Finite Group Databasebeta

← #13150306

Subgroups — #13150306

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6233C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
4#11721253C212yesmaximal, complemented
5#11721241C212yesmaximal
6#11721000C211yesmaximal, characteristic, complemented
7#11720995C212yesmaximal, complemented
8#11720971C211yesmaximal, characteristic, complemented
9#11720929C211yesmaximal, characteristic, complemented
10#11720919C212yesmaximal, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10594991—31nomaximal
13#10589737—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×C9×Dic6C2211yescharacteristic
19C2×(C3⋊D9)⋊C4C2212yes—
20C2×#6189C2212yes—
21#88551C2211yescharacteristic
22#88536C2211yescharacteristic
23#88534C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C9×#217C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2212yes—
30C3×C4⋊Dic9C2211yescharacteristic
31#88445C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C4×#638—31no—
38C2×C3×C4⋊Dic3—31no—
39C3×#924D311yescharacteristic
40#63964—32no—
41#63952—32no—
42#63919—31no—
43#63900—31no—
44#63895—32no—
45#63871—31no—
46#63847—32no—
47#63843—31no—
48C9×C22⋊Q8—31no—
49#63508—31no—
50Q8×Dic9—31no—
51C4×#611—31no—
52C4×#611—61no—
53C2×C4⋊Dic9D311yescharacteristic, complemented
54#63459—32no—
55C4⋊C4×D9—31no—
56#63451—32no—
57#63450—31no—
58#611⋊C4—32no—
59#63440—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65#6233Q814yescomplemented
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9—42no—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C9×Dic6—22no—
76(C3⋊D9)⋊C4—42no—
77#6189—42no—
78#5635—91no—
79#5511—91no—
80C22×C32×C4C2×D311yescharacteristic
81C2×#638—31no—
82C2×C4×C3⋊D3—31no—
83C22×C3×Dic3—32no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×#627—32no—
86C2×#624—32no—
87#3757—31no—
88#3752—31no—
89#3750—31no—
90C4×C3⋊Dic3—31no—
91C3×#217C2×D312yes—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×Dic3⋊C4C2×D312yes—
97#3724—32no—
98#3724—32no—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101#3722—32no—
102Dic32—32no—
103C2×Q8×C9—31no—
104C22×C4×C9—21no—
105C22×C4×C9C2×D311yescharacteristic
106C2×#611—31no—
107C2×#611—61no—
108C22×Dic9C2×D312yes—
109C2×C4×D9—31no—
110C2×C4×D9—32no—
111C2×C4×D9—61no—
112C2×Dic18—32no—
113C9×C4⋊C4—32no—
114C9×C4⋊C4—31no—
115C9×C22⋊C4—32no—
116#3676—31no—
117#3676—61no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—61no—
121C4⋊Dic9—32no—
122C4⋊Dic9C2×D311yescharacteristic
123C4⋊Dic9C2×D311yescharacteristic
124C4⋊Dic9C2×D312yes—
125Dic9⋊C4—31no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—61no—
130C4×Dic9—31no—
131C4×Dic9—32no—
132C4×Dic9—61no—
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9C2×Q812yes—
136C22×C3×C9C2411yescharacteristic
137C2×C3⋊D9—21no—
138C2×C3⋊D9—21no—
139C2×C3⋊D9C2×Q812yes—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3⋊Dic9—21no—
143C3⋊Dic9—21no—
144C3⋊Dic9C2×Q812yes—
145C9×Dic3—22no—
146C9×Dic3—22no—
147C3×Dic9—22no—
148C3×Dic9—22no—
149C3×C22⋊Q8—31no—
150C2×C3×C4⋊C4—91no—
151#946—91no—
152Q8×Dic3—91no—
153C4×#67—91no—
154C4×#67—91no—
155C4×#67—92no—
156C4×#67—181no—
157C2×C4⋊Dic3—31no—
158#924—92no—
159#924D911yescharacteristic, complemented
160#897—92no—
161#897—91no—
162#896—92no—
163D3×C4⋊C4—91no—
164D3×C4⋊C4—91no—
165#889—92no—
166#889—92no—
167#888—91no—
168#888—91no—
169#887—91no—
170#67⋊C4—92no—
171#878—92no—
172C23×C32C22×D311yescharacteristic
173C22×C3⋊D3—31no—
