Finite Group Databasebeta

← #13148886

Subgroups — #13148886

Order 1728.

758 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721746C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic
3C9×#887C211yesmaximal, characteristic, complemented
4Dic3×D36C212yesmaximal, complemented
5Dic3×D36C211yesmaximal, characteristic, complemented
6#11720981C212yesmaximal, complemented
7#11720974C212yesmaximal, complemented
8C4×Dic3×D9C211yesmaximal, characteristic, complemented
9#11720951C211yesmaximal, characteristic, complemented
10#11720949C212yesmaximal, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10594426—31nomaximal
13#10589682—31nomaximal
14C2×C3×D36C2212yes—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×Dic3×D9C2211yescharacteristic
19C2×Dic3×D9C2211yescharacteristic
20C2×Dic3×D9C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2212yes—
24#88534C2212yes—
25C4×C3⋊Dic9C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C4×C9×Dic3C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34#88442C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2211yescharacteristic
37#64099—31no—
38C3×#895—31no—
39C3×#887D311yescharacteristic
40Dic3×D12—32no—
41Dic3×D12—31no—
42#63884—32no—
43#63877—32no—
44C4×D3×Dic3—31no—
45#63863—31no—
46#63861—32no—
47#63856—31no—
48C4×Q8×C9—31no—
49#63509—31no—
50#63508—31no—
51#63460—32no—
52#63457D311yescharacteristic
53#63456—32no—
54#63452—31no—
55#63452—61no—
56C4×D36—31no—
57C4×D36—32no—
58#63434—32no—
59C42×D9—31no—
60C22×C3×D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C3×D36—24no—
72C3×D36—22no—
73C3×C4×D9—22no—
74C9×Dic6—22no—
75Dic3×D9—22no—
76Dic3×D9—22no—
77Dic3×D9—24no—
78#5600—91no—
79#5580—91no—
80C2×C3×D12—31no—
81C2×C3×D12—32no—
82C2×C3×C4×D3—31no—
83C2×C3×Dic6C2×D311yescharacteristic
84C2×D3×Dic3—31no—
85C2×D3×Dic3—31no—
86C2×D3×Dic3—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—32no—
90#3750—32no—
91C4×C3⋊Dic3—31no—
92C3×(C2×D3)⋊C4—32no—
93C3×Dic3⋊C4C2×D312yes—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D312yes—
96#3724—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—31no—
102Dic32—31no—
103C2×Q8×C9—31no—
104C2×D36C2×D311yescharacteristic
105C2×D36C2×D312yes—
106C2×C4×D9—31no—
107C2×C4×D9—31no—
108C2×C4×D9—32no—
109C2×C4×D9C2×D311yescharacteristic
110C9×C4⋊C4—21no—
111C9×C4⋊C4—32no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—31no—
114C42×C9—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4C2×D312yes—
120C4⋊Dic9—31no—
121C4⋊Dic9—32no—
122C4⋊Dic9—62no—
123C4⋊Dic9—61no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—62no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C22×C3×C9C2411yescharacteristic
132C2×C3×D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—22no—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9#4112yes—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9—21no—
142C3⋊Dic9—22no—
143C9×Dic3—21no—
144C9×Dic3—22no—
145C9×Dic3#4112yes—
146C3×Dic9—21no—
147C3×#121—91no—
148C3×C4×Q8—31no—
149#947—91no—
150#946—91no—
151#936—91no—
152#935—92no—
153D4×Dic3—92no—
154D4×Dic3—91no—
155#924—91no—
156C2×C4×Dic3—91no—
157#898—92no—
158#895—31no—
159#894—92no—
160#890—91no—
161#890—91no—
162#890—181no—
163#887D911yescharacteristic, complemented
164#879—92no—
165#877—92no—
