Finite Group Databasebeta

← #13148830

Subgroups — #13148830

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721753C211yesmaximal, characteristic, complemented
4C3×#63459C211yesmaximal, characteristic, complemented
5C9×#887C211yesmaximal, characteristic, complemented
6C4×#6192C211yesmaximal, characteristic
7#11720995C212yesmaximal, complemented
8#11720991C212yesmaximal, complemented
9#11720977C212yesmaximal
10#11720970C212yesmaximal, complemented
11#11720932C211yesmaximal, characteristic, complemented
12#10594383—31nomaximal
13Q8×D36—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6192C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×Dic3×D9C2212yes—
22C2×#6189C2212yes—
23C2×#6189C2212yes—
24(C3⋊D9)⋊Q8—24no—
25D9×Dic6—24no—
26C3×C9×C4⋊C4C2211yescharacteristic
27#88536C2212yes—
28C4⋊(C3⋊Dic9)C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C9×Dic3⋊C4C2212yes—
31C4×C9×Dic3C2212yes—
32C4×C9×Dic3C2211yescharacteristic
33C3×C4⋊Dic9C2212yes—
34#88445C2211yescharacteristic
35(C2×D9)⋊Dic3C2211yescharacteristic
36Dic3⋊Dic9C2211yescharacteristic
37Dic3⋊Dic9C2212yes—
38Dic3⋊Dic9C2212yes—
39C2×#3797—31no—
40C2×D3×Dic6—31no—
41#64106—31no—
42C3×#897—31no—
43C3×#887D311yescharacteristic
44#63918—32no—
45C4×#626—31no—
46#63895—32no—
47#63893—32no—
48#63880—32no—
49#63866—31no—
50C2×Q8×D9—31no—
51C2×Q8×D9—31no—
52C4×Q8×C9—31no—
53#63459—31no—
54#63459—32no—
55#63459—32no—
56#63459—61no—
57#63459D311yescharacteristic, complemented
58C4×D36—31no—
59C4×D36—32no—
60#63431—31no—
61#63431—32no—
62C22×C3⋊D9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C4×C3⋊D9—22no—
69#6226—22no—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2312yes—
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C3×C4×D9—22no—
76C9×Dic6D414yescomplemented
77C3×Dic18—22no—
78#6192Q814yescomplemented
79(C3⋊D9)⋊C4—24no—
80Dic3×D9—24no—
81#6189—24no—
82#6189—24no—
83#5632—91no—
84Q8×D12—91no—
85C2×C4×C3⋊D3—31no—
86C2×#634—31no—
87C2×C3×C4×D3—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×#627—32no—
91C2×#627—32no—
92C2×#626—31no—
93C2×#624—32no—
94C2×D3×Dic3—32no—
95#3797—64no—
96D3×Dic6—64no—
97C32×C4⋊C4C2×D311yescharacteristic
98#3752—32no—
99C4⋊(C3⋊Dic3)—32no—
100C3×(C2×D3)⋊C4—32no—
101C3×C4⋊Dic3—32no—
102C3×Dic3⋊C4C2×D312yes—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C2×D312yes—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—31no—
108#3722—31no—
109(C2×D3)⋊Dic3—31no—
110C2×Q8×C9—31no—
111Q8×D9—64no—
112Q8×D9—64no—
113C2×D36—31no—
114C2×C4×D9—32no—
115C2×C4×D9—31no—
116C2×C4×D9—32no—
117C2×C4×D9—61no—
118C2×C4×D9C2×D311yescharacteristic
119C2×Dic18—31no—
120C2×Dic18—32no—
121C2×Dic18—32no—
122C2×Dic18—61no—
123C2×Dic18C2×D311yescharacteristic
124C9×C4⋊C4—21no—
125C9×C4⋊C4—32no—
126C9×C4⋊C4C2×D311yescharacteristic
127C42×C9—32no—
128C42×C9—31no—
