Finite Group Databasebeta

← #13176465

Subgroups — #13176465

Order 1728.

1672 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#11723934C211yesmaximal, characteristic, complemented
2C3×#64106C211yesmaximal, characteristic, complemented
3C32×#888C211yesmaximal, characteristic, complemented
4#11723526C211yesmaximal, characteristic
5#11723456C212yesmaximal, complemented
6#11723441C211yesmaximal, characteristic, complemented
7#11723437C211yesmaximal, characteristic, complemented
8#11723421C212yesmaximal, complemented
9#11723410C211yesmaximal, characteristic, complemented
10#11723403C212yesmaximal, complemented
11#11723392C212yesmaximal, complemented
12#10596204—31nomaximal
13#10594343—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C32×Dic6C2212yes—
19C2×#6292C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21#88859C2212yes—
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3752C2212yes—
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C32×C4⋊Dic3C2211yescharacteristic
26C32×Dic3⋊C4C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2211yescharacteristic
29#88807C2212yes—
30#88807C2211yescharacteristic
31#88804C2212yes—
32#88804C2212yes—
33#88804C2212yes—
34#88803C2211yescharacteristic
35#88803C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×C4⋊Q8—31no—
38#64156—32no—
39#64155—32no—
40#64107—32no—
41#64106D311yescharacteristic, complemented
42#64104—31no—
43#64103—32no—
44#64102—31no—
45#64102—34no—
46#64102—64no—
47#64096—31no—
48#64084—31no—
49#64081—31no—
50C3×#897—34no—
51C3×#888D314yescomplemented
52#63909—34no—
53#63907—38no—
54#63871—38no—
55#63864—34no—
56#63855—34no—
57#63853—38no—
58#63848—38no—
59#63843—34no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C4⋊(C3⋊(C3⋊D3))—22no—
67C4×C3⋊(C3⋊D3)—42no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C3×C4×C3⋊D3—22no—
71C3×#634—22no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—42no—
76#6292—42no—
77#5733—94no—
78#5643—91no—
79C2×C32×Q8—32no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—32no—
83C2×C4⋊(C3⋊D3)—64no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—38no—
86C2×C4×C3⋊D3—68no—
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×#634C2×D311yescharacteristic
89C2×C3×C4×D3—34no—
90C2×C3×Dic6—34no—
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—31no—
100#3752—31no—
101#3752—32no—
102#3752—38no—
103#3752—32no—
104#3752—32no—
105#3752—32no—
106#3752—68no—
107#3752C2×D312yes—
108C4⋊(C3⋊Dic3)C2×D312yes—
109#3750—31no—
110#3750—31no—
111#3750—32no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—64no—
116C3×(C2×D3)⋊C4—38no—
117C3×C4⋊Dic3—38no—
118C3×C4⋊Dic3C2×D314yes—
119C3×Dic3⋊C4C2×D318yes—
120C3×C4×Dic3C2×D314yes—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—34no—
124#3722—38no—
125#3722—38no—
126#3722—38no—
127(C2×D3)⋊Dic3—34no—
128(C2×D3)⋊Dic3—34no—
129Dic32—38no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C33×C4—22no—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—21no—
140C3⋊(C3⋊Dic3)—21no—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—21no—
143C32×Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×C4⋊Q8—34no—
147C3×C22⋊Q8—91no—
148#947—98no—
149#947—91no—
150#946—98no—
151Q8×Dic3—91no—
152#930—91no—
153#926—91no—
154#898—98no—
155#898—92no—
156#897—34no—
157#897—92no—
158#895—94no—
159#894—98no—
160#893—94no—
161#893—91no—
162#893—184no—
