Finite Group Databasebeta

← #13147559

Subgroups — #13147559

Order 1728.

694 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721709C211yesmaximal, characteristic
2C9×#869C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721000C211yesmaximal, characteristic
5#11720994C212yesmaximal
6#11720986C212yesmaximal
7#11720971C211yesmaximal, characteristic, complemented
8#11720937C211yesmaximal, characteristic, complemented
9#11720922C211yesmaximal, characteristic
10#11720918C212yesmaximal, complemented
11#11720903C212yesmaximal, complemented
12#10593855—31nomaximal
13#10589831—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×Dic6C2212yes—
16C2×(C3⋊D9)⋊C4C2212yes—
17C2×#6189C2212yes—
18C3×C42×C9C2211yescharacteristic
19#88536C2211yescharacteristic
20#88536C2211yescharacteristic
21#88534C2211yescharacteristic
22#88534C2211yescharacteristic
23C4×C3⋊Dic9C2211yescharacteristic
24C9×C4⋊Dic3C2211yescharacteristic
25C9×C4⋊Dic3C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31#88445C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64081—31no—
38C3×#870—31no—
39C3×#869D311yescharacteristic
40#63919—31no—
41#63900—31no—
42#63895—32no—
43#63888—32no—
44#63872—31no—
45#63849—31no—
46#63847—32no—
47#63839—32no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50#63506—32no—
51#63458—32no—
52#63454—31no—
53C4⋊C4×D9—31no—
54#63452—31no—
55#63451—32no—
56#63450—31no—
57#63434—31no—
58#63434—61no—
59#63432D311yescharacteristic, complemented
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C4×C3⋊D9Q814yescomplemented
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C9×Dic6—42no—
73(C3⋊D9)⋊C4—42no—
74#6189—42no—
75#5729—91no—
76#5689—91no—
77C2×C4×C3⋊D3—31no—
78C2×C3×Dic6C2×D312yes—
79C2×#627—32no—
80C2×#624—32no—
81C32×C42C2×D311yescharacteristic
82#3752—31no—
83#3752—31no—
84#3750—31no—
85#3750—31no—
86C4×C3⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3C2×D312yes—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94#3724—32no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99Dic32—32no—
100C2×Q8×C9—32no—
101C2×C4×D9—31no—
102C2×C4×D9—32no—
103C2×C4×D9—61no—
104C2×Dic18—32no—
105C9×C4⋊C4—31no—
106C9×C4⋊C4—32no—
107C9×C4⋊C4—31no—
108C42×C9—21no—
109C42×C9C2×D311yescharacteristic
110(C2×D9)⋊C4—31no—
111(C2×D9)⋊C4—31no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—61no—
114(C2×D9)⋊C4—61no—
115C4⋊Dic9—32no—
116C4⋊Dic9C2×D311yescharacteristic
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—31no—
119Dic9⋊C4—31no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—61no—
124Dic9⋊C4—61no—
125Dic9⋊C4C2×D312yes—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—31no—
128C4×Dic9—32no—
129C4×Dic9—61no—
130C22×C3×C9C2411yescharacteristic
131C2×C3⋊D9—21no—
132C2×C3⋊D9—21no—
133C2×C3⋊D9C2×Q812yes—
134C3×C4×C9—21no—
135C3×C4×C9—21no—
136C3×C4×C9C2×Q812yes—
137C3⋊Dic9—21no—
138C3⋊Dic9—21no—
139C3⋊Dic9C2×Q812yes—
140C9×Dic3—22no—
141C9×Dic3—22no—
142C3×Dic9—22no—
143C3×Dic9—22no—
144C3×C4⋊Q8—31no—
145C3×#125—91no—
146#946—92no—
147#944—92no—
148#897—92no—
149#896—92no—
150#896—92no—
151#892—91no—
152#892—91no—
153D3×C4⋊C4—91no—
154D3×C4⋊C4—91no—
155#890—91no—
156#890—91no—
157#889—92no—
158#889—92no—
159#888—91no—
160#888—92no—
161#888—91no—
162#872—91no—
163#872—91no—
164#872—181no—
165#870—31no—
166#869D911yescharacteristic, complemented
167C22×C3⋊D3—31no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C4×C3⋊D3—34no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3C22×D312yes—
178C2×C3×Dic3C22×D312yes—
179C3×Dic6—42no—
180#627—122no—
181#624—122no—
182C22×D9—31no—
183C22×D9—61no—
184Q8×C9—122no—
185C2×C4×C9—21no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—32no—
189C2×C4×C9—32no—
190C2×C4×C9C22×D311yescharacteristic
191C2×C4×C9C22×D311yescharacteristic
192C2×C4×C9C22×D311yescharacteristic
193C2×Dic9—31no—
194C2×Dic9—31no—
195C2×Dic9—31no—
196C2×Dic9—61no—
197C2×Dic9—61no—
198C2×Dic9—61no—
199C2×Dic9C22×D312yes—
200C2×Dic9C22×D312yes—
201C4×D9—34no—
202C4×D9—64no—
203C4×D9—122no—
204Dic18—122no—
205#544—271noSylow
206C2×C3×C9C22×Q811yescharacteristic
207C2×C3×C9H(5,𝔽2)11yescharacteristic
208C2×C3×C92−1+411yescharacteristic
209C3⋊D9—22no—
210C2×C3×Q8—32no—
211C2×C4×D3—91no—
212C2×C4×D3—91no—
213C2×C4×D3—92no—
214C2×C4×D3—92no—
215C2×C4×D3—181no—
