Finite Group Databasebeta

← #13151010

Subgroups — #13151010

Order 1728.

796 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
1D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C3×#63500C211yesmaximal, characteristic
3C9×#936C211yesmaximal, characteristic, complemented
4#611⋊Dic3C212yesmaximal, complemented
5#11721269C212yesmaximal, complemented
6#11721248C212yesmaximal, complemented
7#11721246C212yesmaximal, complemented
8C4×Dic3×D9C211yesmaximal, characteristic, complemented
9#11720944C211yesmaximal, characteristic, complemented
10#11720927C211yesmaximal, characteristic, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10595173—31nomaximal
13#10589782—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C3×#611C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×Dic3×D9C2211yescharacteristic
20C2×Dic3×D9C2211yescharacteristic
21#88551C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4×C3⋊Dic9C2211yescharacteristic
24C9×#217C2212yes—
25C9×#217C2212yes—
26C3×#3676C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29(C2×D9)⋊Dic3C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37D4×C3⋊Dic3—31no—
38C3×#938—31no—
39C3×#936D311yescharacteristic
40#67⋊Dic3—32no—
41#63969—32no—
42#63957—32no—
43#63956—32no—
44C4×D3×Dic3—31no—
45#63869—31no—
46#63856—31no—
47#63843—31no—
48C9×#124—31no—
49#63508—31no—
50Q8×Dic9—31no—
51#63500D311yescharacteristic
52D4×Dic9—31no—
53D4×Dic9—61no—
54#63445—32no—
55#611⋊C4—32no—
56#63441—32no—
57#63439—32no—
58C42×D9—31no—
59#63432—31no—
60C23×C3×C9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C3×D4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9—24no—
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68C3×#611—24no—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C3×C4×D9—22no—
75C9×Dic6—22no—
76Dic3×D9—22no—
77Dic3×D9—22no—
78#5680—91no—
79#5600—91no—
80C2×C32×D4C2×D311yescharacteristic
81C22×C3⋊Dic3—32no—
82C2×C3×#67—32no—
83C2×C3×C4×D3—31no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×D3×Dic3—31no—
86C2×D3×Dic3—31no—
87#3757—32no—
88C4⋊(C3⋊Dic3)—31no—
89C4×C3⋊Dic3—31no—
90C3×#217—32no—
91C3×#217C2×D312yes—
92C3×#217C2×D312yes—
93C3×C4⋊Dic3—31no—
94C3×C4×Dic3C2×D311yescharacteristic
95#3724—32no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101Dic32—31no—
102Dic32—31no—
103C2×Q8×C9—31no—
104C2×D4×C9—21no—
105C2×D4×C9C2×D311yescharacteristic
106C2×#611C2×D312yes—
107C22×Dic9—32no—
108C22×Dic9—62no—
109C2×C4×D9—31no—
110C2×C4×D9—31no—
111C2×C4×D9C2×D311yescharacteristic
112C9×C22⋊C4—32no—
113C9×C22⋊C4—32no—
114C42×C9—31no—
115#3676—32no—
116#3676—62no—
117#3676C2×D312yes—
118(C2×D9)⋊C4—32no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126C4×Dic9—31no—
127C4×Dic9—32no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C22×C3×C9—22no—
132C22×C3×C9—22no—
133C22×C3×C9—22no—
134C22×C3×C9C2411yescharacteristic
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9#4112yes—
138C3×C4×C9#4112yes—
139C3⋊Dic9—21no—
140C3⋊Dic9—22no—
141C3⋊Dic9#4112yes—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3—22no—
145C3×Dic9—21no—
146C3×Dic9—22no—
147C3×#124—31no—
148C3×#121—91no—
149#946—91no—
150Q8×Dic3—91no—
151#938—31no—
152#936D911yescharacteristic, complemented
153#935—92no—
154D4×Dic3—91no—
