Finite Group Databasebeta

← #13151137

Subgroups — #13151137

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88662C212yesmaximal, complemented
2#11721840C211yesmaximal, characteristic, complemented
3C9×#938C211yesmaximal, characteristic, complemented
4C3×D4×Dic9C211yesmaximal, characteristic, complemented
5#11721282C212yesmaximal, complemented
6#11721242C212yesmaximal, complemented
7#11721241C212yesmaximal, complemented
8#11721038C211yesmaximal, characteristic, complemented
9C4×D3×Dic9C211yesmaximal, characteristic, complemented
10#11720992C211yesmaximal, characteristic, complemented
11#11720971C211yesmaximal, characteristic, complemented
12#10595131—31nomaximal
13#10589698—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C9×#67C2212yes—
19C22×C3×Dic9C2212yes—
20C2×C4×D3×C9C2211yescharacteristic
21C2×#6195C2212yes—
22#88662—28no—
23C2×D3×Dic9C2211yescharacteristic
24C2×D3×Dic9C2211yescharacteristic
25C2×(C3⋊D9)⋊C4C2212yes—
26C2×#6189C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28C9×#217C2212yes—
29C3×#3676C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33#88445C2212yes—
34#88445C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3804—32no—
39#64149—31no—
40C3×#938D311yescharacteristic
41C3×D4×Dic3—31no—
42#63971—32no—
43#63953—32no—
44#63952—32no—
45#63919—31no—
46#63909—31no—
47#63894—31no—
48C4×D3×Dic3—31no—
49C2×#3699—32no—
50C9×#121—31no—
51#63502—31no—
52#63502—61no—
53#63501—32no—
54D4×Dic9D311yescharacteristic, complemented
55#63492—31no—
56C2×C4×Dic9—31no—
57#63455—31no—
58#63450—31no—
59#63447—32no—
60#611⋊C4—32no—
61C23×C3×C9C2312yes—
62C22×C3⋊D9C2312yes—
63C22×D3×C9C2311yescharacteristic
64C3×D4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66#6233—24no—
67#6233—24no—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C4⋊(C3⋊D9)—22no—
71C9×#67—24no—
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2311yescharacteristic
74C2×C3×Dic9—24no—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C4×D3×C9—22no—
79#6195—24no—
80D3×Dic9—22no—
81D3×Dic9D414yescomplemented
82(C3⋊D9)⋊C4—24no—
83#6189—24no—
84#5596—91no—
85#5548—91no—
86C2×C32×D4C2×D311yescharacteristic
87C2×#638—32no—
88C2×#638—32no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C3×#67C2×D312yes—
91C22×C3×Dic3—32no—
92C2×C3×C4×D3C2×D311yescharacteristic
93C2×#627—32no—
94C2×#626—32no—
95C2×#624—32no—
96#3804—68no—
97C2×D3×Dic3—31no—
98C2×D3×Dic3—31no—
99C4×C3⋊Dic3—31no—
100C3×#217—32no—
101C3×#217C2×D312yes—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4×Dic3—31no—
105#3724—32no—
106#3722—32no—
107#3722—32no—
108Dic32—31no—
109Dic32—31no—
110C2×D4×C9—21no—
111C2×D4×C9—32no—
112C2×D4×C9C2×D311yescharacteristic
113C22×C4×C9—31no—
114C2×#611—32no—
115C2×#611—32no—
116C2×#611—32no—
117C2×#611—62no—
118C2×#611—62no—
119C22×Dic9—31no—
120C22×Dic9—31no—
121C22×Dic9C2×D312yes—
122#3699—68no—
123C2×D36—31no—
