Finite Group Databasebeta

← #13175151

Subgroups — #13175151

Order 1728.

1432 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic, complemented
2C3×C4×#634C211yesmaximal, characteristic, complemented
3C32×#872C211yesmaximal, characteristic
4#11723454C211yesmaximal, characteristic, complemented
5#11723448C212yesmaximal, complemented
6#11723445C212yesmaximal, complemented
7#11723443C211yesmaximal, characteristic, complemented
8#11723432C212yesmaximal
9#11723428C212yesmaximal
10#11723426C211yesmaximal, characteristic
11#11723425C211yesmaximal, characteristic
12#10595989—31nomaximal
13#10593880—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×C4×D3C2211yescharacteristic
17C33×C42C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19#88857C2212yes—
20C4×C3⋊(C3⋊Dic3)C2212yes—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750C2212yes—
23C3×C4×C3⋊Dic3C2212yes—
24C32×(C2×D3)⋊C4C2211yescharacteristic
25C32×(C2×D3)⋊C4C2211yescharacteristic
26C32×Dic3⋊C4C2211yescharacteristic
27C32×Dic3⋊C4C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33Dic3⋊(C3⋊Dic3)C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×#117—31no—
38#64133—31no—
39#64133—31no—
40#64098—32no—
41#64098—32no—
42#64088—32no—
43#64088—32no—
44#64079—31no—
45#64078—31no—
46C4×#634—31no—
47C4×#634—34no—
48C4×#634—64no—
49C4×#634D311yescharacteristic, complemented
50C3×#872D314yes—
51C3×C4×Dic6—34no—
52#63870—38no—
53#63869—34no—
54#63867—38no—
55#63866—34no—
56#63862—38no—
57#63859—38no—
58#63857—34no—
59#63856—34no—
60C22×C32×D3C2311yescharacteristic
61C2×C33×C4C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×#634—22no—
70C2×C32×Dic3C2311yescharacteristic
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C32×C4×D3—22no—
74#5583—91no—
75#5518—94no—
76C22×C32×C4—31no—
77C2×#634—31no—
78C2×#634—34no—
79C2×#634—64no—
80C2×#634C2×D311yescharacteristic
81C2×C3×C4×D3C2×D314yes—
82C2×C3×Dic6—34no—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—31no—
85C32×C22⋊C4—31no—
86C32×C22⋊C4—31no—
87C32×C42—24no—
88C32×C42—31no—
89C32×C42C2×D314yes—
90C32×C42C2×D311yescharacteristic
91#3757—32no—
92#3757—32no—
93C4⋊(C3⋊Dic3)—31no—
94C4⋊(C3⋊Dic3)—34no—
95C4⋊(C3⋊Dic3)—32no—
96C4⋊(C3⋊Dic3)—32no—
97C4⋊(C3⋊Dic3)—64no—
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99#3750—32no—
100#3750—38no—
101#3750—32no—
102#3750—32no—
103#3750—32no—
104#3750—32no—
105#3750—68no—
106#3750C2×D312yes—
107C4×C3⋊Dic3—32no—
108C4×C3⋊Dic3—38no—
109C4×C3⋊Dic3—68no—
110C4×C3⋊Dic3C2×D312yes—
111C3×(C2×D3)⋊C4C2×D314yes—
112C3×(C2×D3)⋊C4C2×D314yes—
113C3×C4⋊Dic3—34no—
114C3×Dic3⋊C4—38no—
115C3×Dic3⋊C4C2×D314yes—
116C3×Dic3⋊C4C2×D314yes—
117C3×C4×Dic3—38no—
118C3×C4×Dic3C2×D314yes—
119#3724—38no—
120#3724—38no—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—38no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125(C2×D3)⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127C22×C33C2411yescharacteristic
128C2×C32×D3—21no—
129C2×C32×D3—21no—
130C2×C32×D3—21no—
131C33×C4—21no—
132C33×C4—21no—
133C33×C4#4112yes—
134C3⋊(C3⋊Dic3)—22no—
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—22no—
138C32×Dic3—21no—
139C32×Dic3—21no—
140C32×Dic3—21no—
141C3×C4×Q8—91no—
142C3×#117—34no—
143#946—91no—