174C2×C32×C4—22no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177#638—34no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C4×C3⋊D3—62no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—122no—
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D312yes—
187C3×Dic6—22no—
188#627—122no—
189#624—122no—
190C23×C9—21no—
191C23×C9C22×D311yescharacteristic
192C22×D9—31no—
193C22×D9—61no—
194Q8×C9—62no—
195C2×C4×C9—22no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—32no—
199C2×C4×C9—32no—
200C2×C4×C9—42no—
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203#611—34no—
204#611—64no—
205C2×Dic9—31no—
206C2×Dic9—31no—
207C2×Dic9—31no—
208C2×Dic9—42no—
209C2×Dic9—61no—
210C2×Dic9—61no—
211C2×Dic9—61no—
212C2×Dic9C22×D312yes—
213C2×Dic9C22×D312yes—
214C4×D9—62no—
215C4×D9—122no—
216C4×D9—122no—
217Dic18—122no—
218#554—271noSylow
219C2×C3×C9—22no—
220C2×C3×C9C22×Q811yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223C3⋊D9—22no—
224C2×C3×Q8—31no—
225C22×C3×C4—21no—
226C22×C3×C4—92no—
227C22×C3×C4C2×D911yescharacteristic
228C22×C3×C4D3211yescharacteristic
229C2×#67—91no—
230C2×#67—91no—
231C2×#67—181no—
232C22×Dic3—32no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×C4×D3—181no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6C2×D911yescharacteristic
241C3×C4⋊C4—31no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C22⋊C4—32no—
247#217—91no—
248#217—91no—
249#217—181no—
250#217C2×D912yes—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—91no—
253(C2×D3)⋊C4—91no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—181no—
256C4⋊Dic3—31no—
257C4⋊Dic3—32no—
258C4⋊Dic3—31no—
259C4⋊Dic3—92no—
260C4⋊Dic3—92no—
261C4⋊Dic3C2×D911yescharacteristic
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—91no—
266Dic3⋊C4—91no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—181no—
271Dic3⋊C4C2×D912yes—
272C4×Dic3—92no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—92no—
276C4×Dic3—181no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32D3×Q812yes—
280C22×C32C23×D311yescharacteristic
281C2×C3⋊D3—32no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—61no—
284C32×C4—21no—
285C32×C4#23712yes—
286C3⋊Dic3—32no—
287C3⋊Dic3—61no—
288C3⋊Dic3—61no—
289C3×Dic3—22no—
290C3×Dic3—22no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C22×C9—21no—
294C22×C9—21no—
295C22×C9—21no—
296C22×C9—22no—
297C22×C9—41no—
298C22×C9—41no—
299C22×C9D3×Q812yes—
300C22×C9C23×D311yescharacteristic
301C2×D9—32no—
302C2×D9—62no—
303C2×D9—61no—
304C2×D9—61no—
305C2×D9—121no—
306C2×D9—121no—
307C4×C9—21no—
308C4×C9—22no—
309C4×C9—41no—
310C4×C9—62no—
311C4×C9—62no—
312C4×C9#23512yes—
313Dic9—22no—
314Dic9—22no—
315Dic9—32no—
316Dic9—62no—
317Dic9—61no—
318Dic9—61no—
319Dic9—121no—
320Dic9—121no—
321C4⋊Q8—271no—
322#125—272no—
323C22⋊Q8—31no—
324C22⋊Q8—272no—
325C22⋊Q8—272no—
326C22⋊Q8—271no—
327C4×Q8—271no—
328C4×D4—272no—
329C4×D4—271no—
330C2×C4⋊C4—91no—
331C2×C4⋊C4—271no—
332C3×C9#55411yescharacteristic, complemented, Sylow
333C23×C3—21no—
334C23×C3C22×D911yescharacteristic
335C23×C3C2×D3211yescharacteristic
336C22×D3—91no—
337C22×D3—91no—
338C22×D3—181no—
339C3×Q8—62no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—42no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—362no—
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×D911yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354#67—94no—
355#67—94no—
356#67—184no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—91no—
365C2×Dic3—122no—
366C2×Dic3—181no—
367C2×Dic3—181no—
