Finite Group Databasebeta

← #13150388

Subgroups — #13150388

Order 1728.

1100 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721289C212yesmaximal, complemented
6#11721282C212yesmaximal, complemented
7#11721041C212yesmaximal, complemented
8#11721040C211yesmaximal, characteristic, complemented
9#11721014C212yesmaximal, complemented
10#11721004C211yesmaximal, characteristic, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10594919—31nomaximal
13#10589701—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16#88736—24no—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C4×C3⋊D9C2212yes—
19C2×#6226C2211yescharacteristic
20C2×C9×#67C2212yes—
21C2×C3×#611C2211yescharacteristic
22C2×C9×D12C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×#6195C2212yes—
25C2×#6195C2212yes—
26C2×#6194C2212yes—
27C2×D3×Dic9C2212yes—
28C2×#6192C2212yes—
29#88638—24no—
30C3×#3676C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C9×C4⋊Dic3C2211yescharacteristic
34C3×Dic9⋊C4C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36#88445C2212yes—
37#88442C2212yes—
38Dic3×Dic9C2212yes—
39C2×#3828—31no—
40C2×#3795—31no—
41C3×#930D311yescharacteristic
42C3×C4×#67—31no—
43#63979—32no—
44#63971—32no—
45#63912—32no—
46#63911—31no—
47#63909—31no—
48#63881—32no—
49#63844—31no—
50C2×#3699—31no—
51C2×#3697—31no—
52C2×#3697—61no—
53C9×#121—31no—
54#63502—32no—
55#63502—31no—
56#63501—32no—
57C4×#611D311yescharacteristic, complemented
58#63457—31no—
59#63449—31no—
60#63446—32no—
61#611⋊C4—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69#6233—24no—
70C2×C3⋊Dic9C2312yes—
71C4⋊(C3⋊D9)—22no—
72C4×C3⋊D9—24no—
73#6226D414yescomplemented
74C9×#67—42no—
75C3×#611—22no—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C9×D12—22no—
81C3×C4×D9—22no—
82#6195—24no—
83#6195—42no—
84#6194—24no—
85D3×Dic9—24no—
86#6192—42no—
87#5599—91no—
88#5553—91no—
89C22×C32×C4C2×D311yescharacteristic
90C2×#638—32no—
91#3828—64no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C4×C3⋊D3—32no—
94C2×#634—31no—
95C2×C3×#67—31no—
96C2×C3×#67C2×D312yes—
97C2×C3×D12C2×D311yescharacteristic
98C2×C3×C4×D3—31no—
99C2×#626—32no—
100C2×#626—32no—
101C2×#626—32no—
102C2×#625—32no—
103C2×D3×Dic3—32no—
104#3795—64no—
105C3×#217—31no—
106C3×(C2×D3)⋊C4—31no—
107C3×(C2×D3)⋊C4C2×D312yes—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×Dic3⋊C4—31no—
110C3×C4×Dic3—31no—
111#3724—32no—
112#3722—32no—
113Dic32—32no—
114C2×D4×C9—32no—
115C2×D4×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611—32no—
120C2×#611—32no—
121C2×#611—32no—
122C2×#611—62no—
123C2×#611C2×D311yescharacteristic
124C22×Dic9—32no—
125#3699—64no—
126#3697—64no—
127#3697—68no—
128C2×D36—31no—
129C2×D36—32no—
130C2×D36—61no—
131C2×C4×D9—32no—
132C2×C4×D9—62no—
133C2×C4×D9C2×D311yescharacteristic
134C2×Dic18—31no—
135C2×Dic18—61no—
136C9×C4⋊C4—31no—
137C9×C22⋊C4—32no—
138#3676C2×D311yescharacteristic
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4C2×D311yescharacteristic
141Dic9⋊C4—32no—
