Finite Group Databasebeta

← #13177463

Subgroups — #13177463

Order 1728.

1780 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×#938C211yesmaximal, characteristic
4#67⋊(C3⋊Dic3)C212yesmaximal, complemented
5#11723614C212yesmaximal, complemented
6#11723598C212yesmaximal, complemented
7#11723597C212yesmaximal, complemented
8C4×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9#11723447C211yesmaximal, characteristic, complemented
10#11723425C211yesmaximal, characteristic, complemented
11#11723410C211yesmaximal, characteristic, complemented
12#10596071—31nomaximal
13#10595173—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×C4×D3C2211yescharacteristic
19C2×D3×C3⋊Dic3C2211yescharacteristic
20C2×D3×C3⋊Dic3C2211yescharacteristic
21#88864C2212yes—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3757C2212yes—
25C3×#3757C2212yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29#88807C2212yes—
30#88806C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34Dic3×C3⋊Dic3C2211yescharacteristic
35Dic3×C3⋊Dic3C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×#121—31no—
38#64145D311yescharacteristic, complemented
39#64144—32no—
40D4×C3⋊Dic3—31no—
41D4×C3⋊Dic3—32no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44#64133—31no—
45C2×C4×C3⋊Dic3—31no—
46#64099—31no—
47#64096—31no—
48#64088—32no—
49#64086—32no—
50C3×#938D314yes—
51C3×#936—34no—
52#67⋊Dic3—38no—
53#63969—38no—
54#63957—38no—
55#63956—38no—
56C4×D3×Dic3—34no—
57#63869—34no—
58#63856—34no—
59#63843—34no—
60C23×C33C2312yes—
61C22×C32×D3C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)—24no—
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C3×#634—22no—
72C32×#67—24no—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C32×C4×D3—22no—
76D3×C3⋊Dic3—22no—
77D3×C3⋊Dic3—22no—
78#5680—91no—
79#5600—94no—
80C2×C32×D4—24no—
81C2×C32×D4—32no—
82C2×C32×D4C2×D314yes—
83C2×C32×D4C2×D311yescharacteristic
84C22×C32×C4—31no—
85C22×C3⋊Dic3—32no—
86C22×C3⋊Dic3—38no—
87C22×C3⋊Dic3—31no—
88C22×C3⋊Dic3—31no—
89C22×C3⋊Dic3—68no—
90C2×#634C2×D311yescharacteristic
91C2×C3×#67C2×D318yes—
92C2×C3×C4×D3C2×D314yes—
93C2×C3×Dic6—34no—
94C2×D3×Dic3—34no—
95C2×D3×Dic3—34no—
96C32×C4⋊C4—31no—
97C32×C22⋊C4—32no—
98#3757—32no—
99#3757—38no—
100#3757—32no—
101#3757—68no—
102#3757C2×D312yes—
103#3757C2×D312yes—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—34no—
106C4⋊(C3⋊Dic3)—32no—
107C4⋊(C3⋊Dic3)—64no—
108#3750—32no—
109#3750—32no—
110#3750—32no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—34no—
113C4×C3⋊Dic3—31no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3—64no—
117C4×C3⋊Dic3C2×D311yescharacteristic
118C3×#217—38no—
119C3×#217—38no—
120C3×#217C2×D318yes—
121C3×C4⋊Dic3C2×D314yes—
122C3×C4×Dic3—34no—
123#3724—38no—
124#3724—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128Dic32—38no—
129Dic32—34no—
130Dic32—34no—
131C22×C33—22no—
132C22×C33—22no—
133C22×C33—22no—
134C22×C33C2411yescharacteristic
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C2×C32×D3#4112yes—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—21no—
140C3⋊(C3⋊Dic3)—22no—
141C3⋊(C3⋊Dic3)#4112yes—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—22no—
145C32×Dic3—21no—
146C32×Dic3—22no—
147C3×#124—91no—
148C3×#121—34no—
149#946—91no—
150Q8×Dic3—91no—
151#938C3⋊D311yescharacteristic
152#936—34no—
153#935—98no—
154D4×Dic3—98no—
155D4×Dic3—94no—
156D4×Dic3—91no—
157D4×Dic3—184no—
158#924—94no—
159C2×C4×Dic3—94no—
