Finite Group Databasebeta

← #13176805

Subgroups — #13176805

Order 1728.

1660 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6312C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C32×#924C211yesmaximal, characteristic, complemented
4#11723608C212yesmaximal
5#11723595C212yesmaximal, complemented
6#11723456C211yesmaximal, characteristic, complemented
7#11723450C211yesmaximal, characteristic, complemented
8#11723413C211yesmaximal, characteristic, complemented
9#11723403C212yesmaximal, complemented
10#11723401C212yesmaximal, complemented
11#11723397C211yesmaximal, characteristic, complemented
12#10596113—31nomaximal
13#10594825—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yes—
18C2×C32×Dic6C2211yescharacteristic
19#88864C2211yescharacteristic
20#88859C2211yescharacteristic
21#88857C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2212yes—
29#88807C2212yes—
30#88807C2212yes—
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38#64155—31no—
39Q8×C3⋊Dic3—31no—
40C4×#638—31no—
41C4×#638—34no—
42C4×#638—64no—
43C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
44#64107—32no—
45#64101—31no—
46#64099—32no—
47#64099—31no—
48#64095—32no—
49#64086—32no—
50C2×C3×C4⋊Dic3—34no—
51C3×#924D314yescomplemented
52#63966—38no—
53#63955—38no—
54#63872—34no—
55#63871—34no—
56#63849—34no—
57#63848—38no—
58#63846—38no—
59#63843—34no—
60C23×C33C2311yescharacteristic
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65#6312—22no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4×C3⋊(C3⋊D3)—22no—
70C2×C3×C3⋊Dic3—24no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C32×Dic6—22no—
76#5642—94no—
77#5513—91no—
78C2×C32×Q8—31no—
79C22×C32×C4—24no—
80C22×C32×C4C2×D314yes—
81C22×C32×C4C2×D311yescharacteristic
82C2×#638—31no—
83C2×#638—34no—
84C2×#638—64no—
85C22×C3⋊Dic3C2×D312yes—
86C2×C4×C3⋊D3—31no—
87C2×C4×C3⋊D3—34no—
88C2×C4×C3⋊D3—64no—
89C22×C3×Dic3—38no—
90C2×C3×Dic6C2×D314yes—
91C32×C4⋊C4—31no—
92C32×C4⋊C4—32no—
93C32×C22⋊C4—32no—
94#3757—31no—
95#3757—34no—
96#3757—64no—
97#3752—31no—
98#3752—34no—
99#3752—32no—
100#3752—32no—
101#3752—64no—
102C4⋊(C3⋊Dic3)—32no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)C2×D312yes—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750—31no—
108#3750—34no—
109#3750—32no—
110#3750—32no—
111#3750—64no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—32no—
116C4×C3⋊Dic3—64no—
117C3×#217C2×D318yes—
118C3×C4⋊Dic3—34no—
119C3×C4⋊Dic3—38no—
120C3×C4⋊Dic3—34no—
121C3×C4⋊Dic3C2×D314yes—
122C3×Dic3⋊C4C2×D318yes—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127#3722—38no—
128#3722—38no—
129Dic32—38no—
130Dic32—38no—
131C22×C33—21no—
132C22×C33—21no—
133C22×C33#4112yes—
134C22×C33C2411yescharacteristic
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—21no—
138C33×C4—21no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—21no—
141C3⋊(C3⋊Dic3)—21no—
142C3⋊(C3⋊Dic3)—21no—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—22no—
145C32×Dic3—22no—
146C32×Dic3—22no—
147C3×C22⋊Q8—34no—
148C2×C3×C4⋊C4—91no—
149#946—94no—
150Q8×Dic3—94no—
151#929—92no—
152C4×#67—94no—
153C4×#67—91no—
154C4×#67—184no—
155#926—92no—
156C2×C4⋊Dic3—34no—
157#924C3⋊D311yescharacteristic, complemented
158#898—98no—
159#898—92no—
160#897—91no—
161#892—94no—
162#892—91no—
163#890—98no—
164#890—94no—
165#890—91no—
166#889—92no—
167#887—91no—
168#886—98no—
169#877—98no—
170C23×C32—24no—
171C23×C32C22×D314yes—
172C23×C32C22×D311yescharacteristic
173C22×C3⋊D3—31no—
174C22×C3⋊D3—34no—
