Finite Group Databasebeta

← #13150925

Subgroups — #13150925

Order 1728.

712 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721836C211yesmaximal, characteristic
2D4×C9×Dic3C211yesmaximal, characteristic
3C3×#63498C211yesmaximal, characteristic
4#11721249C212yesmaximal, complemented
5#11721247C212yesmaximal, complemented
6#11721246C212yesmaximal, complemented
7#11721244C212yesmaximal, complemented
8C4×#6189C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#11720913C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10595273—31nomaximal
13#10589644—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C22×C9×Dic3C2212yes—
17C2×C3×Dic18C2211yescharacteristic
18C2×#6189C2211yescharacteristic
19#88551C2212yes—
20#88551C2212yes—
21C4×C3⋊Dic9C2211yescharacteristic
22C9×#217C2212yes—
23C3×#3676C2212yes—
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4×Dic9C2211yescharacteristic
28Dic3⋊Dic9C2211yescharacteristic
29Dic3⋊Dic9C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2211yescharacteristic
32#88442C2212yes—
33Dic9⋊Dic3C2211yescharacteristic
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64145—31no—
38C3×#936—31no—
39C3×D4×Dic3D311yescharacteristic
40#63964—32no—
41#63956—32no—
42#63956—32no—
43#63955—32no—
44C4×#627—31no—
45#63866—31no—
46#63844—31no—
47#63843—31no—
48C4×D4×C9—31no—
49#63498—31no—
50#63498—61no—
51#63498D311yescharacteristic
52#63488—32no—
53#63486—32no—
54#63449—31no—
55#63449—31no—
56#63441—32no—
57#63441—32no—
58#63431—31no—
59C4×Dic18—31no—
60C23×C3×C9C2312yes—
61C3×D4×C9—22no—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3—24no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C3×Dic18—22no—
75#6189—22no—
76#5671—91no—
77#5542—91no—
78C2×C32×D4C2×D311yescharacteristic
79C2×#634—31no—
80C22×C3×Dic3C2×D312yes—
81C2×C3×Dic6—31no—
82C2×#627—31no—
83#3757—32no—
84#3757—32no—
85C4×C3⋊Dic3—31no—
86C3×#217—32no—
87C3×#217—32no—
88C3×#217C2×D312yes—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4×Dic3—31no—
91C3×C4×Dic3C2×D311yescharacteristic
92#3724—32no—
93#3724—31no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—31no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—31no—
99Dic32—32no—
100Dic32—32no—
101C2×D4×C9—21no—
102C2×D4×C9C2×D311yescharacteristic
103C22×C4×C9—32no—
104C2×Dic18—31no—
105C2×Dic18—31no—
106C2×Dic18—61no—
107C2×Dic18C2×D311yescharacteristic
108C9×C4⋊C4—31no—
109C9×C22⋊C4—32no—
110C42×C9—31no—
111#3676—32no—
112#3676—32no—
113#3676—62no—
114#3676—62no—
115#3676C2×D312yes—
116#3676C2×D312yes—
117C4⋊Dic9—31no—
118C4⋊Dic9—32no—
119C4⋊Dic9—32no—
120Dic9⋊C4—31no—
121Dic9⋊C4—32no—
122Dic9⋊C4—31no—
123Dic9⋊C4—32no—
124C4×Dic9—31no—
125C4×Dic9—32no—
126C4×Dic9—32no—
127C4×Dic9—61no—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9—22no—
130C22×C3×C9—22no—
131C22×C3×C9—22no—
132C22×C3×C9C2411yescharacteristic
133C3×C4×C9#4112yes—
134C3⋊Dic9—21no—
135C3⋊Dic9—22no—
136C3⋊Dic9—21no—
137C9×Dic3—22no—
138C9×Dic3—21no—
139C9×Dic3#4112yes—
140C3×Dic9—21no—
141C3×Dic9—22no—
142C3×Dic9—21no—
143C3×#124—91no—
144C3×C4×D4—31no—
145Q8×Dic3—91no—
146#944—91no—
147#936—31no—
148#936—91no—
149#936—91no—
150#936—181no—
151D4×Dic3D911yescharacteristic