129(C2×D9)⋊C4—31no—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—31no—
132(C2×D9)⋊C4—62no—
133(C2×D9)⋊C4C2×D312yes—
134C4⋊Dic9—32no—
135C4⋊Dic9—32no—
136C4⋊Dic9—31no—
137C4⋊Dic9—32no—
138C4⋊Dic9—62no—
139C4⋊Dic9C2×D312yes—
140C22×C3×C9C2411yescharacteristic
141C2×C3⋊D9—21no—
142C2×C3⋊D9—21no—
143C2×C3⋊D9C2×Q812yescomplemented
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×Q812yescomplemented
147C3×C4×C9—22no—
148C3×C4×C9C2×D412yes—
149C3⋊Dic9—21no—
150C3⋊Dic9—22no—
151C9×Dic3—21no—
152C9×Dic3C2×D414yes—
153C9×Dic3C2×Q812yescomplemented
154C3×Dic9—21no—
155C3×Dic9—22no—
156C2×D3×Q8—91no—
157C2×D3×Q8—91no—
158C2×D3×Q8—91no—
159C22×Dic6—91no—
160C3×C22⋊Q8—91no—
161C3×C4×Q8—31no—
162#944—91no—
163C4×#67—91no—
164#897—31no—
165#897—92no—
166#897—92no—
167#897—91no—
168#897—91no—
169#897—181no—
170#896—92no—
171#888—92no—
172#887D911yescharacteristic, complemented
173#67⋊C4—92no—
174#878—92no—
175C4×D12—92no—
176C4×D12—91no—
177#869—92no—
178#869—91no—
179C22×C3⋊D3—31no—
180C22×C3×D3—31no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—32no—
185C4×C3⋊D3—62no—
186#634—62no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3C22×D311yescharacteristic
190C2×C3×Dic3C22×D312yes—
191C2×C3×Dic3C22×D311yescharacteristic
192C3×C4×D3—62no—
193C3×Dic6—62no—
194C3×Dic6D1214yes—
195#627—64no—
196#627—64no—
197#626—34no—
198#624—64no—
199D3×Dic3—64no—
200C22×D9—31no—
201C22×D9—61no—
202C22×D9C22×D311yescharacteristic
203Q8×C9—34no—
204C2×C4×C9—21no—
205C2×C4×C9—22no—
206C2×C4×C9—31no—
207C2×C4×C9—32no—
208C2×C4×C9—31no—
209C2×C4×C9C22×D312yes—
210C2×C4×C9C22×D311yescharacteristic
211C2×Dic9—31no—
212C2×Dic9—32no—
213C2×Dic9—61no—
214C2×Dic9—62no—
215C2×Dic9C22×D312yes—
216C2×Dic9C22×D311yescharacteristic
217D36—34no—
218C4×D9—22no—
219C4×D9—64no—
220C4×D9—64no—
221C4×D9—62no—
222C4×D9—122no—
223Dic18—22no—
224Dic18—64no—
225Dic18—64no—
226Dic18—62no—
227Dic18—122no—
228D4×Q8—271noSylow
229C2×C3×C9C22×D411yescharacteristic
230C2×C3×C9C22×Q811yescharacteristic
231C2×C3×C92−1+411yescharacteristic
232C3⋊D9—22no—
233C3×D9—22no—
234C2×C3×Q8—31no—
235C2×C3×Q8—91no—
236C22×C3×C4—91no—
237C2×#67—91no—
238C22×Dic3—92no—
239D3×Q8—184no—
240D3×Q8—184no—
241D3×Q8—184no—
242C2×D12—91no—
243C2×C4×D3—31no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—92no—
249C2×C4×D3—181no—
250C2×Dic6—31no—
251C2×Dic6—92no—
252C2×Dic6—92no—
253C2×Dic6—91no—
254C2×Dic6—91no—
255C2×Dic6—92no—
256C2×Dic6—92no—
257C2×Dic6—181no—
258C2×Dic6—184no—
259C2×Dic6C2×D911yescharacteristic
260C3×C4⋊C4—21no—
261C3×C4⋊C4—32no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4C2×D911yescharacteristic
264C3×C4⋊C4D3211yescharacteristic
265C3×C22⋊C4—92no—
266C3×C42—32no—
267C3×C42—31no—
268#217—91no—
269(C2×D3)⋊C4—32no—
270(C2×D3)⋊C4—91no—
271(C2×D3)⋊C4—91no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—91no—
