Finite Group Databasebeta

← #13176449

Subgroups — #13176449

Order 1728.

1592 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723938C211yesmaximal, characteristic, complemented
2C3×#64102C211yesmaximal, characteristic, complemented
3C32×#888C211yesmaximal, characteristic, complemented
4#11723532C211yesmaximal, characteristic
5#11723452C211yesmaximal, characteristic, complemented
6#11723439C211yesmaximal, characteristic, complemented
7#11723435C212yesmaximal, complemented
8#11723417C212yesmaximal, complemented
9#11723408C212yesmaximal, complemented
10#11723406C211yesmaximal, characteristic, complemented
11#11723392C212yesmaximal, complemented
12#10596204—31nomaximal
13#10594341—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yes—
18C2×C32×Dic6C2212yes—
19C2×#6292C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21#88859C2212yes—
22C4⋊(C3⋊(C3⋊Dic3))C2212yes—
23C3×#3752C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C32×C4⋊Dic3C2211yescharacteristic
26C32×Dic3⋊C4C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88806C2212yes—
29#88806C2211yescharacteristic
30#88806C2211yescharacteristic
31#88804C2211yescharacteristic
32#88804C2211yescharacteristic
33#88803C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4⋊Q8—31no—
38#64156—32no—
39#64155—32no—
40#64107—32no—
41#64106—31no—
42#64106—34no—
43#64106—64no—
44#64104—31no—
45#64103—32no—
46#64102D311yescharacteristic, complemented
47#64096—31no—
48#64084—31no—
49#64081—31no—
50C3×#893—34no—
51C3×#888D314yescomplemented
52#63916—34no—
53#63907—38no—
54#63874—34no—
55#63861—38no—
56#63860—34no—
57#63856—38no—
58#63850—38no—
59#63844—34no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C4×C3⋊(C3⋊D3)—22no—
67#6308—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4⋊(C3⋊D3)—22no—
71C3×C4×C3⋊D3—42no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—42no—
76#6292—42no—
77#5733—94no—
78#5561—91no—
79C2×C32×Q8—32no—
80C2×C4⋊(C3⋊D3)—32no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—64no—
85C2×C4×C3⋊D3C2×D312yes—
86C2×#634—31no—
87C2×#634—34no—
88C2×#634—64no—
89C2×C3×D12—34no—
90C2×C3×C4×D3—38no—
91C2×C3×Dic6C2×D318yes—
92C2×#626—38no—
93C32×C4⋊C4—24no—
94C32×C4⋊C4—31no—
95C32×C4⋊C4—32no—
96C32×C4⋊C4C2×D314yes—
97C32×C4⋊C4C2×D311yescharacteristic
98C32×C42—31no—
99#3752—32no—
100#3752—32no—
101#3752—32no—
102#3752—32no—
103#3752—38no—
104#3752—31no—
105#3752—31no—
106#3752—68no—
107#3752C2×D312yes—
108C4⋊(C3⋊Dic3)—32no—
109C4⋊(C3⋊Dic3)—38no—
110C4⋊(C3⋊Dic3)—68no—
111#3750—31no—
112#3750—31no—
113#3750—32no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×(C2×D3)⋊C4—38no—
117C3×C4⋊Dic3C2×D314yes—
118C3×Dic3⋊C4C2×D318yes—
119C3×C4×Dic3—34no—
120C3×C4×Dic3C2×D314yes—
121#3724—38no—
122#3724—34no—
123#3724—34no—
124#3722—34no—
125#3722—34no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129Dic32—38no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—21no—
132C2×C3⋊(C3⋊D3)—21no—
133C2×C3⋊(C3⋊D3)—21no—
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C33×C4—22no—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—22no—
140C3⋊(C3⋊Dic3)—21no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—21no—
143C32×Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×C4⋊Q8—34no—
147C3×C4×D4—91no—
148#947—98no—
149#946—98no—
150#938—91no—
151#936—91no—
152#929—92no—
153#924—92no—
154#898—98no—
155#897—94no—
156#897—91no—
157#897—184no—
158#895—94no—