152#926—92no—
153#924—92no—
154#887—91no—
155#887—91no—
156#879—92no—
157#879—92no—
158#879—92no—
159#879—92no—
160#878—92no—
161#877—92no—
162#869—91no—
163C4×Dic6—91no—
164C4×Dic6—91no—
165C4×Dic6—91no—
166C23×C32C22×D312yes—
167C32×D4—22no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—31no—
172#634—62no—
173C2×C3×Dic3—24no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D312yes—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C3×Dic6—62no—
181#627—62no—
182C23×C9—22no—
183C23×C9C22×D312yes—
184D4×C9—22no—
185D4×C9—42no—
186C2×C4×C9—21no—
187C2×C4×C9—31no—
188C2×C4×C9—32no—
189C2×C4×C9—31no—
190C2×C4×C9—64no—
191C2×C4×C9C22×D311yescharacteristic
192C2×Dic9—31no—
193C2×Dic9—31no—
194C2×Dic9—32no—
195C2×Dic9—61no—
196C2×Dic9—61no—
197C2×Dic9—62no—
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D312yes—
200C2×Dic9C22×D311yescharacteristic
201Dic18—22no—
202Dic18—62no—
203Dic18—62no—
204Dic18—122no—
205#556—271noSylow
206C2×C3×C9—42no—
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C2×C3×Q8—91no—
211C2×C3×D4—21no—
212C2×C3×D4C2×D911yescharacteristic
213C2×C3×D4D3211yescharacteristic
214C22×C3×C4—32no—
215C22×Dic3C2×D912yes—
216C2×Dic6—31no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C2×Dic6—91no—
221C2×Dic6—181no—
222C3×C4⋊C4—31no—
223C3×C22⋊C4—32no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—92no—
226C3×C42—31no—
227C3×C42—91no—
228#217—32no—
229#217—32no—
230#217—92no—
231#217—92no—
232#217—92no—
233#217—92no—
234#217—182no—
235#217—182no—
236#217C2×D912yes—
237C4⋊Dic3—91no—
238C4⋊Dic3—92no—
239C4⋊Dic3—92no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3C2×D911yescharacteristic
243Dic3⋊C4—91no—
244Dic3⋊C4—91no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4—91no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252C4×Dic3—31no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—91no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—181no—
260C4×Dic3C2×D911yescharacteristic
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32C23×D311yescharacteristic
265C32×C4#23512yes—
266C3⋊Dic3—62no—
267C3⋊Dic3—61no—
268C3⋊Dic3—61no—
269C3×Dic3—22no—
270C3×Dic3—21no—
271C3×Dic3—62no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3#23712yes—
275C22×C9—22no—
276C22×C9—22no—
277C22×C9—22no—
278C22×C9—21no—
279C22×C9—42no—
280C22×C9—42no—
281C22×C9—42no—
282C22×C9C23×D311yescharacteristic
283C4×C9—22no—
284C4×C9—32no—
285C4×C9—62no—
286C4×C9—61no—
287C4×C9#23712yes—
288Dic9—22no—
289Dic9—21no—
290Dic9—21no—
291Dic9—62no—
292Dic9—61no—
293Dic9—61no—
294Dic9—122no—
295Dic9—121no—
296Dic9—121no—
297C4⋊Q8—271no—
298#126—272no—
299#126—272no—
300#124—91no—
301#124—271no—
302C22⋊Q8—272no—
303C4×Q8—271no—
304C4×Q8—271no—
305C4×Q8—271no—
306C4×D4—31no—
307#117—272no—
308C3×C9#55611yescharacteristic, complemented, Sylow
309C23×C3—22no—
310C23×C3C22×D912yes—
311C23×C3C2×D3212yes—
312C3×Q8—182no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—42no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—32no—
319C2×C3×C4—31no—
320C2×C3×C4—64no—
321C2×C3×C4—91no—
322C2×C3×C4—92no—
323C2×C3×C4—91no—
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C2×D3211yescharacteristic