144#944—91no—
145#924—94no—
146#924—94no—
147#898—92no—
148#897—91no—
149#890—91no—
150#889—98no—
151#889—98no—
152#889—92no—
153#879—98no—
154#879—98no—
155#876—92no—
156#872C3⋊D311yescharacteristic
157#870—94no—
158#870—92no—
159#869—94no—
160C4×Dic6—34no—
161C4×Dic6—94no—
162C4×Dic6—91no—
163C4×Dic6—184no—
164C23×C32—31no—
165C22×C3×D3C22×D314yes—
166C2×C32×C4—24no—
167C2×C32×C4—24no—
168C2×C32×C4—24no—
169C2×C32×C4—31no—
170C2×C32×C4—31no—
171C2×C32×C4—31no—
172C2×C32×C4—62no—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D314yes—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—38no—
182C2×C3⋊Dic3—38no—
183C2×C3⋊Dic3—68no—
184C2×C3⋊Dic3—68no—
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187#634—22no—
188#634—616no—
189#634—68no—
190#634—62no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3C22×D314yes—
194C2×C3×Dic3C22×D314yes—
195C2×C3×Dic3C22×D314yes—
196C3×C4×D3—28no—
197C3×Dic6—68no—
198#541—271noSylow
199C2×C33C2×#4111yescharacteristic
200C2×C332−1+411yescharacteristic
201C2×C332−1+411yescharacteristic
202C32×D3—41no—
203C2×C3×Q8—91no—
204C22×C3×C4—34no—
205C2×C4×D3C2×C3⋊D311yescharacteristic
206C2×Dic6—34no—
207C2×Dic6—94no—
208C2×Dic6—91no—
209C2×Dic6—184no—
210C3×C4⋊C4—34no—
211C3×C4⋊C4—34no—
212C3×C4⋊C4—92no—
213C3×C4⋊C4—91no—
214C3×C22⋊C4—34no—
215C3×C22⋊C4—34no—
216C3×C42—24no—
217C3×C42—34no—
218C3×C42—92no—
219C3×C42D3214yes—
220C3×C42C2×C3⋊D311yescharacteristic
221#217—98no—
222#217—98no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
226(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
227C4⋊Dic3—34no—
228C4⋊Dic3—98no—
229C4⋊Dic3—98no—
230C4⋊Dic3—94no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3—184no—
235Dic3⋊C4—38no—
236Dic3⋊C4—98no—
237Dic3⋊C4—98no—
238Dic3⋊C4—98no—
239Dic3⋊C4—98no—
240Dic3⋊C4—98no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—188no—
247Dic3⋊C4C2×C3⋊D311yescharacteristic
248Dic3⋊C4C2×C3⋊D311yescharacteristic
249C4×Dic3—38no—
250C4×Dic3—98no—
251C4×Dic3—92no—
252C4×Dic3—188no—
253C4×Dic3C2×C3⋊D311yescharacteristic
254C22×C32—24no—
255C22×C32—61no—
256C22×C32—61no—
257C22×C32—61no—
258C22×C32C23×D314yes—
259C22×C32C23×D311yescharacteristic
260C2×C3×D3—24no—
261C2×C3×D3—24no—
262C2×C3×D3—24no—
263C32×C4—28no—
264C32×C4—21no—
265C32×C4—24no—
266C32×C4—21no—
267C32×C4—28no—
268C32×C4—24no—
269C32×C4—44no—
270C32×C4—61no—
271C32×C4—61no—
272C32×C4—61no—
273C32×C4#23518yes—
274C32×C4#23512yes—
275C3⋊Dic3—22no—
276C3⋊Dic3—22no—
277C3⋊Dic3—616no—
278C3⋊Dic3—68no—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3⋊Dic3—68no—
282C3⋊Dic3—128no—
283C3×Dic3—24no—
284C3×Dic3—24no—
285C3×Dic3—24no—
286C3×Dic3—68no—
287C3×Dic3—68no—
288C4⋊Q8—271no—
289#126—272no—
290#126—272no—
291#125—271no—
292#125—272no—
293#125—272no—
294C22⋊Q8—271no—
295C22⋊Q8—271no—
296C4×Q8—91no—
297C4×Q8—271no—
298#117—31no—
299C33#54111yescharacteristic, complemented, Sylow
300C23×C3—34no—
301C22×D3C22×C3⋊D311yescharacteristic
302C3×Q8—182no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—34no—
307C2×C3×C4—34no—
308C2×C3×C4—34no—
309C2×C3×C4—68no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4C2×D3214yes—
313C2×C3×C4C2×D3214yes—
314C2×C3×C4C2×D3214yes—
315C2×C3×C4C22×C3⋊D311yescharacteristic
