Finite Group Databasebeta

← #13175366

Subgroups — #13175366

Order 1728.

1600 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723913C211yesmaximal, characteristic, complemented
2C3×C4×#634C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723536C211yesmaximal, characteristic, complemented
5#11723531C211yesmaximal, characteristic
6#11723449C211yesmaximal, characteristic, complemented
7#11723434C211yesmaximal, characteristic, complemented
8#11723428C212yesmaximal, complemented
9#11723419C212yesmaximal, complemented
10#11723415C212yesmaximal, complemented
11#11723414C212yesmaximal, complemented
12#10596032—31nomaximal
13#10593876—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×C32×C4×D3C2212yes—
18C2×#6293C2212yes—
19C2×#6291C2212yes—
20C33×C42C2211yescharacteristic
21#88859C2211yescharacteristic
22#88859C2211yescharacteristic
23#88857C2211yescharacteristic
24#88857C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#3750C2212yes—
28C3×C4×C3⋊Dic3C2212yes—
29C32×(C2×D3)⋊C4C2212yes—
30C32×C4⋊Dic3C2211yescharacteristic
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38#64147—31no—
39#64145—31no—
40#64138—32no—
41#64133—32no—
42#64106—31no—
43#64097—31no—
44#64094—32no—
45#64088—32no—
46#64081—31no—
47#64081—34no—
48#64081—64no—
49C4×#634D311yescharacteristic, complemented
50C3×C4×D12D314yescomplemented
51C3×C4×Dic6—34no—
52#63919—34no—
53#63916—34no—
54#63871—34no—
55#63863—34no—
56#63859—38no—
57#63852—38no—
58#63851—38no—
59#63850—38no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C22×C32×D3C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4×C3⋊(C3⋊D3)—22no—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C3×#634—22no—
72C2×C32×Dic3C2312yes—
73C32×D12—22no—
74C32×C4×D3—42no—
75#6293—42no—
76#6291—42no—
77#5582—91no—
78#5561—94no—
79C2×C32×D4—31no—
80C22×C32×C4—32no—
81C2×#638—32no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—64no—
85C2×#634—32no—
86C2×#634C2×D311yescharacteristic
87C2×C3×D12C2×D314yes—
88C2×C3×C4×D3C2×D318yes—
89C2×C3×Dic6—34no—
90C2×#627—38no—
91C2×#626—38no—
92C32×C4⋊C4—31no—
93C32×C22⋊C4—32no—
94C32×C42—24no—
95C32×C42C2×D314yes—
96C32×C42C2×D311yescharacteristic
97#3757—32no—
98#3757—32no—
99#3757—32no—
100#3752—31no—
101#3752—31no—
102#3752—34no—
103#3752—34no—
104#3752—32no—
105#3752—64no—
106#3752—64no—
107C4⋊(C3⋊Dic3)—32no—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750—31no—
110#3750—31no—
111#3750—34no—
112#3750—34no—
113#3750—32no—
114#3750—64no—
115#3750—64no—
116#3750C2×D312yes—
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—34no—
119C4×C3⋊Dic3—64no—
120C4×C3⋊Dic3C2×D312yes—
121C3×(C2×D3)⋊C4C2×D318yes—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3C2×D314yes—
124C3×Dic3⋊C4—38no—
125C3×C4×Dic3—38no—
126#3724—38no—
127Dic3⋊Dic3—38no—
128#3722—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131(C2×D3)⋊Dic3—38no—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—21no—
134C2×C3⋊(C3⋊D3)—21no—
135C2×C3⋊(C3⋊D3)—21no—
136C2×C32×D3—22no—
137C2×C32×D3—22no—
138C2×C32×D3—22no—
139C33×C4—21no—
140C33×C4—21no—
141C33×C4#4112yes—
142C3⋊(C3⋊Dic3)—21no—
143C3⋊(C3⋊Dic3)—21no—
144C3⋊(C3⋊Dic3)—21no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3—22no—
147C32×Dic3—22no—
148C3×C4×Q8—91no—
149C3×C4×D4—34no—
150#947—91no—
151#946—91no—
152#938—94no—
153#936—94no—
154#929—98no—
155#924—98no—
156#897—94no—
157#896—92no—
158#895—91no—
159#894—92no—
160#888—94no—
161#888—91no—
162#885—98no—
163#879—98no—
164#876—92no—