124C2×D36—61no—
125C2×C4×D9—32no—
126C2×Dic18—32no—
127C9×C4⋊C4—31no—
128C9×C22⋊C4—32no—
129#3676C2×D312yes—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—32no—
132C4⋊Dic9C2×D311yescharacteristic
133Dic9⋊C4—32no—
134C4×Dic9—31no—
135C4×Dic9—31no—
136C4×Dic9—31no—
137C4×Dic9—61no—
138C4×Dic9C2×D311yescharacteristic
139C22×C3×C9—22no—
140C22×C3×C9—22no—
141C22×C3×C9—22no—
142C22×C3×C9C2411yescharacteristic
143C2×C3⋊D9—22no—
144C2×C3⋊D9—22no—
145C2×C3⋊D9—22no—
146C2×D3×C9—21no—
147C2×D3×C9—21no—
148C2×D3×C9C2×D412yes—
149C3×C4×C9#4112yescomplemented
150C3⋊Dic9C2×D412yes—
151C3⋊Dic9#4112yescomplemented
152C9×Dic3—22no—
153C9×Dic3—21no—
154C3×Dic9—22no—
155C3×Dic9—21no—
156C3×Dic9C2×D412yes—
157C2×#237—92no—
158C2×#235—92no—
159C3×#121—31no—
160C3×C4×D4—91no—
161#940—91no—
162#940—91no—
163#940—181no—
164#939—92no—
165#938D911yescharacteristic, complemented
166D4×Dic3—31no—
167#930—91no—
168C4×#67—92no—
169C2×C4×Dic3—91no—
170#895—91no—
171#893—91no—
172#888—91no—
173#888—91no—
174#885—92no—
175#885—92no—
176#884—92no—
177#67⋊C4—92no—
178C4×D12—91no—
179C42×D3—91no—
180C23×C32C22×D312yes—
181C22×C3⋊D3—32no—
182C22×C3×D3C22×D311yescharacteristic
183C32×D4—22no—
184C2×C32×C4C22×D311yescharacteristic
185#638—64no—
186#638—64no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C4⋊(C3⋊D3)—62no—
190C3×#67—24no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3C22×D311yescharacteristic
196C2×C3×Dic3C22×D312yes—
197C3×C4×D3—22no—
198#627—64no—
199#626—64no—
200#624—64no—
201D3×Dic3—34no—
202D3×Dic3—62no—
203C23×C9—22no—
204C23×C9—31no—
205C23×C9C22×D312yes—
206C22×D9—32no—
207C22×D9—62no—
208D4×C9—22no—
209D4×C9—42no—
210D4×C9—64no—
211C2×C4×C9—21no—
212C2×C4×C9—31no—
213C2×C4×C9—32no—
214C2×C4×C9—62no—
215C2×C4×C9C22×D311yescharacteristic
216#611—64no—
217#611—64no—
218#611—64no—
219#611—124no—
220#611—124no—
221C2×Dic9—24no—
222C2×Dic9—31no—
223C2×Dic9—31no—
224C2×Dic9—34no—
225C2×Dic9—61no—
226C2×Dic9—61no—
227C2×Dic9—62no—
228C2×Dic9C22×D311yescharacteristic
229C2×Dic9C22×D312yes—
230C2×Dic9C22×D311yescharacteristic
231D36—62no—
232D36—122no—
233C4×D9—64no—
234Dic18—64no—
235#532—271noSylow
236C2×C3×C9—42no—
237C2×C3×C9C22×D411yescharacteristic
238C2×C3×C9C2×#4111yescharacteristic
239C2×C3×C9C2×#4111yescharacteristic
240C3⋊D9—42no—
241D3×C9—22no—
242C2×C3×D4—21no—
243C2×C3×D4—32no—
244C2×C3×D4C2×D911yescharacteristic
245C2×C3×D4D3211yescharacteristic
246C22×C3×C4—31no—
247C22×C3×C4—92no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—182no—
254C2×#67—182no—
255C2×#67C2×D912yes—
256C22×Dic3—32no—
257C22×Dic3—91no—
258C22×Dic3—91no—
259#237—188no—
260#235—188no—
261C2×D12—92no—
262C2×D12—91no—
263C2×D12—91no—
264C2×D12—181no—
265C2×C4×D3—91no—
266C2×C4×D3—92no—
267C2×C4×D3—91no—
268C2×C4×D3—92no—
269C2×C4×D3C2×D911yescharacteristic
270C2×Dic6—92no—
271C2×Dic6—92no—
272C3×C4⋊C4—31no—
273C3×C4⋊C4—91no—
274C3×C22⋊C4—32no—
