Finite Group Databasebeta

← #13150381

Subgroups — #13150381

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88662C212yesmaximal, complemented
2#11721819C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
5#11721288C212yesmaximal, complemented
6#11721040C211yesmaximal, characteristic, complemented
7D12×Dic9C211yesmaximal, characteristic, complemented
8#11721013C212yesmaximal, complemented
9#11720992C211yesmaximal, characteristic, complemented
10#11720969C211yesmaximal, characteristic, complemented
11#11720956C212yesmaximal, complemented
12#10594910—31nomaximal
13#10589698—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C9×#67C2212yes—
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×C9×D12C2211yescharacteristic
21C2×#6195C2212yes—
22C2×#6195C2212yes—
23#88662—28no—
24C2×D3×Dic9C2212yes—
25C2×(C3⋊D9)⋊C4C2212yes—
26C2×#6189C2212yes—
27#88536C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C9×(C2×D3)⋊C4C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34C3×C4×Dic9C2211yescharacteristic
35(C2×D3)⋊Dic9C2212yes—
36#88445C2212yes—
37Dic9⋊Dic3C2212yes—
38C2×#3804—32no—
39#64139—31no—
40C3×#930D311yescharacteristic
41C2×C3×C4×Dic3—31no—
42#63978—32no—
43#63918—31no—
44#63911—31no—
45#63894—31no—
46Dic3×D12—31no—
47#63880—32no—
48#63852—32no—
49C2×#3699—32no—
50C9×#121—31no—
51#63502—31no—
52#63501—32no—
53D4×Dic9—31no—
54#63492—31no—
55#63492—61no—
56C2×C4×Dic9D311yescharacteristic, complemented
57#63455—31no—
58#63450—31no—
59#63447—32no—
60#611⋊C4—32no—
61C23×C3×C9C2311yescharacteristic
62C22×C3⋊D9C2312yes—
63C22×D3×C9C2312yes—
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67#6233—24no—
68C2×C3⋊Dic9C2312yes—
69C4⋊(C3⋊D9)—22no—
70C9×#67—24no—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9—22no—
73C2×C3×Dic9D414yescomplemented
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C9×D12—22no—
79#6195—24no—
80#6195—24no—
81D3×Dic9—24no—
82(C3⋊D9)⋊C4—24no—
83#6189—24no—
84#5596—91no—
85#5477—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—32no—
88C2×C4⋊(C3⋊D3)—31no—
89C2×C3×#67C2×D312yes—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C2×C3×D12C2×D311yescharacteristic
93C2×#627—32no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#624—32no—
97#3804—68no—
98C2×D3×Dic3—32no—
99#3752—32no—
100C4⋊(C3⋊Dic3)—31no—
101C3×(C2×D3)⋊C4C2×D312yes—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107Dic3⋊Dic3—32no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110C2×D4×C9—32no—
111C2×D4×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9C2×D311yescharacteristic
114C2×#611—32no—
115C2×#611—32no—
116C2×#611—32no—
117C2×#611—62no—
118C22×Dic9—32no—
119C22×Dic9C2×D311yescharacteristic
120C22×Dic9C2×D311yescharacteristic
121#3699—68no—
122C2×D36—31no—
123C2×D36—61no—
124C2×C4×D9—32no—
125C2×Dic18—32no—
126C9×C4⋊C4—31no—
127C9×C22⋊C4—32no—
128#3676—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—62no—
132C4⋊Dic9—31no—
133C4⋊Dic9—61no—
134Dic9⋊C4—32no—
135C4×Dic9C2×D311yescharacteristic
136C4×Dic9C2×D311yescharacteristic
137C4×Dic9C2×D311yescharacteristic
138C4×Dic9C2×D311yescharacteristic
139C22×C3×C9—21no—
140C22×C3×C9—21no—
141C22×C3×C9C2×D412yes—
142C22×C3×C9C2411yescharacteristic
143C2×C3⋊D9—22no—
144C2×C3⋊D9—22no—
145C2×C3⋊D9—22no—
146C2×D3×C9—22no—
147C2×D3×C9—22no—
148C2×D3×C9—22no—
149C3×C4×C9—21no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—22no—
152C9×Dic3—22no—
153C3×Dic9—21no—
154C3×Dic9C2×D412yes—
155C3×Dic9C2×D412yes—
156C3×Dic9#4112yescomplemented
157C2×#237—92no—
158C2×#235—92no—
159C3×#121—31no—
160C2×C3×C42—91no—
161#940—91no—
162#939—92no—
163D4×Dic3—91no—
164#930—91no—
165#930—91no—
166#930—92no—
167#930—181no—
168#930D911yescharacteristic, complemented
169C2×C4×Dic3—31no—
170#893—91no—
171#888—91no—
172#885—92no—
173#67⋊C4—92no—
174#875—92no—
175#874—91no—