165#875—92no—
166C4×D12C3⋊D311yescharacteristic, complemented
167#872—94no—
168#872—91no—
169#872—184no—
170C4×Dic6—34no—
171C23×C32—32no—
172C22×C3⋊D3—31no—
173C22×C3⋊D3—34no—
174C22×C3⋊D3—64no—
175C22×C3×D3C22×D318yes—
176C32×D4—62no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—32no—
181C2×C32×C4—122no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—122no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—34no—
194C2×C3⋊Dic3—34no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C4×C3⋊D3—616no—
201C4×C3⋊D3—68no—
202C4×C3⋊D3—62no—
203#634—22no—
204#634—122no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3C22×D318yes—
208C3×D12—28no—
209C3×C4×D3—48no—
210C3×Dic6—68no—
211#627—128no—
212#626—128no—
213#542—271noSylow
214C2×C33C2×#4111yescharacteristic
215C2×C33H(5,𝔽2)11yescharacteristic
216C2×C332−1+411yescharacteristic
217C3⋊(C3⋊D3)—41no—
218C32×D3—42no—
219C2×C3×Q8—91no—
220C2×C3×D4—34no—
221C22×C3×C4—38no—
222C2×#67—98no—
223C2×D12—92no—
224C2×D12C2×C3⋊D311yescharacteristic
225C2×C4×D3—94no—
226C2×C4×D3—91no—
227C2×C4×D3—184no—
228C2×C4×D3C2×C3⋊D312yes—
229C2×Dic6—34no—
230C2×Dic6—98no—
231C2×Dic6—92no—
232C3×C4⋊C4—34no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—92no—
235C3×C22⋊C4—38no—
236C3×C42—24no—
237C3×C42—92no—
238C3×C42D3214yes—
239C3×C42C2×C3⋊D311yescharacteristic
240#217—98no—
241#217—98no—
242#217—98no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—94no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—184no—
253(C2×D3)⋊C4—184no—
254(C2×D3)⋊C4C2×C3⋊D312yes—
255C4⋊Dic3—34no—
256C4⋊Dic3—98no—
257C4⋊Dic3C2×C3⋊D311yescharacteristic
258Dic3⋊C4—38no—
259Dic3⋊C4—98no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262Dic3⋊C4—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—184no—
267Dic3⋊C4—184no—
268C4×Dic3—38no—
269C4×Dic3—94no—
270C4×Dic3—91no—
271C4×Dic3—184no—
272C22×C32—24no—
273C22×C32—62no—
274C22×C32—62no—
275C22×C32—62no—
276C22×C32C23×D314yes—
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—68no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—61no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—64no—
284C2×C3⋊D3—64no—
285C2×C3⋊D3—124no—
286C2×C3⋊D3—124no—
287C2×C3×D3—28no—
288C2×C3×D3—28no—
289C2×C3×D3—28no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—28no—
293C32×C4—24no—
294C32×C4—24no—
295C32×C4—44no—
296C32×C4—44no—
297C32×C4—62no—
298C32×C4#23518yes—
299C32×C4#23512yes—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3—68no—
303C3⋊Dic3—64no—
304C3⋊Dic3—61no—
305C3⋊Dic3—61no—
306C3⋊Dic3—61no—
307C3⋊Dic3—64no—
308C3⋊Dic3—64no—
309C3⋊Dic3—124no—
310C3⋊Dic3—124no—
311C3×Dic3—28no—
312C3×Dic3—68no—
313C3×Dic3—68no—
314C4⋊Q8—271no—
315#126—272no—
316#124—272no—
317#124—271no—
318#123—272no—
319C22⋊Q8—271no—
320C22⋊Q8—272no—
321#121—271no—
322C4×Q8—91no—
323C4×D4—31no—
324#117—271no—
325C33#54211yescharacteristic, complemented, Sylow
326C23×C3—38no—
327C22×D3—94no—
328C22×D3—91no—
329C22×D3—184no—
330C22×D3C22×C3⋊D312yes—
331C3×Q8—182no—
332C3×D4—68no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—38no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—128no—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C2×D3214yes—
342C2×C3×C4C2×D3214yes—
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D311yescharacteristic
346#67—368no—
347C2×Dic3—38no—
348C2×Dic3—38no—
349C2×Dic3—94no—
350C2×Dic3—94no—
351C2×Dic3—94no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—184no—
356C2×Dic3—184no—
357C2×Dic3—184no—
