Finite Group Databasebeta

← #13150322

Subgroups — #13150322

Order 1728.

976 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721271C212yesmaximal, complemented
6#11721256C212yesmaximal, complemented
7#11721004C211yesmaximal, characteristic, complemented
8D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#11720985C212yesmaximal, complemented
10#11720983C212yesmaximal, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10594894—31nomaximal
13#10589731—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2211yescharacteristic
19C3×#3697—24no—
20C2×C3×D36C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×C3×Dic18C2211yescharacteristic
25C2×#6192C2212yes—
26C2×#6192C2212yes—
27C2×(C3⋊D9)⋊C4C2212yes—
28C2×Dic3×D9C2212yes—
29#88635—24no—
30#88536C2212yes—
31C4⋊(C3⋊Dic9)C2211yescharacteristic
32C9×#217C2212yes—
33C9×C4⋊Dic3C2211yescharacteristic
34C9×Dic3⋊C4C2212yes—
35#88445C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37#88442C2212yes—
38Dic9⋊Dic3C2212yes—
39C2×C3×#235—31no—
40C2×#3799—31no—
41#64139—31no—
42C3×#924D311yescharacteristic
43#63971—32no—
44#63958—32no—
45#63909—31no—
46D3×C4⋊Dic3—31no—
47#63888—32no—
48#63886—32no—
49#63856—31no—
50C2×#3701—31no—
51C2×#3697D311yescharacteristic, complemented
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63492—31no—
55#63492—61no—
56#63458—32no—
57#63457—32no—
58#63457—31no—
59C4⋊C4×D9—31no—
60#63446—32no—
61#63446—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69#6233—42no—
70C2×C3⋊Dic9C2312yes—
71C4⋊(C3⋊D9)—22no—
72C3×#611—22no—
73C3×#611—22no—
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C3×D36—22no—
79C3×C4×D9—22no—
80C3×C4×D9D414yescomplemented
81C9×Dic6—22no—
82C3×Dic18—22no—
83#6192—24no—
84#6192—42no—
85(C3⋊D9)⋊C4—24no—
86Dic3×D9—42no—
87#5833—91no—
88#5629—91no—
89C22×C32×C4C2×D311yescharacteristic
90C2×#638—32no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C3×#67—31no—
93C2×C3×#67—31no—
94C3×#235—64no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3—31no—
98C2×C3×Dic6—31no—
99C2×C3×Dic6C2×D311yescharacteristic
100C2×#626—32no—
101C2×#626—32no—
102C2×#624—32no—
103C2×D3×Dic3—32no—
104#3799—64no—
105#3752—32no—
106C4⋊(C3⋊Dic3)—31no—
107C3×#217C2×D312yes—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×Dic3⋊C4C2×D312yes—
110#3724—32no—
111Dic3⋊Dic3—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114C2×Q8×C9—31no—
115C22×C4×C9—21no—
116C22×C4×C9C2×D311yescharacteristic
117C2×#611—32no—
118C2×#611—62no—
119C2×#611C2×D311yescharacteristic
120C2×#611C2×D311yescharacteristic
121#3701—64no—
122#3697—24no—
123C2×D36—32no—
124C2×D36—31no—
125C2×D36—32no—
126C2×D36—61no—
127C2×D36C2×D311yescharacteristic
128C2×C4×D9—32no—
129C2×C4×D9—32no—
130C2×C4×D9C2×D311yescharacteristic
131C2×C4×D9C2×D311yescharacteristic
132C2×Dic18C2×D311yescharacteristic
133C9×C4⋊C4—32no—
134C9×C4⋊C4—31no—
135C9×C22⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—62no—
140C4⋊Dic9—31no—
141C4⋊Dic9—61no—
142Dic9⋊C4—32no—
143Dic9⋊C4—32no—
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—22no—
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×C3×D9—21no—
152C2×C3×D9—21no—
153C2×C3×D9—21no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9C2×D412yes—
157C3×C4×C9—21no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—22no—
160C9×Dic3—22no—
161C9×Dic3—22no—
162C3×Dic9—21no—
163C3×Dic9C2×D412yes—
164C2×C3×#41—91no—
165C2×#239—91no—
166C2×#235—31no—
167C2×#235—91no—
168C3×C22⋊Q8—31no—
169#946—91no—
170#946—92no—
171#939—92no—
172#939—92no—
173#938—92no—
174#930—91no—
175#930—91no—
176#930—91no—
177#930—181no—
178C2×C4⋊Dic3—91no—
179#924—91no—
180#924D911yescharacteristic, complemented
181#896—92no—
182#895—91no—
183#895—92no—
184D3×C4⋊C4—91no—
185#884—92no—
186#884—92no—
187C23×C32C22×D311yescharacteristic
188C22×C3⋊D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3—31no—
