Finite Group Databasebeta

← #13177678

Subgroups — #13177678

Order 1728.

1720 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723977C211yesmaximal, characteristic, complemented
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×#940C211yesmaximal, characteristic
4#67×C3⋊Dic3C212yesmaximal, complemented
5#11723614C212yesmaximal, complemented
6#11723602C212yesmaximal, complemented
7#11723600C212yesmaximal, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723499C211yesmaximal, characteristic, complemented
10#11723429C211yesmaximal, characteristic, complemented
11#11723399C211yesmaximal, characteristic, complemented
12#10596184—31nomaximal
13#10595198—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C22×C3×C3⋊Dic3C2212yes—
17C2×C32×#67C2212yes—
18C2×C32×#67C2212yes—
19C2×C32×D12C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C2×D3×C3⋊Dic3C2212yes—
22#88864C2212yes—
23#88864C2212yes—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3757C2212yes—
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2211yescharacteristic
30#88806C2211yescharacteristic
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#127—31no—
38#64145—31no—
39#64145—32no—
40#64145—34no—
41#64145—31no—
42#64145—64no—
43#64144—32no—
44#64144—32no—
45D4×C3⋊Dic3—32no—
46D4×C3⋊Dic3—31no—
47D4×C3⋊Dic3D311yescharacteristic, complemented
48C4×#634—31no—
49C4×#634—31no—
50C3×#940D314yes—
51C3×D4×Dic3—34no—
52Dic3×#67—38no—
53#63969—38no—
54#63961—38no—
55#63959—38no—
56C4×#627—34no—
57#63898—34no—
58#63860—34no—
59#63844—34no—
60C23×C33C2312yes—
61C22×C32×D3C2312yes—
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67#6308—22no—
68C2×C3×C3⋊Dic3—42no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C32×#67—24no—
73C32×#67—42no—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2311yescharacteristic
76C32×D12—22no—
77#6293—22no—
78D3×C3⋊Dic3—42no—
79#5713—94no—
80#5559—91no—
81C2×C32×D4—24no—
82C2×C32×D4—32no—
83C2×C32×D4—32no—
84C2×C32×D4—31no—
85C2×C32×D4C2×D314yes—
86C2×C32×D4C2×D311yescharacteristic
87C22×C3⋊Dic3—32no—
88C22×C3⋊Dic3C2×D312yes—
89C2×#634—31no—
90C2×#634—34no—
91C2×#634—31no—
92C2×#634—64no—
93C2×C3×#67C2×D318yes—
94C2×C3×#67C2×D318yes—
95C22×C3×Dic3—38no—
96C2×C3×D12C2×D314yes—
97C2×#627—34no—
98C2×D3×Dic3—38no—
99C32×C42—31no—
100#3757—32no—
101#3757—32no—
102#3757—38no—
103#3757—38no—
104#3757—32no—
105#3757—32no—
106#3757—32no—
107#3757—68no—
108#3757—68no—
109#3757C2×D312yes—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)C2×D311yescharacteristic
112#3750—32no—
113#3750—31no—
114#3750—31no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3—34no—
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3—64no—
119C4×C3⋊Dic3C2×D311yescharacteristic
120C3×#217—38no—
121C3×C4⋊Dic3—34no—
122C3×C4×Dic3—34no—
123C3×C4×Dic3C2×D314yes—
124#3724—38no—
125#3724—34no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—34no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33—22no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33C2411yescharacteristic
136C2×C32×D3—22no—
137C2×C32×D3—22no—
138C2×C32×D3—22no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—21no—
141C3⋊(C3⋊Dic3)—22no—
142C3⋊(C3⋊Dic3)—21no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—22no—
145C3×C3⋊Dic3—21no—
146C32×Dic3—21no—
147C32×Dic3#4112yes—
148C3×#127—34no—
149C3×C4×D4—91no—
150#940C3⋊D311yescharacteristic
151#936—94no—
152#936—98no—
153#936—94no—
154#936—91no—