159#894—98no—
160#893—34no—
161#888C3⋊D311yescharacteristic, complemented
162#887—94no—
163#885—92no—
164#879—92no—
165#875—94no—
166#872—94no—
167#872—91no—
168C4×Dic6—91no—
169C22×C3⋊D3—31no—
170C22×C3⋊D3—34no—
171C22×C3⋊D3—64no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—38no—
174C32×Q8—122no—
175C2×C32×C4—24no—
176C2×C32×C4—28no—
177C2×C32×C4—32no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4C22×D318yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—38no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C4⋊(C3⋊D3)—22no—
193C4⋊(C3⋊D3)—122no—
194C4×C3⋊D3—42no—
195C4×C3⋊D3—62no—
196C4×C3⋊D3—68no—
197C4×C3⋊D3—128no—
198#634—62no—
199#634—68no—
200#634—128no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3C22×D318yes—
204C2×C3×Dic3C22×D314yes—
205C2×C3×Dic3C22×D314yes—
206C3×D12—68no—
207C3×C4×D3—128no—
208C3×Dic6—48no—
209#626—128no—
210#542—271noSylow
211C2×C33C2×#4111yescharacteristic
212C2×C33H(5,𝔽2)11yescharacteristic
213C2×C332−1+411yescharacteristic
214C3⋊(C3⋊D3)—41no—
215C3×C3⋊D3—42no—
216C2×C3×Q8—38no—
217C2×C3×D4—91no—
218C22×C3×C4—92no—
219C2×#67—92no—
220C2×D12—34no—
221C2×D12—98no—
222C2×C4×D3—38no—
223C2×C4×D3—94no—
224C2×C4×D3—91no—
225C2×C4×D3—184no—
226C2×Dic6—94no—
227C2×Dic6—91no—
228C2×Dic6—184no—
229C2×Dic6C2×C3⋊D312yes—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4D3214yes—
234C3×C4⋊C4C2×C3⋊D311yescharacteristic
235C3×C22⋊C4—92no—
236C3×C42—34no—
237C3×C42—91no—
238#217—92no—
239#217—92no—
240#217—92no—
241(C2×D3)⋊C4—38no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—188no—
252C4⋊Dic3—98no—
253C4⋊Dic3—92no—
254C4⋊Dic3—188no—
255C4⋊Dic3C2×C3⋊D311yescharacteristic
256Dic3⋊C4—94no—
257Dic3⋊C4—94no—
258Dic3⋊C4—98no—
259Dic3⋊C4—92no—
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4C2×C3⋊D312yes—
263C4×Dic3—34no—
264C4×Dic3—98no—
265C4×Dic3—92no—
266C4×Dic3C2×C3⋊D311yescharacteristic
267C22×C32—24no—
268C22×C32C23×D314yes—
269C22×C32C23×D311yescharacteristic
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3⋊D3—68no—
274C2×C3⋊D3—61no—
275C2×C3⋊D3—61no—
276C2×C3⋊D3—64no—
277C2×C3⋊D3—61no—
278C2×C3⋊D3—64no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—124no—
281C2×C3⋊D3—124no—
282C2×C3×D3—68no—
283C2×C3×D3—68no—
284C2×C3×D3—68no—
285C32×C4—22no—
286C32×C4—28no—
287C32×C4—28no—
288C32×C4—48no—
289C32×C4—62no—
290C32×C4—61no—
291C32×C4—61no—
292C32×C4#23512yes—
293C32×C4#23918yes—
294C3⋊Dic3—21no—
295C3⋊Dic3—21no—
296C3⋊Dic3—62no—
297C3⋊Dic3—61no—
298C3⋊Dic3—68no—
299C3⋊Dic3—64no—
300C3⋊Dic3—128no—
301C3⋊Dic3—124no—
302C3×Dic3—28no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—64no—
306C3×Dic3—64no—
307C4⋊Q8—31no—
308#126—272no—
309#124—271no—
310#124—272no—
311#123—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314#121—271no—
315C4×Q8—271no—
316C4×D4—91no—
317#117—271no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—92no—
320C22×D3—38no—
321C22×D3—94no—
322C22×D3—91no—
323C22×D3—184no—
324C3×Q8—128no—
325C3×D4—182no—
326C2×C3×C4—24no—
327C2×C3×C4—28no—
328C2×C3×C4—38no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—362no—
334C2×C3×C4C2×D3218yes—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C22×C3⋊D312yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—362no—
339C2×Dic3—34no—
340C2×Dic3—34no—
341C2×Dic3—98no—
342C2×Dic3—94no—
343C2×Dic3—92no—
344C2×Dic3—91no—
345C2×Dic3—184no—
346C2×Dic3—188no—
347C2×Dic3C22×C3⋊D312yes—