142Dic9⋊C4C2×D311yescharacteristic
143C4×Dic9—32no—
144C4×Dic9C2×D311yescharacteristic
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9—21no—
148C22×C3×C9C2411yescharacteristic
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×C3⋊D9—22no—
152C2×D3×C9—22no—
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9—21no—
158C3×C4×C9—21no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9C2×D414yes—
161C9×Dic3—22no—
162C3×Dic9—21no—
163C3×Dic9—21no—
164C3×Dic9#4112yescomplemented
165C2×#237—91no—
166C2×#235—91no—
167C2×#235—91no—
168C2×#235—91no—
169C2×#235—181no—
170C3×#121—31no—
171C3×C4×D4—91no—
172#940—91no—
173#940—92no—
174#940—92no—
175#939—92no—
176#930—91no—
177#930D911yescharacteristic, complemented
178C4×#67—31no—
179#895—91no—
180#893—92no—
181#887—91no—
182#884—92no—
183#884—92no—
184#884—92no—
185#67⋊C4—92no—
186#874—91no—
187C4×Dic6—91no—
188C23×C32C22×D311yescharacteristic
189C22×C3⋊D3—32no—
190C22×C3×D3—31no—
191C22×C3×D3C22×D312yes—
192C2×C32×C4—22no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195#638—64no—
196C2×C3⋊Dic3—32no—
197C4⋊(C3⋊D3)—62no—
198C4×C3⋊D3—64no—
199#634—34no—
200C3×#67—42no—
201C3×#67—62no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3C22×D312yes—
206C3×D12—22no—
207C3×C4×D3—62no—
208#626—64no—
209#626—122no—
210#626—122no—
211#625—64no—
212D3×Dic3—64no—
213C23×C9—21no—
214C23×C9—32no—
215C23×C9C22×D311yescharacteristic
216C22×D9—32no—
217C22×D9—62no—
218C22×D9C22×D311yescharacteristic
219D4×C9—62no—
220D4×C9—122no—
221C2×C4×C9—24no—
222C2×C4×C9—22no—
223C2×C4×C9—21no—
224C2×C4×C9—21no—
225C2×C4×C9—32no—
226C2×C4×C9C22×D311yescharacteristic
227C2×C4×C9C22×D311yescharacteristic
228#611—22no—
229#611—64no—
230#611—64no—
231#611—68no—
232#611—64no—
233#611—122no—
234C2×Dic9—32no—
235C2×Dic9—62no—
236C2×Dic9—64no—
237C2×Dic9C22×D311yescharacteristic
238C2×Dic9C22×D311yescharacteristic
239C2×Dic9C22×D311yescharacteristic
240D36—64no—
241D36—62no—
242D36—122no—
243C4×D9—22no—
244C4×D9—64no—
245C4×D9—68no—
246Dic18—34no—
247Dic18—64no—
248#550—271noSylow
249C2×C3×C9—41no—
250C2×C3×C9C22×D411yescharacteristic
251C2×C3×C9C2×#4111yescharacteristic
252C2×C3×C9H(5,𝔽2)11yescharacteristic
253C3⋊D9—42no—
254D3×C9—42no—
255C3×D9—41no—
256C2×C3×D4—32no—
257C2×C3×D4—31no—
258C2×C3×D4—91no—
259C22×C3×C4—21no—
260C22×C3×C4—91no—
261C22×C3×C4C2×D911yescharacteristic
262C22×C3×C4D3211yescharacteristic
263C2×#67—31no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—92no—
271C2×#67—182no—
272C2×#67C2×D912yes—
273C22×Dic3—92no—
274#237—184no—
275#235—184no—
276#235—184no—
277#235—184no—
278#235—188no—
279C2×D12—92no—
280C2×D12—91no—
281C2×D12—91no—
282C2×D12—92no—
283C2×D12—92no—
284C2×D12—181no—
285C2×D12C2×D911yescharacteristic
286C2×C4×D3—31no—
287C2×C4×D3—92no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—182no—
291C2×Dic6—91no—
292C2×Dic6—91no—
293C2×Dic6—181no—
294C3×C4⋊C4—31no—
295C3×C4⋊C4—91no—
296C3×C22⋊C4—32no—
297C3×C22⋊C4—91no—
298C3×C22⋊C4—91no—
299C3×C42—91no—
300#217—31no—
