Finite Group Databasebeta

← #13175347

Subgroups — #13175347

Order 1728.

1696 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic, complemented
2C3×#64084C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723529C211yesmaximal, characteristic
5#11723528C212yesmaximal, complemented
6#11723523C212yesmaximal, complemented
7#11723513C212yesmaximal, complemented
8#11723506C212yesmaximal, complemented
9#11723501C211yesmaximal, characteristic, complemented
10#11723434C211yesmaximal, characteristic, complemented
11#11723410C211yesmaximal, characteristic, complemented
12#10596030—31nomaximal
13#10593981—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6290C2212yes—
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C33×C42C2211yescharacteristic
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24#88857C2211yescharacteristic
25#88857C2211yescharacteristic
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3752C2212yes—
29C3×#3752C2212yes—
30C32×(C2×D3)⋊C4C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32#88807C2212yes—
33#88803C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64149—31no—
39#64145—31no—
40#64138—32no—
41C4×#638—32no—
42#64102—31no—
43#64096—31no—
44#64094—32no—
45#64092—32no—
46#64084D311yescharacteristic, complemented
47C4×#634—31no—
48C4×#634—34no—
49C4×#634—64no—
50C3×#875—34no—
51C3×C4×D12D314yescomplemented
52#63914—34no—
53#63913—38no—
54#63905—38no—
55#63898—34no—
56#63891—38no—
57#63879—38no—
58#63863—34no—
59#63843—34no—
60C22×C3×C3⋊D3C2312yes—
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
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69#6308—22no—
70C2×C3×C3⋊Dic3C2312yes—
71C3×C4⋊(C3⋊D3)—22no—
72C3×#634—22no—
73C2×C32×Dic3C2312yes—
74C32×D12—22no—
75C32×C4×D3—42no—
76#6290—24no—
77#6290—42no—
78Dic3×C3⋊D3—42no—
79#5681—91no—
80#5559—94no—
81C2×C32×D4—31no—
82C22×C32×C4—32no—
83C2×#638—32no—
84C2×#638—32no—
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×C4×C3⋊D3—32no—
87C2×#634—31no—
88C2×#634—34no—
89C2×#634—64no—
90C2×#634C2×D311yescharacteristic
91C2×C3×D12—34no—
92C2×C3×D12C2×D314yes—
93C2×C3×C4×D3C2×D318yes—
94C2×C3×Dic6—34no—
95C2×#625—38no—
96C2×#625—38no—
97C2×D3×Dic3—38no—
98C32×C4⋊C4—31no—
99C32×C22⋊C4—32no—
100C32×C42—24no—
101C32×C42C2×D314yes—
102C32×C42C2×D311yescharacteristic
103#3757—32no—
104#3757—32no—
105#3752—32no—
106#3752C2×D312yes—
107#3752C2×D312yes—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—64no—
111#3750—31no—
112#3750—31no—
113#3750—34no—
114#3750—34no—
115#3750—32no—
116#3750—64no—
117#3750—64no—
118C4×C3⋊Dic3—31no—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—34no—
121C4×C3⋊Dic3—34no—
122C4×C3⋊Dic3—32no—
123C4×C3⋊Dic3—64no—
124C4×C3⋊Dic3—64no—
125C3×(C2×D3)⋊C4—38no—
126C3×(C2×D3)⋊C4—38no—
127C3×(C2×D3)⋊C4C2×D318yes—
128C3×C4⋊Dic3C2×D314yes—
129Dic3⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131(C2×D3)⋊Dic3—38no—
132(C2×D3)⋊Dic3—38no—
133Dic32—38no—
134C22×C33C2411yescharacteristic
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C32×D3—22no—
139C2×C32×D3—22no—
140C2×C32×D3—22no—
141C33×C4—21no—
142C33×C4—21no—
143C33×C4#4112yes—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—21no—
146C3⋊(C3⋊Dic3)—21no—
147C3⋊(C3⋊Dic3)#4112yes—
148C3×C3⋊Dic3—22no—
149C32×Dic3—22no—
150C3×#124—91no—
151C3×C4×D4—34no—
152#947—91no—
153Q8×Dic3—91no—
154#940—94no—
155#940—91no—
156#936—94no—
157D4×Dic3—91no—
158#929—98no—
159C4×#67—98no—
160#893—94no—
161#887—94no—
162#886—92no—
163#885—98no—
164#885—92no—
165#883—98no—
