Finite Group Databasebeta

← #13150437

Subgroups — #13150437

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
2C4×C9×#67C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic, complemented
4#67⋊Dic9C212yesmaximal
5#11721267C212yesmaximal, complemented
6#11721023C212yesmaximal, complemented
7D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9#11720960C211yesmaximal, characteristic, complemented
10#11720914C212yesmaximal, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594993—31nomaximal
13#10589645—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C9×#67C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×D3×Dic9C2212yes—
20C2×#6189C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2212yes—
24C9×#217C2211yescharacteristic
25C3×#3676C2212yes—
26C9×(C2×D3)⋊C4C2211yescharacteristic
27C9×Dic3⋊C4C2211yescharacteristic
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C2×C4⋊(C3⋊Dic3)—31no—
38C3×C4×#67D311yescharacteristic
39C3×#924—31no—
40#63982—32no—
41#67⋊Dic3—32no—
42#63918—31no—
43#63893—32no—
44D3×C4⋊Dic3—31no—
45#63856—31no—
46#63845—32no—
47Dic3×Dic6—31no—
48C4×D4×C9—31no—
49D4×Dic9—32no—
50C2×C4⋊Dic9—31no—
51C2×C4⋊Dic9—31no—
52C2×C4⋊Dic9—61no—
53#63486—31no—
54#63486D311yescharacteristic, complemented
55#63452—32no—
56#63440—32no—
57#63440—32no—
58#63431—31no—
59C4×Dic18—31no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9—42no—
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68C9×#67Q814yescomplemented
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2312yes—
74C4×D3×C9—22no—
75C3×Dic18—22no—
76D3×Dic9—42no—
77#6189—42no—
78#5635—91no—
79#5543—91no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C3⋊Dic3—32no—
82C2×C3×#67C2×D311yescharacteristic
83C2×C3×C4×D3C2×D311yescharacteristic
84C2×C3×Dic6—31no—
85C2×#627—32no—
86C2×D3×Dic3—32no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—32no—
89C4⋊(C3⋊Dic3)—31no—
90C3×#217—32no—
91C3×#217C2×D311yescharacteristic
92C3×(C2×D3)⋊C4C2×D311yescharacteristic
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C2×D4×C9—31no—
104C22×C4×C9—21no—
105C22×C4×C9—31no—
106C22×C4×C9C2×D311yescharacteristic
107C22×Dic9—32no—
108C22×Dic9—32no—
109C22×Dic9—62no—
110C2×Dic18—32no—
111C2×Dic18C2×D311yescharacteristic
112C9×C4⋊C4—31no—
113C9×C22⋊C4—31no—
114C9×C22⋊C4—31no—
115C42×C9—31no—
116#3676—32no—
117#3676C2×D312yes—
118C4⋊Dic9—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—32no—
122C4⋊Dic9—32no—
123C4⋊Dic9—61no—
124C4⋊Dic9—61no—
125C4⋊Dic9—62no—
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4C2×D312yes—
130C4×Dic9—32no—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9C2×Q812yes—
134C22×C3×C9C2411yescharacteristic
135C2×D3×C9—21no—
136C2×D3×C9—21no—
137C2×D3×C9C2×Q812yes—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—22no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3C2×Q812yes—
145C3×Dic9—22no—
146C3×Dic9—22no—
147C3×C22⋊Q8—91no—
148C3×C4×D4—31no—
149#946—91no—
150Q8×Dic3—91no—
151D4×Dic3—92no—
152C4×#67D911yescharacteristic
153C2×C4⋊Dic3—91no—
154C2×C4⋊Dic3—91no—
155C2×C4⋊Dic3—91no—
156C2×C4⋊Dic3—181no—
157#924—31no—
158#924—91no—
159#897—92no—
160D3×C4⋊C4—91no—
161#890—92no—
162#889—92no—
163#888—91no—
164#67⋊C4—92no—
165#878—92no—
166#878—92no—
167#878—92no—
168#869—91no—
169C4×Dic6—91no—
170C23×C32C22×D311yescharacteristic
