Finite Group Databasebeta

← #13176499

Subgroups — #13176499

Order 1728.

1552 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723928C211yesmaximal, characteristic, complemented
2C3×#64096C211yesmaximal, characteristic, complemented
3C32×#893C211yesmaximal, characteristic
4C4×#6293C211yesmaximal, characteristic, complemented
5#11723443C211yesmaximal, characteristic, complemented
6#11723434C211yesmaximal, characteristic
7#11723432C212yesmaximal, complemented
8#11723429C211yesmaximal, characteristic
9#11723428C212yesmaximal, complemented
10#11723423C212yesmaximal, complemented
11#11723415C212yesmaximal, complemented
12#10596013—31nomaximal
13#10594335—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×C32×C4×D3C2212yes—
18C2×#6293C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20#88857C2212yes—
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2212yes—
23C3×#3750C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C3×C4×C3⋊Dic3C2212yes—
26C32×(C2×D3)⋊C4C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2211yescharacteristic
29#88806C2211yescharacteristic
30Dic3⋊(C3⋊Dic3)C2211yescharacteristic
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38#64145—31no—
39#64145—31no—
40#64135—32no—
41#64133—32no—
42#64096—31no—
43#64096—34no—
44#64096—64no—
45#64096D311yescharacteristic, complemented
46#64088—32no—
47#64088—32no—
48#64078—31no—
49C4×#634—31no—
50C3×#893D314yes—
51C3×#887—34no—
52C4×#627—34no—
53#63866—34no—
54#63863—34no—
55#63862—38no—
56#63860—34no—
57#63859—38no—
58#63855—38no—
59#63851—38no—
60C22×C32×D3C2312yes—
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—22no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C32×D12—22no—
74C32×C4×D3—24no—
75#6293—22no—
76#5568—91no—
77#5542—94no—
78C2×C32×D4—31no—
79C22×C32×C4—32no—
80C2×#634—31no—
81C2×#634—34no—
82C2×#634—31no—
83C2×#634—64no—
84C2×#634C2×D311yescharacteristic
85C2×C3×D12C2×D314yes—
86C2×C3×C4×D3C2×D318yes—
87C2×C3×Dic6—34no—
88C2×#627—34no—
89C32×C4⋊C4—24no—
90C32×C4⋊C4C2×D314yes—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C22⋊C4—32no—
93C32×C42—31no—
94#3757—32no—
95#3757—32no—
96#3757—32no—
97#3757—32no—
98C4⋊(C3⋊Dic3)—31no—
99C4⋊(C3⋊Dic3)—32no—
100C4⋊(C3⋊Dic3)—32no—
101#3750—32no—
102#3750—38no—
103#3750—31no—
104#3750—31no—
105#3750—68no—
106#3750C2×D312yes—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—32no—
109C4×C3⋊Dic3—38no—
110C4×C3⋊Dic3—34no—
111C4×C3⋊Dic3—68no—
112C4×C3⋊Dic3—64no—
113C4×C3⋊Dic3C2×D311yescharacteristic
114C4×C3⋊Dic3C2×D312yes—
115C3×(C2×D3)⋊C4C2×D318yes—
116C3×Dic3⋊C4—38no—
117C3×C4×Dic3—38no—
118C3×C4×Dic3—34no—
119C3×C4×Dic3C2×D314yes—
120#3724—34no—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—34no—
125(C2×D3)⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129C22×C33C2411yescharacteristic
130C2×C32×D3—22no—
131C2×C32×D3—22no—
132C2×C32×D3—22no—
133C33×C4—22no—
134C33×C4#4112yes—
135C3⋊(C3⋊Dic3)—22no—
136C3⋊(C3⋊Dic3)—21no—
137C3⋊(C3⋊Dic3)—21no—
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3—22no—
140C3×C3⋊Dic3—21no—
141C32×Dic3—21no—
142C32×Dic3#4112yes—
143C3×C4×Q8—91no—
144C3×C4×D4—34no—
145#947—91no—
146#944—91no—
147#936—94no—
148#936—94no—
149#926—98no—
150#924—98no—
