Finite Group Databasebeta

← #13175418

Subgroups — #13175418

Order 1728.

2308 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89026C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3#11723913C211yesmaximal, characteristic, complemented
4C3×#64083C211yesmaximal, characteristic, complemented
5C32×#875C211yesmaximal, characteristic, complemented
6#11723541C214yesmaximal, complemented
7#11723529C211yesmaximal, characteristic, complemented
8#11723519C214yesmaximal, complemented
9#11723436C211yesmaximal, characteristic, complemented
10#10596145—31nomaximal
11#10594010—34nomaximal
12C22×D3×C3⋊D3C2212yes—
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C3×C4⋊(C3⋊D3)C2214yescomplemented
17C2×C32×D12C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic, complemented
19C2×#6292C2212yes—
20C2×#6290C2212yes—
21C2×#6290C2214yescomplemented
22C2×Dic3×C3⋊D3C2212yescomplemented
23#89026—24no—
24#89017—24no—
25C33×C42C2211yescharacteristic, complemented
26#88859C2212yes—
27#88857C2212yescomplemented
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29C32×(C2×D3)⋊C4C2214yescomplemented
30#88803C2214yescomplemented
31C2×#3832—31no—
32C2×D4×C3⋊D3—31no—
33C2×#3802—34no—
34C2×#3796—34no—
35C32×#124—31no—
36#64156—31no—
37#64149—31no—
38#64093—34no—
39#64091—34no—
40#64083D311yescharacteristic, complemented
41#64081—31no—
42#64081—34no—
43#64081—64no—
44C3×#875D314yescomplemented
45C3×#874—34no—
46#63922—316no—
47#63914—34no—
48#63885—316no—
49#63863—34no—
50C22×C3⋊(C3⋊D3)C2311yescharacteristic
51C22×C3×C3⋊D3C2312yes—
52C22×C3×C3⋊D3C2312yes—
53C22×C32×D3C2312yes—
54C2×D3×C3⋊D3—24no—
55C2×D3×C3⋊D3—44no—
56C2×C33×C4C2312yes—
57C2×C33×C4C2311yescharacteristic
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C4×C3⋊(C3⋊D3)D414yescomplemented
61C3×C4⋊(C3⋊D3)—22no—
62C3×C4⋊(C3⋊D3)—22no—
63C3×C4⋊(C3⋊D3)—44no—
64C2×C32×Dic3C2312yes—
65C32×D12—22no—
66C32×Dic6—22no—
67#6292—24no—
68#6290—24no—
69#6290—44no—
70Dic3×C3⋊D3—24no—
71#5715—91no—
72#5674—94no—
73C23×C3⋊D3—32no—
74C22×D32—38no—
75C2×C32×Q8—31no—
76C2×C32×D4—31no—
77C2×#638—32no—
78C2×#638—34no—
79#3832—64no—
80D4×C3⋊D3—64no—
81C2×C4⋊(C3⋊D3)—32no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
84C2×C4⋊(C3⋊D3)C2×D314yescomplemented
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—34no—
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3—64no—
89C2×C3×D12—34no—
90C2×C3×D12—34no—
91C2×C3×D12—316no—
92C2×C3×D12C2×D314yes—
93C2×C3×Dic6C2×D314yescomplemented
94C2×#626—38no—
95C2×#625—38no—
96C2×#625—316no—
97C2×D3×Dic3—38no—
98#3802—616no—
99#3796—616no—
100C32×C22⋊C4—34no—
101C32×C42—24no—
102C32×C42C2×D314yescomplemented
103C32×C42C2×D311yescharacteristic, complemented
104#3752—34no—
105#3752—32no—
106#3752—38no—