163#888C3⋊D311yescharacteristic, complemented
164#887—94no—
165#885—92no—
166#883—92no—
167#875—94no—
168#872—94no—
169C22×C3⋊D3—32no—
170C22×C3⋊D3—38no—
171C22×C3⋊D3—68no—
172C22×C3⋊D3C22×D311yescharacteristic
173C22×C3×D3—34no—
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—31no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—34no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—64no—
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D311yescharacteristic
192C4⋊(C3⋊D3)—616no—
193C4⋊(C3⋊D3)—62no—
194C4⋊(C3⋊D3)—68no—
195C4⋊(C3⋊D3)—122no—
196C4×C3⋊D3—22no—
197C4×C3⋊D3—122no—
198C4×C3⋊D3—128no—
199C4×C3⋊D3—1216no—
200#634—22no—
201C2×C3×Dic3—38no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3C22×D318yes—
204C2×C3×Dic3C22×D314yes—
205C2×C3×Dic3C22×D314yes—
206C3×C4×D3—68no—
207C3×Dic6—48no—
208C3×Dic6—68no—
209#626—128no—
210#542—271noSylow
211C2×C33C2×#4111yescharacteristic
212C2×C33H(5,𝔽2)11yescharacteristic
213C2×C332−1+411yescharacteristic
214C3⋊(C3⋊D3)—42no—
215C3×C3⋊D3—41no—
216C2×C3×Q8—38no—
217C2×C3×Q8—91no—
218C22×C3×C4—91no—
219C2×#67—92no—
220C2×D12—94no—
221C2×D12—91no—
222C2×D12—98no—
223C2×D12—184no—
224C2×C4×D3—34no—
225C2×C4×D3—98no—
226C2×C4×D3—92no—
227C2×C4×D3—188no—
228C2×Dic6—34no—
229C2×Dic6C2×C3⋊D312yes—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4D3214yes—
235C3×C4⋊C4C2×C3⋊D311yescharacteristic
236C3×C22⋊C4—92no—
237C3×C42—34no—
238#217—91no—
239#217—91no—
240(C2×D3)⋊C4—38no—
241(C2×D3)⋊C4—98no—
242(C2×D3)⋊C4—98no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—94no—
245(C2×D3)⋊C4—94no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—188no—
252C4⋊Dic3—38no—
253C4⋊Dic3—92no—
254C4⋊Dic3—91no—
255C4⋊Dic3—91no—
256C4⋊Dic3C2×C3⋊D311yescharacteristic
257Dic3⋊C4—94no—
258Dic3⋊C4—94no—
259Dic3⋊C4—98no—
260Dic3⋊C4C2×C3⋊D312yes—
261C4×Dic3—98no—
262C4×Dic3—94no—
263C4×Dic3—91no—
264C4×Dic3—92no—
265C4×Dic3—184no—
266C4×Dic3C2×C3⋊D311yescharacteristic
267C22×C32—24no—
268C22×C32C23×D314yes—
269C22×C32C23×D311yescharacteristic
270C2×C3⋊D3—21no—
271C2×C3⋊D3—21no—
272C2×C3⋊D3—21no—
273C2×C3⋊D3—616no—
274C2×C3⋊D3—62no—
275C2×C3⋊D3—68no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—68no—
280C2×C3⋊D3—128no—
281C2×C3⋊D3—128no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C32×C4—22no—
286C32×C4—28no—
287C32×C4—28no—
288C32×C4—48no—
289C32×C4—61no—
290C32×C4—62no—
291C32×C4—61no—
292C32×C4#23712yes—
293C32×C4#23918yes—
294C3⋊Dic3—22no—
295C3⋊Dic3—21no—
296C3⋊Dic3—68no—
297C3⋊Dic3—64no—
298C3⋊Dic3—61no—
299C3⋊Dic3—61no—
300C3⋊Dic3—64no—
301C3⋊Dic3—124no—
302C3×Dic3—28no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—68no—
306C3×Dic3—64no—
307C4⋊Q8—31no—
308#126—272no—
309#124—271no—
310#124—272no—
311#123—272no—
312C22⋊Q8—91no—
313C22⋊Q8—272no—
314#121—271no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—271no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C22×D3—34no—
321C22×D3—98no—
322C22×D3—92no—
323C22×D3—188no—
324C3×Q8—128no—
325C3×Q8—182no—
326C2×C3×C4—24no—
327C2×C3×C4—28no—
328C2×C3×C4—38no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—92no—
332C2×C3×C4—91no—
333C2×C3×C4—182no—
334C2×C3×C4C2×D3218yes—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C22×C3⋊D312yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—362no—
339C2×Dic3—38no—
340C2×Dic3—34no—
341C2×Dic3—94no—
342C2×Dic3—94no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—184no—