316C2×C3×C4C22×C3⋊D311yescharacteristic
317C2×C3×C4C22×C3⋊D311yescharacteristic
318C2×Dic3—38no—
319C2×Dic3—38no—
320C2×Dic3—98no—
321C2×Dic3—98no—
322C2×Dic3—92no—
323C2×Dic3—92no—
324C2×Dic3—188no—
325C2×Dic3—188no—
326C2×Dic3C22×C3⋊D311yescharacteristic
327C2×Dic3C22×C3⋊D311yescharacteristic
328C2×Dic3C22×C3⋊D311yescharacteristic
329C4×D3—22no—
330Dic6—68no—
331Dic6—1816no—
332Dic6—188no—
333Dic6—182no—
334C2×C32—24no—
335C2×C32—24no—
336C2×C32—24no—
337C2×C32—121no—
338C2×C32C2×#23514yes—
339C2×C32C2×#23511yescharacteristic
340C2×C32#100711yescharacteristic
341C2×C32#101014yes—
342C2×C32#101014yes—
343C2×C32#101011yescharacteristic
344C3×D3—44no—
345C2×Q8—91no—
346C2×Q8—271no—
347C22×C4—31no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—91no—
351C4⋊C4—92no—
352C4⋊C4—272no—
353C4⋊C4—272no—
354C4⋊C4—272no—
355C4⋊C4—272no—
356C4⋊C4—272no—
357C4⋊C4—272no—
358C4⋊C4—271no—
359C4⋊C4—272no—
360C22⋊C4—31no—
361C22⋊C4—31no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C42—31no—
365C42—92no—
366C42—272no—
367C42D3×C3⋊D311yescharacteristic
368C22×C3—24no—
369C22×C3—64no—
370C22×C3—64no—
371C22×C3—64no—
372C22×C3C22×D3214yes—
373C22×C3C23×C3⋊D311yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C3×C4—28no—
378C3×C4—24no—
379C3×C4—24no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—28no—
383C3×C4—44no—
384C3×C4—64no—
385C3×C4—64no—
386C3×C4—64no—
387C3×C4—182no—
388C3×C4—182no—
389C3×C4#379818yes—
390C3×C4#382812yes—
391Dic3—21no—
392Dic3—21no—
393Dic3—21no—
394Dic3—68no—
395Dic3—68no—
396Dic3—1816no—
397Dic3—188no—
398Dic3—188no—
399Dic3—182no—
400Dic3—182no—
401Dic3—368no—
402C32—24no—
403C32#551814yescomplemented
404C32#558311yescharacteristic, complemented
405C23—31no—
406Q8—182no—
407Q8—542no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—62no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—272no—
415C2×C4—272no—
416C2×C4C2×D3×C3⋊D311yescharacteristic
417C2×C4C2×D3×C3⋊D311yescharacteristic
418C2×C4C2×D3×C3⋊D311yescharacteristic
419C2×C3—24no—
420C2×C3—24no—
421C2×C3—24no—
422C2×C3—124no—
423C2×C3#6429714yes—
424C2×C3#6429814yes—
425C2×C3C2×#379814yes—
426C2×C3C2×#382811yescharacteristic
427C2×C3#6436711yescharacteristic
428C2×C3#6436711yescharacteristic
429D3—41no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22C22×D3×C3⋊D311yescharacteristic
434C4—21no—
435C4—21no—
436C4—61no—
437C4—61no—
438C4—61no—
439C4—182no—
440C4—182no—
441C4—542no—
442C4—542no—
443C4#8902012yes—
444C3—24no—
445C3#1059388014yescomplemented
446C3#1059598911yescharacteristic, complemented
447C2—121no—
448C2#1172456211yescharacteristic
449C2#1172456311yescharacteristic
450C2C2×#8902011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC22×C32×D3C23
hypercenterC42D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
2. C3×C4×#634C2semidirect
4. #11723454C2semidirect
5. #11723448C2semidirect
6. #11723445C2semidirect
7. #11723443C2semidirect
49. C4×#634D3semidirect
299. C33#541semidirect
403. C32#5518semidirect
404. C32#5583semidirect
445. C3#10593880semidirect
446. C3#10595989semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×C4×#6342
C32×#8722
#117234542
#117234482
#117234452
#117234432
#117234322
#117234282
#117234262
#117234252
#105959893
#105938803