171C22×C3×D3C22×D311yescharacteristic
172C2×C32×C4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—122no—
178C3×#67Dic614yes—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C3×C4×D3—22no—
185C3×Dic6—62no—
186#627—122no—
187D3×Dic3—122no—
188C23×C9—21no—
189C23×C9—31no—
190C23×C9C22×D311yescharacteristic
191D4×C9—34no—
192C2×C4×C9—24no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—31no—
197C2×C4×C9—31no—
198C2×C4×C9—31no—
199C2×C4×C9—62no—
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202C2×Dic9—32no—
203C2×Dic9—32no—
204C2×Dic9—62no—
205C2×Dic9—62no—
206C2×Dic9—122no—
207C2×Dic9—122no—
208C2×Dic9—124no—
209C2×Dic9C22×D312yes—
210C2×Dic9C22×D312yes—
211Dic18—22no—
212Dic18—122no—
213#554—271noSylow
214C2×C3×C9—22no—
215C2×C3×C9C22×Q811yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9H(5,𝔽2)11yescharacteristic
218D3×C9—22no—
219C2×C3×Q8—91no—
220C2×C3×D4—31no—
221C22×C3×C4—21no—
222C22×C3×C4—31no—
223C22×C3×C4C2×D911yescharacteristic
224C22×C3×C4D3211yescharacteristic
225C2×#67C2×D911yescharacteristic
226C22×Dic3—92no—
227C22×Dic3—92no—
228C22×Dic3—92no—
229C22×Dic3—182no—
230C2×C4×D3—92no—
231C2×C4×D3C2×D911yescharacteristic
232C2×Dic6—31no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—92no—
238C3×C22⋊C4—31no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—92no—
241C3×C42—31no—
242#217—32no—
243#217—92no—
244#217C2×D911yescharacteristic
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4C2×D911yescharacteristic
247C4⋊Dic3—31no—
248C4⋊Dic3—92no—
249C4⋊Dic3—92no—
250C4⋊Dic3—91no—
251C4⋊Dic3—92no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—91no—
255C4⋊Dic3—92no—
256C4⋊Dic3—92no—
257C4⋊Dic3—181no—
258C4⋊Dic3—182no—
259C4⋊Dic3—181no—
260Dic3⋊C4—32no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4C2×D911yescharacteristic
267C4×Dic3—92no—
268C4×Dic3—92no—
269C4×Dic3C2×D911yescharacteristic
270C22×C32—21no—
271C22×C32—21no—
272C22×C32C2×Dic612yes—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—21no—
275C2×C3×D3—21no—
276C2×C3×D3C2×Dic612yes—
277C32×C4—21no—
278C32×C4#23712yes—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3C2×Dic612yes—
286C22×C9—22no—
287C22×C9—22no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—21no—
291C22×C9—22no—
292C22×C9—32no—
293C22×C9—61no—
294C22×C9—61no—
295C22×C9D3×Q812yes—
296C22×C9C23×D311yescharacteristic
297C4×C9—22no—
298C4×C9—21no—
299C4×C9—22no—
300C4×C9—32no—
301C4×C9—61no—
302C4×C9—61no—
303C4×C9#23712yes—
304Dic9—22no—
305Dic9—22no—
306Dic9—62no—
307Dic9—62no—
308Dic9—122no—
309Dic9—122no—
310C4⋊Q8—271no—
311#125—272no—
312C22⋊Q8—91no—
313C22⋊Q8—272no—
314C22⋊Q8—272no—
315C22⋊Q8—271no—
316C4×Q8—271no—
317C4×D4—31no—
318C4×D4—272no—
319C2×C4⋊C4—271no—
320C2×C4⋊C4—271no—
321C3×C9#55411yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3—31no—
324C23×C3C22×D911yescharacteristic
325C23×C3C2×D3211yescharacteristic
326C22×D3C22×D911yescharacteristic
327C3×Q8—182no—
328C3×D4—34no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—62no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344#67Dic1814yescomplemented
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—182no—
352C2×Dic3—182no—
353C2×Dic3—362no—
354C2×Dic3—362no—
355C2×Dic3—362no—
356C2×Dic3—364no—
357C2×Dic3C22×D911yescharacteristic