176C4×D12—91no—
177C4×D12—92no—
178C4×D12—91no—
179#869—91no—
180C23×C32C22×D311yescharacteristic
181C22×C3⋊D3—32no—
182C22×C3×D3C22×D312yes—
183C2×C32×C4—22no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186#638—64no—
187C2×C3⋊Dic3—32no—
188C4⋊(C3⋊D3)—62no—
189C3×#67—24no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3C22×D312yes—
197C3×D12—22no—
198#627—64no—
199#626—64no—
200#626—64no—
201#624—64no—
202D3×Dic3—64no—
203C23×C9—21no—
204C23×C9—32no—
205C23×C9C22×D311yescharacteristic
206C22×D9—32no—
207C22×D9—62no—
208D4×C9—64no—
209D4×C9—62no—
210C2×C4×C9—22no—
211C2×C4×C9—21no—
212C2×C4×C9—21no—
213C2×C4×C9—32no—
214C2×C4×C9—42no—
215C2×C4×C9C22×D311yescharacteristic
216C2×C4×C9C22×D311yescharacteristic
217#611—64no—
218#611—64no—
219#611—64no—
220#611—124no—
221C2×Dic9—22no—
222C2×Dic9—32no—
223C2×Dic9—62no—
224C2×Dic9—64no—
225C2×Dic9D1214yescomplemented
226C2×Dic9C22×D311yescharacteristic
227C2×Dic9C22×D311yescharacteristic
228C2×Dic9C22×D311yescharacteristic
229C2×Dic9C22×D311yescharacteristic
230D36—62no—
231D36—122no—
232C4×D9—64no—
233Dic18—64no—
234#532—271noSylow
235C2×C3×C9—22no—
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238C2×C3×C9C2×#4111yescharacteristic
239C3⋊D9—42no—
240D3×C9—42no—
241C2×C3×D4—32no—
242C2×C3×D4—31no—
243C22×C3×C4—21no—
244C22×C3×C4—91no—
245C22×C3×C4—91no—
246C22×C3×C4C2×D911yescharacteristic
247C22×C3×C4D3211yescharacteristic
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—92no—
252C2×#67—182no—
253C2×#67C2×D912yes—
254C22×Dic3—31no—
255C22×Dic3—31no—
256C22×Dic3—92no—
257#237—188no—
258#235—188no—
259C2×D12—92no—
260C2×D12—92no—
261C2×D12—91no—
262C2×D12—91no—
263C2×D12—181no—
264C2×D12C2×D911yescharacteristic
265C2×C4×D3—92no—
266C2×C4×D3—92no—
267C2×C4×D3—92no—
268C2×Dic6—92no—
269C2×Dic6—92no—
270C3×C4⋊C4—31no—
271C3×C22⋊C4—32no—
272C3×C42—91no—
273C3×C42—91no—
274C3×C42—91no—
275C3×C42—91no—
276#217—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—182no—
283(C2×D3)⋊C4C2×D912yes—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287C4⋊Dic3—181no—
288C4⋊Dic3C2×D911yescharacteristic
289Dic3⋊C4—92no—
290C4×Dic3—31no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32D3×D412yes—
297C22×C32C23×D311yescharacteristic
298C2×C3⋊D3—62no—
299C2×C3⋊D3—62no—
300C2×C3⋊D3—62no—
301C2×C3×D3—22no—
302C2×C3×D3—22no—
303C2×C3×D3—22no—
304C32×C4—21no—
305C32×C4#23712yes—
306C3⋊Dic3—62no—
307C3×Dic3—22no—
308C3×Dic3—32no—
309C3×Dic3—32no—
310C3×Dic3—32no—
311C3×Dic3—61no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—21no—
315C22×C9—22no—
316C22×C9—41no—
317C22×C9—41no—
318C22×C9—62no—
319C22×C9—62no—
320C22×C9—62no—
321C22×C9C2×D1212yes—
322C22×C9C23×D311yescharacteristic
323C2×D9—62no—
324C2×D9—62no—
325C2×D9—62no—
326C2×D9—122no—
327C2×D9—122no—
328C2×D9—122no—
329C4×C9—21no—
330C4×C9—22no—
331C4×C9—41no—
332C4×C9—62no—
333C4×C9#23512yescomplemented
334Dic9—21no—
335Dic9—62no—
336Dic9—122no—
337Dic9C2×D1212yes—
338Dic9C2×D1212yes—
339Dic9#23512yescomplemented
340C2×#41—272no—
341C4⋊Q8—271no—
342#127—271no—
343#124—272no—
344#121—31no—
345#121—272no—
346#121—271no—
347C4×D4—272no—
348C4×D4—271no—
349C4×D4—271no—
350C2×C42—91no—
351C3×C9#53211yescharacteristic, complemented, Sylow
352C23×C3—21no—
353C23×C3—32no—
354C23×C3C22×D911yescharacteristic
355C23×C3C2×D3211yescharacteristic
356C22×D3—92no—
357C22×D3—92no—
358C22×D3—182no—
359C22×D3C22×D912yes—
360C3×D4—64no—
361C3×D4—62no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—32no—
367C2×C3×C4—42no—
368C2×C3×C4—94no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4—182no—
374C2×C3×C4C22×D911yescharacteristic
375C2×C3×C4C22×D911yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—24no—
379#67—184no—
380#67—184no—
381#67—184no—
382#67—184no—
383#67—364no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—34no—