326C2×Dic3—24no—
327C2×Dic3—31no—
328C2×Dic3—31no—
329C2×Dic3—32no—
330C2×Dic3—91no—
331C2×Dic3—91no—
332C2×Dic3—92no—
333C2×Dic3—91no—
334C2×Dic3—92no—
335C2×Dic3—91no—
336C2×Dic3—181no—
337C2×Dic3—181no—
338C2×Dic3—182no—
339C2×Dic3C22×D912yes—
340C2×Dic3C22×D911yescharacteristic
341C2×Dic3C22×D911yescharacteristic
342Dic6—62no—
343Dic6—182no—
344Dic6—182no—
345Dic6—182no—
346Dic6—182no—
347Dic6—362no—
348C2×C32—42no—
349C2×C32C2×#23511yescharacteristic
350C2×C32C2×#23711yescharacteristic
351C2×C32#101011yescharacteristic
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—42no—
356C2×C9—82no—
357C2×C9C2×#23711yescharacteristic
358C2×C9D3×#4111yescharacteristic
359C2×C9#101011yescharacteristic
360C2×Q8—91no—
361C2×Q8—271no—
362C2×Q8—271no—
363C2×D4D3×D911yescharacteristic
364C22×C4—32no—
365C4⋊C4—31no—
366C4⋊C4—271no—
367C4⋊C4—272no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—271no—
371C4⋊C4—271no—
372C4⋊C4—272no—
373C22⋊C4—32no—
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—31no—
379C42—91no—
380C42—272no—
381C42—272no—
382C42—271no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3C23×D911yescharacteristic
394C22×C3C22×D3211yescharacteristic
395C3×C4—22no—
396C3×C4—32no—
397C3×C4—61no—
398C3×C4—62no—
399C3×C4—182no—
400C3×C4—181no—
401C3×C4—181no—
402C3×C4#369712yes—
403C3×C4#380412yes—
404Dic3—22no—
405Dic3—21no—
406Dic3—61no—
407Dic3—61no—
408Dic3—62no—
409Dic3—181no—
410Dic3—181no—
411Dic3—182no—
412Dic3—182no—
413Dic3—181no—
414Dic3—181no—
415Dic3—362no—
416Dic3—361no—
417Dic3—361no—
418Dic3#369912yes—
419C32#554211yescharacteristic
420C9—21no—
421C9#567111yescharacteristic, complemented
422C23C2×D3×D912yes—
423Q8—182no—
424Q8—542no—
425Q8—542no—
426D4—22no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—32no—
430C2×C4—64no—
431C2×C4—91no—
432C2×C4—91no—
433C2×C4—92no—
434C2×C4—271no—
435C2×C4—271no—
436C2×C4—272no—
437C2×C4C2×D3×D911yescharacteristic
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—42no—
442C2×C3—42no—
443C2×C3—82no—
444C2×C3C2×#369711yescharacteristic
445C2×C3C2×#369911yescharacteristic
446C2×C3#6371611yescharacteristic
447C2×C3#6430911yescharacteristic
448C2×C3D3×#23711yescharacteristic
449C2×C3C2×#380411yescharacteristic
450C22—22no—
451C22—22no—
452C22—22no—
453C22C22×D3×D911yescharacteristic
454C4—32no—
455C4—62no—
456C4—61no—
457C4—181no—
458C4—182no—
459C4—181no—
460C4—541no—
461C4—542no—
462C4—541no—
463C4#8865812yes—
464C3—21no—
465C3#1058964411yescharacteristic, complemented
466C3#1059527311yescharacteristic
467C2—42no—
468C2#1172249411yescharacteristic
469C2D3×#369911yescharacteristic
470C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11721249C2semidirect
5. #11721247C2semidirect
6. #11721246C2semidirect
7. #11721244C2semidirect
8. C4×#6189C2semidirect
9. #11720932C2semidirect
10. #11720913C2semidirect
11. #11720910C2semidirect
308. C3×C9#556semidirect
421. C9#5671semidirect
465. C3#10589644semidirect

Maximal subgroups

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

subgroupindex
#117218362
D4×C9×Dic32
C3×#634982
#117212492
#117212472
#117212462
#117212442
C4×#61892
#117209322
#117209132
#117209102
#105952733
#105896443