107#3752—68no—
108#3750—32no—
109#3750—38no—
110#3750—68no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—34no—
113C4×C3⋊Dic3—64no—
114C3×(C2×D3)⋊C4C2×D3116yescomplemented
115(C2×D3)⋊Dic3—316no—
116C22×C33C2411yescharacteristic
117C2×C3⋊(C3⋊D3)—22no—
118C2×C3⋊(C3⋊D3)C2×D412yes—
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—24no—
122C2×C3×C3⋊D3—24no—
123C2×C32×D3—22no—
124C2×C32×D3—24no—
125D3×C3⋊D3—48no—
126C33×C4—22no—
127C33×C4C2×D412yes—
128C3⋊(C3⋊Dic3)—22no—
129C3⋊(C3⋊Dic3)C2×D412yes—
130C32×Dic3—22no—
131C2×#239—94no—
132C2×#237—91no—
133C2×D3×D4—94no—
134C2×D3×D4—91no—
135C3×#127—91no—
136C3×#124—34no—
137#947—94no—
138#940—94no—
139#940—91no—
140#939—94no—
141#937—94no—
142#936—91no—
143#884—916no—
144#882—916no—
145#875C3⋊D311yescharacteristic, complemented
146#874—34no—
147#872—94no—
148#872—91no—
149#872—184no—
150C23×C32—32no—
151C22×C3⋊D3—31no—
152C22×C3⋊D3—34no—
153C22×C3⋊D3—64no—
154C22×C3⋊D3—64no—
155C22×C3⋊D3—124no—
156C22×C3⋊D3C22×D312yes—
157C22×C3⋊D3C22×D312yes—
158C22×C3×D3—38no—
159C22×C3×D3—38no—
160C22×C3×D3C22×D318yes—
161C2×D32—616no—
162C2×D32—1216no—
163C32×Q8—62no—
164C32×D4—62no—
165C2×C32×C4—24no—
166C2×C32×C4—28no—
167C2×C32×C4—32no—
168C2×C32×C4C22×D318yes—
169C2×C32×C4C22×D314yes—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172#638—64no—
173#638—124no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—34no—
177C2×C3⋊Dic3—38no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—68no—
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)—22no—
182C4⋊(C3⋊D3)—44no—
183C4⋊(C3⋊D3)—64no—
184C4×C3⋊D3—34no—
185C4×C3⋊D3—316no—
186C4×C3⋊D3—616no—
187C4×C3⋊D3—64no—
188C2×C3×Dic3C22×D318yes—
189C3×D12—28no—
190C3×D12—68no—
191C3×D12—68no—
192C3×D12—1216no—
193C3×Dic6—28no—
194#626—616no—
195#625—616no—
196#625—1216no—
197D3×Dic3—616no—
198#535—271noSylow
199C2×C33C22×D411yescharacteristic
200C2×C33H(5,𝔽2)12yes—
201C3⋊(C3⋊D3)—22no—
202C3×C3⋊D3—42no—
203C3×C3⋊D3—42no—
204C32×D3—42no—
205C23×D3—98no—
206C23×D3—92no—
207C2×C3×Q8—34no—
208C2×C3×D4—34no—
209C2×C3×D4—91no—
210C2×C3×D4—91no—
211C2×C3×D4—94no—
212C2×#67—98no—
213C2×#67—92no—
214C2×#67—916no—
215C2×#67—94no—
216C2×#67—92no—
217C22×Dic3—92no—
218#239—1816no—
219#237—184no—
220D3×D4—184no—
221D3×D4—1816no—
222C2×D12—34no—
223C2×D12—34no—
224C2×D12—316no—
225C2×D12—98no—
226C2×D12C2×C3⋊D311yescharacteristic
227C2×C4×D3—94no—
228C2×C4×D3—91no—
229C2×C4×D3—98no—
230C2×C4×D3—184no—
231C2×Dic6C2×C3⋊D311yescharacteristic, complemented
232C3×C22⋊C4—316no—
233C3×C42—24no—
234C3×C42D3214yescomplemented
235C3×C42C2×C3⋊D311yescharacteristic, complemented
236#217—94no—
237(C2×D3)⋊C4—916no—
238(C2×D3)⋊C4—98no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—188no—
