Finite Group Databasebeta

← #13175973

Subgroups — #13175973

Order 1728.

1376 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723931C211yesmaximal, characteristic, complemented
2C3×#64096C211yesmaximal, characteristic, complemented
3C32×#897C211yesmaximal, characteristic
4#11723451C211yesmaximal, characteristic, complemented
5#11723447C212yesmaximal, complemented
6#11723430C212yesmaximal
7#11723428C212yesmaximal
8#11723425C211yesmaximal, characteristic
9#11723423C211yesmaximal, characteristic
10#11723405C212yesmaximal, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596096—31nomaximal
13#10594345—34nomaximal
14C2×C3×#634C2211yescharacteristic
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic
17C33×C4⋊C4C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2212yes—
20#88857C2212yes—
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750C2212yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24C3×C4×C3⋊Dic3C2212yes—
25C32×(C2×D3)⋊C4C2212yes—
26C32×C4⋊Dic3C2212yes—
27#88806C2212yes—
28#88806C2212yes—
29#88806C2212yes—
30Dic3⋊(C3⋊Dic3)C2212yes—
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
34(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38Q8×C3⋊Dic3—31no—
39#64153—31no—
40#64135—31no—
41#64133—31no—
42#64099—31no—
43#64099—32no—
44#64099—34no—
45#64099—64no—
46#64098—32no—
47#64096D311yescharacteristic, complemented
48#64088—32no—
49#64088—32no—
50C3×#897D314yes—
51C3×#887—34no—
52#63873—34no—
53#63869—38no—
54#63861—38no—
55#63859—38no—
56#63856—34no—
57#63855—34no—
58#63847—38no—
59Dic3×Dic6—34no—
60C22×C32×D3C2311yescharacteristic
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2312yes—
69C3×#634—22no—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2311yescharacteristic
72C32×C4×D3—22no—
73C32×Dic6—22no—
74#5625—94no—
75#5583—91no—
76C2×C32×Q8—31no—
77C22×C32×C4—31no—
78C2×#634C2×D311yescharacteristic
79C2×C3×C4×D3C2×D314yes—
80C2×C3×Dic6—34no—
81C2×C3×Dic6C2×D314yes—
82C32×C4⋊C4—24no—
83C32×C4⋊C4—32no—
84C32×C4⋊C4C2×D314yes—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C22⋊C4—32no—
87#3757—32no—
88#3757—31no—
89#3757—31no—
90C4⋊(C3⋊Dic3)—32no—
91C4⋊(C3⋊Dic3)—31no—
92C4⋊(C3⋊Dic3)—34no—
93C4⋊(C3⋊Dic3)—38no—
94C4⋊(C3⋊Dic3)—32no—
95C4⋊(C3⋊Dic3)—31no—
96C4⋊(C3⋊Dic3)—31no—
97C4⋊(C3⋊Dic3)—68no—
98C4⋊(C3⋊Dic3)—64no—
99#3750—32no—
100#3750—38no—
101#3750—32no—
102#3750—32no—
103#3750—32no—
104#3750—68no—
105#3750C2×D312yes—
106C4×C3⋊Dic3—31no—
107C4×C3⋊Dic3—34no—
108C4×C3⋊Dic3—32no—
109C4×C3⋊Dic3—64no—
110C4×C3⋊Dic3C2×D312yes—
111C4×C3⋊Dic3C2×D311yescharacteristic
112C3×(C2×D3)⋊C4C2×D318yes—
113C3×C4⋊Dic3C2×D318yes—
114C3×Dic3⋊C4—38no—
115C3×C4×Dic3—34no—
116C3×C4×Dic3—38no—
117#3724—38no—
118#3724—38no—
119#3724—38no—
120Dic3⋊Dic3—38no—
121Dic3⋊Dic3—34no—
122Dic3⋊Dic3—34no—
123(C2×D3)⋊Dic3—34no—
124(C2×D3)⋊Dic3—34no—
125(C2×D3)⋊Dic3—38no—
126Dic32—38no—
127C22×C33C2411yescharacteristic
128C2×C32×D3—21no—
129C2×C32×D3—21no—
130C2×C32×D3—21no—
131C33×C4—22no—
132C33×C4#4112yes—
133C3⋊(C3⋊Dic3)—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3⋊(C3⋊Dic3)—21no—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—21no—
138C3×C3⋊Dic3—21no—
139C32×Dic3—22no—
140C32×Dic3—21no—
141C3×C22⋊Q8—34no—
