Finite Group Databasebeta

← #13150434

Subgroups — #13150434

Order 1728.

758 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
1#11721813C211yesmaximal, characteristic, complemented
2C4×C9×#67C211yesmaximal, characteristic
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
4#67×Dic9C212yesmaximal
5#11721253C212yesmaximal, complemented
6#11721026C212yesmaximal, complemented
7D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9#11720961C211yesmaximal, characteristic, complemented
10#11720916C212yesmaximal, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10594988—31nomaximal
13#10589645—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×C4×D3×C9C2211yescharacteristic
19C2×D3×Dic9C2212yes—
20C2×#6189C2212yes—
21#88551C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C9×#217C2211yescharacteristic
25C9×(C2×D3)⋊C4C2211yescharacteristic
26C9×Dic3⋊C4C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×C4×#67D311yescharacteristic
39C2×C3×C4⋊Dic3—31no—
40Dic3×#67—32no—
41#63964—32no—
42#63918—31no—
43#63899—32no—
44D3×C4⋊Dic3—31no—
45#63857—31no—
46#63846—32no—
47#63844—31no—
48C4×D4×C9—31no—
49D4×Dic9—32no—
50C2×C4⋊Dic9—31no—
51C2×C4⋊Dic9D311yescharacteristic, complemented
52#63486—31no—
53#63486—31no—
54#63486—61no—
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⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C9×#67Q814yescomplemented
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9—42no—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C4×D3×C9—22no—
76D3×Dic9—42no—
77#6189—42no—
78#5543—91no—
79#5511—91no—
80C22×C32×C4C2×D311yescharacteristic
81C2×#634—31no—
82C2×C3×#67C2×D311yescharacteristic
83C22×C3×Dic3—32no—
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×#627—32no—
86C2×D3×Dic3—32no—
87#3757—32no—
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C3×#217C2×D311yescharacteristic
91C3×(C2×D3)⋊C4C2×D311yescharacteristic
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
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×Dic9C2×D312yes—
109C2×Dic18—31no—
110C2×Dic18—32no—
111C2×Dic18—61no—
112C9×C4⋊C4—31no—
113C9×C22⋊C4—31no—
114C9×C22⋊C4—31no—
115C42×C9—31no—
116#3676—32no—
117#3676—32no—
118#3676—62no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—32no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D311yescharacteristic
124C4⋊Dic9C2×D311yescharacteristic
125C4⋊Dic9C2×D312yes—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—62no—
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×C4×D4—31no—
148C2×C3×C4⋊C4—91no—
149D4×Dic3—92no—
150C4×#67—92no—
151C4×#67D911yescharacteristic
152C2×C4⋊Dic3—31no—
153C2×C4⋊Dic3—91no—
154#924—91no—
155#924—91no—
156#924—91no—
157#924—92no—
158#924—181no—
159#897—91no—
160#896—92no—
161D3×C4⋊C4—91no—
162#890—92no—
163#889—92no—
164#888—91no—
165#887—91no—
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—
177#634—62no—
178C3×#67Dic614yes—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—122no—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C2×C3×Dic3C22×D311yescharacteristic
185C3×C4×D3—22no—
186#627—122no—
187D3×Dic3—122no—
188C23×C9—21no—
189C23×C9—31no—
190C23×C9C22×D311yescharacteristic
191D4×C9—34no—
192C2×C4×C9—22no—
193C2×C4×C9—21no—
194C2×C4×C9—21no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—31no—
198C2×C4×C9—42no—
199C2×C4×C9—62no—
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202C2×Dic9—32no—
203C2×Dic9—32no—
204C2×Dic9—42no—
205C2×Dic9—62no—
206C2×Dic9—62no—
207C2×Dic9—122no—
208C2×Dic9C22×D312yes—
209C2×Dic9C22×D312yes—
210Dic18—62no—
211Dic18—122no—
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×D4—31no—
220C22×C3×C4—21no—
221C22×C3×C4—31no—
222C22×C3×C4—92no—
223C22×C3×C4C2×D911yescharacteristic
224C22×C3×C4D3211yescharacteristic
225C2×#67C2×D911yescharacteristic
226C22×Dic3—32no—
227C22×Dic3—92no—
228C2×C4×D3—92no—
229C2×C4×D3C2×D911yescharacteristic
230C2×Dic6—91no—
231C2×Dic6—91no—
232C2×Dic6—92no—
233C2×Dic6—92no—
234C2×Dic6—181no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—31no—
241C3×C42—31no—