151#898—92no—
152#896—92no—
153#893C3⋊D311yescharacteristic
154#887—34no—
155#887—94no—
156#887—91no—
157#887—184no—
158#879—98no—
159#879—98no—
160#876—92no—
161#875—91no—
162#872—92no—
163#869—94no—
164C4×Dic6—94no—
165C4×Dic6—91no—
166C23×C32—32no—
167C22×C3×D3C22×D318yes—
168C32×D4—62no—
169C2×C32×C4—24no—
170C2×C32×C4—28no—
171C2×C32×C4—31no—
172C2×C32×C4—31no—
173C2×C32×C4—64no—
174C2×C32×C4C22×D318yes—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—38no—
183C2×C3⋊Dic3—34no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—68no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D312yes—
190#634—22no—
191#634—62no—
192#634—616no—
193#634—68no—
194#634—62no—
195C2×C3×Dic3—38no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C22×D314yes—
200C3×D12—28no—
201C3×C4×D3—216no—
202C3×Dic6—68no—
203#627—68no—
204#556—271noSylow
205C2×C33C2×#4111yescharacteristic
206C2×C33C2×#4111yescharacteristic
207C2×C332−1+411yescharacteristic
208C32×D3—42no—
209C2×C3×Q8—91no—
210C2×C3×D4—34no—
211C22×C3×C4—38no—
212C2×D12C2×C3⋊D311yescharacteristic
213C2×C4×D3C2×C3⋊D312yes—
214C2×Dic6—34no—
215C2×Dic6—94no—
216C2×Dic6—94no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—184no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4D3214yes—
223C3×C4⋊C4C2×C3⋊D311yescharacteristic
224C3×C22⋊C4—38no—
225C3×C42—34no—
226C3×C42—91no—
227C3×C42—92no—
228#217—98no—
229#217—98no—
230#217—98no—
231#217—98no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4C2×C3⋊D312yes—
237C4⋊Dic3—94no—
238C4⋊Dic3—98no—
239C4⋊Dic3—98no—
240C4⋊Dic3—91no—
241Dic3⋊C4—38no—
242Dic3⋊C4—94no—
243Dic3⋊C4—94no—
244Dic3⋊C4—98no—
245Dic3⋊C4—92no—
246Dic3⋊C4—91no—
247Dic3⋊C4—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—188no—
251C4×Dic3—34no—
252C4×Dic3—38no—
253C4×Dic3—94no—
254C4×Dic3—98no—
255C4×Dic3—91no—
256C4×Dic3—92no—
257C4×Dic3—188no—
258C4×Dic3—184no—
259C4×Dic3C2×C3⋊D311yescharacteristic
260C22×C32—24no—
261C22×C32—62no—
262C22×C32—62no—
263C22×C32—62no—
264C22×C32C23×D314yes—
265C22×C32C23×D311yescharacteristic
266C2×C3×D3—28no—
267C2×C3×D3—28no—
268C2×C3×D3—28no—
269C32×C4—22no—
270C32×C4—28no—
271C32×C4—28no—
272C32×C4—32no—
273C32×C4—48no—
274C32×C4—61no—
275C32×C4#23518yes—
276C32×C4#23512yes—
277C3⋊Dic3—22no—
278C3⋊Dic3—21no—
279C3⋊Dic3—21no—
280C3⋊Dic3—616no—
281C3⋊Dic3—68no—
282C3⋊Dic3—68no—
283C3⋊Dic3—62no—
284C3⋊Dic3—61no—
285C3⋊Dic3—61no—
286C3⋊Dic3—64no—
287C3⋊Dic3—64no—
288C3⋊Dic3—124no—
289C3×Dic3—24no—
290C3×Dic3—68no—
291C3×Dic3—64no—
292C3×Dic3—64no—
293C3×Dic3#23718yes—
294C4⋊Q8—271no—
295#126—272no—
296#126—272no—
297#124—271no—
298#124—271no—
299C22⋊Q8—272no—
300C4×Q8—91no—
301C4×Q8—271no—
302C4×Q8—271no—
303C4×D4—31no—
304#117—272no—
305C33#55611yescharacteristic, complemented, Sylow
306C23×C3—38no—
307C22×D3C22×C3⋊D312yes—
308C3×Q8—182no—
309C3×D4—68no—
310C2×C3×C4—24no—
311C2×C3×C4—28no—
312C2×C3×C4—34no—
313C2×C3×C4—34no—
314C2×C3×C4—616no—
315C2×C3×C4—91no—
316C2×C3×C4—92no—
317C2×C3×C4—91no—
318C2×C3×C4C2×D3218yes—
319C2×C3×C4C2×D3214yes—
320C2×C3×C4C22×C3⋊D312yes—
321C2×C3×C4C22×C3⋊D311yescharacteristic
322C2×Dic3—34no—
323C2×Dic3—34no—
324C2×Dic3—38no—
325C2×Dic3—94no—
326C2×Dic3—94no—
327C2×Dic3—98no—
328C2×Dic3—91no—
