Finite Group Databasebeta

← #13178623

Subgroups — #13178623

Order 1728.

2498 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89049C214yesmaximal, complemented
3C2×C4⋊#1113C211yesmaximal, characteristic, complemented
4C3×#64149C212yesmaximal, complemented
5#11723728C214yesmaximal, complemented
6#11723697C212yesmaximal, complemented
7C4×#6294C211yesmaximal, characteristic, complemented
8#10595300—31nomaximal
9D3×#940—32nomaximal
10C23×C3×C3⋊D3C2212yes—
11C22×#1113C2212yes—
12C2×C33×D4C2211yescharacteristic, complemented
13C2×C3×#638C2218yescomplemented
14C2×C3×#638C2212yescomplemented
15C3×D4×C3⋊D3—24no—
16C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19#89049—216no—
20C2×#6295C2212yescomplemented
21C2×#6295C2218yescomplemented
22C2×#6294C2212yes—
23C4⋊#1113—24no—
24C3×C4×C3⋊Dic3C2212yescomplemented
25#88813C2212yescomplemented
26C2×D4×C3⋊D3D311yescharacteristic, complemented
27C2×C3×D3×D4—32no—
28C2×C3×D3×D4—61no—
29C2×#3811—34no—
30C2×D3×#67—38no—
31C2×#3802—31no—
32C2×D3×D12—32no—
33#64149D312yescomplemented
34C3×#940—32no—
35C3×#940—32no—
36C3×#940—62no—
37#63979—38no—
38#63979—34no—
39#63914—32no—
40#63911—32no—
41#63911—32no—
42C4×#624—31no—
43C4×D3×Dic3—32no—
44C23×C33C2312yes—
45C22×C3×C3⋊D3—24no—
46C22×C3×C3⋊D3—28no—
47C22×C3×C3⋊D3C2314yes—
48C22×C3×C3⋊D3C2312yes—
49C22×C3×C3⋊D3C2311yescharacteristic
50C2×#1113—24no—
51C2×#1113—28no—
52C33×D4—22no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—24no—
55C3×#638—216no—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×C4⋊(C3⋊D3)—22no—
59C3×C4⋊(C3⋊D3)—24no—
60C3×C4×C3⋊D3D414yescomplemented
61#6295—24no—
62#6295—216no—
63#6294D418yescomplemented
64C2×#940—91no—
65D3×#127—92no—
66C23×C3⋊D3C2×D312yes—
67C23×C3×D3—34no—
68C23×C3×D3—62no—
69C22×D32—34no—
70C22×D32—32no—
71C2×C32×D4—21no—
72C2×C32×D4—22no—
73C2×C32×D4—41no—
74C2×C32×D4C2×D311yescharacteristic, complemented
75C2×C32×D4C2×D312yescomplemented
76C2×#638C2×D318yescomplemented
77C2×#638C2×D312yescomplemented
78D4×C3⋊D3—24no—
79C2×C4⋊(C3⋊D3)C2×D312yescomplemented
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C3×#67—38no—
83C2×C3×#67—38no—
84C2×C3×#67—34no—
85C2×C3×#67—62no—
86C2×C3×#67—68no—
87C3×D3×D4—68no—
88C3×D3×D4—68no—
89C2×C3×D12—32no—
90C2×C3×D12—32no—
91C2×C3×D12—32no—
92C2×C3×D12—61no—
93C2×C3×D12—62no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—61no—
96#3811—616no—
97D3×#67—632no—
98C2×#626—34no—
99C2×#626—38no—
100C2×#626—38no—
101C2×#625—32no—
102C2×#625—38no—
103C2×#624—32no—
104C2×D3×Dic3—34no—
105#3802—64no—
106D3×D12—68no—
107C4×C3⋊Dic3C2×D312yescomplemented
108C3×C4×Dic3—32no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—62no—
111Dic32—34no—
112Dic32—32no—
113C22×C33—22no—
114C22×C33—24no—
115C22×C33C2411yescharacteristic
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3—24no—
118C2×C3×C3⋊D3—24no—
119C2×C3×C3⋊D3—28no—
120C2×C3×C3⋊D3—48no—
121C2×C3×C3⋊D3C2×D414yes—
122C2×C3×C3⋊D3C2×D412yes—
123#1113—48no—
124C33×C4C2×D412yescomplemented
125C3×C3⋊Dic3C2×D418yescomplemented
126C3×C3⋊Dic3C2×D412yescomplemented