275C3×C22⋊C4—92no—
276C3×C42—91no—
277#217—32no—
278#217C2×D912yes—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283C4⋊Dic3—31no—
284C4⋊Dic3C2×D911yescharacteristic
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—31no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—181no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32C23×D311yescharacteristic
299C2×C3⋊D3—62no—
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3×D3—21no—
303C2×C3×D3—21no—
304C2×C3×D3D3×D412yes—
305C32×C4#23712yes—
306C3⋊Dic3—32no—
307C3⋊Dic3—32no—
308C3×Dic3—22no—
309C3×Dic3—21no—
310C3×Dic3—32no—
311C3×Dic3—62no—
312C3×Dic3—61no—
313C22×C9—22no—
314C22×C9—22no—
315C22×C9—22no—
316C22×C9—21no—
317C22×C9—32no—
318C22×C9—42no—
319C22×C9—42no—
320C22×C9—42no—
321C22×C9—61no—
322C22×C9—61no—
323C22×C9C23×D311yescharacteristic
324C2×D9—62no—
325C2×D9—62no—
326C2×D9—62no—
327C2×D9—122no—
328C2×D9—122no—
329C2×D9—122no—
330C4×C9—22no—
331C4×C9—61no—
332C4×C9—62no—
333C4×C9#23512yescomplemented
334Dic9—22no—
335Dic9—21no—
336Dic9—32no—
337Dic9—32no—
338Dic9—62no—
339Dic9—62no—
340Dic9D3×D412yes—
341C2×#41—272no—
342C4⋊Q8—271no—
343#127—271no—
344#124—272no—
345#121—31no—
346#121—272no—
347#121—271no—
348C4×D4—91no—
349C4×D4—271no—
350C4×D4—272no—
351C2×C42—271no—
352C3×C9#53211yescharacteristic, complemented, Sylow
353C23×C3—22no—
354C23×C3—31no—
355C23×C3C22×D912yes—
356C23×C3C2×D3212yes—
357C22×D3—92no—
358C22×D3—92no—
359C22×D3—182no—
360C22×D3C22×D911yescharacteristic
361C3×D4—22no—
362C3×D4—22no—
363C3×D4—42no—
364C3×D4—64no—
365C2×C3×C4—21no—
366C2×C3×C4—31no—
367C2×C3×C4—32no—
368C2×C3×C4—62no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—184no—
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375#67—24no—
376#67—184no—
377#67—184no—
378#67—184no—
379#67—184no—
380#67—184no—
381#67—364no—
382#67—364no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—32no—
386C2×Dic3—64no—
387C2×Dic3—94no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—182no—
395C2×Dic3C22×D911yescharacteristic
396C2×Dic3C22×D912yes—
397D12—184no—
398D12—182no—
399D12—182no—
400D12—362no—
401C4×D3—22no—
402C4×D3—94no—
403C4×D3—184no—
404C4×D3—184no—
405C4×D3—182no—
406Dic6—184no—
407Dic6—184no—
408C2×C32—42no—
409C2×C32C2×D3×D411yescharacteristic
410C2×C32C2×#23711yescharacteristic
411C2×C32D3×#4111yescharacteristic
412C3⋊D3—122no—
413C3×D3—22no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—21no—
417C2×C9—42no—
418C2×C9—62no—
419C2×C9—82no—
420C2×C9C2×#23511yescharacteristic
421C2×C9C2×D3×D411yescharacteristic
422C2×C9D3×#4111yescharacteristic
423D9—122no—
424D9—242no—
425#41—548no—
426C2×Q8—272no—
427C2×D4—32no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—271no—
432C2×D4D3×D911yescharacteristic
433C22×C4—31no—
434C22×C4—92no—
435C22×C4—272no—
436C22×C4—271no—
437C22×C4—271no—
438C4⋊C4—31no—
439C4⋊C4—91no—
440C4⋊C4—272no—
