Finite Group Databasebeta

← #13150196

Subgroups — #13150196

Order 1728.

934 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic, complemented
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic
4C9×#924C211yesmaximal, characteristic, complemented
5#11721253C212yesmaximal, complemented
6#11721241C212yesmaximal
7#11720985C212yesmaximal, complemented
8#11720936C211yesmaximal, characteristic, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#11720910C211yesmaximal, characteristic, complemented
11#11720904C212yesmaximal, complemented
12#10594803—31nomaximal
13#10589734—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18—24no—
20C2×C3×Dic18C2212yes—
21C2×C3×Dic18C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×(C3⋊D9)⋊C4C2212yes—
24C2×#6189C2212yes—
25(C3⋊D9)⋊Q8—24no—
26#88551C2211yescharacteristic
27#88536C2211yescharacteristic
28#88534C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C9×#217C2212yes—
31C9×C4⋊Dic3C2211yescharacteristic
32C9×Dic3⋊C4C2212yes—
33#88445C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6—31no—
40C2×#3797—31no—
41C4×#638—31no—
42C3×#924D311yescharacteristic
43#63964—32no—
44#63952—32no—
45#63888—32no—
46#63873—31no—
47#63871—31no—
48#63843—31no—
49#63839—32no—
50C2×Q8×D9—31no—
51C22×Dic18D311yescharacteristic, complemented
52C9×C22⋊Q8—31no—
53#63506—31no—
54C4×#611—31no—
55C4×#611—61no—
56#63459—31no—
57#63458—32no—
58#63450—32no—
59#63449—31no—
60#611⋊C4—32no—
61#63440—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67#6233Q814yescomplemented
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C4×C3⋊D9—22no—
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9—24no—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2312yes—
77C9×Dic6—22no—
78C3×Dic18—22no—
79C3×Dic18—24no—
80C3×Dic18D414yescomplemented
81(C3⋊D9)⋊C4—24no—
82#6189—24no—
83Q8×#67—91no—
84#5632—91no—
85C22×C32×C4C2×D311yescharacteristic
86C2×#638—31no—
87C2×C4×C3⋊D3—31no—
88C22×C3×Dic3—32no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—64no—
93C2×C3×Dic6C2×D311yescharacteristic
94C2×#627—32no—
95C2×#624—32no—
96#3797—64no—
97#3757—31no—
98#3752—31no—
99#3750—31no—
100C4×C3⋊Dic3—31no—
101C3×#217C2×D312yes—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×Dic3⋊C4C2×D312yes—
104#3724—32no—
105#3724—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108#3722—32no—
109Dic32—32no—
110C2×Q8×C9—31no—
111C22×C4×C9—21no—
112C22×C4×C9C2×D311yescharacteristic
113C2×#611—31no—
114C2×#611—61no—
115C22×Dic9C2×D312yes—
116Q8×D9—64no—
117C2×C4×D9—31no—
118C2×C4×D9—32no—
119C2×C4×D9—61no—
120C2×Dic18—24no—
121C2×Dic18—32no—
122C2×Dic18C2×D311yescharacteristic
123C2×Dic18C2×D312yes—
124C2×Dic18C2×D311yescharacteristic
125C9×C4⋊C4—32no—
126C9×C4⋊C4—31no—
127C9×C22⋊C4—32no—
128#3676—31no—
129#3676—61no—
130(C2×D9)⋊C4—31no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—61no—
