Finite Group Databasebeta

← #13177428

Subgroups — #13177428

Order 1728.

2948 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C32×#936C211yesmaximal, characteristic, complemented
5#11723639C214yesmaximal, complemented
6#11723624C214yesmaximal, complemented
7#11723517C211yesmaximal, characteristic, complemented
8#11723452C211yesmaximal, characteristic, complemented
9#11723440C211yesmaximal, characteristic, complemented
10#10596145—31nomaximal
11#10595168—34nomaximal
12C23×C3⋊(C3⋊D3)C2212yes—
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2212yes—
15D4×C3⋊(C3⋊D3)—24no—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#638C2214yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C2×C32×Dic6C2211yescharacteristic, complemented
21C2×#6292C2212yes—
22C2×#6292C2214yescomplemented
23C2×Dic3×C3⋊D3C2212yescomplemented
24#89017—24no—
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26C32×#217C2214yescomplemented
27C32×C4×Dic3C2211yescharacteristic, complemented
28#88806C2212yescomplemented
29#88804C2212yes—
30#88804C2214yescomplemented
31C2×#3832—31no—
32C2×D4×C3⋊D3—31no—
33C2×D4×C3⋊D3—34no—
34C2×D4×C3⋊D3—64no—
35C2×#3796—34no—
36C32×#124—31no—
37#64156—31no—
38#64149D311yescharacteristic, complemented
39#64093—34no—
40#64091—34no—
41#64083—31no—
42#64081—31no—
43C3×#940—34no—
44C3×#936D314yescomplemented
45#63981—316no—
46#63971—316no—
47#63911—34no—
48#63874—34no—
49#63864—34no—
50C23×C33C2312yes—
51C22×C3⋊(C3⋊D3)—24no—
52C22×C3⋊(C3⋊D3)—44no—
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C3⋊(C3⋊D3)C2311yescharacteristic
55C22×C3×C3⋊D3C2312yes—
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58#6312—24no—
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C4⋊(C3⋊(C3⋊D3))—22no—
61C4×C3⋊(C3⋊D3)D414yescomplemented
62C3×#638—44no—
63C2×C3×C3⋊Dic3C2312yes—
64C3×C4⋊(C3⋊D3)—22no—
65C2×C32×Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67C32×Dic6—22no—
68#6292—24no—
69#6292—44no—
70Dic3×C3⋊D3—24no—
71#5715—91no—
72#5674—94no—
73C23×C3⋊D3—32no—
74C23×C3⋊D3—38no—
75C23×C3⋊D3—68no—
76C2×C32×Q8—31no—
77C2×C32×D4—24no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D311yescharacteristic
80C2×#638—32no—
81C2×#638—38no—
82C2×#638—68no—
83C2×#638C2×D314yescomplemented
84#3832—64no—
85D4×C3⋊D3—64no—
86D4×C3⋊D3—616no—
87D4×C3⋊D3—632no—
88C2×C4⋊(C3⋊D3)—32no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4⋊(C3⋊D3)—34no—
91C2×C4⋊(C3⋊D3)—34no—
92C2×C4⋊(C3⋊D3)—64no—
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
94C2×C4×C3⋊D3—31no—
95C2×C4×C3⋊D3—34no—
96C2×C4×C3⋊D3—32no—
97C2×C4×C3⋊D3—64no—
98C2×C3×#67—316no—
99C2×C3×D12—34no—
100C2×C3×Dic6C2×D314yescomplemented
101C2×#626—38no—
102C2×#626—316no—
103C2×D3×Dic3—38no—
104#3796—616no—
105C32×C22⋊C4—34no—
106C32×C42—31no—
107#3752—32no—
108#3752—34no—
109#3750—32no—
110C4×C3⋊Dic3C2×D311yescharacteristic, complemented