301(C2×D3)⋊C4—31no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4C2×D912yes—
305C4⋊Dic3C2×D911yescharacteristic
306Dic3⋊C4—31no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309C4×Dic3—31no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—21no—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—62no—
317C2×C3⋊D3—62no—
318C2×C3⋊D3—62no—
319C2×C3×D3—22no—
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C32×C4—21no—
326C32×C4D3×D412yes—
327C3⋊Dic3—34no—
328C3×Dic3—22no—
329C3×Dic3—32no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C22×C9—22no—
333C22×C9—21no—
334C22×C9—21no—
335C22×C9—21no—
336C22×C9—21no—
337C22×C9—41no—
338C22×C9—41no—
339C22×C9—62no—
340C22×C9—62no—
341C22×C9—62no—
342C22×C9C23×D311yescharacteristic
343C2×D9—21no—
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—64no—
347C2×D9—62no—
348C2×D9—62no—
349C2×D9—62no—
350C2×D9—122no—
351C2×D9—122no—
352C4×C9—22no—
353C4×C9—21no—
354C4×C9—22no—
355C4×C9—62no—
356C4×C9D3×D412yes—
357Dic9—21no—
358Dic9—21no—
359Dic9—34no—
360Dic9—64no—
361Dic9#23512yescomplemented
362C2×#41—271no—
363C2×#41—271no—
364#127—271no—
365#127—272no—
366#121—31no—
367#121—272no—
368#121—272no—
369#121—271no—
370C4×Q8—271no—
371C4×D4—91no—
372C4×D4—272no—
373C3×C9#55011yescharacteristic, complemented, Sylow
374C23×C3—21no—
375C23×C3—32no—
376C23×C3—91no—
377C23×C3C22×D911yescharacteristic
378C23×C3C2×D3211yescharacteristic
379C22×D3—31no—
380C22×D3—92no—
381C22×D3—92no—
382C22×D3—182no—
383C22×D3C22×D912yes—
384C3×D4—62no—
385C3×D4—122no—
386C3×D4—182no—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—32no—
393C2×C3×C4—91no—
394C2×C3×C4—91no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4C22×D911yescharacteristic
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401#67—42no—
402#67—62no—
403#67—184no—
404#67—184no—
405#67—184no—
406#67—184no—
407#67—184no—
408#67—188no—
409#67—362no—
410#67—362no—
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—92no—
415C2×Dic3—92no—
416C2×Dic3—182no—
417C2×Dic3—184no—
418C2×Dic3C22×D912yes—
419D12—22no—
420D12—184no—
421D12—184no—
422D12—182no—
423D12—182no—
424D12—362no—
425D12—362no—
426C4×D3—62no—
427C4×D3—184no—
428C4×D3—184no—
429C4×D3—184no—
430C4×D3—188no—
431Dic6—94no—
432Dic6—94no—
433Dic6—184no—
434C2×C32—41no—
435C2×C32C2×D3×D411yescharacteristic
436C2×C32#100411yescharacteristic
437C2×C32D3×#4111yescharacteristic
438C3⋊D3—122no—
439C3×D3—42no—
440C3×D3—121no—
441C2×C9—21no—
442C2×C9—21no—
443C2×C9—21no—
444C2×C9—42no—
445C2×C9—41no—
446C2×C9—122no—
447C2×C9C2×#23511yescharacteristic
448C2×C9C2×D3×D411yescharacteristic
449C2×C9#100911yescharacteristic
450D9—41no—
451D9—124no—
452D9—122no—
453#41—544no—
454#41—544no—
455C2×Q8—271no—
456C2×D4—32no—
457C2×D4—31no—
458C2×D4—91no—
459C2×D4—272no—
460C2×D4—272no—
461C2×D4—272no—
462C2×D4—272no—
463C2×D4—272no—
464C2×D4—271no—
465C22×C4—91no—
466C22×C4—272no—
467C22×C4—272no—
468C22×C4D3×D911yescharacteristic
469C4⋊C4—31no—
470C4⋊C4—91no—
471C4⋊C4—272no—