175C22×C3⋊D3—64no—
176C32×Q8—62no—
177C2×C32×C4—22no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—28no—
181C2×C32×C4—32no—
182C2×C32×C4—32no—
183C2×C32×C4—48no—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—616no—
189#638—62no—
190#638—68no—
191C2×C3⋊Dic3—24no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—34no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D312yes—
203C4×C3⋊D3—62no—
204C4×C3⋊D3—68no—
205C4×C3⋊D3—128no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—616no—
209C2×C3×Dic3C22×D318yes—
210C2×C3×Dic3C22×D318yes—
211C3×Dic6—28no—
212#552—271noSylow
213C2×C33—22no—
214C2×C33C2×#4111yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C332−1+411yescharacteristic
217C3⋊(C3⋊D3)—41no—
218C2×C3×Q8—34no—
219C22×C3×C4—24no—
220C22×C3×C4—92no—
221C22×C3×C4D3214yes—
222C22×C3×C4C2×C3⋊D311yescharacteristic
223C2×#67—94no—
224C2×#67—91no—
225C2×#67—184no—
226C22×Dic3—38no—
227C2×C4×D3—94no—
228C2×C4×D3—91no—
229C2×C4×D3—184no—
230C2×Dic6C2×C3⋊D311yescharacteristic
231C3×C4⋊C4—34no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—91no—
236C3×C22⋊C4—38no—
237#217—94no—
238#217—91no—
239#217—184no—
240#217C2×C3⋊D312yes—
241(C2×D3)⋊C4—98no—
242(C2×D3)⋊C4—98no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—91no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—184no—
248C4⋊Dic3—38no—
249C4⋊Dic3—34no—
250C4⋊Dic3—34no—
251C4⋊Dic3—98no—
252C4⋊Dic3—98no—
253C4⋊Dic3—92no—
254C4⋊Dic3—92no—
255C4⋊Dic3C2×C3⋊D311yescharacteristic
256Dic3⋊C4—98no—
257Dic3⋊C4—98no—
258Dic3⋊C4—94no—
259Dic3⋊C4—91no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—184no—
263Dic3⋊C4C2×C3⋊D312yes—
264C4×Dic3—98no—
265C4×Dic3—98no—
266C4×Dic3—94no—
267C4×Dic3—91no—
268C4×Dic3—92no—
269C4×Dic3—92no—
270C4×Dic3—184no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—24no—
274C22×C32—28no—
275C22×C32—24no—
276C22×C32—24no—
277C22×C32—44no—
278C22×C32—44no—
279C22×C32#23712yes—
280C22×C32#23918yes—
281C22×C32C23×D314yes—
282C22×C32C23×D311yescharacteristic
283C2×C3⋊D3—68no—
284C2×C3⋊D3—61no—
285C2×C3⋊D3—61no—
286C2×C3⋊D3—64no—
287C2×C3⋊D3—61no—
288C2×C3⋊D3—64no—
289C2×C3⋊D3—64no—
290C2×C3⋊D3—124no—
291C2×C3⋊D3—124no—
292C32×C4—21no—
293C32×C4—28no—
294C32×C4—24no—
295C32×C4—44no—
296C32×C4—62no—
297C32×C4—62no—
298C32×C4#23512yes—
299C32×C4#23718yes—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3—68no—
303C3⋊Dic3—61no—
304C3⋊Dic3—61no—
305C3⋊Dic3—64no—
306C3⋊Dic3—61no—
307C3⋊Dic3—64no—
308C3⋊Dic3—64no—
309C3⋊Dic3—124no—
310C3⋊Dic3—124no—
311C3×Dic3—28no—
312C3×Dic3—28no—
313C3×Dic3—68no—
314C3×Dic3—68no—
315#126—272no—
316#125—272no—
317#125—271no—
318#123—272no—
319C22⋊Q8—31no—
320C22⋊Q8—271no—
321C4×Q8—271no—
322C4×D4—271no—
323#117—272no—
324#117—271no—
325C2×C4⋊C4—91no—
326C33#55211yescharacteristic, complemented, Sylow
327C23×C3—24no—
328C23×C3C2×D3214yes—
329C23×C3C22×C3⋊D311yescharacteristic
330C22×D3—94no—
331C22×D3—91no—
332C22×D3—184no—
333C3×Q8—68no—
334C2×C3×C4—28no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—38no—
339C2×C3×C4—38no—
340C2×C3×C4—48no—
341C2×C3×C4—92no—
342C2×C3×C4—92no—
343C2×C3×C4—184no—
344C2×C3×C4C2×D3214yes—
345C2×C3×C4C2×D3214yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347C2×C3×C4C22×C3⋊D311yescharacteristic
348#67—1816no—
349#67—188no—
350#67—182no—
351C2×Dic3—38no—
352C2×Dic3—38no—
353C2×Dic3—616no—
354C2×Dic3—94no—
355C2×Dic3—94no—
356C2×Dic3—94no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—184no—
361C2×Dic3—184no—
362C2×Dic3—184no—