166C4×D12—92no—
167C4×D12—91no—
168#872—92no—
169C42×D3—91no—
170C22×C3×D3—31no—
171C22×C3×D3—32no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—32no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D312yes—
180C2×C3×Dic3C22×D311yescharacteristic
181C3×D12—64no—
182C3×D12—62no—
183C3×C4×D3—62no—
184C3×Dic6—22no—
185D3×Dic3—62no—
186D3×Dic3—62no—
187D3×Dic3—64no—
188C22×D9C22×D312yes—
189C22×D9C22×D311yescharacteristic
190Q8×C9—62no—
191C2×C4×C9—21no—
192C2×C4×C9—22no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—32no—
196C2×C4×C9C22×D312yes—
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—31no—
199C2×Dic9—31no—
200C2×Dic9—32no—
201C2×Dic9—61no—
202C2×Dic9—61no—
203C2×Dic9—62no—
204C2×Dic9C22×D311yescharacteristic
205D36—24no—
206D36—22no—
207C4×D9—22no—
208C4×D9—62no—
209C4×D9—64no—
210C4×D9—62no—
211#529—271noSylow
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C3×D9—22no—
216C3×D9—42no—
217C2×C3×Q8—31no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—91no—
221C22×Dic3—92no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C2×D12—32no—
225C2×D12—31no—
226C2×C4×D3—31no—
227C2×C4×D3—91no—
228C2×C4×D3—91no—
229C2×C4×D3—92no—
230C2×Dic6C2×D911yescharacteristic
231C3×C4⋊C4—21no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4C2×D911yescharacteristic
234C3×C4⋊C4D3211yescharacteristic
235C3×C22⋊C4—92no—
236C3×C42—31no—
237C3×C42—32no—
238#217—92no—
239#217—92no—
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—32no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3—91no—
249C4⋊Dic3—92no—
250C4⋊Dic3—91no—
251C4⋊Dic3—181no—
252C4⋊Dic3—182no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—182no—
258Dic3⋊C4C2×D912yes—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—181no—
266C4×Dic3C2×D912yes—
267C4×Dic3C2×D911yescharacteristic
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—32no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C32×C4—22no—
276C32×C4#23512yes—
277C3⋊Dic3—61no—
278C3⋊Dic3—62no—
279C3⋊Dic3—61no—
280C3×Dic3—22no—
281C3×Dic3—21no—
282C3×Dic3—61no—
283C3×Dic3#23912yes—
284C22×C9—21no—
285C22×C9C23×D311yescharacteristic
286C2×D9—21no—
287C2×D9—21no—
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9#23712yes—
292C4×C9—22no—
293C4×C9—22no—
294C4×C9—32no—
295C4×C9—42no—
296C4×C9—61no—
297C4×C9—62no—
298C4×C9#23712yes—
299Dic9—21no—
300Dic9—61no—
301Dic9—61no—
302Dic9—62no—
303Dic9—121no—
304Dic9—121no—
305Dic9—122no—
306#126—272no—
307#125—271no—
308#124—271no—
309#123—272no—
310C22⋊Q8—271no—
311#121—91no—
312C4×Q8—31no—
313C4×D4—271no—
314C4×D4—272no—
315#117—272no—
316C2×C42—271no—
317C3×C9#52911yescharacteristic, complemented, Sylow
318C23×C3—91no—
319C23×C3—92no—
320C22×D3—31no—
321C22×D3—32no—
322C3×Q8—62no—
323C3×D4—184no—
324C3×D4—182no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—32no—
330C2×C3×C4—91no—
331C2×C3×C4—182no—
332C2×C3×C4C22×D912yes—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C2×D3212yes—
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—31no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—92no—
340C2×Dic3—91no—
341C2×Dic3—91no—
342C2×Dic3—92no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—182no—
346C2×Dic3—182no—
347C2×Dic3—182no—
348C2×Dic3—184no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D912yes—