388C2×Dic3—31no—
389C2×Dic3—62no—
390C2×Dic3—92no—
391C2×Dic3—92no—
392C2×Dic3—182no—
393C2×Dic3—184no—
394C2×Dic3C22×D912yes—
395D12—22no—
396D12—184no—
397D12—184no—
398D12—182no—
399D12—182no—
400D12—362no—
401C4×D3—184no—
402C4×D3—184no—
403C4×D3—184no—
404Dic6—184no—
405Dic6—184no—
406C2×C32—22no—
407C2×C32C2×D3×D411yescharacteristic
408C2×C32C2×#23711yescharacteristic
409C2×C32D3×#4111yescharacteristic
410C3⋊D3—122no—
411C3×D3—42no—
412C2×C9—22no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—42no—
417C2×C9—122no—
418C2×C9C22×D1211yescharacteristic
419C2×C9C2×#23511yescharacteristic
420C2×C9C2×#23511yescharacteristic
421D9—122no—
422D9—242no—
423#41—548no—
424C2×Q8—272no—
425C2×D4—32no—
426C2×D4—31no—
427C2×D4—272no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4—271no—
431C22×C4—91no—
432C22×C4—91no—
433C22×C4—272no—
434C22×C4—272no—
435C22×C4D3×D911yescharacteristic
436C4⋊C4—31no—
437C4⋊C4—272no—
438C4⋊C4—271no—
439C22⋊C4—32no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C42—91no—
444C42—91no—
445C42—91no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3D4×D912yes—
459C22×C3C23×D911yescharacteristic
460C22×C3D3×D1212yes—
461C22×C3C22×D3211yescharacteristic
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—182no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—362no—
472C2×D3—362no—
473C2×D3—362no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—62no—
479C3×C4—92no—
480C3×C4—92no—
481C3×C4—92no—
482C3×C4—181no—
483C3×C4#369912yescomplemented
484C3×C4#380412yes—
485Dic3—22no—
486Dic3—32no—
487Dic3—32no—
488Dic3—32no—
489Dic3—61no—
490Dic3—182no—
491Dic3—182no—
492Dic3—362no—
493C32#559611yescharacteristic
494C9—21no—
495C9#547711yescharacteristic, complemented
496C23—32no—
497C23—272no—
498C23C2×D3×D911yescharacteristic
499Q8—544no—
500D4—64no—
501D4—62no—
502D4—544no—
503D4—544no—
504D4—544no—
505D4—542no—
506C2×C4—22no—
507C2×C4—32no—
508C2×C4—91no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—94no—
512C2×C4—91no—
513C2×C4—182no—
514C2×C4—272no—
515C2×C4—544no—
516C2×C4—544no—
517C2×C4C2×D3×D911yescharacteristic
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—122no—
526C2×C3C2×D4×D911yescharacteristic
527C2×C3C2×#369911yescharacteristic
528C2×C3#41×D911yescharacteristic
529C2×C3C2×D3×D1211yescharacteristic
530C2×C3D3×#23511yescharacteristic
531C2×C3C2×#380411yescharacteristic
532D3—42no—
533D3—362no—
534D3—362no—
535D3—722no—
536C22—21no—
537C22—21no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—542no—
542C22—542no—
543C22—542no—
544C22D9×D1212yes—
545C22C22×D3×D911yescharacteristic
546C4—21no—
547C4—62no—
548C4—92no—
549C4—92no—
550C4—92no—
551C4—181no—
552C4—542no—
553C4#8866212yescomplemented
554C3—21no—
555C3#1058969811yescharacteristic, complemented
556C3#1059491011yescharacteristic
557C2—22no—
558C2—122no—
559C2—1082no—
560C2C2×D9×D1211yescharacteristic
561C2D9×#23511yescharacteristic
562C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88662C2semidirect
2. #11721819C2semidirect
3. C9×#930C2semidirect
4. C2×C3×C4×Dic9C2semidirect
5. #11721288C2semidirect
6. #11721040C2semidirect
7. D12×Dic9C2semidirect
8. #11721013C2semidirect
9. #11720992C2semidirect
10. #11720969C2semidirect
11. #11720956C2semidirect
56. C2×C4×Dic9D3semidirect
73. C2×C3×Dic9D4semidirect
150. C3×C4×C9#41semidirect
156. C3×Dic9#41semidirect
168. #930D9semidirect
225. C2×Dic9D12semidirect
333. C4×C9#235semidirect
339. Dic9#235semidirect
351. C3×C9#532semidirect
483. C3×C4#3699semidirect
495. C9#5477semidirect
553. C4#88662semidirect
555. C3#10589698semidirect

Maximal subgroups

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

subgroupindex
C2×#886622
#117218192
C9×#9302
C2×C3×C4×Dic92
#117212882
#117210402
D12×Dic92
#117210132
#117209922
#117209692
#117209562
#105949103
#105896983