Finite Group Databasebeta

← #13147516

Subgroups — #13147516

Order 1728.

740 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic
3C4×C9×Dic6C211yesmaximal, characteristic, complemented
4#11720999C212yesmaximal, complemented
5Dic3×D36C211yesmaximal, characteristic
6Dic3×D36C211yesmaximal, characteristic
7#11720984C212yesmaximal, complemented
8D9×Dic3⋊C4C212yesmaximal, complemented
9#11720970C211yesmaximal, characteristic, complemented
10#11720939C212yesmaximal, complemented
11#11720927C211yesmaximal, characteristic, complemented
12#10593929—31nomaximal
13#10589681—31nomaximal
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9C2212yes—
16C2×C9×Dic6C2211yescharacteristic
17C2×Dic3×D9C2212yes—
18C2×Dic3×D9C2212yes—
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24#88534C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2211yescharacteristic
28C9×Dic3⋊C4C2211yescharacteristic
29C4×C9×Dic3C2211yescharacteristic
30C4×C9×Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37#64079—31no—
38C3×C4×D12—31no—
39C3×C4×Dic6D311yescharacteristic
40#63918—31no—
41#63899—32no—
42Dic3×D12—31no—
43Dic3×D12—31no—
44#63887—32no—
45D3×Dic3⋊C4—32no—
46#63869—31no—
47#63851—32no—
48C4×Q8×C9—31no—
49#63508—32no—
50#63458—32no—
51#63458—32no—
52C4⋊C4×D9—32no—
53C4×D36—31no—
54C4×D36—31no—
55C4×D36D311yescharacteristic
56#63432—31no—
57#63432—31no—
58#63432—61no—
59#63431—31no—
60C22×C3×D9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66C2×C9×Dic3C2311yescharacteristic
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C3×D36Q814yescomplemented
72C3×C4×D9—42no—
73C9×Dic6—22no—
74Dic3×D9—24no—
75Dic3×D9—42no—
76#6189—42no—
77#5579—91no—
78#5539—91no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—32no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#627—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—31no—
88#3750—32no—
89#3750—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4C2×D311yescharacteristic
94C3×Dic3⋊C4C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×Q8×C9—31no—
103C2×D36C2×D311yescharacteristic
104C2×C4×D9—32no—
105C2×C4×D9—32no—
106C2×C4×D9C2×D312yes—
107C2×Dic18—32no—
108C9×C4⋊C4—31no—
109C9×C4⋊C4—31no—
110C9×C4⋊C4—31no—
111C42×C9—21no—
112C42×C9—31no—
113C42×C9—31no—
114C42×C9C2×D311yescharacteristic
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4C2×D312yes—
118C4⋊Dic9—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D311yescharacteristic
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129Dic9⋊C4—62no—
130C22×C3×C9C2411yescharacteristic
131C2×C3×D9—22no—
132C2×C3×D9—22no—
133C2×C3×D9C2×Q814yes—
134C3×C4×C9—21no—
135C3×C4×C9—21no—
136C3×C4×C9C2×Q812yes—
137C3⋊Dic9—22no—
138C3⋊Dic9—22no—
139C9×Dic3—21no—
140C9×Dic3—21no—
141C9×Dic3—21no—
142C9×Dic3#4112yes—
143C3×Dic9—22no—
144C3×C4×Q8—31no—
145C3×C4×D4—91no—
146#946—92no—
147D4×Dic3—91no—
148D4×Dic3—91no—
149#924—92no—
150C2×Dic3⋊C4—92no—
151#896—92no—
152#896—92no—
153D3×C4⋊C4—92no—
154#890—91no—
155#888—91no—
156#878—92no—
157#878—92no—
158C4×D12—31no—
159C4×D12—91no—
160C4×D12—91no—
161#870—91no—
162#870—91no—
163#870—91no—
164#870—181no—
165#869—91no—
166C4×Dic6D911yescharacteristic, complemented
167C22×C3×D3—32no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D311yescharacteristic
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C3×D12—34no—
179C3×C4×D3—122no—
180C3×Dic6—22no—
181#627—122no—
182D3×Dic3—64no—
183D3×Dic3—122no—
184C22×D9C22×D312yes—
185Q8×C9—62no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—31no—
190C2×C4×C9—31no—
191C2×C4×C9—31no—
192C2×C4×C9—31no—
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196C2×Dic9—32no—
197C2×Dic9—32no—
198C2×Dic9—62no—
199C2×Dic9—62no—
200C2×Dic9C22×D312yes—
201D36Dic614yescomplemented
202C4×D9—42no—
203C4×D9—64no—
204C4×D9—122no—
205Dic18—122no—
206#554—271noSylow
207C2×C3×C9C22×Q811yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210C3×D9—24no—
211C2×C3×Q8—31no—
212C2×C3×D4—91no—
213C22×C3×C4—92no—
214C22×Dic3—92no—
215C22×Dic3—92no—
216C2×D12—31no—
217C2×C4×D3—32no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×Dic6—92no—
221C2×Dic6—92no—
222C2×Dic6C2×D911yescharacteristic
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—91no—
227C3×C22⋊C4—92no—
228C3×C42—21no—
229C3×C42—31no—
230C3×C42—31no—
231C3×C42C2×D911yescharacteristic
232C3×C42D3211yescharacteristic
233#217—92no—
234#217—92no—
235(C2×D3)⋊C4—32no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—181no—
246C4⋊Dic3—181no—
247C4⋊Dic3C2×D911yescharacteristic
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—182no—
257Dic3⋊C4—182no—
