Finite Group Databasebeta

← #13151161

Subgroups — #13151161

Order 1728.

778 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#11721836C211yesmaximal, characteristic, complemented
2C9×#938C211yesmaximal, characteristic
3C3×D4×Dic9C211yesmaximal, characteristic, complemented
4#67×Dic9C212yesmaximal, complemented
5#11721280C212yesmaximal, complemented
6#11721246C212yesmaximal, complemented
7#11721245C212yesmaximal, complemented
8C4×D3×Dic9C211yesmaximal, characteristic, complemented
9#11720961C211yesmaximal, characteristic, complemented
10#11720937C211yesmaximal, characteristic, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10595172—31nomaximal
13#10589702—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2212yes—
17C22×C3×Dic9C2212yes—
18C2×C4×D3×C9C2211yescharacteristic
19C2×D3×Dic9C2211yescharacteristic
20C2×D3×Dic9C2211yescharacteristic
21#88551C2212yes—
22#88551C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C9×#217C2212yes—
25C3×#3676C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29(C2×D3)⋊Dic9C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32Dic9⋊Dic3C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2211yescharacteristic
37#64145—31no—
38C3×#938D311yescharacteristic
39C3×D4×Dic3—31no—
40Dic3×#67—32no—
41#63969—32no—
42#63956—32no—
43#63955—32no—
44C4×D3×Dic3—31no—
45#63872—31no—
46#63857—31no—
47#63844—31no—
48C9×#121—31no—
49#63498—31no—
50#63498—61no—
51#63497—32no—
52D4×Dic9—32no—
53D4×Dic9D311yescharacteristic, complemented
54#63486—31no—
55C2×C4×Dic9—31no—
56#63452—31no—
57#63449—31no—
58#63441—32no—
59#63439—32no—
60C23×C3×C9C2312yes—
61C22×D3×C9C2311yescharacteristic
62C3×D4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C9×#67—24no—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9—24no—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C4×D3×C9—22no—
76D3×Dic9—22no—
77D3×Dic9—22no—
78#5600—91no—
79#5557—91no—
80C2×C32×D4C2×D311yescharacteristic
81C2×#634—31no—
82C2×C3×#67C2×D312yes—
83C22×C3×Dic3—32no—
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×D3×Dic3—31no—
86C2×D3×Dic3—31no—
87#3757—32no—
88#3757—32no—
89C4×C3⋊Dic3—31no—
90C3×#217—32no—
91C3×#217C2×D312yes—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3—31no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101Dic32—31no—
102Dic32—31no—
103C2×D4×C9—21no—
104C2×D4×C9—32no—
105C2×D4×C9C2×D311yescharacteristic
106C22×C4×C9—31no—
107C22×Dic9—31no—
108C22×Dic9—31no—
109C22×Dic9C2×D312yes—
110C2×Dic18—31no—
111C2×Dic18—61no—
112C9×C4⋊C4—31no—
113C9×C22⋊C4—32no—
114#3676—32no—
115#3676—32no—
116#3676—32no—
117#3676—62no—
118#3676—62no—
119#3676C2×D312yes—
120C4⋊Dic9—32no—
121C4⋊Dic9C2×D311yescharacteristic
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125C4×Dic9—31no—
126C4×Dic9—32no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—61no—
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9—22no—
132C22×C3×C9—22no—
133C22×C3×C9—22no—
134C22×C3×C9C2411yescharacteristic
135C2×D3×C9—21no—
136C2×D3×C9—21no—
137C2×D3×C9#4112yes—
138C3×C4×C9#4112yes—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C3⋊Dic9—22no—
142C9×Dic3—21no—
143C9×Dic3—22no—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×Dic9#4112yes—
147C3×#121—31no—
148C3×C4×D4—91no—
149#938D911yescharacteristic
150#936—91no—
151#936—91no—
152#936—181no—
153#935—92no—
154D4×Dic3—31no—
155D4×Dic3—92no—
156C4×#67—92no—
