Finite Group Databasebeta

← #13150385

Subgroups — #13150385

Order 1728.

1034 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3C4×C9×#67C211yesmaximal, characteristic, complemented
4C3×#63486C211yesmaximal, characteristic, complemented
5#11721282C212yesmaximal, complemented
6#11721242C212yesmaximal, complemented
7#11721038C211yesmaximal, characteristic, complemented
8#11721014C212yesmaximal, complemented
9#11720952C211yesmaximal, characteristic, complemented
10#11720923C212yesmaximal, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10594916—31nomaximal
13#10589654—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16#88736—24no—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C4×C3⋊D9C2212yes—
19C2×#6226C2211yescharacteristic
20C2×C9×#67C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×#6195C2212yes—
24C2×D3×Dic9C2212yes—
25C2×#6189C2212yes—
26D3×Dic18—24no—
27C9×#217C2211yescharacteristic
28C3×#3676C2212yes—
29C9×(C2×D3)⋊C4C2211yescharacteristic
30C9×Dic3⋊C4C2211yescharacteristic
31C4×C9×Dic3C2211yescharacteristic
32C3×C4⋊Dic9C2211yescharacteristic
33C3×Dic9⋊C4C2212yes—
34#88445C2212yes—
35#88445C2212yes—
36#88445C2212yes—
37#88442C2212yes—
38Dic3×Dic9C2212yes—
39C2×#3828—31no—
40C2×D3×Dic6—31no—
41C3×C4×#67D311yescharacteristic
42C3×#924—31no—
43#63971—32no—
44#63953—32no—
45#63909—31no—
46#63881—32no—
47#63865—32no—
48#63864—31no—
49#63844—31no—
50C2×#3697—31no—
51C2×#3697—61no—
52C22×Dic18—31no—
53C4×D4×C9—31no—
54#63501—32no—
55#63492—31no—
56#63486D311yescharacteristic, complemented
57#63458—32no—
58#63447—32no—
59#611⋊C4—32no—
60#63437—31no—
61C4×Dic18—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233—24no—
69C2×C3⋊Dic9C2312yes—
70C4⋊(C3⋊D9)—22no—
71C4×C3⋊D9—24no—
72#6226D414yescomplemented
73C9×#67—22no—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2312yes—
79C4×D3×C9—22no—
80C3×Dic18—22no—
81#6195—24no—
82D3×Dic9—24no—
83#6189—24no—
84#5633—91no—
85#5552—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—32no—
88#3828—64no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4×C3⋊D3—32no—
91C2×#634—31no—
92C2×C3×#67C2×D311yescharacteristic
93C2×C3×C4×D3C2×D311yescharacteristic
94C2×C3×Dic6—31no—
95C2×#627—32no—
96C2×#626—32no—
97C2×D3×Dic3—32no—
98D3×Dic6—64no—
99C3×#217—32no—
100C3×#217C2×D311yescharacteristic
101C3×(C2×D3)⋊C4C2×D311yescharacteristic
102C3×C4⋊Dic3—31no—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4C2×D311yescharacteristic
105C3×C4×Dic3C2×D311yescharacteristic
106#3724—32no—
107#3722—32no—
108#3722—32no—
109#3722—32no—
110Dic32—32no—
111C2×D4×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9—31no—
114C22×C4×C9C2×D311yescharacteristic
115C2×#611—32no—
116C2×#611—32no—
117C2×#611—62no—
118C22×Dic9—32no—
119#3697—64no—
120#3697—68no—
121C2×D36—31no—
122C2×D36—61no—
123C2×C4×D9—32no—
124C2×C4×D9—62no—
125C2×Dic18—31no—
126C2×Dic18—32no—
127C2×Dic18—64no—
128C2×Dic18—61no—
129C2×Dic18C2×D311yescharacteristic
130C9×C4⋊C4—31no—
131C9×C22⋊C4—31no—
132C9×C22⋊C4—31no—
133C42×C9—31no—
134#3676C2×D312yes—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138C4⋊Dic9C2×D311yescharacteristic
139Dic9⋊C4—32no—
140Dic9⋊C4C2×D312yes—
141C4×Dic9—32no—
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9—21no—
145C22×C3×C9C2411yescharacteristic
146C2×C3⋊D9—22no—
147C2×C3⋊D9—22no—
148C2×C3⋊D9—22no—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C2×D3×C9#4112yescomplemented
152C3×C4×C9—21no—
153C3×C4×C9C2×D412yes—
154C3⋊Dic9C2×D414yes—
155C9×Dic3—21no—
156C9×Dic3—21no—
157C9×Dic3—21no—
158C3×Dic9—22no—
159C3×Dic9—22no—
160C2×D3×Q8—91no—
161C2×#235—91no—
162C2×#235—91no—
163C2×#235—181no—
164C22×Dic6—91no—
165C3×C22⋊Q8—91no—
166C3×C4×D4—31no—
167#947—91no—
168#939—92no—
169#930—91no—
170C4×#67D911yescharacteristic, complemented
171#924—31no—
172#896—92no—
173#896—92no—
174#895—91no—
175#893—92no—
176#887—91no—
177#885—92no—
178#885—92no—
179#884—92no—
180#67⋊C4—92no—
181#875—91no—