111C3×#217C2×D3116yescomplemented
112C3×C4×Dic3—34no—
113C3×C4×Dic3C2×D314yescomplemented
114#3724—38no—
115#3722—316no—
116#3722—38no—
117C22×C33—22no—
118C22×C33—24no—
119C22×C33C2411yescharacteristic
120C2×C3⋊(C3⋊D3)—22no—
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)—24no—
123C2×C3⋊(C3⋊D3)—48no—
124C2×C3⋊(C3⋊D3)C2×D412yes—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—24no—
127C33×C4C2×D412yes—
128C3⋊(C3⋊Dic3)C2×D412yes—
129C3×C3⋊Dic3—22no—
130C32×Dic3—22no—
131C32×Dic3—22no—
132C2×#239—94no—
133C2×#237—91no—
134C2×D3×D4—94no—
135C2×D3×D4—91no—
136C2×D3×D4—184no—
137C3×#127—91no—
138C3×#124—34no—
139#947—94no—
140#940—34no—
141#940—91no—
142#939—94no—
143#937—94no—
144#936C3⋊D311yescharacteristic, complemented
145#884—916no—
146#882—916no—
147#875—91no—
148#874—94no—
149#872—94no—
150#872—91no—
151C23×C32—28no—
152C23×C32C22×D318yes—
153C23×C32C22×D312yes—
154C22×C3⋊D3—31no—
155C22×C3⋊D3—32no—
156C22×C3⋊D3—34no—
157C22×C3⋊D3—38no—
158C22×C3⋊D3—64no—
159C22×C3⋊D3—68no—
160C22×C3⋊D3—64no—
161C22×C3⋊D3—616no—
162C22×C3⋊D3—632no—
163C22×C3⋊D3—124no—
164C22×C3⋊D3—1216no—
165C22×C3⋊D3—1232no—
166C22×C3⋊D3C22×D312yes—
167C22×C3×D3—38no—
168C32×Q8—62no—
169C32×D4—216no—
170C32×D4—28no—
171C32×D4—22no—
172C2×C32×C4—24no—
173C2×C32×C4—32no—
174C2×C32×C4—32no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D311yescharacteristic
177#638—44no—
178#638—64no—
179#638—616no—
180#638—632no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3C22×D312yes—
185C4⋊(C3⋊D3)—22no—
186C4⋊(C3⋊D3)—616no—
187C4⋊(C3⋊D3)—64no—
188C4⋊(C3⋊D3)—68no—
189C4⋊(C3⋊D3)—62no—
190C4⋊(C3⋊D3)—124no—
191C4×C3⋊D3—34no—
192C4×C3⋊D3—316no—
193C4×C3⋊D3—616no—
194C4×C3⋊D3—64no—
195C3×#67—1216no—
196C2×C3×Dic3—38no—
197C2×C3×Dic3C22×D318yes—
198C2×C3×Dic3C22×D318yes—
199C3×D12—68no—
200C3×Dic6—28no—
201#626—616no—
202#626—1216no—
203D3×Dic3—616no—
204#535—271noSylow
205C2×C33—42no—
206C2×C33C22×D411yescharacteristic
207C2×C33H(5,𝔽2)12yes—
208C3⋊(C3⋊D3)—22no—
209C3⋊(C3⋊D3)—42no—
210C3×C3⋊D3—42no—
211C23×D3—98no—
212C23×D3—92no—
213C23×D3—188no—
214C2×C3×Q8—34no—
215C2×C3×D4—24no—
216C2×C3×D4—94no—
217C2×C3×D4—91no—
218C2×C3×D4D3214yes—
219C2×C3×D4C2×C3⋊D311yescharacteristic
220C2×#67—316no—
221C2×#67—98no—
222C2×#67—92no—
223C2×#67—92no—
224C2×#67—94no—
225C2×#67—188no—
226C22×Dic3—92no—
227#239—1816no—
228#237—184no—
229D3×D4—1816no—
230D3×D4—184no—
231D3×D4—1832no—
232C2×D12—34no—
233C2×D12—98no—
234C2×D12—94no—
235C2×D12—91no—
236C2×D12—916no—
237C2×D12—184no—
238C2×C4×D3—94no—
239C2×C4×D3—91no—
240C2×C4×D3—98no—
241C2×C4×D3—184no—
242C2×Dic6C2×C3⋊D311yescharacteristic, complemented
243C3×C22⋊C4—316no—
244C3×C42—34no—
245C3×C42—91no—
246#217C2×C3⋊D314yescomplemented
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—916no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—94no—