358C2×Dic3C22×C3⋊D312yes—
359D12—22no—
360D12—362no—
361C4×D3—42no—
362C4×D3—1816no—
363C4×D3—188no—
364C4×D3—182no—
365Dic6—68no—
366Dic6—362no—
367Dic6—368no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—122no—
372C2×C32C2×#23514yes—
373C2×C32C2×#23511yescharacteristic
374C2×C32#100414yes—
375C2×C32#100711yescharacteristic
376C2×C32#100911yescharacteristic
377C2×C32#101014yes—
378C3⋊D3—128no—
379C3⋊D3—124no—
380C3⋊D3—121no—
381C3×D3—48no—
382C2×Q8—91no—
383C2×Q8—272no—
384C2×D4—31no—
385C2×D4—272no—
386C22×C4—32no—
387C22×C4—271no—
388C4⋊C4—31no—
389C4⋊C4—91no—
390C4⋊C4—92no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C22⋊C4—32no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—271no—
401C22⋊C4—271no—
402C42—92no—
403C42—271no—
404C42D3×C3⋊D311yescharacteristic
405C22×C3—24no—
406C22×C3—68no—
407C22×C3—68no—
408C22×C3—68no—
409C22×C3C22×D3214yes—
410C22×C3C23×C3⋊D311yescharacteristic
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—188no—
415C2×D3—184no—
416C2×D3—184no—
417C2×D3—184no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—364no—
422C2×D3—364no—
423C3×C4—24no—
424C3×C4—24no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—28no—
428C3×C4—44no—
429C3×C4—44no—
430C3×C4—68no—
431C3×C4—182no—
432C3×C4—182no—
433C3×C4#379818yes—
434C3×C4#382812yes—
435Dic3—22no—
436Dic3—68no—
437Dic3—68no—
438Dic3—188no—
439Dic3—184no—
440Dic3—184no—
441Dic3—184no—
442Dic3—181no—
443Dic3—181no—
444Dic3—181no—
445Dic3—364no—
446Dic3—364no—
447C32—24no—
448C32#556114yescomplemented
449C32#558211yescharacteristic, complemented
450C23—32no—
451C23—271no—
452Q8—182no—
453Q8—1082no—
454D4—62no—
455D4—1082no—
456C2×C4—32no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—122no—
460C2×C4—271no—
461C2×C4—271no—
462C2×C4—271no—
463C2×C4—542no—
464C2×C4C2×D3×C3⋊D311yescharacteristic
465C2×C4C2×D3×C3⋊D311yescharacteristic
466C2×C4C2×D3×C3⋊D311yescharacteristic
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—24no—
470C2×C3—128no—
471C2×C3#6429714yes—
472C2×C3C2×#379814yes—
473C2×C3#6430714yes—
474C2×C3C2×#382811yescharacteristic
475C2×C3#6436111yescharacteristic
476C2×C3#6436711yescharacteristic
477D3—42no—
478D3—368no—
479D3—364no—
480D3—361no—
481C22—62no—
482C22—62no—
483C22—62no—
484C22—541no—
485C22—541no—
486C22—541no—
487C22C22×D3×C3⋊D311yescharacteristic
488C4—21no—
489C4—21no—
490C4—62no—
491C4—182no—
492C4—182no—
493C4—541no—
494C4—541no—
495C4—541no—
496C4#8902012yes—
497C3—24no—
498C3#1059387614yescomplemented
499C3#1059603211yescharacteristic, complemented
500C2—122no—
501C2—1081no—
502C2#1172456211yescharacteristic
503C2C2×#8902011yescharacteristic
504C2#1172457911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723531C2
hypercenterC42D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723913C2semidirect
2. C3×C4×#634C2semidirect
3. C32×C4×D12C2semidirect
4. #11723536C2semidirect
6. #11723449C2semidirect
7. #11723434C2semidirect
8. #11723428C2semidirect
9. #11723419C2semidirect
10. #11723415C2semidirect
11. #11723414C2semidirect
49. C4×#634D3semidirect
50. C3×C4×D12D3semidirect
166. C4×D12C3⋊D3semidirect
325. C33#542semidirect
448. C32#5561semidirect
449. C32#5582semidirect
498. C3#10593876semidirect
499. C3#10596032semidirect

Maximal subgroups

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

subgroupindex
#117239132
C3×C4×#6342
C32×C4×D122
#117235362
#117235312
#117234492
#117234342
#117234282
#117234192
#117234152
#117234142
#105960323
#105938763