142C3×C4×Q8—91no—
143#946—91no—
144Q8×Dic3—94no—
145#944—94no—
146#944—91no—
147#926—94no—
148#924—94no—
149#898—92no—
150#897C3⋊D311yescharacteristic
151#890—98no—
152#890—94no—
153#890—91no—
154#890—184no—
155#889—98no—
156#889—92no—
157#887—34no—
158#879—98no—
159#879—98no—
160#876—92no—
161#872—91no—
162#870—92no—
163C4×Dic6—91no—
164C23×C32—31no—
165C22×C3×D3C22×D314yes—
166C32×Q8—62no—
167C2×C32×C4—24no—
168C2×C32×C4—28no—
169C2×C32×C4—32no—
170C2×C32×C4—31no—
171C2×C32×C4—62no—
172C2×C32×C4C22×D318yes—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—32no—
179C2×C3⋊Dic3—38no—
180C2×C3⋊Dic3—34no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—64no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—68no—
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188#634—22no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3C22×D318yes—
193C2×C3×Dic3C22×D314yes—
194C3×C4×D3—28no—
195C3×Dic6—28no—
196C3×Dic6—68no—
197#541—271noSylow
198C2×C33C2×#4111yescharacteristic
199C2×C332−1+411yescharacteristic
200C2×C332−1+411yescharacteristic
201C32×D3—41no—
202C2×C3×Q8—34no—
203C2×C3×Q8—91no—
204C22×C3×C4—34no—
205C2×C4×D3C2×C3⋊D311yescharacteristic
206C2×Dic6—34no—
207C2×Dic6C2×C3⋊D311yescharacteristic
208C3×C4⋊C4—24no—
209C3×C4⋊C4—38no—
210C3×C4⋊C4—92no—
211C3×C4⋊C4D3214yes—
212C3×C4⋊C4C2×C3⋊D311yescharacteristic
213C3×C22⋊C4—38no—
214C3×C42—91no—
215C3×C42—92no—
216#217—98no—
217#217—94no—
218#217—94no—
219(C2×D3)⋊C4—91no—
220(C2×D3)⋊C4—91no—
221(C2×D3)⋊C4—92no—
222(C2×D3)⋊C4C2×C3⋊D312yes—
223C4⋊Dic3—98no—
224C4⋊Dic3—94no—
225C4⋊Dic3—94no—
226C4⋊Dic3—98no—
227C4⋊Dic3—94no—
228C4⋊Dic3—91no—
229C4⋊Dic3—92no—
230C4⋊Dic3—184no—
231C4⋊Dic3—188no—
232C4⋊Dic3C2×C3⋊D312yes—
233Dic3⋊C4—38no—
234Dic3⋊C4—98no—
235Dic3⋊C4—98no—
236Dic3⋊C4—98no—
237Dic3⋊C4—98no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—91no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—91no—
244Dic3⋊C4—92no—
245Dic3⋊C4—188no—
246C4×Dic3—38no—
247C4×Dic3—34no—
248C4×Dic3—98no—
249C4×Dic3—94no—
250C4×Dic3—91no—
251C4×Dic3—92no—
252C4×Dic3—184no—
253C22×C32—24no—
254C22×C32—61no—
255C22×C32—61no—
256C22×C32—61no—
257C22×C32C23×D314yes—
258C22×C32C23×D311yescharacteristic
259C2×C3×D3—24no—
260C2×C3×D3—24no—
261C2×C3×D3—24no—
262C32×C4—22no—
263C32×C4—28no—
264C32×C4—28no—
265C32×C4—48no—
266C32×C4—61no—
267C32×C4—62no—
268C32×C4#23512yes—
269C32×C4#23718yes—
270C3⋊Dic3—22no—
271C3⋊Dic3—21no—
272C3⋊Dic3—21no—
273C3⋊Dic3—61no—
274C3⋊Dic3—62no—
275C3⋊Dic3—61no—
276C3⋊Dic3—68no—
277C3⋊Dic3—64no—
278C3⋊Dic3—64no—
279C3⋊Dic3—124no—
280C3⋊Dic3—124no—
281C3⋊Dic3—128no—
282C3×Dic3—28no—
283C3×Dic3—24no—
284C3×Dic3—68no—
285C3×Dic3—64no—
286C3×Dic3—64no—
287C4⋊Q8—271no—
288#126—272no—
289#126—272no—
290#125—272no—
291#125—272no—
292#125—271no—
293C22⋊Q8—31no—
294C22⋊Q8—271no—
295C4×Q8—91no—
296C4×Q8—271no—
297#117—271no—
298C33#54111yescharacteristic, complemented, Sylow
299C23×C3—34no—
300C22×D3C22×C3⋊D311yescharacteristic
301C3×Q8—68no—
302C3×Q8—182no—
303C2×C3×C4—24no—
304C2×C3×C4—28no—
305C2×C3×C4—38no—
306C2×C3×C4—34no—
307C2×C3×C4—68no—
308C2×C3×C4—91no—
309C2×C3×C4—91no—
310C2×C3×C4—92no—
311C2×C3×C4C2×D3218yes—
312C2×C3×C4C2×D3214yes—
313C2×C3×C4C22×C3⋊D312yes—