241(C2×D3)⋊C4C2×C3⋊D314yescomplemented
242Dic3⋊C4—98no—
243Dic3⋊C4—92no—
244Dic3⋊C4—188no—
245C4×Dic3—94no—
246C4×Dic3—91no—
247C4×Dic3—184no—
248C22×C32—24no—
249C22×C32—62no—
250C22×C32—64no—
251C22×C32C23×D314yes—
252C22×C32C23×D311yescharacteristic
253C2×C3⋊D3—22no—
254C2×C3⋊D3—22no—
255C2×C3⋊D3—24no—
256C2×C3⋊D3—24no—
257C2×C3⋊D3—32no—
258C2×C3⋊D3—38no—
259C2×C3⋊D3—68no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—68no—
262C2×C3⋊D3—128no—
263C2×C3⋊D3—128no—
264C2×C3×D3—28no—
265C2×C3×D3—216no—
266C2×C3×D3—68no—
267C2×C3×D3—68no—
268C2×C3×D3—616no—
269C2×C3×D3—616no—
270D32—1232no—
271C32×C4—22no—
272C32×C4—28no—
273C32×C4—28no—
274C32×C4—48no—
275C32×C4—62no—
276C32×C4D3×D418yes—
277C32×C4D3×D412yes—
278C3⋊Dic3—32no—
279C3⋊Dic3—38no—
280C3⋊Dic3—68no—
281C3⋊Dic3—62no—
282C3⋊Dic3—68no—
283C3⋊Dic3—128no—
284C3×Dic3—28no—
285C2×#41—271no—
286C22×D4—271no—
287#127—91no—
288#127—271no—
289#124—31no—
290#124—271no—
291#121—274no—
292#120—274no—
293#117—271no—
294C33#53511yescharacteristic, complemented, Sylow
295C23×C3—38no—
296C23×C3—92no—
297C23×C3—92no—
298C22×D3—38no—
299C22×D3—38no—
300C22×D3—94no—
301C22×D3—91no—
302C22×D3—184no—
303C22×D3—184no—
304C22×D3—1816no—
305C22×D3—364no—
306C22×D3—3616no—
307C22×D3C22×C3⋊D312yes—
308C3×Q8—68no—
309C3×D4—68no—
310C3×D4—182no—
311C3×D4—182no—
312C3×D4—364no—
313C2×C3×C4—24no—
314C2×C3×C4—28no—
315C2×C3×C4—38no—
316C2×C3×C4C2×D3218yes—
317C2×C3×C4C2×D3214yes—
318C2×C3×C4C22×C3⋊D312yes—
319C2×C3×C4C22×C3⋊D311yescharacteristic
320#67—184no—
321#67—184no—
322#67—1816no—
323#67—364no—
324#67—3616no—
325C2×Dic3—94no—
326C2×Dic3—91no—
327C2×Dic3—98no—
328C2×Dic3—92no—
329C2×Dic3—184no—
330C2×Dic3—188no—
331C2×Dic3—184no—
332C2×Dic3C22×C3⋊D312yes—
333D12—22no—
334D12—68no—
335D12—68no—
336D12—1216no—
337D12—1816no—
338C4×D3—916no—
339C4×D3—94no—
340C4×D3—1816no—
341C4×D3—1816no—
342Dic6—22no—
343C2×C32—24no—
344C2×C32—28no—
345C2×C32—122no—
346C2×C32C2×D3×D414yes—
347C2×C32C2×D3×D411yescharacteristic
348C2×C32#100412yes—
349C2×C32#100918yes—
350C3⋊D3—42no—
351C3⋊D3—42no—
352C3⋊D3—62no—
353C3⋊D3—68no—
354C3⋊D3—128no—
355C3×D3—48no—
356C3×D3—128no—
357C3×D3—128no—
358C24—272no—
359#41—544no—
360C2×Q8—31no—
361C2×D4—31no—
362C2×D4—91no—
363C2×D4—91no—
364C2×D4—94no—
365C2×D4—272no—
366C2×D4—272no—
367C2×D4—274no—
368C2×D4—544no—
369C22×C4—272no—
370C22×C4—271no—
371C4⋊C4—272no—
372C22⋊C4—34no—
373C22⋊C4—274no—
374C22⋊C4—272no—
375C42—271no—
376C42D3×C3⋊D311yescharacteristic, complemented
377C22×C3—24no—
378C22×C3—68no—
379C22×C3—616no—
380C22×C3—182no—
381C22×C3—182no—
382C22×C3—184no—
383C22×C3—184no—
384C22×C3C22×D3214yes—
385C22×C3C23×C3⋊D311yescharacteristic
386C2×D3—22no—
387C2×D3—24no—
388C2×D3—68no—
389C2×D3—68no—
390C2×D3—616no—