358C2×Dic3C22×D911yescharacteristic
359C2×Dic3C22×D911yescharacteristic
360C4×D3—22no—
361C4×D3—362no—
362Dic6—62no—
363Dic6—362no—
364Dic6—362no—
365C2×C32—22no—
366C2×C32C22×Dic611yescharacteristic
367C2×C32C2×#23711yescharacteristic
368C2×C32#100911yescharacteristic
369C3×D3—22no—
370C2×C9—24no—
371C2×C9—22no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—62no—
376C2×C9C2×#23711yescharacteristic
377C2×C9C2×D3×Q811yescharacteristic
378C2×C9#100911yescharacteristic
379C2×Q8—91no—
380C2×Q8—272no—
381C2×D4—31no—
382C22×C4—31no—
383C22×C4—272no—
384C22×C4—272no—
385C22×C4D3×D911yescharacteristic
386C4⋊C4—31no—
387C4⋊C4—91no—
388C4⋊C4—92no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—271no—
394C4⋊C4—272no—
395C4⋊C4—271no—
396C22⋊C4—31no—
397C22⋊C4—31no—
398C22⋊C4—92no—
399C22⋊C4—272no—
400C42—31no—
401C42—272no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—32no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3C2×Dic1812yes—
414C22×C3C23×D911yescharacteristic
415C22×C3D3×Dic612yes—
416C22×C3C22×D3211yescharacteristic
417C2×D3—21no—
418C2×D3—21no—
419C2×D3C2×Dic1812yes—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—22no—
424C3×C4—32no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—182no—
428C3×C4—182no—
429C3×C4#369912yes—
430C3×C4#380512yes—
431Dic3—21no—
432Dic3—21no—
433Dic3—62no—
434Dic3—62no—
435Dic3—182no—
436Dic3—182no—
437Dic3—182no—
438Dic3—182no—
439Dic3—362no—
440Dic3—362no—
441Dic3C2×Dic1812yes—
442C32#554311yescharacteristic
443C9—21no—
444C9#563511yescharacteristic, complemented
445C23—31no—
446C23C2×D3×D911yescharacteristic
447Q8—182no—
448Q8—1082no—
449D4—34no—
450C2×C4—22no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—62no—
455C2×C4—92no—
456C2×C4—92no—
457C2×C4—272no—
458C2×C4—272no—
459C2×C4—1082no—
460C2×C4—1082no—
461C2×C4C2×D3×D911yescharacteristic
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—24no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—62no—
470C2×C3C22×Dic1811yescharacteristic
471C2×C3C2×#369911yescharacteristic
472C2×C3#6371511yescharacteristic
473C2×C3C2×D3×Dic611yescharacteristic
474C2×C3#6430811yescharacteristic
475C2×C3C2×#380511yescharacteristic
476D3—22no—
477C22—21no—
478C22—21no—
479C22—32no—
480C22—61no—
481C22—61no—
482C22D3×Dic1812yes—
483C22C22×D3×D911yescharacteristic
484C4—21no—
485C4—32no—
486C4—61no—
487C4—61no—
488C4—182no—
489C4—182no—
490C4—542no—
491C4—542no—
492C4#8866312yes—
493C3—21no—
494C3#1058964511yescharacteristic, complemented
495C3#1059499311yescharacteristic
496C2—22no—
497C2—62no—
498C2C2×D3×Dic1811yescharacteristic
499C2#1172247811yescharacteristic
500C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#67C2×D3
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
1. C2×C4⋊(C3⋊Dic9)C2semidirect
3. C3×#63486C2semidirect
5. #11721267C2semidirect
6. #11721023C2semidirect
7. D3×C4⋊Dic9C2semidirect
8. #11720970C2semidirect
9. #11720960C2semidirect
10. #11720914C2semidirect
11. Dic3×Dic18C2semidirect
54. #63486D3semidirect
68. C9×#67Q8semidirect
321. C3×C9#554semidirect
344. #67Dic18semidirect
444. C9#5635semidirect
494. C3#10589645semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊Dic9)2
C4×C9×#672
C3×#634862
#67⋊Dic92
#117212672
#117210232
D3×C4⋊Dic92
#117209702
#117209602
#117209142
Dic3×Dic182
#105949933
#105896453