351C2×Dic3C22×D911yescharacteristic
352D12—64no—
353D12—62no—
354C4×D3—62no—
355C4×D3—182no—
356C4×D3—184no—
357C4×D3—182no—
358Dic6—22no—
359C2×C32C2×#23511yescharacteristic
360C2×C32C2×#23911yescharacteristic
361C2×C32D3×#4111yescharacteristic
362C3×D3—62no—
363C3×D3—122no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9C2×#23711yescharacteristic
368C2×C9C2×#23711yescharacteristic
369C2×C9D3×#4111yescharacteristic
370D9—22no—
371D9—42no—
372C2×Q8—31no—
373C2×D4—92no—
374C2×D4—91no—
375C22×C4—91no—
376C22×C4—271no—
377C22×C4—271no—
378C22×C4—272no—
379C4⋊C4—32no—
380C4⋊C4—272no—
381C4⋊C4—272no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C4⋊C4D3×D911yescharacteristic
385C22⋊C4—92no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C42—31no—
391C42—32no—
392C42—271no—
393C42—271no—
394C42—271no—
395C22×C3—21no—
396C22×C3—92no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3—182no—
402C22×C3C23×D911yescharacteristic
403C22×C3C22×D3211yescharacteristic
404C2×D3—32no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—62no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—32no—
414C3×C4—42no—
415C3×C4—62no—
416C3×C4—61no—
417C3×C4—181no—
418C3×C4#369712yes—
419C3×C4#379512yes—
420Dic3—22no—
421Dic3—21no—
422Dic3—61no—
423Dic3—181no—
424Dic3—181no—
425Dic3—182no—
426Dic3—181no—
427Dic3—182no—
428Dic3—181no—
429Dic3—361no—
430Dic3—361no—
431Dic3—362no—
432Dic3#370112yes—
433C32#558011yescharacteristic
434C9—21no—
435C9#560011yescharacteristic, complemented
436C23—91no—
437C23—92no—
438Q8—62no—
439D4—184no—
440D4—182no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—32no—
444C2×C4—91no—
445C2×C4—182no—
446C2×C4—271no—
447C2×C4—271no—
448C2×C4—272no—
449C2×C4—542no—
450C2×C4—544no—
451C2×C4—542no—
452C2×C4C2×D3×D912yes—
453C2×C4C2×D3×D911yescharacteristic
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—182no—
458C2×C3—362no—
459C2×C3C2×#369711yescharacteristic
460C2×C3C2×#370111yescharacteristic
461C2×C3#41×D911yescharacteristic
462C2×C3C2×#379511yescharacteristic
463C2×C3D3×#23711yescharacteristic
464C2×C3D3×#23911yescharacteristic
465D3—62no—
466D3—122no—
467C22—92no—
468C22—181no—
469C22—181no—
470C22—182no—
471C22—182no—
472C22—182no—
473C22C22×D3×D911yescharacteristic
474C4—22no—
475C4—32no—
476C4—61no—
477C4—62no—
478C4—181no—
479C4—541no—
480C4—541no—
481C4—542no—
482C4#8864112yes—
483C3—21no—
484C3#1058968211yescharacteristic, complemented
485C3#1059442611yescharacteristic
486C2—182no—
487C2—362no—
488C2C2×#8864111yescharacteristic
489C2D9×#23711yescharacteristic
490C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC3×#63457C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721746C2semidirect
3. C9×#887C2semidirect
4. Dic3×D36C2semidirect
5. Dic3×D36C2semidirect
6. #11720981C2semidirect
7. #11720974C2semidirect
8. C4×Dic3×D9C2semidirect
9. #11720951C2semidirect
10. #11720949C2semidirect
11. #11720944C2semidirect
163. #887D9semidirect
317. C3×C9#529semidirect
435. C9#5600semidirect
484. C3#10589682semidirect

Maximal subgroups

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

subgroupindex
#117217462
C3×#634572
C9×#8872
Dic3×D362
#117209812
#117209742
C4×Dic3×D92
#117209512
#117209492
#117209442
#105944263
#105896823