127C22×C3×D4—91no—
128C22×#67—94no—
129C2×D3×D4—32no—
130C2×D3×D4—61no—
131C2×D3×D4—92no—
132C2×D3×D4—98no—
133C22×D12—91no—
134C3×#127—92no—
135#940—32no—
136#940—32no—
137#940—62no—
138#940—92no—
139#940—92no—
140#940—98no—
141#940—94no—
142C2×C4×Dic3—91no—
143#874—92no—
144C42×D3—92no—
145C23×C32—22no—
146C23×C32—24no—
147C23×C32—42no—
148C23×C32C22×D314yes—
149C23×C32C22×D312yes—
150C22×C3⋊D3—24no—
151C22×C3⋊D3—28no—
152C22×C3⋊D3C22×D314yes—
153C22×C3⋊D3C22×D312yes—
154C22×C3⋊D3C22×D311yescharacteristic
155C22×C3×D3—32no—
156C22×C3×D3—34no—
157C22×C3×D3—34no—
158C22×C3×D3—34no—
159C22×C3×D3—61no—
160C22×C3×D3—62no—
161C22×C3×D3—64no—
162C22×C3×D3—616no—
163C22×C3×D3—68no—
164C22×C3×D3—68no—
165C22×C3×D3—128no—
166C2×D32—68no—
167C2×D32—616no—
168C2×D32—68no—
169C2×D32—64no—
170C32×D4—22no—
171C32×D4—24no—
172C32×D4—24no—
173C32×D4—44no—
174C32×D4—44no—
175C2×C32×C4—22no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180#638—24no—
181#638—216no—
182C2×C3⋊Dic3C22×D314yes—
183C2×C3⋊Dic3C22×D311yescharacteristic
184C4⋊(C3⋊D3)—22no—
185C4⋊(C3⋊D3)—24no—
186C4×C3⋊D3#6714yescomplemented
187C3×#67—616no—
188C3×#67—616no—
189C3×#67—68no—
190C3×#67—68no—
191C3×#67—1216no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—61no—
197C3×D12—64no—
198C3×D12—64no—
199C3×D12—64no—
200C3×D12—64no—
201C3×D12—124no—
202C3×C4×D3—38no—
203C3×C4×D3—64no—
204#626—68no—
205#626—616no—
206#626—616no—
207#625—616no—
208#625—64no—
209#624—38no—
210D3×Dic3—316no—
211C2×#127—271noSylow
212C2×C33—42no—
213C2×C33C22×D412yes—
214C2×C33C22×D411yescharacteristic
215C3×C3⋊D3—42no—
216C3×C3⋊D3—44no—
217C3×C3⋊D3#12714yescomplemented
218C24×C3—92no—
219C23×D3—34no—
220C23×D3—62no—
221C23×D3—92no—
222C23×D3—94no—
223C2×C3×D4—21no—
224C2×C3×D4—22no—
225C2×C3×D4—41no—
226C2×C3×D4—92no—
227C2×C3×D4—98no—
228C2×C3×D4—91no—
229C2×C3×D4—92no—
230C2×C3×D4—184no—
231C2×C3×D4D3211yescharacteristic, complemented
232C2×C3×D4D3212yescomplemented
233C22×C3×C4—91no—
234C2×#67—38no—
235C2×#67—38no—
236C2×#67—34no—
237C2×#67—62no—
238C2×#67—68no—
239C2×#67—98no—
240C2×#67—94no—
241C2×#67—98no—
242C2×#67—1816no—
243C22×Dic3—92no—
244D3×D4—68no—
245D3×D4—68no—
246D3×D4—188no—
247D3×D4—1832no—
248C2×D12—32no—
249C2×D12—32no—
250C2×D12—32no—
251C2×D12—61no—
252C2×D12—62no—
253C2×D12—98no—
254C2×D12—92no—
255C2×D12—184no—
256C2×C4×D3—32no—
257C2×C4×D3—61no—
258C2×C4×D3—94no—
259C3×C42—92no—
260C4×Dic3—32no—
261C4×Dic3—32no—
262C4×Dic3—62no—
263C4×Dic3—94no—
264C4×Dic3—92no—
265C22×C32—22no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—28no—
269C22×C32—24no—
270C22×C32—22no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—44no—
274C22×C32—44no—
275C22×C32—44no—
276C22×C32—48no—
277C22×C32—84no—
278C22×C32C23×D312yes—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—22no—
281C2×C3⋊D3—24no—
282C2×C3⋊D3—24no—
283C2×C3⋊D3—28no—
284C2×C3⋊D3—48no—
285C2×C3⋊D3D3×D414yes—
286C2×C3⋊D3C2×#6712yes—
287C2×C3×D3—38no—
288C2×C3×D3—34no—
289C2×C3×D3—62no—
290C2×C3×D3—64no—
291C2×C3×D3—64no—
292C2×C3×D3—68no—
293C2×C3×D3—68no—