251Dic3⋊C4—98no—
252Dic3⋊C4—92no—
253C4×Dic3—34no—
254C4×Dic3C2×C3⋊D311yescharacteristic, complemented
255C22×C32—216no—
256C22×C32—28no—
257C22×C32—22no—
258C22×C32—24no—
259C22×C32—24no—
260C22×C32—216no—
261C22×C32—416no—
262C22×C32C23×D314yes—
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—22no—
265C2×C3⋊D3—24no—
266C2×C3⋊D3—32no—
267C2×C3⋊D3—38no—
268C2×C3⋊D3—68no—
269C2×C3⋊D3—616no—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—68no—
272C2×C3⋊D3—616no—
273C2×C3⋊D3—68no—
274C2×C3⋊D3—62no—
275C2×C3⋊D3—64no—
276C2×C3⋊D3—616no—
277C2×C3⋊D3—1216no—
278C2×C3⋊D3—1264no—
279C2×C3⋊D3—1232no—
280C2×C3⋊D3—128no—
281C2×C3×D3—68no—
282C2×C3×D3—616no—
283C32×C4—28no—
284C32×C4—62no—
285C32×C4—62no—
286C32×C4D3×D418yes—
287C32×C4D3×D412yes—
288C3⋊Dic3—22no—
289C3⋊Dic3—32no—
290C3⋊Dic3—38no—
291C3⋊Dic3—68no—
292C3×Dic3—28no—
293C3×Dic3—28no—
294C3×Dic3—68no—
295C2×#41—271no—
296C22×D4—271no—
297#127—91no—
298#127—271no—
299#124—31no—
300#124—271no—
301#121—274no—
302#120—274no—
303#117—271no—
304C33#53511yescharacteristic, complemented, Sylow
305C23×C3—28no—
306C23×C3—92no—
307C23×C3C2×D3218yes—
308C23×C3C22×C3⋊D312yes—
309C22×D3—38no—
310C22×D3—94no—
311C22×D3—91no—
312C22×D3—98no—
313C22×D3—92no—
314C22×D3—184no—
315C22×D3—188no—
316C22×D3—1816no—
317C22×D3—184no—
318C22×D3—1832no—
319C22×D3—3616no—
320C22×D3—364no—
321C22×D3—3632no—
322C3×Q8—68no—
323C3×D4—216no—
324C3×D4—28no—
325C3×D4—22no—
326C3×D4—182no—
327C3×D4—364no—
328C2×C3×C4—24no—
329C2×C3×C4—38no—
330C2×C3×C4—38no—
331C2×C3×C4—92no—
332C2×C3×C4C2×D3214yes—
333C2×C3×C4C22×C3⋊D311yescharacteristic
334#67—1216no—
335#67—184no—
336#67—1816no—
337#67—184no—
338#67—1832no—
339#67—364no—
340C2×Dic3—38no—
341C2×Dic3—94no—
342C2×Dic3—91no—
343C2×Dic3—184no—
344C2×Dic3—184no—
345C2×Dic3C22×C3⋊D312yes—
346C2×Dic3C22×C3⋊D312yes—
347D12—68no—
348D12—1816no—
349D12—1816no—
350D12—188no—
351D12—182no—
352D12—3616no—
353C4×D3—916no—
354C4×D3—94no—
355C4×D3—1816no—
356C4×D3—1816no—
357Dic6—22no—
358C2×C32—28no—
359C2×C32—24no—
360C2×C32—416no—
361C2×C32—48no—
362C2×C32—42no—
363C2×C32C2×D3×D414yes—
364C2×C32C2×D3×D411yescharacteristic
365C2×C32#100412yes—
366C2×C32#100918yes—
367C3⋊D3—42no—
368C3⋊D3—616no—
369C3⋊D3—68no—
370C3⋊D3—62no—
371C3⋊D3—1216no—
372C3⋊D3—128no—
373C3⋊D3—122no—
374C3×D3—128no—
375C24—272no—
376#41—544no—
377C2×Q8—31no—
378C2×D4—94no—
379C2×D4—91no—
380C2×D4—272no—
381C2×D4—274no—
382C2×D4—272no—
383C2×D4—271no—
384C2×D4—544no—
385C2×D4D3×C3⋊D311yescharacteristic
386C22×C4—272no—
387C22×C4—271no—
388C4⋊C4—272no—
389C22⋊C4—34no—
390C22⋊C4—272no—
391C22⋊C4—274no—
392C42—31no—
393C42—91no—
394C22×C3—216no—
395C22×C3—216no—
396C22×C3—28no—
397C22×C3—22no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—416no—
401C22×C3—182no—
402C22×C3—184no—
403C22×C3C22×D3214yes—
404C22×C3C23×C3⋊D311yescharacteristic