182C4×Dic6—91no—
183C23×C32C22×D311yescharacteristic
184C22×C3⋊D3—32no—
185C22×C3×D3C22×D311yescharacteristic
186C2×C32×C4—22no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189#638—64no—
190C2×C3⋊Dic3—32no—
191C4⋊(C3⋊D3)—62no—
192C4×C3⋊D3—64no—
193#634—34no—
194C3×#67—22no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3C22×D311yescharacteristic
198C2×C3×Dic3C22×D311yescharacteristic
199C2×C3×Dic3C22×D311yescharacteristic
200C3×C4×D3—22no—
201C3×Dic6—62no—
202#627—64no—
203#626—64no—
204D3×Dic3—64no—
205C23×C9—21no—
206C23×C9—31no—
207C23×C9C22×D311yescharacteristic
208C22×D9—32no—
209C22×D9—62no—
210D4×C9—62no—
211C2×C4×C9—24no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—31no—
216C2×C4×C9—31no—
217C2×C4×C9—31no—
218C2×C4×C9—62no—
219C2×C4×C9C22×D311yescharacteristic
220C2×C4×C9C22×D311yescharacteristic
221#611—64no—
222#611—64no—
223#611—68no—
224C2×Dic9—32no—
225C2×Dic9—62no—
226C2×Dic9—64no—
227C2×Dic9C22×D312yes—
228C2×Dic9C22×D312yes—
229D36—64no—
230D36—62no—
231C4×D9—64no—
232C4×D9—68no—
233Dic18—22no—
234Dic18—34no—
235Dic18—64no—
236Dic18—64no—
237#548—271noSylow
238C2×C3×C9—41no—
239C2×C3×C9C22×D411yescharacteristic
240C2×C3×C9C2×#4111yescharacteristic
241C2×C3×C92−1+411yescharacteristic
242C3⋊D9—42no—
243D3×C9—22no—
244C2×C3×Q8—91no—
245C2×C3×D4—31no—
246C22×C3×C4—21no—
247C22×C3×C4—31no—
248C22×C3×C4C2×D911yescharacteristic
249C22×C3×C4D3211yescharacteristic
250C2×#67—92no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—182no—
254C2×#67C2×D911yescharacteristic
255C22×Dic3—92no—
256D3×Q8—184no—
257#235—184no—
258#235—184no—
259#235—188no—
260C2×D12—92no—
261C2×D12—91no—
262C2×D12—91no—
263C2×D12—181no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—92no—
267C2×C4×D3—182no—
268C2×C4×D3C2×D911yescharacteristic
269C2×Dic6—31no—
270C2×Dic6—91no—
271C2×Dic6—91no—
272C2×Dic6—92no—
273C2×Dic6—92no—
274C2×Dic6—181no—
275C2×Dic6—184no—
276C3×C4⋊C4—31no—
277C3×C4⋊C4—91no—
278C3×C4⋊C4—92no—
279C3×C22⋊C4—31no—
280C3×C22⋊C4—31no—
281C3×C22⋊C4—92no—
282C3×C42—31no—
283#217—32no—
284#217C2×D911yescharacteristic
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4C2×D911yescharacteristic
292C4⋊Dic3—31no—
293Dic3⋊C4—32no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4C2×D911yescharacteristic
297C4×Dic3—92no—
298C4×Dic3—92no—
299C4×Dic3C2×D911yescharacteristic
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32C23×D311yescharacteristic
304C2×C3⋊D3—62no—
305C2×C3⋊D3—62no—
306C2×C3⋊D3—62no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3#23512yes—
310C32×C4—21no—
311C32×C4D3×D412yes—
312C3⋊Dic3—34no—
313C3×Dic3—21no—
314C3×Dic3—21no—
315C3×Dic3—21no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C22×C9—22no—
319C22×C9—21no—
320C22×C9—21no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—32no—
324C22×C9—41no—
325C22×C9—41no—
326C22×C9—61no—
327C22×C9—61no—
328C22×C9C23×D311yescharacteristic
329C2×D9—64no—
330C2×D9—62no—
331C2×D9—62no—
332C2×D9—62no—
333C2×D9—122no—
334C2×D9—122no—
335C4×C9—22no—
336C4×C9—21no—
337C4×C9—22no—
338C4×C9—61no—
339C4×C9—61no—
340C4×C9—61no—
341C4×C9D3×D412yes—
342Dic9—22no—
343Dic9—22no—
344Dic9—34no—
345Dic9—64no—
346C2×#41—271no—
347C22×Q8—271no—
348#124—271no—
349#124—272no—
350C22⋊Q8—91no—
351C22⋊Q8—272no—
352#121—272no—
353#121—271no—
354C4×Q8—271no—
355C4×D4—31no—
356C4×D4—272no—
357C3×C9#54811yescharacteristic, complemented, Sylow
358C23×C3—21no—
359C23×C3—31no—
360C23×C3C22×D911yescharacteristic
361C23×C3C2×D3211yescharacteristic
362C22×D3—92no—
363C22×D3—92no—
364C22×D3—182no—
365C22×D3C22×D911yescharacteristic
366C3×Q8—182no—
367C3×D4—62no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—62no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4C22×D911yescharacteristic
380C2×C3×C4C22×D911yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383#67—22no—
384#67—184no—
385#67—184no—
386#67—184no—
387#67—188no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—92no—