133C4⋊Dic9—32no—
134Dic9⋊C4—31no—
135Dic9⋊C4—32no—
136Dic9⋊C4—32no—
137Dic9⋊C4—32no—
138Dic9⋊C4—61no—
139C4×Dic9—31no—
140C4×Dic9—32no—
141C4×Dic9—61no—
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9C2×Q812yescomplemented
145C22×C3×C9C2411yescharacteristic
146C2×C3⋊D9—21no—
147C2×C3⋊D9—21no—
148C2×C3⋊D9C2×Q812yescomplemented
149C3×C4×C9—21no—
150C3×C4×C9C2×D412yes—
151C3⋊Dic9—21no—
152C3⋊Dic9—21no—
153C3⋊Dic9C2×Q812yescomplemented
154C9×Dic3—22no—
155C9×Dic3—22no—
156C3×Dic9—22no—
157C3×Dic9C2×D414yes—
158C22×C3×Q8—91no—
159C2×D3×Q8—91no—
160C2×D3×Q8—91no—
161C22×Dic6—31no—
162C3×C22⋊Q8—31no—
163#946—91no—
164#946—92no—
165Q8×Dic3—91no—
166#944—91no—
167#944—91no—
168#944—92no—
169C4×#67—91no—
170C4×#67—91no—
171C4×#67—92no—
172C4×#67—181no—
173#924—92no—
174#924D911yescharacteristic, complemented
175#897—91no—
176#896—92no—
177#888—92no—
178#887—91no—
179#67⋊C4—92no—
180#878—92no—
181C23×C32C22×D311yescharacteristic
182C22×C3⋊D3—31no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186#638—34no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190C4×C3⋊D3—62no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C22×D312yes—
196C3×Dic6—22no—
197C3×Dic6—34no—
198C3×Dic6—64no—
199C3×Dic6—62no—
200#627—64no—
201#624—64no—
202C23×C9—21no—
203C23×C9C22×D311yescharacteristic
204C22×D9—31no—
205C22×D9—61no—
206Q8×C9—62no—
207C2×C4×C9—22no—
208C2×C4×C9—21no—
209C2×C4×C9—21no—
210C2×C4×C9—32no—
211C2×C4×C9—32no—
212C2×C4×C9—42no—
213C2×C4×C9C22×D311yescharacteristic
214C2×C4×C9C22×D311yescharacteristic
215#611—34no—
216#611—64no—
217C2×Dic9—24no—
218C2×Dic9—31no—
219C2×Dic9—31no—
220C2×Dic9—31no—
221C2×Dic9—61no—
222C2×Dic9—61no—
223C2×Dic9—61no—
224C2×Dic9C22×D312yes—
225C2×Dic9C22×D312yes—
226C4×D9—62no—
227C4×D9—64no—
228C4×D9—122no—
229Dic18—22no—
230Dic18—24no—
231Dic18—64no—
232Dic18#6714yescomplemented
233D4×Q8—271noSylow
234C2×C3×C9—22no—
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C22×Q811yescharacteristic
237C2×C3×C92−1+411yescharacteristic
238C3⋊D9—22no—
239C2×C3×Q8—31no—
240C2×C3×Q8—91no—
241C2×C3×Q8—92no—
242C2×C3×Q8—91no—
243C2×C3×Q8—184no—
244C22×C3×C4—21no—
245C22×C3×C4—92no—
246C22×C3×C4C2×D911yescharacteristic
247C22×C3×C4D3211yescharacteristic
248C2×#67—91no—
249C2×#67—91no—
250C2×#67—181no—
251C22×Dic3—32no—
252D3×Q8—184no—
253D3×Q8—184no—
254C2×C4×D3—92no—
255C2×C4×D3—91no—
256C2×C4×D3—91no—
257C2×C4×D3—92no—
258C2×C4×D3—181no—
259C2×Dic6—31no—
260C2×Dic6—32no—
261C2×Dic6—31no—
262C2×Dic6—64no—
263C2×Dic6—92no—
264C2×Dic6—92no—
265C2×Dic6C2×D911yescharacteristic
266C3×C4⋊C4—31no—
267C3×C4⋊C4—32no—
268C3×C22⋊C4—32no—
269#217—91no—
270#217—91no—
271#217—181no—
272#217C2×D912yes—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—91no—
275(C2×D3)⋊C4—91no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—181no—
278C4⋊Dic3—92no—
279C4⋊Dic3—92no—
280C4⋊Dic3C2×D911yescharacteristic
