Finite Group Databasebeta

← #13148807

Subgroups — #13148807

Order 1728.

986 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)⋊Q8C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic
3C9×#887C211yesmaximal, characteristic, complemented
4C3×#63449C211yesmaximal, characteristic, complemented
5#11720980C212yesmaximal
6C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic
7C4×(C3⋊D9)⋊C4C212yesmaximal
8Dic6×Dic9C211yesmaximal, characteristic, complemented
9Dic3×Dic18C211yesmaximal, characteristic, complemented
10#11720901C212yesmaximal, complemented
11#11720900C212yesmaximal, complemented
12#10594363—31nomaximal
13C4×Q8×D9—31nomaximal
14C2×C4×C3⋊D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2211yescharacteristic
19C2×(C3⋊D9)⋊C4C2212yes—
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×#6189C2212yes—
22(C3⋊D9)⋊Q8C418yescomplemented
23C3×C9×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25#88534C2212yes—
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30C3×C4×Dic9C2212yes—
31C3×C4×Dic9C2211yescharacteristic
32Dic3⋊Dic9C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2212yes—
38C2×#3797—31no—
39C4⋊C4×C3⋊D3—31no—
40C3×#887—31no—
41C3×#887D311yescharacteristic
42#63883—32no—
43C4×#624—31no—
44C4×#624—32no—
45Dic3×Dic6—31no—
46Dic3×Dic6—31no—
47#63837—32no—
48#63837—32no—
49C2×Q8×D9—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52C4⋊C4×D9—31no—
53C4⋊C4×D9—32no—
54C4⋊C4×D9—61no—
55#63449—32no—
56#63449D311yescharacteristic, complemented
57C42×D9—32no—
58C42×D9—31no—
59C4×Dic18—31no—
60C4×Dic18—32no—
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9—24no—
67C4×C3⋊D9C2×C414yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C9×Dic6C2×C414yescomplemented
75C3×Dic18C2×C414yescomplemented
76(C3⋊D9)⋊C4—22no—
77(C3⋊D9)⋊C4C2×C418yes—
78(C3⋊D9)⋊C4Q814yescomplemented
79#6189C2×C418yescomplemented
80C4×D3×Q8—91no—
81C4×D3×Q8—91no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—32no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#627—32no—
87C2×#624—32no—
88C2×#624—31no—
89C2×#624—31no—
90#3797—38no—
91C32×C4⋊C4C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)—31no—
93#3750—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4C2×D312yes—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3C2×D312yes—
99C3×C4×Dic3C2×D311yescharacteristic
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic32—31no—
103Dic32—31no—
104Dic32—32no—
105Dic32—32no—
106C2×Q8×C9—31no—
107Q8×D9—38no—
108C2×C4×D9—31no—
109C2×C4×D9—31no—
110C2×C4×D9—32no—
111C2×C4×D9—32no—
112C2×C4×D9—31no—
113C2×C4×D9—61no—
114C2×C4×D9—62no—
115C2×Dic18—32no—
116C2×Dic18C2×D311yescharacteristic
117C9×C4⋊C4—21no—
118C9×C4⋊C4—32no—
119C9×C4⋊C4C2×D311yescharacteristic
120C42×C9—32no—
121C42×C9—31no—
122C4⋊Dic9—31no—
123C4⋊Dic9—32no—
124C4⋊Dic9—61no—
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—62no—
128Dic9⋊C4C2×D312yes—
129C4×Dic9—32no—
130C4×Dic9—31no—
131C4×Dic9—32no—
132C4×Dic9—31no—
133C4×Dic9C2×D311yescharacteristic
134C4×Dic9C2×D312yes—
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9C22×C412yes—
137C2×C3⋊D9C2×Q812yes—
138C2×C3⋊D9#4112yes—
139C3×C4×C9—22no—
140C3×C4×C9C22×C412yes—
141C3⋊Dic9—22no—
142C3⋊Dic9C22×C412yes—
143C9×Dic3—21no—
144C9×Dic3C22×C414yes—
145C9×Dic3C2×Q812yescomplemented
146C3×Dic9—21no—
147C3×Dic9C22×C414yes—
148C3×Dic9C2×Q812yescomplemented
149C2×D3×Q8—91no—
150C2×D3×Q8—91no—
151C3×C4×Q8—31no—
152C3×C4×Q8—91no—
153Q8×Dic3—91no—
154Q8×Dic3—91no—
155D3×C4⋊C4—92no—
156D3×C4⋊C4—91no—
157D3×C4⋊C4—91no—
158D3×C4⋊C4—92no—
159D3×C4⋊C4—181no—
160#887—31no—
161#887—92no—
162#887—92no—
163#887D911yescharacteristic, complemented