314C2×C3×C4C22×C3⋊D311yescharacteristic
315C2×Dic3—34no—
316C2×Dic3—34no—
317C2×Dic3—38no—
318C2×Dic3—94no—
319C2×Dic3—98no—
320C2×Dic3—94no—
321C2×Dic3—91no—
322C2×Dic3—92no—
323C2×Dic3—91no—
324C2×Dic3—184no—
325C2×Dic3—184no—
326C2×Dic3—188no—
327C2×Dic3C22×C3⋊D312yes—
328C2×Dic3C22×C3⋊D311yescharacteristic
329C4×D3—22no—
330Dic6—22no—
331Dic6—68no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—24no—
335C2×C32—121no—
336C2×C32C2×#23511yescharacteristic
337C2×C32C2×#23714yes—
338C2×C32#100714yes—
339C2×C32#100711yescharacteristic
340C2×C32#101014yes—
341C2×C32#101011yescharacteristic
342C3×D3—44no—
343C2×Q8—31no—
344C2×Q8—91no—
345C22×C4—31no—
346C4⋊C4—32no—
347C4⋊C4—92no—
348C4⋊C4—271no—
349C4⋊C4—271no—
350C4⋊C4—272no—
351C4⋊C4—272no—
352C4⋊C4—272no—
353C4⋊C4—272no—
354C4⋊C4—271no—
355C4⋊C4—272no—
356C4⋊C4—272no—
357C4⋊C4D3×C3⋊D311yescharacteristic
358C22⋊C4—32no—
359C22⋊C4—271no—
360C22⋊C4—271no—
361C22⋊C4—272no—
362C42—91no—
363C42—92no—
364C42—272no—
365C42—271no—
366C22×C3—24no—
367C22×C3—64no—
368C22×C3—64no—
369C22×C3—64no—
370C22×C3C22×D3214yes—
371C22×C3C23×C3⋊D311yescharacteristic
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—21no—
375C3×C4—28no—
376C3×C4—22no—
377C3×C4—28no—
378C3×C4—48no—
379C3×C4—64no—
380C3×C4—68no—
381C3×C4—182no—
382C3×C4—181no—
383C3×C4—181no—
384C3×C4#379518yes—
385C3×C4#383012yes—
386Dic3—22no—
387Dic3—21no—
388Dic3—68no—
389Dic3—64no—
390Dic3—64no—
391Dic3—184no—
392Dic3—188no—
393Dic3—184no—
394Dic3—182no—
395Dic3—181no—
396Dic3—181no—
397Dic3—364no—
398Dic3—364no—
399Dic3—368no—
400C32—24no—
401C32#558311yescharacteristic, complemented
402C32#562514yescomplemented
403C23—31no—
404Q8—62no—
405Q8—182no—
406C2×C4—32no—
407C2×C4—31no—
408C2×C4—62no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—92no—
412C2×C4—271no—
413C2×C4—272no—
414C2×C4—271no—
415C2×C4C2×D3×C3⋊D312yes—
416C2×C4C2×D3×C3⋊D311yescharacteristic
417C2×C3—24no—
418C2×C3—24no—
419C2×C3—24no—
420C2×C3—124no—
421C2×C3C2×#379514yes—
422C2×C3#6430914yes—
423C2×C3#6431614yes—
424C2×C3C2×#383011yescharacteristic
425C2×C3#6436411yescharacteristic
426C2×C3#6436711yescharacteristic
427D3—41no—
428C22—61no—
429C22—61no—
430C22—61no—
431C22C22×D3×C3⋊D311yescharacteristic
432C4—22no—
433C4—61no—
434C4—62no—
435C4—182no—
436C4—181no—
437C4—181no—
438C4—541no—
439C4—542no—
440C4—541no—
441C4#8901412yes—
442C3—24no—
443C3#1059434514yescomplemented
444C3#1059609611yescharacteristic, complemented
445C2—121no—
446C2C2×#8901411yescharacteristic
447C2#1172458211yescharacteristic
448C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C32×D3C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723931C2semidirect
2. C3×#64096C2semidirect
4. #11723451C2semidirect
5. #11723447C2semidirect
10. #11723405C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
47. #64096D3semidirect
298. C33#541semidirect
401. C32#5583semidirect
402. C32#5625semidirect
443. C3#10594345semidirect
444. C3#10596096semidirect

Maximal subgroups

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

subgroupindex
#117239312
C3×#640962
C32×#8972
#117234512
#117234472
#117234302
#117234282
#117234252
#117234232
#117234052
Dic6×C3⋊Dic32
#105960963
#105943453