363C2×Dic3C22×C3⋊D312yes—
364C2×Dic3C22×C3⋊D312yes—
365C4×D3—188no—
366C4×D3—182no—
367C4×D3—368no—
368Dic6—22no—
369C2×C32—22no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—28no—
374C2×C32—48no—
375C2×C32C2×#23511yescharacteristic
376C2×C32C2×#23714yes—
377C2×C32C2×#23711yescharacteristic
378C2×C32C2×#23914yes—
379C2×C32#100711yescharacteristic
380C2×C32#101014yes—
381C3⋊D3—128no—
382C3⋊D3—124no—
383C3⋊D3—121no—
384C2×Q8—31no—
385C2×D4—271no—
386C22×C4—92no—
387C22×C4—271no—
388C22×C4D3×C3⋊D311yescharacteristic
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—91no—
392C4⋊C4—91no—
393C4⋊C4—92no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—272no—
397C4⋊C4—272no—
398C4⋊C4—271no—
399C22⋊C4—32no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—271no—
403C22⋊C4—271no—
404C42—272no—
405C42—272no—
406C42—271no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—24no—
412C22×C3—28no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3#379618yes—
416C22×C3#383212yes—
417C22×C3C22×D3214yes—
418C22×C3C23×C3⋊D311yescharacteristic
419C2×D3—188no—
420C2×D3—184no—
421C2×D3—184no—
422C2×D3—184no—
423C2×D3—181no—
424C2×D3—181no—
425C2×D3—181no—
426C2×D3—364no—
427C2×D3—364no—
428C3×C4—24no—
429C3×C4—21no—
430C3×C4—28no—
431C3×C4—44no—
432C3×C4—68no—
433C3×C4—68no—
434C3×C4—182no—
435C3×C4—182no—
436C3×C4#380418yes—
437C3×C4#383012yes—
438Dic3—22no—
439Dic3—22no—
440Dic3—68no—
441Dic3—68no—
442Dic3—188no—
443Dic3—184no—
444Dic3—184no—
445Dic3—184no—
446Dic3—181no—
447Dic3—181no—
448Dic3—181no—
449Dic3—364no—
450Dic3—364no—
451C32—24no—
452C32#551311yescharacteristic, complemented
453C32#564214yescomplemented
454C23—271no—
455C23C2×D3×C3⋊D311yescharacteristic
456Q8—62no—
457D4—542no—
458C2×C4—22no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—92no—
462C2×C4—92no—
463C2×C4—184no—
464C2×C4—271no—
465C2×C4—271no—
466C2×C4—271no—
467C2×C4—542no—
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C4C2×D3×C3⋊D311yescharacteristic
470C2×C3—28no—
471C2×C3—22no—
472C2×C3—24no—
473C2×C3—24no—
474C2×C3—24no—
475C2×C3—48no—
476C2×C3C2×#379614yes—
477C2×C3#6429714yes—
478C2×C3C2×#380414yes—
479C2×C3C2×#383011yescharacteristic
480C2×C3C2×#383211yescharacteristic
481C2×C3#6436711yescharacteristic
482D3—368no—
483D3—364no—
484D3—361no—
485C22—21no—
486C22—21no—
487C22—541no—
488C22—541no—
489C22—541no—
490C22#8901712yes—
491C22C22×D3×C3⋊D311yescharacteristic
492C4—21no—
493C4—62no—
494C4—62no—
495C4—182no—
496C4—182no—
497C4—541no—
498C4—541no—
499C4—541no—
500C4#8902812yes—
501C3—24no—
502C3#1059482514yescomplemented
503C3#1059611311yescharacteristic, complemented
504C2—22no—
505C2—1081no—
506C2#1172456211yescharacteristic
507C2C2×#8901711yescharacteristic
508C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC2×#6312C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C32×#924C2semidirect
5. #11723595C2semidirect
6. #11723456C2semidirect
7. #11723450C2semidirect
8. #11723413C2semidirect
9. #11723403C2semidirect
10. #11723401C2semidirect
11. #11723397C2semidirect
43. C2×C4⋊(C3⋊Dic3)D3semidirect
51. C3×#924D3semidirect
157. #924C3⋊D3semidirect
326. C33#552semidirect
452. C32#5513semidirect
453. C32#5642semidirect
502. C3#10594825semidirect
503. C3#10596113semidirect

Maximal subgroups

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

subgroupindex
C4×#63122
C2×C3×C4⋊(C3⋊Dic3)2
C32×#9242
#117236082
#117235952
#117234562
#117234502
#117234132
#117234032
#117234012
#117233972
#105961133
#105948253