Finite Group Databasebeta

← #13148760

Subgroups — #13148760

Order 1728.

934 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C3⋊D9)⋊Q8C212yesmaximal, complemented
2#11721749C211yesmaximal, characteristic
3C9×#888C211yesmaximal, characteristic, complemented
4C3×#63450C211yesmaximal, characteristic, complemented
5#11720993C211yesmaximal, characteristic
6#11720992C211yesmaximal, characteristic
7#11720986C212yesmaximal, complemented
8#11720985C212yesmaximal, complemented
9#11720971C211yesmaximal, characteristic, complemented
10C4×#6189C211yesmaximal, characteristic, complemented
11#11720923C212yesmaximal, complemented
12#10594379—31nomaximal
13#10589829—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×Dic6C2212yes—
17C2×C3×Dic18C2212yes—
18C2×(C3⋊D9)⋊C4C2212yes—
19C2×(C3⋊D9)⋊C4C2212yes—
20C2×#6189C2212yes—
21C2×#6189C2211yescharacteristic
22(C3⋊D9)⋊Q8—28no—
23C3×C9×C4⋊C4C2211yescharacteristic
24#88536C2212yes—
25C4×C3⋊Dic9C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31C3×C4×Dic9C2211yescharacteristic
32#88445C2212yes—
33#88445C2212yes—
34Dic3⋊Dic9C2211yescharacteristic
35#88442C2211yescharacteristic
36#88442C2212yes—
37Dic9⋊Dic3C2211yescharacteristic
38C2×#3797—32no—
39#64102—31no—
40C3×#888—31no—
41C3×#888D311yescharacteristic
42#63919—31no—
43C4×#627—31no—
44#63894—31no—
45#63894—31no—
46#63888—32no—
47#63888—32no—
48#63865—32no—
49C2×Q8×D9—32no—
50C9×C4⋊Q8—31no—
51#63508—32no—
52#63458—32no—
53#63458—32no—
54#63455—31no—
55#63455—31no—
56#63455—61no—
57#63450—31no—
58#63450D311yescharacteristic, complemented
59C4×D36—31no—
60C4×Dic18—31no—
61C22×C3⋊D9C2312yes—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C4⋊(C3⋊D9)Q814yescomplemented
67C4×C3⋊D9—24no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C9×Dic6—24no—
75C3×Dic18—24no—
76(C3⋊D9)⋊C4—24no—
77(C3⋊D9)⋊C4—24no—
78#6189—24no—
79#6189D414yescomplemented
80#5727—91no—
81#5727—91no—
82C2×C4⋊(C3⋊D3)—31no—
83C2×C4×C3⋊D3—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6C2×D312yes—
86C2×#627—31no—
87C2×#627—32no—
88C2×#624—32no—
89C2×#624—32no—
90#3797—68no—
91C32×C4⋊C4C2×D311yescharacteristic
92#3752—32no—
93C4×C3⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4C2×D312yes—
98C3×C4×Dic3—31no—
99C3×C4×Dic3C2×D311yescharacteristic
100#3724—31no—
101#3724—32no—
102Dic3⋊Dic3—31no—
103Dic3⋊Dic3—31no—
104#3722—32no—
105#3722—32no—
106C2×Q8×C9—32no—
107Q8×D9—68no—
108C2×D36—31no—
109C2×D36—61no—
110C2×C4×D9—32no—
111C2×C4×D9—32no—
112C2×C4×D9—32no—
113C2×C4×D9—62no—
114C2×Dic18—31no—
115C2×Dic18—32no—
116C2×Dic18C2×D312yes—
117C9×C4⋊C4—21no—
118C9×C4⋊C4—32no—
119C9×C4⋊C4—31no—
120C9×C4⋊C4C2×D311yescharacteristic
121C42×C9—31no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—62no—
126C4⋊Dic9—31no—
127C4⋊Dic9C2×D311yescharacteristic
128Dic9⋊C4—32no—
129Dic9⋊C4—31no—
130Dic9⋊C4—31no—
131Dic9⋊C4C2×D312yes—
132C4×Dic9—31no—
133C4×Dic9—61no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—22no—
137C2×C3⋊D9—22no—
138C2×C3⋊D9C2×Q814yescomplemented
139C3×C4×C9—22no—
140C3×C4×C9C2×Q812yescomplemented
141C3⋊Dic9—21no—
142C3⋊Dic9C2×D412yes—
143C9×Dic3—22no—
144C9×Dic3—21no—
145C9×Dic3C2×D412yes—
146C3×Dic9—22no—
147C3×Dic9—21no—
148C3×Dic9C2×D412yes—
149C2×D3×Q8—92no—
150C2×D3×Q8—92no—
151C3×C4⋊Q8—31no—
152C3×C4⋊Q8—91no—
153#946—92no—
154#946—92no—
155#896—92no—
156#896—92no—
157#896—92no—
158#896—92no—
159#893—91no—