191C2×C32×C4—22no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—122no—
195C2×C3⋊Dic3—32no—
196C4⋊(C3⋊D3)—62no—
197C3×#67—62no—
198C3×#67—62no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3C22×D312yes—
202C2×C3×Dic3C22×D312yes—
203C3×D12—62no—
204C3×C4×D3—34no—
205C3×C4×D3—62no—
206C3×Dic6—22no—
207C3×Dic6—62no—
208#626—64no—
209#626—122no—
210#624—64no—
211D3×Dic3—122no—
212C23×C9—21no—
213C23×C9C22×D311yescharacteristic
214C22×D9—32no—
215C22×D9—62no—
216C22×D9C22×D311yescharacteristic
217C22×D9C22×D311yescharacteristic
218Q8×C9—62no—
219C2×C4×C9—22no—
220C2×C4×C9—21no—
221C2×C4×C9—21no—
222C2×C4×C9—32no—
223C2×C4×C9—32no—
224C2×C4×C9—42no—
225C2×C4×C9C22×D311yescharacteristic
226C2×C4×C9C22×D311yescharacteristic
227#611—22no—
228#611—22no—
229#611—122no—
230#611—124no—
231C2×Dic9—32no—
232C2×Dic9—62no—
233C2×Dic9C22×D311yescharacteristic
234C2×Dic9C22×D311yescharacteristic
235D36—22no—
236D36—64no—
237D36—62no—
238D36—122no—
239D36—122no—
240C4×D9—22no—
241C4×D9—64no—
242C4×D9—122no—
243C4×D9#6714yescomplemented
244Dic18—22no—
245#537—271noSylow
246C2×C3×C9—41no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9H(5,𝔽2)11yescharacteristic
249C2×C3×C92−1+411yescharacteristic
250C3⋊D9—42no—
251C3×D9—22no—
252C3×D9—41no—
253C3×#41—184no—
254C2×C3×Q8—31no—
255C2×C3×Q8—91no—
256C2×C3×D4—91no—
257C2×C3×D4—91no—
258C2×C3×D4—91no—
259C22×C3×C4—21no—
260C22×C3×C4—91no—
261C22×C3×C4—91no—
262C22×C3×C4C2×D911yescharacteristic
263C22×C3×C4D3211yescharacteristic
264C2×#67—31no—
265C2×#67—31no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—182no—
271C22×Dic3—92no—
272#239—184no—
273#235—64no—
274#235—184no—
275C2×D12—31no—
276C2×D12—92no—
277C2×D12—91no—
278C2×D12—91no—
279C2×D12—92no—
280C2×D12—181no—
281C2×C4×D3—31no—
282C2×C4×D3—31no—
283C2×C4×D3—92no—
284C2×C4×D3—92no—
285C2×C4×D3—92no—
286C2×Dic6—31no—
287C2×Dic6C2×D911yescharacteristic
288C3×C4⋊C4—31no—
289C3×C4⋊C4—32no—
290C3×C22⋊C4—32no—
291#217—92no—
292#217C2×D912yes—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—182no—
299C4⋊Dic3—91no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—181no—
303C4⋊Dic3C2×D911yescharacteristic
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4C2×D912yes—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—62no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3×D3—32no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C32×C4—21no—
322C32×C4D3×D412yes—
323C3⋊Dic3—62no—
324C3×Dic3—22no—
325C3×Dic3—22no—
326C3×Dic3—32no—
327C3×Dic3—61no—
328C22×C9—22no—
329C22×C9—21no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—21no—
333C22×C9—41no—
334C22×C9—41no—
335C22×C9C23×D311yescharacteristic
336C2×D9—21no—
337C2×D9—21no—
338C2×D9—21no—
339C2×D9—21no—
340C2×D9—21no—
341C2×D9—62no—
342C2×D9—62no—
343C2×D9—62no—
344C2×D9—122no—
345C2×D9—122no—
346C2×D9—122no—
347C2×D9C2×#6712yes—
348C4×C9—21no—
349C4×C9—22no—
350C4×C9—41no—
351C4×C9—62no—
352C4×C9—62no—
353C4×C9C2×#6712yes—
354Dic9—21no—
355Dic9—62no—
356Dic9—122no—
357Dic9C2×#6712yes—
358C2×#41—91no—
359C2×#41—271no—
360C22⋊Q8—31no—
361C22⋊Q8—272no—
362C22⋊Q8—271no—
363#121—272no—
364#121—272no—
365#121—271no—
366#121—272no—
367#121—271no—
368C2×C4⋊C4—271no—
369C3×C9#53711yescharacteristic, complemented, Sylow
370C23×C3—21no—
371C23×C3—91no—
372C23×C3—91no—
373C23×C3C22×D911yescharacteristic
374C23×C3C2×D3211yescharacteristic
375C22×D3—31no—
376C22×D3—31no—
377C22×D3—92no—
378C22×D3—92no—
379C22×D3—182no—
380C3×Q8—62no—
381C3×Q8—182no—
382C3×D4—182no—
383C3×D4—182no—
384C3×D4—182no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—32no—
390C2×C3×C4—32no—
391C2×C3×C4—42no—
392C2×C3×C4—91no—
393C2×C3×C4—94no—
394C2×C3×C4—91no—
395C2×C3×C4—182no—
396C2×C3×C4C22×D911yescharacteristic
397C2×C3×C4C22×D911yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400#67—62no—
401#67—62no—
402#67—184no—
403#67—362no—