155#936—184no—
156#935—98no—
157#935—98no—
158D4×Dic3—34no—
159D4×Dic3—94no—
160D4×Dic3—98no—
161#929—92no—
162C4×#67—92no—
163#893—91no—
164#887—91no—
165#885—92no—
166#883—92no—
167#875—91no—
168C4×Dic6—94no—
169C4×Dic6—94no—
170C4×Dic6—91no—
171C23×C32—28no—
172C23×C32—32no—
173C23×C32C22×D318yes—
174C23×C32C22×D312yes—
175C22×C3×D3C22×D318yes—
176C32×D4—22no—
177C32×D4—28no—
178C32×D4—48no—
179C32×D4—64no—
180C32×D4—62no—
181C32×D4—122no—
182C2×C32×C4—24no—
183C2×C32×C4—31no—
184C2×C32×C4—31no—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—34no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—42no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—68no—
197C2×C3⋊Dic3—122no—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201#634—62no—
202#634—62no—
203#634—68no—
204#634—128no—
205C3×#67—216no—
206C3×#67—48no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—128no—
211C2×C3×Dic3C22×D314yes—
212C2×C3×Dic3C22×D314yes—
213C3×D12—28no—
214#627—68no—
215D3×Dic3—128no—
216#559—271noSylow
217C2×C33—42no—
218C2×C33C2×#4111yescharacteristic
219C2×C33C2×#4111yescharacteristic
220C2×C33H(5,𝔽2)11yescharacteristic
221C32×D3—42no—
222C2×C3×D4—24no—
223C2×C3×D4—38no—
224C2×C3×D4—38no—
225C2×C3×D4—34no—
226C2×C3×D4D3214yes—
227C2×C3×D4C2×C3⋊D311yescharacteristic
228C22×C3×C4—92no—
229C2×#67C2×C3⋊D312yes—
230C2×#67C2×C3⋊D312yes—
231C22×Dic3—38no—
232C22×Dic3—98no—
233C2×D12C2×C3⋊D311yescharacteristic
234C2×C4×D3—92no—
235C2×Dic6—94no—
236C2×Dic6—94no—
237C2×Dic6—91no—
238C2×Dic6—91no—
239C2×Dic6—184no—
240C3×C4⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C42—34no—
243C3×C42—91no—
244#217—38no—
245#217—98no—
246#217—98no—
247#217—98no—
248#217—98no—
249#217—98no—
250#217—92no—
251#217—92no—
252#217—188no—
253#217—188no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257C4⋊Dic3—34no—
258C4⋊Dic3—94no—
259C4⋊Dic3—91no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262Dic3⋊C4—98no—
263Dic3⋊C4—91no—
264Dic3⋊C4—91no—
265Dic3⋊C4—92no—
266C4×Dic3—34no—
267C4×Dic3—98no—
268C4×Dic3—94no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—184no—
272C4×Dic3C2×C3⋊D311yescharacteristic
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—24no—
277C22×C32—28no—
278C22×C32—28no—
279C22×C32—28no—
280C22×C32—48no—
281C22×C32—48no—
282C22×C32—48no—
283C22×C32—62no—
284C22×C32—62no—
285C22×C32—62no—
286C22×C32C23×D314yes—
287C22×C32C23×D311yescharacteristic
288C2×C3×D3—28no—
289C2×C3×D3—28no—
290C2×C3×D3—28no—
291C32×C4—28no—
292C32×C4—32no—
293C32×C4—61no—
294C32×C4#23512yes—
295C32×C4#23718yes—
296C3⋊Dic3—21no—
297C3⋊Dic3—22no—
298C3⋊Dic3—21no—
299C3⋊Dic3—62no—
300C3⋊Dic3—61no—
301C3⋊Dic3—61no—
302C3⋊Dic3—64no—
303C3⋊Dic3—68no—
304C3⋊Dic3—64no—
305C3⋊Dic3—128no—
306C3⋊Dic3—124no—
307C3⋊Dic3—124no—
308C3×Dic3—24no—
309C3×Dic3—64no—
310C3×Dic3—68no—
311C3×Dic3—64no—
312C3×Dic3#23718yes—
313#127—31no—
314#124—271no—
315#124—272no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—271no—
323C4×D4—272no—
324C33#55911yescharacteristic, complemented, Sylow
325C23×C3—28no—
326C23×C3—38no—
327C23×C3C2×D3218yes—
328C23×C3C22×C3⋊D312yes—
329C22×D3C22×C3⋊D312yes—
330C3×D4—28no—
331C3×D4—22no—
332C3×D4—48no—
333C3×D4—616no—
334C3×D4—68no—
335C3×D4—128no—
336C2×C3×C4—24no—
337C2×C3×C4—34no—
338C2×C3×C4—34no—
339C2×C3×C4—92no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—362no—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345#67—24no—
346#67—42no—