329C2×Dic3—92no—
330C2×Dic3—91no—
331C2×Dic3—184no—
332C2×Dic3—188no—
333C2×Dic3—184no—
334C2×Dic3C22×C3⋊D311yescharacteristic
335C2×Dic3C22×C3⋊D311yescharacteristic
336D12—22no—
337C4×D3—24no—
338Dic6—68no—
339Dic6—188no—
340Dic6—1816no—
341Dic6—188no—
342Dic6—182no—
343Dic6—182no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32—122no—
348C2×C32C2×#23514yes—
349C2×C32C2×#23511yescharacteristic
350C2×C32C2×#23714yes—
351C2×C32#100711yescharacteristic
352C2×C32D3×#4111yescharacteristic
353C2×C32#101014yes—
354C3×D3—48no—
355C2×Q8—91no—
356C2×Q8—271no—
357C2×Q8—271no—
358C2×D4—31no—
359C22×C4—32no—
360C4⋊C4—92no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—271no—
364C4⋊C4—271no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4D3×C3⋊D311yescharacteristic
368C22⋊C4—32no—
369C22⋊C4—272no—
370C22⋊C4—272no—
371C22⋊C4—272no—
372C22⋊C4—272no—
373C42—31no—
374C42—92no—
375C42—91no—
376C42—272no—
377C42—271no—
378C22×C3—24no—
379C22×C3—68no—
380C22×C3—68no—
381C22×C3—68no—
382C22×C3C22×D3214yes—
383C22×C3C23×C3⋊D311yescharacteristic
384C2×D3—22no—
385C2×D3—22no—
386C2×D3—22no—
387C3×C4—28no—
388C3×C4—22no—
389C3×C4—28no—
390C3×C4—38no—
391C3×C4—48no—
392C3×C4—64no—
393C3×C4—182no—
394C3×C4—181no—
395C3×C4—181no—
396C3×C4#379818yes—
397C3×C4#382812yes—
398Dic3—21no—
399Dic3—68no—
400Dic3—64no—
401Dic3—64no—
402Dic3—1816no—
403Dic3—188no—
404Dic3—188no—
405Dic3—184no—
406Dic3—184no—
407Dic3—182no—
408Dic3—181no—
409Dic3—181no—
410Dic3—364no—
411Dic3#383012yes—
412C32—24no—
413C32#554214yescomplemented
414C32#556811yescharacteristic, complemented
415C23—32no—
416Q8—182no—
417Q8—542no—
418Q8—542no—
419D4—62no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—64no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—92no—
426C2×C4—271no—
427C2×C4—272no—
428C2×C4—271no—
429C2×C4C2×D3×C3⋊D312yes—
430C2×C4C2×D3×C3⋊D311yescharacteristic
431C2×C3—24no—
432C2×C3—24no—
433C2×C3—24no—
434C2×C3—128no—
435C2×C3C2×#379814yes—
436C2×C3D3×#23714yes—
437C2×C3#6431514yes—
438C2×C3C2×#382811yescharacteristic
439C2×C3C2×#383011yescharacteristic
440C2×C3#6436711yescharacteristic
441D3—42no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22C22×D3×C3⋊D311yescharacteristic
446C4—22no—
447C4—32no—
448C4—61no—
449C4—182no—
450C4—181no—
451C4—181no—
452C4—542no—
453C4—541no—
454C4—541no—
455C4#8902012yes—
456C3—24no—
457C3#1059433514yescomplemented
458C3#1059601311yescharacteristic, complemented
459C2—122no—
460C2C2×#8902011yescharacteristic
461C2D3×#383011yescharacteristic
462C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C32×D12C22
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
1. #11723928C2semidirect
2. C3×#64096C2semidirect
4. C4×#6293C2semidirect
5. #11723443C2semidirect
7. #11723432C2semidirect
9. #11723428C2semidirect
10. #11723423C2semidirect
11. #11723415C2semidirect
45. #64096D3semidirect
305. C33#556semidirect
413. C32#5542semidirect
414. C32#5568semidirect
457. C3#10594335semidirect
458. C3#10596013semidirect

Maximal subgroups

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

subgroupindex
#117239282
C3×#640962
C32×#8932
C4×#62932
#117234432
#117234342
#117234322
#117234292
#117234282
#117234232
#117234152
#105960133
#105943353