346C2×Dic3—184no—
347C2×Dic3C22×C3⋊D312yes—
348C2×Dic3C22×C3⋊D311yescharacteristic
349C2×Dic3C22×C3⋊D311yescharacteristic
350D12—1816no—
351D12—188no—
352D12—182no—
353D12—368no—
354C4×D3—68no—
355C4×D3—368no—
356C4×D3—362no—
357C4×D3—3616no—
358Dic6—42no—
359Dic6—68no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32C2×#23711yescharacteristic
364C2×C32#100414yes—
365C2×C32C2×#23914yes—
366C2×C32#100714yes—
367C2×C32#100711yescharacteristic
368C2×C32#100911yescharacteristic
369C3⋊D3—41no—
370C3⋊D3—1216no—
371C3⋊D3—122no—
372C3⋊D3—128no—
373C3×D3—124no—
374C2×Q8—32no—
375C2×Q8—91no—
376C2×D4—272no—
377C2×D4—271no—
378C22×C4—91no—
379C22×C4—272no—
380C4⋊C4—31no—
381C4⋊C4—32no—
382C4⋊C4—92no—
383C4⋊C4—271no—
384C4⋊C4—271no—
385C4⋊C4—272no—
386C4⋊C4D3×C3⋊D311yescharacteristic
387C22⋊C4—92no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—271no—
392C22⋊C4—271no—
393C22⋊C4—272no—
394C42—31no—
395C42—272no—
396C42—271no—
397C22×C3—24no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3C22×D3214yes—
402C22×C3C23×C3⋊D311yescharacteristic
403C2×D3—64no—
404C2×D3—64no—
405C2×D3—64no—
406C2×D3—1816no—
407C2×D3—188no—
408C2×D3—188no—
409C2×D3—188no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—182no—
413C2×D3—368no—
414C2×D3—368no—
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—182no—
424C3×C4#379618yes—
425C3×C4#383212yes—
426Dic3—21no—
427Dic3—22no—
428Dic3—21no—
429Dic3—68no—
430Dic3—64no—
431Dic3—188no—
432Dic3—184no—
433Dic3—184no—
434Dic3—181no—
435Dic3—181no—
436Dic3—364no—
437C32—24no—
438C32#564311yescharacteristic, complemented
439C32#573314yescomplemented
440C23—91no—
441C23—272no—
442Q8—122no—
443Q8—182no—
444D4—542no—
445D4—1082no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—92no—
450C2×C4—91no—
451C2×C4—182no—
452C2×C4—271no—
453C2×C4—271no—
454C2×C4—1082no—
455C2×C4C2×D3×C3⋊D312yes—
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—361no—
461C2×C3C2×#379614yes—
462C2×C3#6431414yes—
463C2×C3#6431614yes—
464C2×C3#6436111yescharacteristic
465C2×C3C2×#383211yescharacteristic
466C2×C3#6436411yescharacteristic
467D3—124no—
468D3—3616no—
469D3—368no—
470D3—362no—
471C22—181no—
472C22—181no—
473C22—181no—
474C22—542no—
475C22—542no—
476C22—542no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—22no—
479C4—62no—
480C4—61no—
481C4—61no—
482C4—181no—
483C4—182no—
484C4—541no—
485C4—541no—
486C4#8901712yes—
487C3—24no—
488C3#1059434314yescomplemented
489C3#1059620411yescharacteristic, complemented
490C2—361no—
491C2—1082no—
492C2C2×#8901711yescharacteristic
493C2#1172458811yescharacteristic
494C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radical#11723526C2
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. #11723934C2semidirect
2. C3×#64106C2semidirect
3. C32×#888C2semidirect
5. #11723456C2semidirect
6. #11723441C2semidirect
7. #11723437C2semidirect
8. #11723421C2semidirect
9. #11723410C2semidirect
10. #11723403C2semidirect
11. #11723392C2semidirect
41. #64106D3semidirect
51. C3×#888D3semidirect
163. #888C3⋊D3semidirect
318. C33#542semidirect
438. C32#5643semidirect
439. C32#5733semidirect
488. C3#10594343semidirect
489. C3#10596204semidirect

Maximal subgroups

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

subgroupindex
#117239342
C3×#641062
C32×#8882
#117235262
#117234562
#117234412
#117234372
#117234212
#117234102
#117234032
#117233922
#105962043
#105943433