157#926—92no—
158#924—91no—
159C2×C4×Dic3—91no—
160#897—91no—
161#890—91no—
162#890—91no—
163#887—91no—
164#883—92no—
165#879—92no—
166#879—92no—
167#877—92no—
168C42×D3—91no—
169C4×Dic6—91no—
170C23×C32C22×D312yes—
171C22×C3×D3C22×D311yescharacteristic
172C32×D4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177#634—62no—
178C3×#67—24no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3C22×D311yescharacteristic
184C2×C3×Dic3C22×D312yes—
185C3×C4×D3—22no—
186D3×Dic3—62no—
187D3×Dic3—62no—
188C23×C9—22no—
189C23×C9—31no—
190C23×C9C22×D312yes—
191D4×C9—22no—
192D4×C9—42no—
193D4×C9—64no—
194C2×C4×C9—21no—
195C2×C4×C9—31no—
196C2×C4×C9—32no—
197C2×C4×C9—62no—
198C2×C4×C9C22×D311yescharacteristic
199C2×Dic9—24no—
200C2×Dic9—31no—
201C2×Dic9—31no—
202C2×Dic9—32no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9—62no—
206C2×Dic9—62no—
207C2×Dic9—62no—
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D312yes—
210C2×Dic9C22×D311yescharacteristic
211Dic18—62no—
212Dic18—122no—
213#529—271noSylow
214C2×C3×C9—42no—
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218D3×C9—22no—
219C2×C3×D4—21no—
220C2×C3×D4—32no—
221C2×C3×D4C2×D911yescharacteristic
222C2×C3×D4D3211yescharacteristic
223C22×C3×C4—31no—
224C22×C3×C4—92no—
225C2×#67C2×D912yes—
226C22×Dic3—32no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3C2×D911yescharacteristic
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—181no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—91no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—92no—
239C3×C42—91no—
240#217—32no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245#217—92no—
246#217—182no—
247#217—182no—
248#217C2×D912yes—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—31no—
251C4⋊Dic3—92no—
252C4⋊Dic3—92no—
253C4⋊Dic3—92no—
254C4⋊Dic3C2×D911yescharacteristic
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260C4×Dic3—31no—
261C4×Dic3—91no—
262C4×Dic3—92no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—92no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3—181no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—21no—
275C2×C3×D3—21no—
276C2×C3×D3#23712yes—
277C32×C4#23712yes—
278C3⋊Dic3—61no—
279C3⋊Dic3—62no—
280C3⋊Dic3—61no—
281C3×Dic3—21no—
282C3×Dic3—22no—
283C3×Dic3—32no—
284C3×Dic3—61no—
285C3×Dic3—62no—
286C22×C9—22no—
287C22×C9—22no—
288C22×C9—22no—
289C22×C9—21no—
290C22×C9—32no—
291C22×C9—42no—
292C22×C9—42no—
293C22×C9—42no—
294C22×C9—61no—
295C22×C9—61no—
296C22×C9C23×D311yescharacteristic
297C4×C9—22no—
298C4×C9—61no—
299C4×C9—62no—
300C4×C9#23512yes—
301Dic9—22no—
302Dic9—21no—
303Dic9—62no—
304Dic9—61no—
305Dic9—61no—
306Dic9—122no—
307Dic9—121no—
308Dic9—121no—
309Dic9#23712yes—
310#126—272no—
311#125—271no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—31no—
316C4×Q8—271no—
317C4×D4—91no—
318C4×D4—272no—
319#117—272no—
320C2×C42—271no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—22no—
323C23×C3—31no—
324C23×C3C22×D912yes—
325C23×C3C2×D3212yes—
326C22×D3C22×D911yescharacteristic
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—42no—
330C3×D4—64no—
331C2×C3×C4—21no—
332C2×C3×C4—31no—
333C2×C3×C4—32no—
334C2×C3×C4—62no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—184no—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341#67—24no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—64no—
346C2×Dic3—91no—