155D4×Dic3—91no—
156D4×Dic3—92no—
157D4×Dic3—181no—
158#924—91no—
159C2×C4×Dic3—91no—
160#890—91no—
161#887—91no—
162#883—92no—
163#67⋊C4—92no—
164#879—92no—
165#879—92no—
166#877—92no—
167#877—92no—
168C42×D3—91no—
169#870—91no—
170C23×C32C22×D312yes—
171C22×C3×D3—31no—
172C32×D4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—64no—
178C3×#67—64no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C3×C4×D3—62no—
185C3×Dic6—22no—
186D3×Dic3—62no—
187D3×Dic3—62no—
188C23×C9—22no—
189C23×C9C22×D312yes—
190C22×D9C22×D311yescharacteristic
191Q8×C9—62no—
192D4×C9—24no—
193D4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—32no—
198C2×C4×C9C22×D311yescharacteristic
199#611—24no—
200C2×Dic9—31no—
201C2×Dic9—31no—
202C2×Dic9—32no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9—62no—
206C2×Dic9—64no—
207C2×Dic9—124no—
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D312yes—
210C4×D9—22no—
211C4×D9—62no—
212C4×D9—62no—
213#529—271noSylow
214C2×C3×C9—42no—
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C3×D9—22no—
219C2×C3×Q8—31no—
220C2×C3×D4—21no—
221C2×C3×D4—92no—
222C2×C3×D4C2×D911yescharacteristic
223C2×C3×D4D3211yescharacteristic
224C22×C3×C4—91no—
225C2×#67—32no—
226C22×Dic3—92no—
227C22×Dic3—92no—
228C22×Dic3—91no—
229C22×Dic3—91no—
230C22×Dic3—182no—
231C2×C4×D3—31no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×Dic6C2×D911yescharacteristic
235C3×C4⋊C4—91no—
236C3×C22⋊C4—32no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—92no—
239C3×C42—31no—
240#217—32no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—182no—
245#217C2×D912yes—
246#217C2×D912yes—
247(C2×D3)⋊C4—92no—
248C4⋊Dic3—31no—
249C4⋊Dic3—92no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—92no—
253C4⋊Dic3—181no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—92no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—92no—
267C4×Dic3—91no—
268C4×Dic3—181no—
269C4×Dic3C2×D911yescharacteristic
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C32×C4#23712yes—
278C3⋊Dic3—32no—
279C3⋊Dic3—61no—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—61no—
285C3×Dic3—62no—
286C22×C9—24no—
287C22×C9—22no—
288C22×C9—22no—
289C22×C9—22no—
290C22×C9—21no—
291C22×C9—42no—
292C22×C9—42no—
293C22×C9C23×D311yescharacteristic
294C2×D9—21no—
295C2×D9—21no—
296C2×D9#23712yes—
297C4×C9—22no—
298C4×C9—61no—
299C4×C9—62no—
300C4×C9—61no—
301C4×C9#23712yes—
302Dic9—21no—
303Dic9—22no—
304Dic9—32no—
305Dic9—62no—
306Dic9—61no—
307Dic9—62no—
308Dic9—121no—
309Dic9—122no—
310#126—272no—
311#125—271no—
312#124—31no—
313#123—272no—
314C22⋊Q8—271no—
315#121—91no—
316C4×Q8—271no—
317C4×D4—272no—
318C4×D4—271no—
319#117—272no—
320C2×C42—271no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—22no—
323C23×C3—91no—
324C23×C3C22×D912yes—
325C23×C3C2×D3212yes—
326C22×D3—31no—
327C3×Q8—62no—
328C3×D4—24no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—184no—
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—32no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—182no—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341#67—64no—
342C2×Dic3—31no—
343C2×Dic3—32no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—92no—
348C2×Dic3—91no—
349C2×Dic3—92no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—182no—