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—91no—
285Dic3⋊C4—91no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—181no—
290Dic3⋊C4C2×D912yes—
291C4×Dic3—92no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—92no—
295C4×Dic3—181no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32D3×Q812yes—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—32no—
301C2×C3⋊D3—61no—
302C2×C3⋊D3—61no—
303C32×C4—21no—
304C32×C4D3×D412yes—
305C3⋊Dic3—32no—
306C3⋊Dic3—61no—
307C3⋊Dic3—61no—
308C3×Dic3—22no—
309C3×Dic3—22no—
310C3×Dic3—34no—
311C3×Dic3—62no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—21no—
315C22×C9—22no—
316C22×C9—41no—
317C22×C9—41no—
318C22×C9D3×Q812yescomplemented
319C22×C9C23×D311yescharacteristic
320C2×D9—32no—
321C2×D9—62no—
322C2×D9—61no—
323C2×D9—61no—
324C2×D9—121no—
325C2×D9—121no—
326C4×C9—21no—
327C4×C9—22no—
328C4×C9—41no—
329C4×C9—62no—
330C4×C9—62no—
331C4×C9C2×#6712yes—
332Dic9—22no—
333Dic9—32no—
334Dic9—62no—
335Dic9—61no—
336Dic9—61no—
337Dic9—121no—
338Dic9—121no—
339Dic9C2×#6714yes—
340C22×Q8—91no—
341C22×Q8—271no—
342C4⋊Q8—272no—
343C4⋊Q8—271no—
344C22⋊Q8—31no—
345C22⋊Q8—272no—
346C22⋊Q8—272no—
347C22⋊Q8—271no—
348C4×Q8—271no—
349C4×D4—272no—
350C4×D4—271no—
351C3×C9D4×Q811yescharacteristic, complemented, Sylow
352C23×C3—21no—
353C23×C3C22×D911yescharacteristic
354C23×C3C2×D3211yescharacteristic
355C22×D3—91no—
356C22×D3—91no—
357C22×D3—181no—
358C3×Q8—62no—
359C3×Q8—94no—
360C3×Q8—184no—
361C3×Q8—182no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—32no—
367C2×C3×C4—32no—
368C2×C3×C4—42no—
369C2×C3×C4—92no—
370C2×C3×C4—92no—
371C2×C3×C4—184no—
372C2×C3×C4C22×D911yescharacteristic
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376#67—94no—
377#67—94no—
378#67—184no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—64no—
382C2×Dic3—91no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×Dic3C22×D912yes—
392C2×Dic3C22×D912yes—
393C4×D3—184no—
394C4×D3—184no—
395C4×D3—182no—
396C4×D3—182no—
397C4×D3—362no—
398Dic6—22no—
399Dic6—34no—
400Dic6—64no—
401Dic6—62no—
402Dic6—184no—
403Dic6—184no—
404C2×C32—22no—
405C2×C32C2×D3×D411yescharacteristic
406C2×C32C2×D3×Q811yescharacteristic
407C2×C32#101011yescharacteristic
408C3⋊D3—62no—
409C2×C9—22no—
410C2×C9—21no—
411C2×C9—21no—
412C2×C9—21no—
413C2×C9—42no—
414C2×C9C2×D3×Q811yescharacteristic
415C2×C9#100711yescharacteristic
416C2×C9C22×#6711yescharacteristic
417D9—62no—
418D9—122no—
419C2×Q8—31no—
420C2×Q8—91no—
421C2×Q8—92no—
422C2×Q8—91no—
423C2×Q8—184no—
424C2×Q8—272no—
425C2×Q8—544no—
426C2×D4—271no—
427C22×C4—92no—
428C22×C4—272no—
429C22×C4—271no—
430C22×C4D3×D911yescharacteristic
431C4⋊C4—31no—
432C4⋊C4—32no—
433C4⋊C4—272no—
434C4⋊C4—272no—
435C4⋊C4—272no—
436C4⋊C4—272no—
437C4⋊C4—271no—
438C22⋊C4—32no—
439C22⋊C4—272no—
440C22⋊C4—271no—
441C22⋊C4—271no—
442C42—272no—
