100714yes—
354C2×C32#100711yescharacteristic
355C2×C32D3×#4114yes—
356C2×C32D3×#4111yescharacteristic
357C3⋊D3—1216no—
358C3⋊D3—128no—
359C3⋊D3—122no—
360C2×Q8—31no—
361C2×Q8—91no—
362C2×Q8—271no—
363C2×D4—271no—
364C22×C4—272no—
365C4⋊C4—32no—
366C4⋊C4—92no—
367C4⋊C4—271no—
368C4⋊C4—272no—
369C4⋊C4—271no—
370C4⋊C4—271no—
371C4⋊C4—272no—
372C4⋊C4D3×C3⋊D311yescharacteristic
373C22⋊C4—272no—
374C22⋊C4—272no—
375C22⋊C4—272no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—32no—
379C42—31no—
380C42—92no—
381C42—91no—
382C42—271no—
383C22×C3—24no—
384C22×C3C22×D3214yes—
385C22×C3C23×C3⋊D311yescharacteristic
386C2×D3—1816no—
387C2×D3—188no—
388C2×D3—188no—
389C2×D3—188no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—368no—
394C2×D3—368no—
395C3×C4—28no—
396C3×C4—22no—
397C3×C4—28no—
398C3×C4—48no—
399C3×C4—68no—
400C3×C4—64no—
401C3×C4—64no—
402C3×C4—182no—
403C3×C4—181no—
404C3×C4—181no—
405C3×C4#379818yes—
406C3×C4#382812yes—
407Dic3—22no—
408Dic3—21no—
409Dic3—21no—
410Dic3—68no—
411Dic3—64no—
412Dic3—64no—
413Dic3—98no—
414Dic3—92no—
415Dic3—188no—
416Dic3—184no—
417Dic3—181no—
418Dic3—364no—
419C32—24no—
420C32#556814yescomplemented
421C32#556811yescharacteristic, complemented
422C23—272no—
423Q8—62no—
424Q8—182no—
425Q8—542no—
426D4—542no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—32no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—271no—
434C2×C4—271no—
435C2×C4—544no—
436C2×C4C2×D3×C3⋊D312yes—
437C2×C4C2×D3×C3⋊D311yescharacteristic
438C2×C3—24no—
439C2×C3—24no—
440C2×C3—24no—
441C2×C3C2×#379814yes—
442C2×C3#6431214yes—
443C2×C3#6431614yes—
444C2×C3C2×#382811yescharacteristic
445C2×C3#6436411yescharacteristic
446C2×C3#41×C3⋊D311yescharacteristic
447D3—3616no—
448D3—368no—
449D3—362no—
450C22—542no—
451C22—542no—
452C22—542no—
453C22C22×D3×C3⋊D311yescharacteristic
454C4—22no—
455C4—62no—
456C4—61no—
457C4—61no—
458C4—182no—
459C4—181no—
460C4—181no—
461C4—272no—
462C4—541no—
463C4#8902012yes—
464C3—24no—
465C3#1059435014yescomplemented
466C3#1059603911yescharacteristic, complemented
467C2—1082no—
468C2C2×#8902011yescharacteristic
469C2#1172458611yescharacteristic
470C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64096C2semidirect
3. C32×#887C2semidirect
4. C4×#6293C2semidirect
5. #11723453C2semidirect
6. #11723452C2semidirect
7. #11723451C2semidirect
8. #11723446C2semidirect
9. #11723442C2semidirect
45. #64096D3semidirect
51. C3×#887D3semidirect
157. #887C3⋊D3semidirect
308. C33#556semidirect
420. C32#5568semidirect
421. C32#5568semidirect
465. C3#10594350semidirect
466. C3#10596039semidirect

Maximal subgroups

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

subgroupindex
#117239342
C3×#640962
C32×#8872
C4×#62932
#117234532
#117234522
#117234512
#117234462
#117234422
#117234412
#117234402
#105960393
#105943503