347C2×Dic3—91no—
348C2×Dic3—92no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—92no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—182no—
355C2×Dic3—182no—
356C2×Dic3—182no—
357C2×Dic3C22×D911yescharacteristic
358C2×Dic3C22×D912yes—
359C4×D3—22no—
360C4×D3—182no—
361C4×D3—182no—
362Dic6—182no—
363Dic6—182no—
364Dic6—362no—
365C2×C32—42no—
366C2×C32C2×#23711yescharacteristic
367C2×C32C2×#23711yescharacteristic
368C2×C32D3×#4111yescharacteristic
369C3×D3—22no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—42no—
374C2×C9—62no—
375C2×C9—82no—
376C2×C9C2×#23511yescharacteristic
377C2×C9C2×#23711yescharacteristic
378C2×C9D3×#4111yescharacteristic
379C2×Q8—271no—
380C2×D4—32no—
381C2×D4D3×D911yescharacteristic
382C22×C4—31no—
383C22×C4—92no—
384C22×C4—271no—
385C22×C4—271no—
386C4⋊C4—31no—
387C4⋊C4—91no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—32no—
393C22⋊C4—92no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—91no—
398C42—271no—
399C42—271no—
400C42—272no—
401C42—271no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—32no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3C23×D911yescharacteristic
416C22×C3C22×D3211yescharacteristic
417C2×D3—21no—
418C2×D3—21no—
419C2×D3#369912yes—
420C3×C4—22no—
421C3×C4—62no—
422C3×C4—61no—
423C3×C4—92no—
424C3×C4—181no—
425C3×C4—182no—
426C3×C4#369912yes—
427C3×C4#380412yes—
428Dic3—21no—
429Dic3—22no—
430Dic3—32no—
431Dic3—62no—
432Dic3—61no—
433Dic3—181no—
434Dic3—181no—
435Dic3—182no—
436Dic3—182no—
437Dic3—181no—
438Dic3—181no—
439Dic3—362no—
440Dic3—361no—
441Dic3—361no—
442C32#560011yescharacteristic
443C9—21no—
444C9#555711yescharacteristic, complemented
445C23—31no—
446C23C2×D3×D912yes—
447Q8—542no—
448D4—22no—
449D4—64no—
450C2×C4—31no—
451C2×C4—32no—
452C2×C4—62no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—92no—
456C2×C4—184no—
457C2×C4—271no—
458C2×C4—271no—
459C2×C4—272no—
460C2×C4—542no—
461C2×C4—542no—
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—42no—
467C2×C3—42no—
468C2×C3—62no—
469C2×C3—82no—
470C2×C3C2×#369911yescharacteristic
471C2×C3C2×#369911yescharacteristic
472C2×C3#41×D911yescharacteristic
473C2×C3D3×#23711yescharacteristic
474C2×C3D3×#23711yescharacteristic
475C2×C3C2×#380411yescharacteristic
476D3—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—32no—
481C22—61no—
482C22—61no—
483C22C22×D3×D911yescharacteristic
484C4—62no—
485C4—61no—
486C4—92no—
487C4—181no—
488C4—182no—
489C4—541no—
490C4—541no—
491C4—542no—
492C4#8866212yes—
493C3—21no—
494C3#1058970211yescharacteristic, complemented
495C3#1059517211yescharacteristic
496C2—42no—
497C2—62no—
498C2D3×#369911yescharacteristic
499C2D9×#23711yescharacteristic
500C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC3×#938D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721836C2semidirect
3. C3×D4×Dic9C2semidirect
4. #67×Dic9C2semidirect
5. #11721280C2semidirect
6. #11721246C2semidirect
7. #11721245C2semidirect
8. C4×D3×Dic9C2semidirect
9. #11720961C2semidirect
10. #11720937C2semidirect
11. #11720913C2semidirect
53. D4×Dic9D3semidirect
321. C3×C9#529semidirect
444. C9#5557semidirect
494. C3#10589702semidirect

Maximal subgroups

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

subgroupindex
#117218362
C9×#9382
C3×D4×Dic92
#67×Dic92
#117212802
#117212462
#117212452
C4×D3×Dic92
#117209612
#117209372
#117209132
#105951723
#105897023