164C42×D3—92no—
165C42×D3—91no—
166C42×D3—91no—
167C42×D3—92no—
168C4×Dic6—92no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C4×Dic6—92no—
172C22×C3⋊D3—31no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—32no—
177C4×C3⋊D3—34no—
178C4×C3⋊D3—64no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D312yes—
184C2×C3×Dic3C22×D311yescharacteristic
185C3×Dic6—34no—
186C3×Dic6C4×D314yes—
187#627—38no—
188#624—38no—
189#624—34no—
190#624—62no—
191C22×D9—31no—
192C22×D9—61no—
193Q8×C9—34no—
194C2×C4×C9—21no—
195C2×C4×C9—22no—
196C2×C4×C9—31no—
197C2×C4×C9—32no—
198C2×C4×C9—31no—
199C2×C4×C9C22×D312yes—
200C2×C4×C9C22×D311yescharacteristic
201C2×Dic9—31no—
202C2×Dic9—32no—
203C2×Dic9—61no—
204C2×Dic9—62no—
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D312yes—
207C2×Dic9C22×D311yescharacteristic
208C4×D9—34no—
209C4×D9—38no—
210C4×D9—34no—
211C4×D9—64no—
212C4×D9—62no—
213C4×D9—64no—
214C4×D9—124no—
215Dic18—38no—
216Dic18C4×D314yescomplemented
217C2×C4×Q8—271noSylow
218C2×C3×C9C23×C411yescharacteristic
219C2×C3×C9C22×Q811yescharacteristic
220C2×C3×C9C2×#4111yescharacteristic
221C3⋊D9C4×Q814yescomplemented
222C2×C3×Q8—31no—
223C2×C3×Q8—91no—
224D3×Q8—98no—
225D3×Q8—98no—
226C2×C4×D3—91no—
227C2×C4×D3—92no—
228C2×C4×D3—91no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—92no—
232C2×C4×D3—91no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×C4×D3—91no—
236C2×C4×D3—181no—
237C2×C4×D3—182no—
238C2×Dic6—31no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C2×Dic6C2×D911yescharacteristic
242C3×C4⋊C4—21no—
243C3×C4⋊C4—32no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4C2×D911yescharacteristic
246C3×C4⋊C4D3211yescharacteristic
247C3×C42—32no—
248C3×C42—31no—
249C3×C42—92no—
250C3×C42—91no—
251C4⋊Dic3—92no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—181no—
256Dic3⋊C4—32no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—182no—
262Dic3⋊C4C2×D912yes—
263C4×Dic3—32no—
264C4×Dic3—31no—
265C4×Dic3—92no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—92no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—91no—
272C4×Dic3—92no—
273C4×Dic3C2×D912yes—
274C4×Dic3C2×D911yescharacteristic
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—32no—
277C2×C3⋊D3—32no—
278C2×C3⋊D3—32no—
279C32×C4—22no—
280C32×C4C2×C4×D312yes—
281C3⋊Dic3—32no—
282C3⋊Dic3—62no—
283C3×Dic3—21no—
284C3×Dic3—34no—
285C3×Dic3—32no—
286C3×Dic3—61no—
287C3×Dic3C2×C4×D314yes—
288C3×Dic3D3×Q812yes—
289C22×C9—21no—
290C22×C9C23×D311yescharacteristic
291C2×D9—32no—
292C2×D9—32no—
293C2×D9—32no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—62no—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—34no—
300C4×C9—32no—
301C4×C9—42no—
302C4×C9—61no—
303C4×C9C2×C4×D312yes—
304Dic9—21no—
305Dic9—32no—
306Dic9—62no—
307Dic9—62no—
308Dic9—122no—
309Dic9C2×C4×D314yes—
310Dic9D3×Q812yescomplemented
311C22×Q8—271no—
312C4×Q8—31no—
313C4×Q8—91no—
314C4×Q8—272no—
315C4×Q8—271no—
316C4×Q8—272no—
317C4×Q8—271no—
318C2×C4⋊C4—272no—
319C2×C4⋊C4—271no—
320C2×C42—272no—
321C2×C42—271no—
322C3×C9C2×C4×Q811yescharacteristic, complemented, Sylow
323C22×D3—91no—
324C22×D3—91no—
325C22×D3—181no—
326C3×Q8—34no—
327C3×Q8—94no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—31no—
331C2×C3×C4—32no—
332C2×C3×C4—31no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—91no—
336C2×C3×C4C22×D912yes—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3212yes—
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—31no—
341C2×Dic3—32no—
342C2×Dic3—31no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—181no—
348C2×Dic3—182no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D912yes—