166#883—92no—
167#67⋊C4—92no—
168#875—34no—
169C4×D12C3⋊D311yescharacteristic, complemented
170C4×Dic6—94no—
171C4×Dic6—91no—
172C4×Dic6—184no—
173C23×C32—32no—
174C22×C3⋊D3C22×D312yes—
175C22×C3×D3—38no—
176C22×C3×D3C22×D318yes—
177C32×D4—62no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—24no—
181C2×C32×C4—32no—
182C2×C32×C4—122no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189#638—64no—
190#638—122no—
191C2×C3⋊Dic3—31no—
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—34no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3—64no—
202C2×C3⋊Dic3—64no—
203C2×C3⋊Dic3C22×D312yes—
204C4⋊(C3⋊D3)—22no—
205C4×C3⋊D3—122no—
206#634—22no—
207#634—62no—
208#634—68no—
209#634—616no—
210C2×C3×Dic3—38no—
211C2×C3×Dic3C22×D318yes—
212C3×D12—28no—
213C3×D12—68no—
214C3×C4×D3—48no—
215C3×Dic6—68no—
216#625—616no—
217#625—128no—
218D3×Dic3—128no—
219#559—271noSylow
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3×C3⋊D3—42no—
224C32×D3—42no—
225C2×C3×Q8—91no—
226C2×C3×D4—34no—
227C2×C3×D4—91no—
228C22×C3×C4—38no—
229C2×#67—98no—
230C2×#67—92no—
231C2×#67—98no—
232C2×#67—92no—
233C22×Dic3—92no—
234C2×D12—34no—
235C2×D12C2×C3⋊D311yescharacteristic
236C2×C4×D3—98no—
237C2×C4×D3C2×C3⋊D312yes—
238C2×Dic6—34no—
239C2×Dic6—94no—
240C2×Dic6—91no—
241C2×Dic6—184no—
242C3×C4⋊C4—34no—
243C3×C22⋊C4—38no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C42—24no—
247C3×C42D3214yes—
248C3×C42C2×C3⋊D311yescharacteristic
249#217—98no—
250#217—98no—
251#217—92no—
252(C2×D3)⋊C4—38no—
253(C2×D3)⋊C4—38no—
254(C2×D3)⋊C4—98no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4C2×C3⋊D312yes—
258C4⋊Dic3—94no—
259C4⋊Dic3—91no—
260C4⋊Dic3—92no—
261C4⋊Dic3—184no—
262C4⋊Dic3C2×C3⋊D311yescharacteristic
263Dic3⋊C4—98no—
264Dic3⋊C4—94no—
265Dic3⋊C4—94no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—184no—
269Dic3⋊C4—184no—
270C4×Dic3—98no—
271C4×Dic3—94no—
272C4×Dic3—94no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—92no—
276C4×Dic3—184no—
277C4×Dic3—184no—
278C22×C32—24no—
279C22×C32—62no—
280C22×C32—62no—
281C22×C32—62no—
282C22×C32C23×D314yes—
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—22no—
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3×D3—28no—
288C2×C3×D3—28no—
289C2×C3×D3—28no—
290C2×C3×D3—68no—
291C2×C3×D3—68no—
292C2×C3×D3—68no—
293C32×C4—28no—
294C32×C4—24no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—28no—
298C32×C4—24no—
299C32×C4—44no—
300C32×C4—62no—
301C32×C4#23518yes—
302C32×C4#23712yes—
303C3⋊Dic3—22no—
304C3⋊Dic3—32no—
305C3⋊Dic3—38no—
306C3⋊Dic3—68no—
307C3⋊Dic3—68no—
308C3⋊Dic3—64no—
309C3⋊Dic3—61no—
310C3⋊Dic3—61no—
311C3⋊Dic3—61no—
312C3⋊Dic3—64no—
313C3⋊Dic3—64no—
314C3⋊Dic3—124no—
315C3⋊Dic3—124no—
316C3×Dic3—28no—
317C3×Dic3—68no—
318#127—271no—
319#124—91no—
320#124—272no—
321#124—271no—
322#123—272no—
323#123—272no—
324C4×Q8—271no—
325C4×Q8—271no—
326C4×D4—31no—
327C4×D4—272no—
328C4×D4—271no—
329C33#55911yescharacteristic, complemented, Sylow
330C23×C3—38no—
331C23×C3—92no—
332C22×D3—38no—
333C22×D3C22×C3⋊D312yes—
334C3×Q8—182no—
335C3×D4—68no—
336C3×D4—182no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—38no—
341C2×C3×C4—92no—
342C2×C3×C4—128no—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C2×D3214yes—
345C2×C3×C4C2×D3214yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347C2×C3×C4C22×C3⋊D311yescharacteristic
348C2×C3×C4C22×C3⋊D311yescharacteristic
349#67—184no—
350#67—1816no—
351#67—362no—
352#67—368no—
353C2×Dic3—38no—
354C2×Dic3—94no—
355C2×Dic3—91no—
356C2×Dic3—94no—