391C2×Dic3—92no—
392C2×Dic3—182no—
393C2×Dic3—184no—
394C2×Dic3C22×D911yescharacteristic
395C2×Dic3C22×D911yescharacteristic
396C2×Dic3C22×D911yescharacteristic
397D12—184no—
398D12—184no—
399D12—182no—
400D12—182no—
401C4×D3—22no—
402C4×D3—184no—
403C4×D3—184no—
404C4×D3—184no—
405C4×D3—188no—
406Dic6—62no—
407Dic6—94no—
408Dic6—94no—
409Dic6—184no—
410Dic6—184no—
411Dic6—184no—
412C2×C32—41no—
413C2×C32C2×#23511yescharacteristic
414C2×C32C2×D3×D411yescharacteristic
415C2×C32#101011yescharacteristic
416C3⋊D3—122no—
417C3×D3—22no—
418C2×C9—21no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—42no—
422C2×C9—41no—
423C2×C9—62no—
424C2×C9C2×D3×D411yescharacteristic
425C2×C9#100711yescharacteristic
426C2×C9D3×#4111yescharacteristic
427D9—124no—
428D9—122no—
429#41—544no—
430C2×Q8—91no—
431C2×Q8—272no—
432C2×Q8—271no—
433C2×Q8—544no—
434C2×D4—31no—
435C2×D4—272no—
436C2×D4—272no—
437C2×D4—271no—
438C22×C4—31no—
439C22×C4—272no—
440C22×C4—272no—
441C22×C4D3×D911yescharacteristic
442C4⋊C4—31no—
443C4⋊C4—92no—
444C4⋊C4—91no—
445C4⋊C4—272no—
446C22⋊C4—31no—
447C22⋊C4—31no—
448C22⋊C4—92no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C22⋊C4—272no—
452C42—31no—
453C42—272no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—32no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3C23×D911yescharacteristic
468C22×C3C22×D3211yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3—184no—
472C2×D3—182no—
473C2×D3—182no—
474C2×D3—182no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—362no—
479C2×D3—362no—
480C2×D3#369712yescomplemented
481C3×C4—22no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—22no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—61no—
488C3×C4—182no—
489C3×C4—182no—
490C3×C4D4×D912yes—
491C3×C4#381112yes—
492Dic3—21no—
493Dic3—21no—
494Dic3—21no—
495Dic3—62no—
496Dic3—62no—
497Dic3—94no—
498Dic3—94no—
499Dic3—184no—
500C32#555211yescharacteristic
501C9—21no—
502C9#563311yescharacteristic, complemented
503C23—31no—
504C23—272no—
505C23C2×D3×D911yescharacteristic
506Q8—182no—
507Q8—274no—
508Q8—544no—
509D4—62no—
510D4—544no—
511D4—544no—
512D4—542no—
513C2×C4—22no—
514C2×C4—31no—
515C2×C4—31no—
516C2×C4—31no—
517C2×C4—62no—
518C2×C4—92no—
519C2×C4—92no—
520C2×C4—272no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4C2×D3×D911yescharacteristic
524C2×C4C2×D3×D911yescharacteristic
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—42no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—62no—
532C2×C3C2×#369711yescharacteristic
533C2×C3C2×D4×D911yescharacteristic
534C2×C3#6371611yescharacteristic
535C2×C3#6429711yescharacteristic
536C2×C3D3×#23511yescharacteristic
537C2×C3C2×#381111yescharacteristic
538D3—22no—
539D3—364no—
540D3—362no—
541D3—362no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—32no—
546C22—61no—
547C22—61no—
548C22—542no—
549C22—542no—
550C22—542no—
551C22C22×D3×D911yescharacteristic
552C4—21no—
553C4—61no—
554C4—61no—
555C4—61no—
556C4—182no—
557C4—182no—
558C4—274no—
559C4#8866812yes—
560C3—21no—
561C3#1058965411yescharacteristic, complemented
562C3#1059491611yescharacteristic
563C2—41no—
564C2—62no—
565C2—1082no—
566C2#1172246411yescharacteristic
567C2D3×#369711yescharacteristic
568C2C2×#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#548
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×D3×Dic18C2semidirect
3. C4×C9×#67C2semidirect
4. C3×#63486C2semidirect
5. #11721282C2semidirect
6. #11721242C2semidirect
7. #11721038C2semidirect
8. #11721014C2semidirect
9. #11720952C2semidirect
10. #11720923C2semidirect
11. #11720912C2semidirect
56. #63486D3semidirect
72. #6226D4semidirect
151. C2×D3×C9#41semidirect
170. C4×#67D9semidirect
357. C3×C9#548semidirect
480. C2×D3#3697semidirect
502. C9#5633semidirect
561. C3#10589654semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×D3×Dic182
C4×C9×#672
C3×#634862
#117212822
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