160#890—94no—
161#887—94no—
162#883—92no—
163#67⋊C4—92no—
164#879—98no—
165#879—92no—
166#877—98no—
167#877—92no—
168C42×D3—91no—
169#870—91no—
170C23×C32—28no—
171C23×C32—31no—
172C23×C32C22×D318yes—
173C23×C32C22×D312yes—
174C22×C3×D3C22×D314yes—
175C32×D4—216no—
176C32×D4—28no—
177C32×D4—22no—
178C32×D4—64no—
179C2×C32×C4—24no—
180C2×C32×C4—31no—
181C2×C32×C4—32no—
182C2×C32×C4—62no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—34no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—64no—
193C2×C3⋊Dic3—68no—
194C2×C3⋊Dic3—62no—
195C2×C3⋊Dic3—62no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—616no—
198C2×C3⋊Dic3—1216no—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202#634—22no—
203C3×#67—216no—
204C2×C3×Dic3—38no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3C22×D314yes—
208C2×C3×Dic3C22×D318yes—
209C3×C4×D3—28no—
210C3×Dic6—68no—
211D3×Dic3—68no—
212D3×Dic3—68no—
213#529—271noSylow
214C2×C33—42no—
215C2×C33C2×#4111yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C2×C33C2×#4111yescharacteristic
218C32×D3—22no—
219C2×C3×Q8—91no—
220C2×C3×D4—24no—
221C2×C3×D4—38no—
222C2×C3×D4D3214yes—
223C2×C3×D4C2×C3⋊D311yescharacteristic
224C22×C3×C4—34no—
225C2×#67C2×C3⋊D312yes—
226C22×Dic3—94no—
227C22×Dic3—94no—
228C22×Dic3—98no—
229C22×Dic3—92no—
230C22×Dic3—188no—
231C2×C4×D3—91no—
232C2×C4×D3—91no—
233C2×C4×D3C2×C3⋊D311yescharacteristic
234C2×Dic6—34no—
235C3×C4⋊C4—34no—
236C3×C22⋊C4—38no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C42—91no—
240#217—38no—
241#217—38no—
242#217—98no—
243#217—98no—
244#217—92no—
245#217—188no—
246#217C2×C3⋊D312yes—
247(C2×D3)⋊C4—92no—
248C4⋊Dic3—98no—
249C4⋊Dic3—94no—
250C4⋊Dic3—91no—
251C4⋊Dic3—92no—
252C4⋊Dic3—184no—
253C4⋊Dic3C2×C3⋊D311yescharacteristic
254Dic3⋊C4—98no—
255Dic3⋊C4—98no—
256Dic3⋊C4—98no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260C4×Dic3—34no—
261C4×Dic3—94no—
262C4×Dic3—98no—
263C4×Dic3—94no—
264C4×Dic3—94no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—92no—
269C4×Dic3—184no—
270C22×C32—216no—
271C22×C32—28no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—24no—
276C22×C32—28no—
277C22×C32—28no—
278C22×C32—32no—
279C22×C32—48no—
280C22×C32—48no—
281C22×C32—61no—
282C22×C32—61no—
283C22×C32C23×D314yes—
284C22×C32C23×D311yescharacteristic
285C2×C3×D3—24no—
286C2×C3×D3—24no—
287C2×C3×D3#23718yes—
288C32×C4—28no—
289C32×C4—61no—
290C32×C4—62no—
291C32×C4#23718yes—
292C32×C4#23712yes—
293C3⋊Dic3—22no—
294C3⋊Dic3—21no—
295C3⋊Dic3—21no—
296C3⋊Dic3—32no—
297C3⋊Dic3—38no—
298C3⋊Dic3—68no—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3⋊Dic3—64no—
302C3⋊Dic3—68no—
303C3⋊Dic3—124no—
304C3⋊Dic3—128no—
305C3×Dic3—24no—
306C3×Dic3—28no—
307C3×Dic3—64no—
308C3×Dic3—68no—
309C3×Dic3—64no—
310#126—272no—
311#125—271no—
312#124—91no—
313#123—272no—
314C22⋊Q8—271no—
315#121—31no—
316C4×Q8—271no—
317C4×D4—272no—
318C4×D4—271no—
319#117—272no—
320C2×C42—271no—
321C33#52911yescharacteristic, complemented, Sylow
322C23×C3—28no—
323C23×C3—34no—
324C23×C3C2×D3218yes—
325C23×C3C22×C3⋊D312yes—
326C22×D3C22×C3⋊D311yescharacteristic
327C3×Q8—182no—
328C3×D4—216no—
329C3×D4—28no—
330C3×D4—22no—
331C3×D4—616no—
332C2×C3×C4—24no—
333C2×C3×C4—34no—
334C2×C3×C4—38no—
335C2×C3×C4—68no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—92no—
339C2×C3×C4C2×D3214yes—
340C2×C3×C4C22×C3⋊D311yescharacteristic
341#67—24no—
342C2×Dic3—34no—
343C2×Dic3—34no—
344C2×Dic3—38no—
345C2×Dic3—94no—
346C2×Dic3—91no—