160#893—91no—
161#893—91no—
162#893—91no—
163#893—181no—
164#888—31no—
165#888—91no—
166#888—91no—
167#888D911yescharacteristic, complemented
168C4×D12—91no—
169C4×D12—91no—
170C4×Dic6—91no—
171C4×Dic6—91no—
172C22×C3⋊D3—32no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C4⋊(C3⋊D3)—34no—
178C4×C3⋊D3—64no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D312yes—
184C2×C3×Dic3C22×D311yescharacteristic
185C3×Dic6—24no—
186C3×Dic6—64no—
187#627—34no—
188#627—64no—
189#624—64no—
190#624—64no—
191C22×D9—32no—
192C22×D9—62no—
193Q8×C9—64no—
194C2×C4×C9—21no—
195C2×C4×C9—22no—
196C2×C4×C9—31no—
197C2×C4×C9—31no—
198C2×C4×C9—32no—
199C2×C4×C9C22×D312yes—
200C2×C4×C9C22×D311yescharacteristic
201C2×Dic9—31no—
202C2×Dic9—31no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D312yes—
207C2×Dic9C22×D311yescharacteristic
208D36—34no—
209D36—64no—
210C4×D9—64no—
211C4×D9—64no—
212C4×D9—64no—
213C4×D9—124no—
214Dic18—24no—
215Dic18—34no—
216Dic18—64no—
217D4×Q8—271noSylow
218C2×C3×C9C22×D411yescharacteristic
219C2×C3×C9C22×Q811yescharacteristic
220C2×C3×C92−1+411yescharacteristic
221C3⋊D9—24no—
222C2×C3×Q8—32no—
223C2×C3×Q8—92no—
224D3×Q8—188no—
225D3×Q8—188no—
226C2×D12—91no—
227C2×D12—91no—
228C2×D12—181no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×C4×D3—182no—
236C2×Dic6—32no—
237C2×Dic6—91no—
238C2×Dic6—91no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C2×Dic6C2×D912yes—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4C2×D911yescharacteristic
248C3×C4⋊C4D3211yescharacteristic
249C3×C42—31no—
250C3×C42—91no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—182no—
258C4⋊Dic3—31no—
259C4⋊Dic3—91no—
260C4⋊Dic3—91no—
261C4⋊Dic3C2×D911yescharacteristic
262Dic3⋊C4—32no—
263Dic3⋊C4—91no—
264Dic3⋊C4—91no—
265Dic3⋊C4—92no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—92no—
269Dic3⋊C4C2×D912yes—
270C4×Dic3—31no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—181no—
274C4×Dic3C2×D911yescharacteristic
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—34no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—62no—
279C32×C4—22no—
280C32×C4D3×Q812yes—
281C3⋊Dic3—32no—
282C3⋊Dic3—61no—
283C3×Dic3—22no—
284C3×Dic3—21no—
285C3×Dic3—32no—
286C3×Dic3—62no—
287C3×Dic3—61no—
288C3×Dic3D3×D412yes—
289C22×C9—21no—
290C22×C9C23×D311yescharacteristic
291C2×D9—34no—
292C2×D9—64no—
293C2×D9—62no—
294C2×D9—62no—
295C2×D9—122no—
296C2×D9—122no—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—32no—
300C4×C9—42no—
301C4×C9—62no—
302C4×C9—61no—
303C4×C9D3×Q812yescomplemented
304Dic9—22no—
305Dic9—21no—
306Dic9—32no—
307Dic9—62no—
308Dic9—61no—
309Dic9—121no—
310Dic9D3×D412yes—
311C22×Q8—272no—
312C4⋊Q8—31no—
313C4⋊Q8—91no—
314C4⋊Q8—271no—
315C22⋊Q8—272no—
316C22⋊Q8—272no—
317C22⋊Q8—272no—
318C4×Q8—271no—
319C4×D4—271no—
320C4×D4—271no—
321C4×D4—271no—
322C3×C9D4×Q811yescharacteristic, complemented, Sylow
323C22×D3—92no—
324C22×D3—92no—
325C22×D3—182no—
326C3×Q8—64no—
327C3×Q8—184no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—32no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4C22×D912yes—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3212yes—
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—32no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D912yes—
351C2×Dic3C22×D911yescharacteristic
352D12—94no—
353D12—94no—
354D12—184no—
355C4×D3—184no—
356C4×D3—184no—