357C2×Dic3—94no—
358C2×Dic3—94no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—184no—
363C2×Dic3—184no—
364C2×Dic3—184no—
365C2×Dic3—184no—
366C2×Dic3—362no—
367C2×Dic3C22×C3⋊D312yes—
368D12—22no—
369D12—68no—
370C4×D3—42no—
371C4×D3—368no—
372Dic6—68no—
373Dic6—188no—
374Dic6—182no—
375Dic6—1816no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—122no—
380C2×C32C2×#23514yes—
381C2×C32C2×#23711yescharacteristic
382C2×C32#100414yes—
383C2×C32D3×#4114yes—
384C2×C32D3×#4111yescharacteristic
385C2×C32#100911yescharacteristic
386C3⋊D3—42no—
387C3×D3—48no—
388C3×D3—128no—
389C2×Q8—91no—
390C2×Q8—271no—
391C2×D4—31no—
392C2×D4—91no—
393C2×D4—272no—
394C2×D4—272no—
395C22×C4—32no—
396C22×C4—272no—
397C4⋊C4—31no—
398C4⋊C4—272no—
399C4⋊C4—271no—
400C4⋊C4—271no—
401C4⋊C4—271no—
402C22⋊C4—32no—
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C22⋊C4—272no—
408C42—272no—
409C42—271no—
410C42—271no—
411C42D3×C3⋊D311yescharacteristic
412C22×C3—24no—
413C22×C3—68no—
414C22×C3—68no—
415C22×C3—68no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—182no—
419C22×C3C22×D3214yes—
420C22×C3C23×C3⋊D311yescharacteristic
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—68no—
425C2×D3—68no—
426C2×D3—68no—
427C3×C4—28no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—28no—
433C3×C4—44no—
434C3×C4—68no—
435C3×C4—182no—
436C3×C4#379518yes—
437C3×C4#382812yes—
438Dic3—22no—
439Dic3—68no—
440Dic3—98no—
441Dic3—92no—
442Dic3—188no—
443Dic3—188no—
444Dic3—184no—
445Dic3—184no—
446Dic3—184no—
447Dic3—181no—
448Dic3—181no—
449Dic3—181no—
450Dic3—364no—
451Dic3—364no—
452C32—24no—
453C32#555914yescomplemented
454C32#568111yescharacteristic, complemented
455C23—32no—
456C23—92no—
457Q8—182no—
458Q8—542no—
459D4—62no—
460D4—182no—
461D4—544no—
462D4—1082no—
463C2×C4—32no—
464C2×C4—92no—
465C2×C4—122no—
466C2×C4—271no—
467C2×C4—271no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—1082no—
471C2×C4C2×D3×C3⋊D311yescharacteristic
472C2×C4C2×D3×C3⋊D311yescharacteristic
473C2×C4C2×D3×C3⋊D311yescharacteristic
474C2×C3—24no—
475C2×C3—24no—
476C2×C3—24no—
477C2×C3—128no—
478C2×C3—362no—
479C2×C3C2×#379514yes—
480C2×C3#6430614yes—
481C2×C3#6430714yes—
482C2×C3C2×#382811yescharacteristic
483C2×C3#6436111yescharacteristic
484C2×C3#41×C3⋊D311yescharacteristic
485D3—42no—
486D3—128no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—182no—
491C22—182no—
492C22—182no—
493C22C22×D3×C3⋊D311yescharacteristic
494C4—21no—
495C4—21no—
496C4—62no—
497C4—182no—
498C4—272no—
499C4—541no—
500C4—541no—
501C4—541no—
502C4#8902112yes—
503C3—24no—
504C3#1059398114yescomplemented
505C3#1059603011yescharacteristic, complemented
506C2—122no—
507C2—362no—
508C2C2×#8902111yescharacteristic
509C2#1172457811yescharacteristic
510C2#1172457911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723529C2
hypercenterC42D3×C3⋊D3
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
2. C3×#64084C2semidirect
3. C32×C4×D12C2semidirect
5. #11723528C2semidirect
6. #11723523C2semidirect
7. #11723513C2semidirect
8. #11723506C2semidirect
9. #11723501C2semidirect
10. #11723434C2semidirect
11. #11723410C2semidirect
46. #64084D3semidirect
51. C3×C4×D12D3semidirect
169. C4×D12C3⋊D3semidirect
329. C33#559semidirect
453. C32#5559semidirect
454. C32#5681semidirect
504. C3#10593981semidirect
505. C3#10596030semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×#640842
C32×C4×D122
#117235292
#117235282
#117235232
#117235132
#117235062
#117235012
#117234342
#117234102
#105960303
#105939813