443C42—271no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3Q8×D912yescomplemented
453C22×C3C23×D911yescharacteristic
454C22×C3#379712yes—
455C22×C3C22×D3211yescharacteristic
456C2×D3—92no—
457C2×D3—92no—
458C2×D3—182no—
459C2×D3—181no—
460C2×D3—181no—
461C2×D3—181no—
462C2×D3—181no—
463C2×D3—361no—
464C2×D3—361no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—22no—
468C3×C4—41no—
469C3×C4—62no—
470C3×C4—62no—
471C3×C4—94no—
472C3×C4—182no—
473C3×C4D4×D912yes—
474C3×C4D3×#6712yes—
475Dic3—22no—
476Dic3—22no—
477Dic3—34no—
478Dic3—62no—
479Dic3—92no—
480Dic3—92no—
481Dic3—182no—
482Dic3—181no—
483Dic3—181no—
484Dic3—181no—
485Dic3—181no—
486Dic3—361no—
487Dic3—361no—
488C32#563211yescharacteristic
489C9—21no—
490C9Q8×#6711yescharacteristic, complemented
491C23—271no—
492C23C2×D3×D911yescharacteristic
493Q8—62no—
494Q8—94no—
495Q8—184no—
496Q8—182no—
497Q8—544no—
498D4—274no—
499C2×C4—22no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—92no—
503C2×C4—92no—
504C2×C4—184no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4—271no—
508C2×C4—544no—
509C2×C4—542no—
510C2×C4C2×D3×D911yescharacteristic
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—22no—
513C2×C3—22no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—42no—
518C2×C3C2×D4×D911yescharacteristic
519C2×C3C2×Q8×D911yescharacteristic
520C2×C3#6371611yescharacteristic
521C2×C3#6429711yescharacteristic
522C2×C3C2×#379711yescharacteristic
523C2×C3C2×D3×#6711yescharacteristic
524D3—182no—
525D3—182no—
526D3—362no—
527C22—21no—
528C22—21no—
529C22—272no—
530C22—541no—
531C22—541no—
532C22(C3⋊D9)⋊Q812yescomplemented
533C22C22×D3×D911yescharacteristic
534C4—21no—
535C4—62no—
536C4—62no—
537C4—94no—
538C4—182no—
539C4—272no—
540C4—541no—
541C4—541no—
542C4D9×#6712yes—
543C3—21no—
544C3#1058973411yescharacteristic, complemented
545C3#1059480311yescharacteristic
546C2—22no—
547C2—542no—
548C2C2×(C3⋊D9)⋊Q811yescharacteristic
549C2#1172246411yescharacteristic
550C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenterC22×C4D3×D9
hypocenterC3×C9D4×Q8
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. C22×C3×Dic18C2semidirect
2. C2×(C3⋊D9)⋊Q8C2semidirect
4. C9×#924C2semidirect
5. #11721253C2semidirect
7. #11720985C2semidirect
8. #11720936C2semidirect
9. #11720930C2semidirect
10. #11720910C2semidirect
11. #11720904C2semidirect
51. C22×Dic18D3semidirect
67. #6233Q8semidirect
80. C3×Dic18D4semidirect
144. C22×C3×C9C2×Q8semidirect
148. C2×C3⋊D9C2×Q8semidirect
153. C3⋊Dic9C2×Q8semidirect
174. #924D9semidirect
232. Dic18#67semidirect
318. C22×C9D3×Q8semidirect
351. C3×C9D4×Q8semidirect
452. C22×C3Q8×D9semidirect
490. C9Q8×#67semidirect
532. C22(C3⋊D9)⋊Q8semidirect
544. C3#10589734semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×(C3⋊D9)⋊Q82
C4×#62332
C9×#9242
#117212532
#117212412
#117209852
#117209362
#117209302
#117209102
#117209042
#105948033
#105897343