294C2×C3×D3—64no—
295C2×C3×D3—68no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—124no—
299C2×C3×D3—124no—
300C2×C3×D3—128no—
301C2×C3×D3—1216no—
302C2×C3×D3—1216no—
303D32—1216no—
304D32—128no—
305C32×C4—24no—
306C32×C4—22no—
307C32×C4—42no—
308C32×C4D3×D414yescomplemented
309C32×C4C2×#6712yescomplemented
310C3⋊Dic3D3×D418yescomplemented
311C3⋊Dic3C2×#6712yescomplemented
312C3×Dic3—38no—
313C3×Dic3—38no—
314C3×Dic3—34no—
315C3×Dic3—62no—
316C3×Dic3—68no—
317C22×D4—91no—
318C22×D4—271no—
319C22×D4—274no—
320#127—92no—
321#127—272no—
322#127—274no—
323C2×C42—271no—
324C33C2×#12711yescharacteristic, complemented, Sylow
325C23×C3—22no—
326C23×C3—24no—
327C23×C3—42no—
328C23×C3—91no—
329C23×C3—92no—
330C23×C3—94no—
331C23×C3—184no—
332C23×C3—188no—
333C23×C3C2×D3212yes—
334C23×C3C2×D3214yes—
335C22×D3—32no—
336C22×D3—34no—
337C22×D3—34no—
338C22×D3—34no—
339C22×D3—61no—
340C22×D3—62no—
341C22×D3—64no—
342C22×D3—68no—
343C22×D3—616no—
344C22×D3—68no—
345C22×D3—128no—
346C22×D3—1816no—
347C22×D3—188no—
348C22×D3—184no—
349C22×D3—188no—
350C3×D4—24no—
351C3×D4—24no—
352C3×D4—22no—
353C3×D4—44no—
354C3×D4—44no—
355C3×D4—184no—
356C3×D4—1816no—
357C3×D4—184no—
358C3×D4—182no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—91no—
363C2×C3×C4—94no—
364C2×C3×C4—94no—
365C2×C3×C4C2×D3212yes—
366C2×C3×C4C2×D3211yescharacteristic
367#67—68no—
368#67—616no—
369#67—616no—
370#67—68no—
371#67—1216no—
372#67—1816no—
373#67—188no—
374#67—1816no—
375C2×Dic3—34no—
376C2×Dic3—34no—
377C2×Dic3—32no—
378C2×Dic3—64no—
379C2×Dic3—61no—
380C2×Dic3—98no—
381D12—64no—
382D12—64no—
383D12—64no—
384D12—64no—
385D12—124no—
386D12—184no—
387D12—1816no—
388C4×D3—38no—
389C4×D3—64no—
390C4×D3—916no—
391C2×C32—24no—
392C2×C32—22no—
393C2×C32—22no—
394C2×C32—21no—
395C2×C32—41no—
396C2×C32—42no—
397C2×C32—42no—
398C2×C32—44no—
399C2×C32—44no—
400C2×C32—84no—
401C2×C32—84no—
402C2×C32C2×D3×D414yes—
403C2×C32C2×D3×D412yes—
404C2×C32C2×D3×D412yes—
405C2×C32C22×#6711yescharacteristic
406C3⋊D3—42no—
407C3⋊D3—44no—
408C3⋊D3#94014yescomplemented
409C3×D3—38no—
410C3×D3—64no—
411C3×D3—124no—
412C3×D3—124no—
413C3×D3—124no—
414C3×D3—124no—
415C3×D3—244no—
416C24—92no—
417C24—272no—
418C2×D4—98no—
419C2×D4—92no—
420C2×D4—92no—
421C2×D4—91no—
422C2×D4—184no—
423C2×D4—278no—
424C2×D4—272no—
425C2×D4—544no—
426C2×D4—5416no—
427C2×D4#111311yescharacteristic, complemented
428C22×C4—91no—
429C22×C4—272no—
430C42—92no—
431C42—272no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—28no—
435C22×C3—24no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—21no—
439C22×C3—44no—
440C22×C3—48no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—41no—
444C22×C3—84no—
445C22×C3—94no—
446C22×C3—92no—
447C22×C3—184no—
448C22×C3—188no—
449C22×C3—184no—
450C22×C3—182no—
451C22×C3—368no—
452C22×C3C22×D3212yes—
453C22×C3C22×D3211yescharacteristic
454C2×D3—38no—
455C2×D3—34no—
456C2×D3—62no—
457C2×D3—64no—
458C2×D3—64no—
459C2×D3—68no—
460C2×D3—68no—
461C2×D3—68no—
462C2×D3—64no—
463C2×D3—64no—
464C2×D3—64no—
465C2×D3—124no—
466C2×D3—124no—