347C2×Dic3—94no—
348C2×Dic3—98no—
349C2×Dic3—91no—
350C2×Dic3—92no—
351C2×Dic3—184no—
352C2×Dic3—184no—
353C2×Dic3—188no—
354C2×Dic3—188no—
355C2×Dic3—188no—
356C2×Dic3—184no—
357C2×Dic3—1816no—
358C2×Dic3—3616no—
359C2×Dic3C22×C3⋊D311yescharacteristic
360C2×Dic3C22×C3⋊D312yes—
361C4×D3—22no—
362C4×D3—182no—
363C4×D3—182no—
364Dic6—68no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—416no—
369C2×C32—42no—
370C2×C32—48no—
371C2×C32—62no—
372C2×C32C2×#23714yes—
373C2×C32C2×#23714yes—
374C2×C32C2×#23711yescharacteristic
375C2×C32D3×#4114yes—
376C2×C32D3×#4111yescharacteristic
377C2×C32D3×#4111yescharacteristic
378C3×D3—28no—
379C2×Q8—91no—
380C2×D4—32no—
381C2×D4D3×C3⋊D311yescharacteristic
382C22×C4—31no—
383C22×C4—271no—
384C22×C4—271no—
385C22×C4—272no—
386C4⋊C4—31no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—32no—
393C22⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—91no—
398C42—271no—
399C42—271no—
400C42—272no—
401C42—271no—
402C22×C3—216no—
403C22×C3—28no—
404C22×C3—28no—
405C22×C3—28no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—24no—
410C22×C3—38no—
411C22×C3—48no—
412C22×C3—48no—
413C22×C3—64no—
414C22×C3—64no—
415C22×C3C22×D3214yes—
416C22×C3C23×C3⋊D311yescharacteristic
417C2×D3—21no—
418C2×D3—21no—
419C2×D3#383012yes—
420C3×C4—28no—
421C3×C4—68no—
422C3×C4—64no—
423C3×C4—181no—
424C3×C4—182no—
425C3×C4—181no—
426C3×C4#380518yes—
427C3×C4#383012yes—
428Dic3—21no—
429Dic3—22no—
430Dic3—64no—
431Dic3—64no—
432Dic3—68no—
433Dic3—98no—
434Dic3—92no—
435Dic3—188no—
436Dic3—184no—
437Dic3—188no—
438Dic3—181no—
439Dic3—182no—
440Dic3—368no—
441Dic3—364no—
442C32—24no—
443C32#560014yescomplemented
444C32#568011yescharacteristic, complemented
445C23—31no—
446C23C2×D3×C3⋊D312yes—
447Q8—182no—
448D4—22no—
449D4—64no—
450C2×C4—31no—
451C2×C4—32no—
452C2×C4—62no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—92no—
456C2×C4—271no—
457C2×C4—271no—
458C2×C4—272no—
459C2×C4—542no—
460C2×C4—542no—
461C2×C4—544no—
462C2×C4C2×D3×C3⋊D311yescharacteristic
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—416no—
467C2×C3—48no—
468C2×C3—42no—
469C2×C3—68no—
470C2×C3D3×#23714yes—
471C2×C3#6431214yes—
472C2×C3C2×#380514yes—
473C2×C3C2×#383011yescharacteristic
474C2×C3C2×#383011yescharacteristic
475C2×C3#41×C3⋊D311yescharacteristic
476D3—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—32no—
481C22—61no—
482C22—61no—
483C22C22×D3×C3⋊D311yescharacteristic
484C4—62no—
485C4—61no—
486C4—181no—
487C4—181no—
488C4—182no—
489C4—272no—
490C4—541no—
491C4—542no—
492C4#8902912yes—
493C3—24no—
494C3#1059517314yescomplemented
495C3#1059607111yescharacteristic, complemented
496C2—42no—
497C2—62no—
498C2D3×#383011yescharacteristic
499C2#1172458611yescharacteristic
500C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC32×#938C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊(C3⋊Dic3)C2semidirect
2. C3×#64145C2semidirect
4. #67⋊(C3⋊Dic3)C2semidirect
5. #11723614C2semidirect
6. #11723598C2semidirect
7. #11723597C2semidirect
8. C4×D3×C3⋊Dic3C2semidirect
9. #11723447C2semidirect
10. #11723425C2semidirect
11. #11723410C2semidirect
38. #64145D3semidirect
321. C33#529semidirect
443. C32#5600semidirect
444. C32#5680semidirect
494. C3#10595173semidirect
495. C3#10596071semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C3×#641452
C32×#9382
#67⋊(C3⋊Dic3)2
#117236142
#117235982
#117235972
C4×D3×C3⋊Dic32
#117234472
#117234252
#117234102
#105960713
#105951733