404#67—362no—
405#67—362no—
406#67—364no—
407C2×Dic3—31no—
408C2×Dic3—31no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—182no—
412C2×Dic3—362no—
413C2×Dic3C22×D912yes—
414C2×Dic3C22×D912yes—
415D12—62no—
416D12—184no—
417D12—182no—
418D12—182no—
419D12—362no—
420D12—362no—
421C4×D3—34no—
422C4×D3—62no—
423C4×D3—184no—
424C4×D3—184no—
425C4×D3—362no—
426Dic6—22no—
427Dic6—62no—
428C2×C32—41no—
429C2×C32C2×D3×D411yescharacteristic
430C2×C32#100711yescharacteristic
431C2×C32#100911yescharacteristic
432C3⋊D3—122no—
433C3×D3—62no—
434C3×D3—121no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—42no—
439C2×C9—41no—
440C2×C9#100911yescharacteristic
441C2×C9#101011yescharacteristic
442C2×C9C22×#6711yescharacteristic
443D9—22no—
444D9—41no—
445D9—122no—
446D9—242no—
447#41—184no—
448#41—544no—
449C2×Q8—31no—
450C2×Q8—91no—
451C2×D4—91no—
452C2×D4—91no—
453C2×D4—91no—
454C2×D4—272no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—271no—
458C22×C4—91no—
459C22×C4—91no—
460C22×C4—272no—
461C22×C4—272no—
462C22×C4D3×D911yescharacteristic
463C4⋊C4—31no—
464C4⋊C4—32no—
465C4⋊C4—272no—
466C4⋊C4—272no—
467C4⋊C4—271no—
468C22⋊C4—32no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—92no—
483C22×C3—181no—
484C22×C3—181no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3C23×D911yescharacteristic
489C22×C3C22×D3211yescharacteristic
490C2×D3—32no—
491C2×D3—61no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C2×D3—362no—
503C2×D3—362no—
504C2×D3—362no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—62no—
510C3×C4—62no—
511C3×C4—92no—
512C3×C4—181no—
513C3×C4D4×D912yes—
514C3×C4D3×#6712yes—
515Dic3—22no—
516Dic3—22no—
517Dic3—32no—
518Dic3—61no—
519Dic3—182no—
520Dic3—182no—
521Dic3—362no—
522C32#562911yescharacteristic
523C9—21no—
524C9#583311yescharacteristic, complemented
525C23—91no—
526C23—91no—
527C23—272no—
528C23C2×D3×D911yescharacteristic
529Q8—62no—
530Q8—182no—
531D4—182no—
532D4—182no—
533D4—182no—
534D4—544no—
535D4—542no—
536D4—1082no—
537D4—1082no—
538C2×C4—22no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—91no—
542C2×C4—94no—
543C2×C4—91no—
544C2×C4—182no—
545C2×C4—272no—
546C2×C4—544no—
547C2×C4—1082no—
548C2×C4C2×D3×D911yescharacteristic
549C2×C4C2×D3×D911yescharacteristic
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—182no—
557C2×C3—361no—
558C2×C3C2×D4×D911yescharacteristic
559C2×C3#6371311yescharacteristic
560C2×C3#6371511yescharacteristic
561C2×C3#6429711yescharacteristic
562C2×C3#6430811yescharacteristic
563C2×C3C2×D3×#6711yescharacteristic
564D3—62no—
565D3—121no—
566D3—362no—
567D3—362no—
568D3—722no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—92no—
573C22—181no—
574C22—181no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22—542no—
579C22—542no—
580C22—542no—
581C22C22×D3×D911yescharacteristic
582C4—21no—
583C4—62no—
584C4—62no—
585C4—92no—
586C4—181no—
587C4—542no—
588C4D9×#6712yes—
589C3—21no—
590C3#1058973111yescharacteristic, complemented
591C3#1059489411yescharacteristic
592C2—41no—
593C2—182no—
594C2—361no—
595C2—1082no—
596C2#1172245811yescharacteristic
597C2#1172247811yescharacteristic
598C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#537
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×C3×#3697C2semidirect
2. C2×#88635C2semidirect
3. #11721819C2semidirect
4. C9×#924C2semidirect
5. #11721271C2semidirect
6. #11721256C2semidirect
7. #11721004C2semidirect
8. D9×C4⋊Dic3C2semidirect
9. #11720985C2semidirect
10. #11720983C2semidirect
11. #11720944C2semidirect
51. C2×#3697D3semidirect
80. C3×C4×D9D4semidirect
180. #924D9semidirect
243. C4×D9#67semidirect
369. C3×C9#537semidirect
524. C9#5833semidirect
590. C3#10589731semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×#886352
#117218192
C9×#9242
#117212712
#117212562
#117210042
D9×C4⋊Dic32
#117209852
#117209832
#117209442
#105948943
#105897313