275(C2×D3)⋊C4—182no—
276C4⋊Dic3—32no—
277C4⋊Dic3—91no—
278C4⋊Dic3—92no—
279C4⋊Dic3—92no—
280C4⋊Dic3—92no—
281C4⋊Dic3—92no—
282C4⋊Dic3—182no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—91no—
286Dic3⋊C4C2×D912yes—
287C4×Dic3C2×D912yes—
288C4×Dic3C2×D911yescharacteristic
289C22×C32C23×D311yescharacteristic
290C2×C3⋊D3—32no—
291C2×C3⋊D3—61no—
292C2×C3⋊D3—61no—
293C2×C3×D3—32no—
294C2×C3×D3—61no—
295C2×C3×D3—61no—
296C32×C4—22no—
297C32×C4C2×D1212yes—
298C3⋊Dic3—62no—
299C3⋊Dic3—61no—
300C3×Dic3—21no—
301C3×Dic3—62no—
302C3×Dic3—61no—
303C3×Dic3C2×D1214yes—
304C3×Dic3D3×Q812yes—
305C22×C9—21no—
306C22×C9C23×D311yescharacteristic
307C2×D9—21no—
308C2×D9—21no—
309C2×D9—32no—
310C2×D9—62no—
311C2×D9—61no—
312C2×D9—61no—
313C2×D9—121no—
314C2×D9—121no—
315C2×D9D3×Q812yescomplemented
316C4×C9—22no—
317C4×C9—22no—
318C4×C9—34no—
319C4×C9—32no—
320C4×C9—42no—
321C4×C9—61no—
322C4×C9D3×D412yes—
323Dic9—22no—
324Dic9—21no—
325Dic9—61no—
326Dic9—62no—
327Dic9—122no—
328Dic9—121no—
329C22×Q8—271no—
330C22×Q8—271no—
331C4⋊Q8—272no—
332C4⋊Q8—271no—
333C22⋊Q8—91no—
334C22⋊Q8—272no—
335C22⋊Q8—272no—
336C22⋊Q8—271no—
337C4×Q8—31no—
338C4×D4—272no—
339C4×D4—271no—
340C3×C9D4×Q811yescharacteristic, complemented, Sylow
341C23×C3—91no—
342C22×D3—31no—
343C22×D3—91no—
344C22×D3—91no—
345C22×D3—181no—
346C3×Q8—34no—
347C3×Q8—182no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—31no—
351C2×C3×C4—32no—
352C2×C3×C4—31no—
353C2×C3×C4—92no—
354C2×C3×C4—91no—
355C2×C3×C4—182no—
356C2×C3×C4C22×D912yes—
357C2×C3×C4C22×D911yescharacteristic
358C2×C3×C4C2×D3212yes—
359C2×C3×C4C2×D3211yescharacteristic
360#67—94no—
361C2×Dic3—31no—
362C2×Dic3—32no—
363C2×Dic3—91no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—91no—
367C2×Dic3—181no—
368C2×Dic3—182no—
369C2×Dic3—184no—
370C2×Dic3C22×D911yescharacteristic
371C2×Dic3C22×D912yes—
372C2×Dic3C22×D911yescharacteristic
373D12—94no—
374C4×D3—62no—
375C4×D3—184no—
376C4×D3—184no—
377C4×D3—182no—
378C4×D3—182no—
379C4×D3—184no—
380C4×D3—362no—
381Dic6—62no—
382Dic6—184no—
383Dic6—184no—
384Dic6—182no—
385Dic6—182no—
386Dic6—184no—
387Dic6—184no—
388Dic6—362no—
389Dic6D3614yescomplemented
390C2×C32C22×D1211yescharacteristic
391C2×C32C2×D3×Q811yescharacteristic
392C2×C32#101011yescharacteristic
393C3⋊D3—62no—
394C3×D3—62no—
395C2×C9—21no—
396C2×C9—21no—
397C2×C9—21no—
398C2×C9C2×D3×D411yescharacteristic
399C2×C9C2×D3×Q811yescharacteristic
400C2×C9#101011yescharacteristic
401D9—22no—
402D9—62no—
403D9—122no—
404C2×Q8—31no—
405C2×Q8—91no—
406C2×Q8—272no—
407C2×Q8—272no—
408C2×Q8—271no—
409C2×Q8—544no—
410C2×Q8—544no—
411C2×D4—271no—
412C22×C4—91no—
413C22×C4—272no—
414C22×C4—272no—
415C22×C4—271no—
416C4⋊C4—32no—
417C4⋊C4—92no—
418C4⋊C4—272no—
419C4⋊C4—271no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4D3×D911yescharacteristic