351C2×Dic3C22×D911yescharacteristic
352C4×D3—94no—
353C4×D3—98no—
354C4×D3—94no—
355C4×D3—94no—
356C4×D3—98no—
357C4×D3—94no—
358C4×D3—184no—
359C4×D3—182no—
360C4×D3—182no—
361C4×D3—184no—
362C4×D3—184no—
363C4×D3—364no—
364Dic6—34no—
365Dic6—98no—
366Dic6—98no—
367Dic6C4×D914yescomplemented
368C2×C32C22×C4×D311yescharacteristic
369C2×C32C2×D3×Q811yescharacteristic
370C2×C32D3×#4111yescharacteristic
371C3⋊D3—34no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9C22×C4×D311yescharacteristic
376C2×C9C2×D3×Q811yescharacteristic
377C2×C9D3×#4111yescharacteristic
378D9—34no—
379D9—64no—
380C2×Q8—31no—
381C2×Q8—91no—
382C2×Q8—272no—
383C2×Q8—278no—
384C22×C4—271no—
385C22×C4—272no—
386C22×C4—271no—
387C22×C4—271no—
388C22×C4—272no—
389C4⋊C4—32no—
390C4⋊C4—92no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C4⋊C4—271no—
395C4⋊C4D3×D911yescharacteristic
396C42—32no—
397C42—31no—
398C42—92no—
399C42—91no—
400C42—272no—
401C42—271no—
402C42—272no—
403C42—271no—
404C22×C3—21no—
405C22×C3C23×D911yescharacteristic
406C22×C3C22×D3211yescharacteristic
407C2×D3—92no—
408C2×D3—92no—
409C2×D3—92no—
410C2×D3—92no—
411C2×D3—92no—
412C2×D3—92no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—34no—
420C3×C4—32no—
421C3×C4—42no—
422C3×C4—61no—
423C3×C4—94no—
424C3×C4—92no—
425C3×C4—181no—
426C3×C4C2×C4×D912yes—
427C3×C4C4×D3212yes—
428Dic3—21no—
429Dic3—34no—
430Dic3—32no—
431Dic3—61no—
432Dic3—92no—
433Dic3—92no—
434Dic3—182no—
435Dic3—182no—
436Dic3—182no—
437Dic3—362no—
438Dic3C2×C4×D914yes—
439Dic3Q8×D912yescomplemented
440C32C4×D3×Q811yescharacteristic
441C9—21no—
442C9C4×D3×Q811yescharacteristic, complemented
443C23—271no—
444Q8—34no—
445Q8—94no—
446Q8—278no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—31no—
450C2×C4—91no—
451C2×C4—92no—
452C2×C4—91no—
453C2×C4—278no—
454C2×C4—274no—
455C2×C4—271no—
456C2×C4—274no—
457C2×C4—272no—
458C2×C4—544no—
459C2×C4—542no—
460C2×C4C2×D3×D912yes—
461C2×C4C2×D3×D911yescharacteristic
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3C22×C4×D911yescharacteristic
466C2×C3C2×Q8×D911yescharacteristic
467C2×C3#41×D911yescharacteristic
468C2×C3C2×C4×D3211yescharacteristic
469C2×C3#6431211yescharacteristic
470C2×C3D32×Q811yescharacteristic
471D3—94no—
472D3—94no—
473D3—184no—
474C22—272no—
475C22—272no—
476C22—272no—
477C22C22×D3×D911yescharacteristic
478C4—22no—
479C4—32no—
480C4—34no—
481C4—61no—
482C4—92no—
483C4—94no—
484C4—181no—
485C4—272no—
486C4—542no—
487C4C4×D3×D912yes—
488C3—21no—
489C3C4×Q8×D911yescharacteristic, complemented
490C3#1059436311yescharacteristic
491C2—274no—
492C2C2×C4×D3×D911yescharacteristic
493C2#1172249811yescharacteristic
494C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
cosocleC22×C3×C9C24
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×C4×Q8
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C9×#887C2semidirect
4. C3×#63449C2semidirect
8. Dic6×Dic9C2semidirect
9. Dic3×Dic18C2semidirect
10. #11720901C2semidirect
11. #11720900C2semidirect
22. (C3⋊D9)⋊Q8C4semidirect
56. #63449D3semidirect
74. C9×Dic6C2×C4semidirect
75. C3×Dic18C2×C4semidirect
78. (C3⋊D9)⋊C4Q8semidirect
79. #6189C2×C4semidirect
145. C9×Dic3C2×Q8semidirect
148. C3×Dic9C2×Q8semidirect
163. #887D9semidirect
216. Dic18C4×D3semidirect
221. C3⋊D9C4×Q8semidirect
310. Dic9D3×Q8semidirect
322. C3×C9C2×C4×Q8semidirect
367. Dic6C4×D9semidirect
439. Dic3Q8×D9semidirect
442. C9C4×D3×Q8semidirect
489. C3C4×Q8×D9semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C4⋊C4×C3⋊D92
C9×#8872
C3×#634492
#117209802
C4×(C3⋊D9)⋊C42
Dic6×Dic92
Dic3×Dic182
#117209012
#117209002
#105943633
C4×Q8×D93