467C2×D3—128no—
468C2×D3—1216no—
469C2×D3—1216no—
470C2×D3—3616no—
471C2×D3—368no—
472C3×C4—22no—
473C3×C4—24no—
474C3×C4—42no—
475C3×C4—98no—
476C3×C4—92no—
477C3×C4D3×#6714yescomplemented
478C3×C4#381112yescomplemented
479Dic3—38no—
480Dic3—38no—
481Dic3—34no—
482Dic3—62no—
483Dic3—68no—
484C32—22no—
485C32—21no—
486C32—41no—
487C32D3×#12712yescomplemented
488C32C2×#94011yescharacteristic, complemented
489C23—91no—
490C23—94no—
491C23—92no—
492C23—184no—
493C23—188no—
494C23—548no—
495C23—544no—
496C23C2×#111312yes—
497D4—22no—
498D4—184no—
499D4—1816no—
500D4—182no—
501D4—184no—
502D4—5416no—
503D4—544no—
504C2×C4—94no—
505C2×C4—91no—
506C2×C4—94no—
507C2×C4—278no—
508C2×C4C2×#111311yescharacteristic
509C2×C3—24no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—22no—
513C2×C3—44no—
514C2×C3—44no—
515C2×C3—42no—
516C2×C3—42no—
517C2×C3—41no—
518C2×C3—84no—
519C2×C3—84no—
520C2×C3—94no—
521C2×C3—364no—
522C2×C3—362no—
523C2×C3D32×D414yes—
524C2×C3D32×D412yes—
525C2×C3C2×D3×#6712yes—
526C2×C3C2×#381111yescharacteristic
527D3—38no—
528D3—64no—
529D3—124no—
530D3—124no—
531D3—124no—
532D3—124no—
533D3—244no—
534C22—24no—
535C22—22no—
536C22—94no—
537C22—92no—
538C22—188no—
539C22—184no—
540C22—182no—
541C22—184no—
542C22—368no—
543C22—1088no—
544C22C22×#111311yescharacteristic
545C4—98no—
546C4—92no—
547C4#8904912yescomplemented
548C3—21no—
549C3—22no—
550C3—41no—
551C3D3×#94012yescomplemented
552C3#1059530011yescharacteristic, complemented
553C2—42no—
554C2—94no—
555C2—362no—
556C2—364no—
557C2D4×#111312yes—
558C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33C2×#127
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C2×#89049C2semidirect
3. C2×C4⋊#1113C2semidirect
4. C3×#64149C2semidirect
5. #11723728C2semidirect
6. #11723697C2semidirect
7. C4×#6294C2semidirect
12. C2×C33×D4C22semidirect
13. C2×C3×#638C22semidirect
14. C2×C3×#638C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×#6295C22semidirect
21. C2×#6295C22semidirect
24. C3×C4×C3⋊Dic3C22semidirect
25. #88813C22semidirect
26. C2×D4×C3⋊D3D3semidirect
33. #64149D3semidirect
60. C3×C4×C3⋊D3D4semidirect
63. #6294D4semidirect
74. C2×C32×D4C2×D3semidirect
75. C2×C32×D4C2×D3semidirect
76. C2×#638C2×D3semidirect
77. C2×#638C2×D3semidirect
79. C2×C4⋊(C3⋊D3)C2×D3semidirect
80. C2×C4⋊(C3⋊D3)C2×D3semidirect
107. C4×C3⋊Dic3C2×D3semidirect
124. C33×C4C2×D4semidirect
125. C3×C3⋊Dic3C2×D4semidirect
126. C3×C3⋊Dic3C2×D4semidirect
186. C4×C3⋊D3#67semidirect
217. C3×C3⋊D3#127semidirect
231. C2×C3×D4D32semidirect
232. C2×C3×D4D32semidirect
308. C32×C4D3×D4semidirect
309. C32×C4C2×#67semidirect
310. C3⋊Dic3D3×D4semidirect
311. C3⋊Dic3C2×#67semidirect
324. C33C2×#127semidirect
408. C3⋊D3#940semidirect
427. C2×D4#1113semidirect
477. C3×C4D3×#67semidirect
478. C3×C4#3811semidirect
487. C32D3×#127semidirect
488. C32C2×#940semidirect
547. C4#89049semidirect
551. C3D3×#940semidirect
552. C3#10595300semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890492
C2×C4⋊#11132
C3×#641492
#117237282
#117236972
C4×#62942
#105953003
D3×#9403