441C22⋊C4—32no—
442C22⋊C4—92no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C42—91no—
446C42—271no—
447C42—271no—
448C42—271no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—32no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—61no—
461C22×C3—61no—
462C22×C3C23×D911yescharacteristic
463C22×C3C22×D3211yescharacteristic
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—362no—
473C2×D3—362no—
474C2×D3—362no—
475C2×D3D4×D912yes—
476C3×C4—22no—
477C3×C4—62no—
478C3×C4—61no—
479C3×C4—92no—
480C3×C4—182no—
481C3×C4—181no—
482C3×C4#369912yescomplemented
483C3×C4#380412yes—
484Dic3—22no—
485Dic3—21no—
486Dic3—32no—
487Dic3—62no—
488Dic3—61no—
489Dic3—92no—
490Dic3—92no—
491Dic3—92no—
492Dic3—92no—
493Dic3—182no—
494Dic3—182no—
495C32#559611yescharacteristic
496C9—21no—
497C9#554811yescharacteristic, complemented
498C23—31no—
499C23—272no—
500C23C2×D3×D912yes—
501Q8—544no—
502D4—22no—
503D4—64no—
504D4—544no—
505D4—544no—
506D4—544no—
507D4—542no—
508C2×C4—31no—
509C2×C4—32no—
510C2×C4—62no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—184no—
515C2×C4—274no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—544no—
519C2×C4—542no—
520C2×C4C2×D3×D911yescharacteristic
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—42no—
526C2×C3—62no—
527C2×C3—82no—
528C2×C3C2×D4×D911yescharacteristic
529C2×C3C2×#369911yescharacteristic
530C2×C3#41×D911yescharacteristic
531C2×C3D32×D411yescharacteristic
532C2×C3#6431211yescharacteristic
533C2×C3C2×#380411yescharacteristic
534D3—22no—
535D3—362no—
536D3—362no—
537D3—722no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—32no—
542C22—61no—
543C22—61no—
544C22—542no—
545C22—542no—
546C22—542no—
547C22C22×D3×D911yescharacteristic
548C4—61no—
549C4—62no—
550C4—92no—
551C4—181no—
552C4—182no—
553C4—272no—
554C4—272no—
555C4#8866212yescomplemented
556C3—21no—
557C3#1058969811yescharacteristic, complemented
558C3#1059513111yescharacteristic
559C2—42no—
560C2—62no—
561C2—1082no—
562C2D3×D4×D911yescharacteristic
563C2#1172249811yescharacteristic
564C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88662C2semidirect
2. #11721840C2semidirect
3. C9×#938C2semidirect
4. C3×D4×Dic9C2semidirect
5. #11721282C2semidirect
6. #11721242C2semidirect
7. #11721241C2semidirect
8. #11721038C2semidirect
9. C4×D3×Dic9C2semidirect
10. #11720992C2semidirect
11. #11720971C2semidirect
54. D4×Dic9D3semidirect
81. D3×Dic9D4semidirect
149. C3×C4×C9#41semidirect
151. C3⋊Dic9#41semidirect
165. #938D9semidirect
333. C4×C9#235semidirect
352. C3×C9#532semidirect
482. C3×C4#3699semidirect
497. C9#5548semidirect
555. C4#88662semidirect
557. C3#10589698semidirect

Maximal subgroups

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

subgroupindex
C2×#886622
#117218402
C9×#9382
C3×D4×Dic92
#117212822
#117212422
#117212412
#117210382
C4×D3×Dic92
#117209922
#117209712
#105951313
#105896983