391C2×D3—616no—
392C2×D3—98no—
393C2×D3—92no—
394C2×D3—188no—
395C2×D3—188no—
396C2×D3—182no—
397C2×D3—368no—
398C2×D3—368no—
399C2×D3—3632no—
400C3×C4—28no—
401C3×C4—22no—
402C3×C4—28no—
403C3×C4—48no—
404C3×C4—68no—
405C3×C4#380218yes—
406C3×C4D4×C3⋊D312yes—
407Dic3—22no—
408Dic3—98no—
409Dic3—92no—
410Dic3—188no—
411Dic3—188no—
412Dic3—182no—
413Dic3—368no—
414C32—24no—
415C32#567414yescomplemented
416C32#571511yescharacteristic, complemented
417C23—32no—
418C23—92no—
419C23—92no—
420C23—271no—
421C23—544no—
422C23—1084no—
423Q8—62no—
424D4—62no—
425D4—182no—
426D4—182no—
427D4—364no—
428D4—544no—
429D4—544no—
430D4—1084no—
431C2×C4—32no—
432C2×C4—271no—
433C2×C4—274no—
434C2×C4—272no—
435C2×C4—544no—
436C2×C4C2×D3×C3⋊D312yes—
437C2×C4C2×D3×C3⋊D311yescharacteristic
438C2×C3—24no—
439C2×C3—28no—
440C2×C3—128no—
441C2×C3—362no—
442C2×C3—362no—
443C2×C3C2×#380214yes—
444C2×C3#6430718yes—
445C2×C3C2×D4×C3⋊D311yescharacteristic
446C2×C3#6436612yes—
447D3—42no—
448D3—128no—
449D3—128no—
450D3—182no—
451D3—188no—
452D3—368no—
453C22—62no—
454C22—64no—
455C22—182no—
456C22—182no—
457C22—184no—
458C22—184no—
459C22—272no—
460C22—542no—
461C22—1088no—
462C22C22×D3×C3⋊D311yescharacteristic
463C4—22no—
464C4—62no—
465C4—272no—
466C4—542no—
467C4#8902612yes—
468C3—24no—
469C3#1059401014yescomplemented
470C3#1059614511yescharacteristic, complemented
471C2—122no—
472C2—362no—
473C2—362no—
474C2—542no—
475C2C2×#8902611yescharacteristic
476C2#1172458012yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89026C2semidirect
2. C2×#89017C2semidirect
3. #11723913C2semidirect
4. C3×#64083C2semidirect
5. C32×#875C2semidirect
6. #11723541C2semidirect
7. #11723529C2semidirect
8. #11723519C2semidirect
9. #11723436C2semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
18. C2×C32×Dic6C22semidirect
21. C2×#6290C22semidirect
22. C2×Dic3×C3⋊D3C22semidirect
25. C33×C42C22semidirect
27. #88857C22semidirect
28. C4×C3⋊(C3⋊Dic3)C22semidirect
29. C32×(C2×D3)⋊C4C22semidirect
30. #88803C22semidirect
40. #64083D3semidirect
44. C3×#875D3semidirect
60. C4×C3⋊(C3⋊D3)D4semidirect
83. C2×C4⋊(C3⋊D3)C2×D3semidirect
84. C2×C4⋊(C3⋊D3)C2×D3semidirect
93. C2×C3×Dic6C2×D3semidirect
102. C32×C42C2×D3semidirect
103. C32×C42C2×D3semidirect
114. C3×(C2×D3)⋊C4C2×D3semidirect
145. #875C3⋊D3semidirect
231. C2×Dic6C2×C3⋊D3semidirect
234. C3×C42D32semidirect
235. C3×C42C2×C3⋊D3semidirect
241. (C2×D3)⋊C4C2×C3⋊D3semidirect
294. C33#535semidirect
376. C42D3×C3⋊D3semidirect
415. C32#5674semidirect
416. C32#5715semidirect
469. C3#10594010semidirect
470. C3#10596145semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C2×#890172
#117239132
C3×#640832
C32×#8752
#117235412
#117235292
#117235192
#117234362
#105961453
#105940103