348C2×Dic3C22×C3⋊D311yescharacteristic
349C2×Dic3C22×C3⋊D311yescharacteristic
350D12—68no—
351D12—368no—
352C4×D3—128no—
353C4×D3—188no—
354C4×D3—182no—
355C4×D3—368no—
356Dic6—42no—
357Dic6—188no—
358Dic6—182no—
359Dic6—368no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32C2×#23511yescharacteristic
364C2×C32#100414yes—
365C2×C32#100411yescharacteristic
366C2×C32C2×#23914yes—
367C2×C32#100714yes—
368C2×C32#101011yescharacteristic
369C3⋊D3—42no—
370C3⋊D3—128no—
371C3⋊D3—124no—
372C3⋊D3—121no—
373C3×D3—128no—
374C2×Q8—32no—
375C2×Q8—271no—
376C2×D4—91no—
377C2×D4—272no—
378C22×C4—92no—
379C22×C4—271no—
380C4⋊C4—31no—
381C4⋊C4—32no—
382C4⋊C4—271no—
383C4⋊C4—271no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C4⋊C4D3×C3⋊D311yescharacteristic
387C22⋊C4—92no—
388C22⋊C4—271no—
389C22⋊C4—271no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—31no—
395C42—91no—
396C42—272no—
397C22×C3—24no—
398C22×C3—182no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3C22×D3214yes—
402C22×C3C23×C3⋊D311yescharacteristic
403C2×D3—68no—
404C2×D3—68no—
405C2×D3—68no—
406C2×D3—188no—
407C2×D3—184no—
408C2×D3—184no—
409C2×D3—184no—
410C2×D3—181no—
411C2×D3—181no—
412C2×D3—181no—
413C2×D3—364no—
414C2×D3—364no—
415C3×C4—28no—
416C3×C4—22no—
417C3×C4—28no—
418C3×C4—48no—
419C3×C4—68no—
420C3×C4—64no—
421C3×C4—64no—
422C3×C4—181no—
423C3×C4—181no—
424C3×C4#379918yes—
425C3×C4#383212yes—
426Dic3—21no—
427Dic3—22no—
428Dic3—21no—
429Dic3—64no—
430Dic3—64no—
431Dic3—188no—
432Dic3—184no—
433Dic3—182no—
434Dic3—181no—
435Dic3—368no—
436Dic3—364no—
437C32—24no—
438C32#556111yescharacteristic, complemented
439C32#573314yescomplemented
440C23—92no—
441C23—271no—
442Q8—122no—
443Q8—542no—
444D4—182no—
445D4—1082no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—272no—
452C2×C4—271no—
453C2×C4—362no—
454C2×C4—542no—
455C2×C4C2×D3×C3⋊D312yes—
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—362no—
461C2×C3C2×#379914yes—
462C2×C3#6431314yes—
463C2×C3#6431514yes—
464C2×C3#6436111yescharacteristic
465C2×C3C2×#383211yescharacteristic
466C2×C3#6436411yescharacteristic
467D3—128no—
468D3—368no—
469D3—364no—
470D3—361no—
471C22—182no—
472C22—182no—
473C22—182no—
474C22—541no—
475C22—541no—
476C22—541no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—22no—
479C4—62no—
480C4—61no—
481C4—61no—
482C4—181no—
483C4—181no—
484C4—541no—
485C4—542no—
486C4#8901812yes—
487C3—24no—
488C3#1059434114yescomplemented
489C3#1059620411yescharacteristic, complemented
490C2—362no—
491C2—1081no—
492C2C2×#8901811yescharacteristic
493C2#1172458711yescharacteristic
494C2#1172459211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radical#11723532C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723938C2semidirect
2. C3×#64102C2semidirect
3. C32×#888C2semidirect
5. #11723452C2semidirect
6. #11723439C2semidirect
7. #11723435C2semidirect
8. #11723417C2semidirect
9. #11723408C2semidirect
10. #11723406C2semidirect
11. #11723392C2semidirect
46. #64102D3semidirect
51. C3×#888D3semidirect
161. #888C3⋊D3semidirect
318. C33#542semidirect
438. C32#5561semidirect
439. C32#5733semidirect
488. C3#10594341semidirect
489. C3#10596204semidirect

Maximal subgroups

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

subgroupindex
#117239382
C3×#641022
C32×#8882
#117235322
#117234522
#117234392
#117234352
#117234172
#117234082
#117234062
#117233922
#105962043
#105943413