423C22⋊C4—92no—
424C22⋊C4—271no—
425C22⋊C4—271no—
426C22⋊C4—272no—
427C42—32no—
428C42—31no—
429C22×C3—21no—
430C22×C3—92no—
431C22×C3—181no—
432C22×C3—181no—
433C22×C3C23×D911yescharacteristic
434C22×C3C22×D3211yescharacteristic
435C2×D3—32no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—92no—
439C2×D3—92no—
440C2×D3—182no—
441C2×D3—181no—
442C2×D3—181no—
443C2×D3—181no—
444C2×D3—181no—
445C2×D3—361no—
446C2×D3—361no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—34no—
451C3×C4—32no—
452C3×C4—42no—
453C3×C4—61no—
454C3×C4—181no—
455C3×C4—182no—
456C3×C4C2×D3612yes—
457C3×C4D3×D1212yes—
458Dic3—21no—
459Dic3—62no—
460Dic3—61no—
461Dic3—181no—
462Dic3—182no—
463Dic3—181no—
464Dic3—182no—
465Dic3—362no—
466Dic3—361no—
467Dic3C2×D3614yes—
468Dic3Q8×D912yescomplemented
469C32Q8×D1211yescharacteristic
470C9—21no—
471C9#563211yescharacteristic, complemented
472C23—91no—
473C23—271no—
474Q8—34no—
475Q8—182no—
476Q8—544no—
477Q8—544no—
478Q8—542no—
479D4—274no—
480C2×C4—31no—
481C2×C4—32no—
482C2×C4—31no—
483C2×C4—92no—
484C2×C4—91no—
485C2×C4—182no—
486C2×C4—271no—
487C2×C4—272no—
488C2×C4—544no—
489C2×C4—544no—
490C2×C4—542no—
491C2×C4C2×D3×D912yes—
492C2×C4C2×D3×D911yescharacteristic
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—182no—
497C2×C3C22×D3611yescharacteristic
498C2×C3C2×Q8×D911yescharacteristic
499C2×C3#6371611yescharacteristic
500C2×C3C2×D3×D1211yescharacteristic
501C2×C3#6430911yescharacteristic
502C2×C3D32×Q811yescharacteristic
503D3—62no—
504D3—182no—
505D3—182no—
506D3—362no—
507C22—92no—
508C22—181no—
509C22—181no—
510C22—272no—
511C22—541no—
512C22—541no—
513C22C22×D3×D911yescharacteristic
514C4—22no—
515C4—32no—
516C4—34no—
517C4—61no—
518C4—181no—
519C4—182no—
520C4—542no—
521C4—541no—
522C4D3×D3612yes—
523C3—21no—
524C3Q8×D3611yescharacteristic, complemented
525C3#1059438311yescharacteristic
526C2—182no—
527C2—542no—
528C2C2×D3×D3611yescharacteristic
529C2#1172249411yescharacteristic
530C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9D4×Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×(C3⋊D9)⋊Q8C2semidirect
2. C2×D9×Dic6C2semidirect
3. #11721753C2semidirect
4. C3×#63459C2semidirect
5. C9×#887C2semidirect
7. #11720995C2semidirect
8. #11720991C2semidirect
10. #11720970C2semidirect
11. #11720932C2semidirect
57. #63459D3semidirect
76. C9×Dic6D4semidirect
78. #6192Q8semidirect
143. C2×C3⋊D9C2×Q8semidirect
146. C2×C3×D9C2×Q8semidirect
153. C9×Dic3C2×Q8semidirect
172. #887D9semidirect
315. C2×D9D3×Q8semidirect
340. C3×C9D4×Q8semidirect
389. Dic6D36semidirect
468. Dic3Q8×D9semidirect
471. C9#5632semidirect
524. C3Q8×D36semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C2×D9×Dic62
#117217532
C3×#634592
C9×#8872
C4×#61922
#117209952
#117209912
#117209772
#117209702
#117209322
#105943833
Q8×D363