353C2×Dic3—182no—
354C2×Dic3—182no—
355C2×Dic3—184no—
356C2×Dic3—184no—
357C2×Dic3—364no—
358C2×Dic3C22×D911yescharacteristic
359C2×Dic3C22×D911yescharacteristic
360C2×Dic3C22×D912yes—
361C4×D3—62no—
362C4×D3—182no—
363C4×D3—182no—
364Dic6—22no—
365C2×C32—42no—
366C2×C32C2×#23711yescharacteristic
367C2×C32D3×#4111yescharacteristic
368C2×C32D3×#4111yescharacteristic
369C3×D3—62no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—44no—
374C2×C9—42no—
375C2×C9C2×#23711yescharacteristic
376C2×C9C2×#23711yescharacteristic
377C2×C9D3×#4111yescharacteristic
378D9—22no—
379C2×Q8—31no—
380C2×D4—92no—
381C2×D4D3×D911yescharacteristic
382C22×C4—91no—
383C22×C4—271no—
384C22×C4—271no—
385C22×C4—272no—
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—32no—
393C22⋊C4—32no—
394C22⋊C4—92no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—31no—
398C42—271no—
399C42—271no—
400C42—272no—
401C42—271no—
402C22×C3—24no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—92no—
413C22×C3—181no—
414C22×C3—181no—
415C22×C3C23×D911yescharacteristic
416C22×C3C22×D3211yescharacteristic
417C2×D3—32no—
418C2×D3—61no—
419C2×D3—61no—
420C3×C4—22no—
421C3×C4—61no—
422C3×C4—61no—
423C3×C4—62no—
424C3×C4—181no—
425C3×C4—182no—
426C3×C4#369912yes—
427C3×C4#380512yes—
428Dic3—22no—
429Dic3—21no—
430Dic3—21no—
431Dic3—62no—
432Dic3—61no—
433Dic3—92no—
434Dic3—92no—
435Dic3—182no—
436Dic3—181no—
437Dic3—182no—
438Dic3—181no—
439Dic3—182no—
440Dic3—362no—
441Dic3—361no—
442C32#568011yescharacteristic
443C9—21no—
444C9#560011yescharacteristic, complemented
445C23—91no—
446C23C2×D3×D912yes—
447Q8—62no—
448D4—22no—
449D4—184no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—32no—
453C2×C4—91no—
454C2×C4—92no—
455C2×C4—182no—
456C2×C4—271no—
457C2×C4—271no—
458C2×C4—272no—
459C2×C4—542no—
460C2×C4—542no—
461C2×C4—544no—
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—44no—
467C2×C3—42no—
468C2×C3—42no—
469C2×C3—182no—
470C2×C3C2×#369911yescharacteristic
471C2×C3#41×D911yescharacteristic
472C2×C3#41×D911yescharacteristic
473C2×C3D3×#23711yescharacteristic
474C2×C3#6431211yescharacteristic
475C2×C3C2×#380511yescharacteristic
476D3—62no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—92no—
481C22—181no—
482C22—181no—
483C22C22×D3×D911yescharacteristic
484C4—61no—
485C4—61no—
486C4—62no—
487C4—182no—
488C4—181no—
489C4—272no—
490C4—541no—
491C4—542no—
492C4#8866312yes—
493C3—21no—
494C3#1058978211yescharacteristic, complemented
495C3#1059517311yescharacteristic
496C2—42no—
497C2—182no—
498C2D9×#23711yescharacteristic
499C2#1172249811yescharacteristic
500C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC3×#63500C2
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. D4×C3⋊Dic9C2semidirect
3. C9×#936C2semidirect
4. #611⋊Dic3C2semidirect
5. #11721269C2semidirect
6. #11721248C2semidirect
7. #11721246C2semidirect
8. C4×Dic3×D9C2semidirect
9. #11720944C2semidirect
10. #11720927C2semidirect
11. #11720911C2semidirect
152. #936D9semidirect
321. C3×C9#529semidirect
444. C9#5600semidirect
494. C3#10589782semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C3×#635002
C9×#9362
#611⋊Dic32
#117212692
#117212482
#117212462
C4×Dic3×D92
#117209442
#117209272
#117209112
#105951733
#105897823