357C4×D3—184no—
358C4×D3—184no—
359C4×D3—184no—
360C4×D3—184no—
361C4×D3—364no—
362Dic6—24no—
363Dic6—64no—
364Dic6—94no—
365Dic6—94no—
366Dic6—184no—
367Dic6—184no—
368C2×C32C2×D3×D411yescharacteristic
369C2×C32C2×D3×Q811yescharacteristic
370C2×C32#100711yescharacteristic
371C3⋊D3—64no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9C2×D3×D411yescharacteristic
376C2×C9C2×D3×Q811yescharacteristic
377C2×C9#100711yescharacteristic
378D9—64no—
379D9—124no—
380C2×Q8—32no—
381C2×Q8—92no—
382C2×Q8—271no—
383C2×Q8—272no—
384C2×Q8—548no—
385C2×D4—271no—
386C22×C4—272no—
387C22×C4—272no—
388C22×C4—272no—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—91no—
392C4⋊C4—92no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C4⋊C4—271no—
396C4⋊C4—272no—
397C4⋊C4D3×D911yescharacteristic
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C42—31no—
402C42—91no—
403C42—271no—
404C22×C3—21no—
405C22×C3C23×D911yescharacteristic
406C22×C3C22×D3211yescharacteristic
407C2×D3—94no—
408C2×D3—94no—
409C2×D3—184no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3—362no—
415C2×D3—362no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—32no—
420C3×C4—42no—
421C3×C4—62no—
422C3×C4—61no—
423C3×C4—92no—
424C3×C4—182no—
425C3×C4—181no—
426C3×C4Q8×D912yescomplemented
427C3×C4#379712yes—
428Dic3—22no—
429Dic3—21no—
430Dic3—32no—
431Dic3—62no—
432Dic3—61no—
433Dic3—92no—
434Dic3—92no—
435Dic3—182no—
436Dic3—181no—
437Dic3—181no—
438Dic3—361no—
439Dic3D4×D912yes—
440C32#572711yescharacteristic
441C9—21no—
442C9#572711yescharacteristic, complemented
443C23—272no—
444Q8—64no—
445Q8—184no—
446Q8—274no—
447Q8—544no—
448D4—274no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—32no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—92no—
455C2×C4—271no—
456C2×C4—271no—
457C2×C4—544no—
458C2×C4—544no—
459C2×C4—544no—
460C2×C4C2×D3×D912yes—
461C2×C4C2×D3×D911yescharacteristic
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3C2×D4×D911yescharacteristic
466C2×C3C2×Q8×D911yescharacteristic
467C2×C3#6371311yescharacteristic
468C2×C3C2×#379711yescharacteristic
469C2×C3D32×D411yescharacteristic
470C2×C3#6431611yescharacteristic
471D3—184no—
472D3—184no—
473D3—364no—
474C22—274no—
475C22—542no—
476C22—542no—
477C22C22×D3×D911yescharacteristic
478C4—22no—
479C4—32no—
480C4—61no—
481C4—62no—
482C4—92no—
483C4—181no—
484C4—182no—
485C4—272no—
486C4—541no—
487C4(C3⋊D9)⋊Q812yescomplemented
488C3—21no—
489C3#1058982911yescharacteristic, complemented
490C3#1059437911yescharacteristic
491C2—544no—
492C2C2×(C3⋊D9)⋊Q811yescharacteristic
493C2D3×D4×D911yescharacteristic
494C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
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
3. C9×#888C2semidirect
4. C3×#63450C2semidirect
7. #11720986C2semidirect
8. #11720985C2semidirect
9. #11720971C2semidirect
10. C4×#6189C2semidirect
11. #11720923C2semidirect
58. #63450D3semidirect
66. C4⋊(C3⋊D9)Q8semidirect
79. #6189D4semidirect
138. C2×C3⋊D9C2×Q8semidirect
140. C3×C4×C9C2×Q8semidirect
167. #888D9semidirect
303. C4×C9D3×Q8semidirect
322. C3×C9D4×Q8semidirect
426. C3×C4Q8×D9semidirect
442. C9#5727semidirect
487. C4(C3⋊D9)⋊Q8semidirect
489. C3#10589829semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
#117217492
C9×#8882
C3×#634502
#117209932
#117209922
#117209862
#117209852
#117209712
C4×#61892
#117209232
#105943793
#105898293