347C2×Dic3—34no—
348C2×Dic3—34no—
349C2×Dic3—38no—
350C2×Dic3—94no—
351C2×Dic3—94no—
352C2×Dic3—98no—
353C2×Dic3—91no—
354C2×Dic3—92no—
355C2×Dic3—91no—
356C2×Dic3—128no—
357C2×Dic3—184no—
358C2×Dic3—184no—
359C2×Dic3—188no—
360C2×Dic3—368no—
361C2×Dic3C22×C3⋊D311yescharacteristic
362C2×Dic3C22×C3⋊D311yescharacteristic
363D12—22no—
364C4×D3—362no—
365Dic6—188no—
366Dic6—188no—
367Dic6—182no—
368Dic6—182no—
369Dic6—368no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—42no—
374C2×C32—48no—
375C2×C32—88no—
376C2×C32—122no—
377C2×C32C2×#23511yescharacteristic
378C2×C32C2×#23714yes—
379C2×C32C2×#23714yes—
380C2×C32#100414yes—
381C2×C32#100411yescharacteristic
382C2×C32D3×#4111yescharacteristic
383C3×D3—48no—
384C2×Q8—271no—
385C2×Q8—271no—
386C2×D4—32no—
387C2×D4—32no—
388C2×D4—31no—
389C2×D4D3×C3⋊D311yescharacteristic
390C22×C4—92no—
391C22×C4—272no—
392C4⋊C4—91no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—31no—
404C42—91no—
405C42—272no—
406C42—271no—
407C22×C3—28no—
408C22×C3—28no—
409C22×C3—28no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—24no—
414C22×C3—48no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3—68no—
418C22×C3—68no—
419C22×C3—68no—
420C22×C3C22×D3214yes—
421C22×C3C23×C3⋊D311yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C3×C4—28no—
426C3×C4—38no—
427C3×C4—64no—
428C3×C4—181no—
429C3×C4—182no—
430C3×C4—181no—
431C3×C4#380418yes—
432C3×C4#383012yes—
433Dic3—21no—
434Dic3—64no—
435Dic3—64no—
436Dic3—68no—
437Dic3—184no—
438Dic3—184no—
439Dic3—188no—
440Dic3—181no—
441Dic3—182no—
442Dic3—181no—
443Dic3—368no—
444Dic3—364no—
445Dic3—364no—
446Dic3#383012yes—
447C32—24no—
448C32#555911yescharacteristic, complemented
449C32#571314yescomplemented
450C23—32no—
451C23C2×D3×C3⋊D312yes—
452Q8—542no—
453Q8—542no—
454D4—22no—
455D4—64no—
456D4—62no—
457D4—122no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—271no—
464C2×C4—272no—
465C2×C4—271no—
466C2×C4—362no—
467C2×C4—1082no—
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—42no—
473C2×C3—48no—
474C2×C3—88no—
475C2×C3—128no—
476C2×C3D3×#23714yes—
477C2×C3#6431314yes—
478C2×C3C2×#380414yes—
479C2×C3C2×#383011yescharacteristic
480C2×C3C2×#383011yescharacteristic
481C2×C3#6436111yescharacteristic
482D3—42no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—62no—
487C22—62no—
488C22—62no—
489C22C22×D3×C3⋊D311yescharacteristic
490C4—32no—
491C4—61no—
492C4—181no—
493C4—181no—
494C4—182no—
495C4—541no—
496C4—541no—
497C4—542no—
498C4#8902812yes—
499C3—24no—
500C3#1059519814yescomplemented
501C3#1059618411yescharacteristic, complemented
502C2—42no—
503C2—122no—
504C2D3×#383011yescharacteristic
505C2#1172458711yescharacteristic
506C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC32×#940C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723977C2semidirect
2. C3×D4×C3⋊Dic3C2semidirect
4. #67×C3⋊Dic3C2semidirect
5. #11723614C2semidirect
6. #11723602C2semidirect
7. #11723600C2semidirect
8. C4×#6293C2semidirect
9. #11723499C2semidirect
10. #11723429C2semidirect
11. #11723399C2semidirect
47. D4×C3⋊Dic3D3semidirect
324. C33#559semidirect
448. C32#5559semidirect
449. C32#5713semidirect
500. C3#10595198semidirect
501. C3#10596184semidirect

Maximal subgroups

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

subgroupindex
#117239772
C3×D4×C3⋊Dic32
C32×#9402
#67×C3⋊Dic32
#117236142
#117236022
#117236002
C4×#62932
#117234992
#117234292
#117233992
#105961843
#105951983