258Dic3⋊C4C2×D911yescharacteristic
259Dic3⋊C4C2×D911yescharacteristic
260C4×Dic3C2×D911yescharacteristic
261C4×Dic3C2×D911yescharacteristic
262C22×C32C23×D311yescharacteristic
263C2×C3×D3—34no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4D3×Q812yes—
269C3⋊Dic3—62no—
270C3⋊Dic3—62no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—21no—
274C3×Dic3—62no—
275C3×Dic3#23512yes—
276C22×C9—21no—
277C22×C9C23×D311yescharacteristic
278C2×D9—22no—
279C2×D9—22no—
280C2×D9C2×Dic614yes—
281C4×C9—21no—
282C4×C9—21no—
283C4×C9—22no—
284C4×C9—32no—
285C4×C9—41no—
286C4×C9—41no—
287C4×C9—61no—
288C4×C9—61no—
289C4×C9—61no—
290C4×C9C2×Dic612yes—
291Dic9—22no—
292Dic9—62no—
293Dic9—62no—
294Dic9—122no—
295Dic9—122no—
296C4⋊Q8—271no—
297#125—271no—
298#125—271no—
299C22⋊Q8—272no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C4×Q8—31no—
303C4×D4—91no—
304C4×D4—271no—
305C4×D4—271no—
306C2×C4⋊C4—272no—
307C3×C9#55411yescharacteristic, complemented, Sylow
308C23×C3—92no—
309C22×D3—32no—
310C3×Q8—62no—
311C3×D4—94no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—92no—
320C2×C3×C4—362no—
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C22×D911yescharacteristic
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×C3×C4C2×D3211yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×Dic3—32no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—182no—
333C2×Dic3—182no—
334C2×Dic3—184no—
335C2×Dic3—362no—
336C2×Dic3C22×D911yescharacteristic
337C2×Dic3C22×D911yescharacteristic
338C2×Dic3C22×D911yescharacteristic
339C2×Dic3C22×D911yescharacteristic
340D12—34no—
341C4×D3—122no—
342C4×D3—184no—
343C4×D3—362no—
344Dic6—22no—
345Dic6—362no—
346Dic6—362no—
347C2×C32C2×#23511yescharacteristic
348C2×C32C2×D3×Q811yescharacteristic
349C2×C32#100911yescharacteristic
350C3×D3—64no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9C22×Dic611yescharacteristic
355C2×C9#100411yescharacteristic
356C2×C9D3×#4111yescharacteristic
357D9—24no—
358C2×Q8—31no—
359C2×Q8—272no—
360C2×D4—91no—
361C22×C4—92no—
362C22×C4—272no—
363C22×C4—272no—
364C4⋊C4—31no—
365C4⋊C4—31no—
366C4⋊C4—31no—
367C4⋊C4—91no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—271no—
373C4⋊C4—271no—
374C4⋊C4—272no—
375C22⋊C4—92no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—31no—
379C42—31no—
380C42D3×D911yescharacteristic
381C22×C3—21no—
382C22×C3—94no—
383C22×C3—182no—
384C22×C3—182no—
385C22×C3C23×D911yescharacteristic
386C22×C3C22×D3211yescharacteristic
387C2×D3—34no—
388C2×D3—62no—
389C2×D3—62no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—22no—
395C3×C4—32no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—182no—
402C3×C4Q8×D912yes—
403C3×C4D3×Dic612yes—
404Dic3—21no—
405Dic3—21no—
406Dic3—21no—
407Dic3—62no—
408Dic3—182no—
409Dic3—182no—
410Dic3—182no—
411Dic3—182no—
412Dic3—362no—
413Dic3—362no—
414Dic3#369712yes—
415C32#557911yescharacteristic
416C9—21no—
417C9#553911yescharacteristic, complemented
418C23—92no—
419Q8—62no—
420Q8—1082no—
421D4—94no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—92no—
427C2×C4—272no—
428C2×C4—272no—
429C2×C4—362no—
430C2×C4—544no—
431C2×C4—1082no—
432C2×C4C2×D3×D911yescharacteristic
433C2×C4C2×D3×D911yescharacteristic
434C2×C4C2×D3×D911yescharacteristic
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—184no—
439C2×C3C2×#369711yescharacteristic
440C2×C3C2×Q8×D911yescharacteristic
441C2×C3#6371511yescharacteristic
442C2×C3C2×D3×Dic611yescharacteristic
443C2×C3D3×#23511yescharacteristic
444C2×C3#6430711yescharacteristic
445D3—64no—
446C22—94no—
447C22—182no—
448C22—182no—
449C22C22×D3×D911yescharacteristic
450C4—21no—
451C4—21no—
452C4—32no—
453C4—61no—
454C4—61no—
455C4—61no—
456C4—182no—
457C4—542no—
458C4—542no—
459C4D9×Dic612yes—
460C3—21no—
461C3#1058968111yescharacteristic, complemented
462C3#1059392911yescharacteristic
463C2—184no—
464C2C2×D9×Dic611yescharacteristic
465C2D3×#369711yescharacteristic
466C2#1172247711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC42D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
3. C4×C9×Dic6C2semidirect
4. #11720999C2semidirect
7. #11720984C2semidirect
8. D9×Dic3⋊C4C2semidirect
9. #11720970C2semidirect
10. #11720939C2semidirect
11. #11720927C2semidirect
71. C3×D36Q8semidirect
166. C4×Dic6D9semidirect
201. D36Dic6semidirect
307. C3×C9#554semidirect
417. C9#5539semidirect
461. C3#10589681semidirect

Maximal subgroups

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

subgroupindex
#117217072
C3×C4×D362
C4×C9×Dic62
#117209992
Dic3×D362
#117209842
D9×Dic3⋊C42
#117209702
#117209392
#117209272
#105939293
#105896813