405C2×D3—616no—
406C2×D3—68no—
407C2×D3—98no—
408C2×D3—92no—
409C2×D3—188no—
410C2×D3—1816no—
411C2×D3—188no—
412C2×D3—182no—
413C2×D3—1816no—
414C2×D3—1816no—
415C2×D3—188no—
416C2×D3—182no—
417C2×D3—184no—
418C2×D3—3616no—
419C2×D3—3664no—
420C2×D3—368no—
421C2×D3—3632no—
422C3×C4—28no—
423C3×C4—68no—
424C3×C4—68no—
425C3×C4—182no—
426C3×C4#381118yes—
427C3×C4D4×C3⋊D312yes—
428Dic3—22no—
429Dic3—22no—
430Dic3—68no—
431Dic3—98no—
432Dic3—92no—
433Dic3—188no—
434C32—24no—
435C32#567414yescomplemented
436C32#571511yescharacteristic, complemented
437C23—92no—
438C23—271no—
439C23—272no—
440C23—544no—
441C23—1084no—
442C23C2×D3×C3⋊D312yes—
443Q8—62no—
444D4—22no—
445D4—182no—
446D4—364no—
447D4—544no—
448D4—544no—
449D4—542no—
450D4—1084no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—92no—
454C2×C4—271no—
455C2×C4—274no—
456C2×C4—544no—
457C2×C4C2×D3×C3⋊D311yescharacteristic
458C2×C3—28no—
459C2×C3—24no—
460C2×C3—416no—
461C2×C3—42no—
462C2×C3—48no—
463C2×C3—362no—
464C2×C3#6431418yes—
465C2×C3C2×#381114yes—
466C2×C3C2×D4×C3⋊D311yescharacteristic
467C2×C3#6436612yes—
468D3—128no—
469D3—1816no—
470D3—182no—
471D3—188no—
472D3—3616no—
473D3—362no—
474D3—368no—
475C22—24no—
476C22—22no—
477C22—184no—
478C22—182no—
479C22—272no—
480C22—542no—
481C22—544no—
482C22—542no—
483C22—1088no—
484C22C22×D3×C3⋊D311yescharacteristic
485C4—62no—
486C4—62no—
487C4—182no—
488C4—272no—
489C4#8903912yes—
490C3—24no—
491C3#1059516814yescomplemented
492C3#1059614511yescharacteristic, complemented
493C2—42no—
494C2—362no—
495C2—542no—
496C2—1082no—
497C2#1172458912yes—
498C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊(C3⋊D3)C2semidirect
2. C2×#89017C2semidirect
3. C3×#64149C2semidirect
4. C32×#936C2semidirect
5. #11723639C2semidirect
6. #11723624C2semidirect
7. #11723517C2semidirect
8. #11723452C2semidirect
9. #11723440C2semidirect
18. C2×C3×#638C22semidirect
19. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×C32×Dic6C22semidirect
22. C2×#6292C22semidirect
23. C2×Dic3×C3⋊D3C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. C32×#217C22semidirect
27. C32×C4×Dic3C22semidirect
28. #88806C22semidirect
30. #88804C22semidirect
38. #64149D3semidirect
44. C3×#936D3semidirect
61. C4×C3⋊(C3⋊D3)D4semidirect
83. C2×#638C2×D3semidirect
93. C2×C4⋊(C3⋊D3)C2×D3semidirect
100. C2×C3×Dic6C2×D3semidirect
110. C4×C3⋊Dic3C2×D3semidirect
111. C3×#217C2×D3semidirect
113. C3×C4×Dic3C2×D3semidirect
144. #936C3⋊D3semidirect
242. C2×Dic6C2×C3⋊D3semidirect
246. #217C2×C3⋊D3semidirect
254. C4×Dic3C2×C3⋊D3semidirect
304. C33#535semidirect
435. C32#5674semidirect
436. C32#5715semidirect
491. C3#10595168semidirect
492. C3#10596145semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×#890172
C3×#641492
C32×#9362
#117236392
#117236242
#117235172
#117234522
#117234402
#105961453
#105951683