242#217—92no—
243#217—92no—
244#217—92no—
245#217—182no—
246#217C2×D911yescharacteristic
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4C2×D911yescharacteristic
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—32no—
252C4⋊Dic3—92no—
253C4⋊Dic3—92no—
254C4⋊Dic3—91no—
255C4⋊Dic3—91no—
256C4⋊Dic3—92no—
257C4⋊Dic3—181no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—182no—
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—21no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—22no—
290C22×C9—32no—
291C22×C9—41no—
292C22×C9—41no—
293C22×C9—61no—
294C22×C9—61no—
295C22×C9D3×Q812yes—
296C22×C9C23×D311yescharacteristic
297C4×C9—21no—
298C4×C9—22no—
299C4×C9—32no—
300C4×C9—41no—
301C4×C9—61no—
302C4×C9—61no—
303C4×C9#23512yes—
304Dic9—22no—
305Dic9—22no—
306Dic9—62no—
307Dic9—62no—
308Dic9—122no—
309Dic9—122no—
310C4⋊Q8—271no—
311#125—272no—
312C22⋊Q8—272no—
313C22⋊Q8—272no—
314C22⋊Q8—271no—
315C22⋊Q8—271no—
316C4×Q8—271no—
317C4×D4—31no—
318C4×D4—272no—
319C2×C4⋊C4—91no—
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×D4—34no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—42no—
336C2×C3×C4—62no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—362no—
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—122no—
352C2×Dic3—182no—
353C2×Dic3—182no—
354C2×Dic3—362no—
355C2×Dic3C22×D911yescharacteristic
356C2×Dic3C22×D911yescharacteristic
357C2×Dic3C22×D911yescharacteristic
358C4×D3—22no—
359C4×D3—362no—
360Dic6—182no—
361Dic6—182no—
362Dic6—362no—
363Dic6—362no—
364Dic6—362no—
365C2×C32—22no—
366C2×C32C22×Dic611yescharacteristic
367C2×C32C2×#23711yescharacteristic
368C2×C32#100911yescharacteristic
369C3×D3—22no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—42no—
375C2×C9—62no—
376C2×C9C2×#23511yescharacteristic
377C2×C9#100411yescharacteristic
378C2×C9C2×D3×Q811yescharacteristic
379C2×Q8—272no—
380C2×Q8—271no—
381C2×D4—31no—
382C22×C4—31no—
383C22×C4—92no—
384C22×C4—272no—
385C22×C4D3×D911yescharacteristic
386C4⋊C4—31no—
387C4⋊C4—91no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C4⋊C4—271no—
395C4⋊C4—272no—
396C22⋊C4—31no—
397C22⋊C4—31no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C42—31no—
401C42—272no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—32no—
409C22×C3—41no—
410C22×C3—41no—
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—21no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—32no—
424C3×C4—41no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—182no—
428C3×C4—182no—
429C3×C4#369912yes—
430C3×C4#380412yes—
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#551111yescharacteristic, complemented
445C23—31no—
446C23C2×D3×D911yescharacteristic
447Q8—542no—
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—362no—
460C2×C4—1082no—
461C2×C4C2×D3×D911yescharacteristic
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3—62no—
470C2×C3C22×Dic1811yescharacteristic
471C2×C3C2×#369911yescharacteristic
472C2×C3#6371511yescharacteristic
473C2×C3C2×D3×Dic611yescharacteristic
474C2×C3#6430711yescharacteristic
475C2×C3C2×#380411yescharacteristic
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#8866212yes—
493C3—21no—
494C3#1058964511yescharacteristic, complemented
495C3#1059498811yescharacteristic
496C2—22no—
497C2—62no—
498C2C2×D3×Dic1811yescharacteristic
499C2#1172247711yescharacteristic
500C2C2×#8866211yescharacteristic
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. #11721813C2semidirect
3. C2×C3×C4⋊Dic9C2semidirect
5. #11721253C2semidirect
6. #11721026C2semidirect
7. D3×C4⋊Dic9C2semidirect
8. #11720970C2semidirect
9. #11720961C2semidirect
10. #11720916C2semidirect
11. #11720912C2semidirect
51. C2×C4⋊Dic9D3semidirect
68. C9×#67Q8semidirect
321. C3×C9#554semidirect
344. #67Dic18semidirect
444. C9#5511semidirect
494. C3#10589645semidirect

Maximal subgroups

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

subgroupindex
#117218132
C4×C9×#672
C2×C3×C4⋊Dic92
#67×Dic92
#117212532
#117210262
D3×C4⋊Dic92
#117209702
#117209612
#117209162
#117209122
#105949883
#105896453