472C22⋊C4—32no—
473C22⋊C4—91no—
474C22⋊C4—91no—
475C22⋊C4—272no—
476C42—91no—
477C42—272no—
478C22×C3—22no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—62no—
489C22×C3—62no—
490C22×C3—62no—
491C22×C3—181no—
492C22×C3—181no—
493C22×C3—181no—
494C22×C3C23×D911yescharacteristic
495C22×C3C22×D3211yescharacteristic
496C2×D3—22no—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—61no—
502C2×D3—184no—
503C2×D3—182no—
504C2×D3—182no—
505C2×D3—182no—
506C2×D3—182no—
507C2×D3—182no—
508C2×D3—182no—
509C2×D3—362no—
510C2×D3—362no—
511C3×C4—22no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—22no—
515C3×C4—62no—
516C3×C4—92no—
517C3×C4—181no—
518C3×C4—181no—
519C3×C4D4×D912yes—
520C3×C4#381112yes—
521Dic3—22no—
522Dic3—32no—
523Dic3—61no—
524Dic3—61no—
525Dic3—94no—
526Dic3—94no—
527Dic3—184no—
528C32#559911yescharacteristic
529C9—21no—
530C9#555311yescharacteristic, complemented
531C23—32no—
532C23—91no—
533C23—272no—
534C23C2×D3×D911yescharacteristic
535Q8—274no—
536D4—62no—
537D4—122no—
538D4—182no—
539D4—544no—
540D4—544no—
541D4—544no—
542D4—542no—
543D4—1082no—
544D4—1082no—
545C2×C4—22no—
546C2×C4—32no—
547C2×C4—91no—
548C2×C4—91no—
549C2×C4—91no—
550C2×C4—182no—
551C2×C4—272no—
552C2×C4—544no—
553C2×C4—544no—
554C2×C4C2×D3×D911yescharacteristic
555C2×C4C2×D3×D911yescharacteristic
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—42no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—122no—
563C2×C3—361no—
564C2×C3C2×D4×D911yescharacteristic
565C2×C3#6371011yescharacteristic
566C2×C3#41×D911yescharacteristic
567C2×C3D3×#23511yescharacteristic
568C2×C3#6430711yescharacteristic
569C2×C3C2×#381111yescharacteristic
570D3—42no—
571D3—121no—
572D3—364no—
573D3—362no—
574D3—362no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—62no—
579C22—62no—
580C22—62no—
581C22—181no—
582C22—181no—
583C22—181no—
584C22—542no—
585C22—542no—
586C22—542no—
587C22C22×D3×D911yescharacteristic
588C4—21no—
589C4—62no—
590C4—92no—
591C4—181no—
592C4—181no—
593C4—274no—
594C4#8866812yes—
595C3—21no—
596C3#1058970111yescharacteristic, complemented
597C3#1059491911yescharacteristic
598C2—41no—
599C2—122no—
600C2—361no—
601C2—1082no—
602C2D9×#23511yescharacteristic
603C2#1172247611yescharacteristic
604C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#550
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×#88736C2semidirect
2. C2×#88638C2semidirect
3. C9×#930C2semidirect
4. C3×C4×#611C2semidirect
5. #11721289C2semidirect
6. #11721282C2semidirect
7. #11721041C2semidirect
8. #11721040C2semidirect
9. #11721014C2semidirect
10. #11721004C2semidirect
11. #11720913C2semidirect
57. C4×#611D3semidirect
73. #6226D4semidirect
164. C3×Dic9#41semidirect
177. #930D9semidirect
361. Dic9#235semidirect
373. C3×C9#550semidirect
530. C9#5553semidirect
596. C3#10589701semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×#886382
C9×#9302
C3×C4×#6112
#117212892
#117212822
#117210412
#117210402
#117210142
#117210042
#117209132
#105949193
#105897013