368C2×Dic3—181no—
369C2×Dic3C22×D912yes—
370C2×Dic3C22×D912yes—
371C4×D3—182no—
372C4×D3—182no—
373C4×D3—362no—
374C4×D3—362no—
375C4×D3—362no—
376Dic6—22no—
377Dic6—362no—
378Dic6—362no—
379C2×C32—22no—
380C2×C32C2×#23711yescharacteristic
381C2×C32C2×D3×Q811yescharacteristic
382C2×C32#100911yescharacteristic
383C3⋊D3—62no—
384C2×C9—22no—
385C2×C9—21no—
386C2×C9—21no—
387C2×C9—21no—
388C2×C9—42no—
389C2×C9C2×#23511yescharacteristic
390C2×C9#100411yescharacteristic
391C2×C9C2×D3×Q811yescharacteristic
392D9—62no—
393D9—122no—
394C2×Q8—31no—
395C2×Q8—272no—
396C2×D4—271no—
397C22×C4—92no—
398C22×C4—272no—
399C22×C4—271no—
400C22×C4D3×D911yescharacteristic
401C4⋊C4—31no—
402C4⋊C4—32no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4—91no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C4⋊C4—272no—
409C4⋊C4—272no—
410C4⋊C4—271no—
411C22⋊C4—32no—
412C22⋊C4—272no—
413C22⋊C4—271no—
414C22⋊C4—271no—
415C42—272no—
416C42—271no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3Q8×D912yes—
426C22×C3C23×D911yescharacteristic
427C22×C3#379712yes—
428C22×C3C22×D3211yescharacteristic
429C2×D3—92no—
430C2×D3—92no—
431C2×D3—182no—
432C2×D3—181no—
433C2×D3—181no—
434C2×D3—181no—
435C2×D3—181no—
436C2×D3—361no—
437C2×D3—361no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—41no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—182no—
445C3×C4—182no—
446C3×C4#369912yes—
447C3×C4#380412yes—
448Dic3—22no—
449Dic3—22no—
450Dic3—62no—
451Dic3—62no—
452Dic3—92no—
453Dic3—92no—
454Dic3—182no—
455Dic3—181no—
456Dic3—181no—
457Dic3—181no—
458Dic3—181no—
459Dic3—361no—
460Dic3—361no—
461C32#563511yescharacteristic
462C9—21no—
463C9#551111yescharacteristic, complemented
464C23—271no—
465C23C2×D3×D911yescharacteristic
466Q8—62no—
467Q8—1082no—
468D4—274no—
469C2×C4—22no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—271no—
475C2×C4—271no—
476C2×C4—271no—
477C2×C4—362no—
478C2×C4—542no—
479C2×C4—1082no—
480C2×C4C2×D3×D911yescharacteristic
481C2×C4C2×D3×D911yescharacteristic
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—42no—
488C2×C3C2×#369911yescharacteristic
489C2×C3C2×Q8×D911yescharacteristic
490C2×C3#6371511yescharacteristic
491C2×C3C2×#379711yescharacteristic
492C2×C3#6430711yescharacteristic
493C2×C3C2×#380411yescharacteristic
494D3—182no—
495D3—182no—
496D3—362no—
497C22—21no—
498C22—21no—
499C22—272no—
500C22—541no—
501C22—541no—
502C22(C3⋊D9)⋊Q812yes—
503C22C22×D3×D911yescharacteristic
504C4—21no—
505C4—62no—
506C4—62no—
507C4—182no—
508C4—182no—
509C4—272no—
510C4—541no—
511C4—541no—
512C4#8866212yes—
513C3—21no—
514C3#1058973711yescharacteristic, complemented
515C3#1059499111yescharacteristic
516C2—22no—
517C2—542no—
518C2C2×(C3⋊D9)⋊Q811yescharacteristic
519C2#1172247711yescharacteristic
520C2C2×#8866211yescharacteristic
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×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C2×C3×C4⋊Dic9C2semidirect
4. #11721253C2semidirect
6. #11721000C2semidirect
7. #11720995C2semidirect
8. #11720971C2semidirect
9. #11720929C2semidirect
10. #11720919C2semidirect
11. #11720911C2semidirect
53. C2×C4⋊Dic9D3semidirect
65. #6233Q8semidirect
159. #924D9semidirect
332. C3×C9#554semidirect
463. C9#5511semidirect
514. C3#10589737semidirect

Maximal subgroups

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

subgroupindex
C4×#62332
C9×#9242
C2×C3×C4⋊Dic92
#117212532
#117212412
#117210002
#117209952
#117209712
#117209292
#117209192
#117209112
#105949913
#105897373