216C2×Dic6—92no—
217C2×Dic6—92no—
218C2×Dic6C2×D912yes—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—92no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—91no—
226C3×C42—21no—
227C3×C42C2×D911yescharacteristic
228C3×C42D3211yescharacteristic
229(C2×D3)⋊C4—92no—
230(C2×D3)⋊C4—91no—
231(C2×D3)⋊C4—91no—
232(C2×D3)⋊C4—91no—
233(C2×D3)⋊C4—91no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240C4⋊Dic3—92no—
241C4⋊Dic3C2×D911yescharacteristic
242C4⋊Dic3C2×D912yes—
243C4⋊Dic3C2×D911yescharacteristic
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—91no—
252Dic3⋊C4—91no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—181no—
257Dic3⋊C4—181no—
258C4×Dic3—92no—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—92no—
262C4×Dic3—181no—
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—32no—
265C2×C3⋊D3—61no—
266C2×C3⋊D3—61no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4D3×Q812yes—
270C3⋊Dic3—32no—
271C3⋊Dic3—61no—
272C3⋊Dic3—61no—
273C3×Dic3—22no—
274C3×Dic3—22no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C22×C9—21no—
278C22×C9C23×D311yescharacteristic
279C2×D9—32no—
280C2×D9—62no—
281C2×D9—61no—
282C2×D9—61no—
283C2×D9—121no—
284C2×D9—121no—
285C4×C9—21no—
286C4×C9—21no—
287C4×C9—22no—
288C4×C9—41no—
289C4×C9—41no—
290C4×C9—62no—
291C4×C9—62no—
292C4×C9D3×Q812yes—
293Dic9—22no—
294Dic9—22no—
295Dic9—32no—
296Dic9—62no—
297Dic9—61no—
298Dic9—61no—
299Dic9—121no—
300Dic9—121no—
301C4⋊Q8—31no—
302C4⋊Q8—271no—
303C4⋊Q8—272no—
304#125—91no—
305#125—272no—
306#125—271no—
307C22⋊Q8—272no—
308C22⋊Q8—272no—
309#117—271no—
310#117—271no—
311C2×C4⋊C4—271no—
312C3×C9#54411yescharacteristic, complemented, Sylow
313C22×D3—91no—
314C22×D3—91no—
315C22×D3—181no—
316C3×Q8—122no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—92no—
323C2×C3×C4—92no—
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C22×D911yescharacteristic
326C2×C3×C4C22×D911yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—91no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3C22×D912yes—
342C2×Dic3C22×D912yes—
343C4×D3—94no—
344C4×D3—94no—
345C4×D3—184no—
346C4×D3—362no—
347C4×D3—362no—
348Dic6—42no—
349Dic6—362no—
350Dic6—362no—
351C2×C32#100411yescharacteristic
352C2×C32C2×D3×Q811yescharacteristic
353C2×C32#100711yescharacteristic
354C3⋊D3—62no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9C2×D3×Q811yescharacteristic
359C2×C9#100911yescharacteristic
360C2×C9#101011yescharacteristic
361D9—62no—
362D9—122no—
363C2×Q8—32no—
364C2×Q8—272no—
365C22×C4—272no—
366C22×C4—271no—
367C4⋊C4—31no—
368C4⋊C4—31no—
369C4⋊C4—32no—
370C4⋊C4—91no—
371C4⋊C4—91no—
372C4⋊C4—92no—
373C4⋊C4—92no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4—271no—
380C22⋊C4—272no—
381C22⋊C4—271no—
382C22⋊C4—271no—
383C42—272no—
384C42—271no—
385C42D3×D911yescharacteristic
386C22×C3—21no—
387C22×C3C23×D911yescharacteristic
388C22×C3C22×D3211yescharacteristic
389C2×D3—92no—
390C2×D3—92no—
391C2×D3—182no—
392C2×D3—181no—
393C2×D3—181no—
394C2×D3—181no—
395C2×D3—181no—
396C2×D3—361no—
397C2×D3—361no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—22no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—62no—
406C3×C4—62no—
407C3×C4—182no—
408C3×C4—182no—
409C3×C4Q8×D912yes—
410C3×C4#379712yes—
411Dic3—22no—
412Dic3—22no—
413Dic3—62no—
414Dic3—62no—
415Dic3—92no—
416Dic3—92no—
417Dic3—182no—
418Dic3—181no—
419Dic3—181no—
420Dic3—181no—
421Dic3—181no—
422Dic3—361no—
423Dic3—361no—
424C32#572911yescharacteristic
425C9—21no—
426C9#568911yescharacteristic, complemented
427C23—271no—
428Q8—122no—
429Q8—1082no—
430C2×C4—32no—
431C2×C4—32no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—271no—
435C2×C4—274no—
436C2×C4—271no—
437C2×C4—271no—
438C2×C4—1082no—
439C2×C4C2×D3×D911yescharacteristic
440C2×C4C2×D3×D911yescharacteristic
441C2×C4C2×D3×D911yescharacteristic
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3#6371011yescharacteristic
446C2×C3C2×Q8×D911yescharacteristic
447C2×C3#6371311yescharacteristic
448C2×C3#6429711yescharacteristic
449C2×C3C2×#379711yescharacteristic
450C2×C3#6430711yescharacteristic
451D3—182no—
452D3—182no—
453D3—362no—
454C22—272no—
455C22—541no—
456C22—541no—
457C22C22×D3×D911yescharacteristic
458C4—21no—
459C4—21no—
460C4—62no—
461C4—62no—
462C4—182no—
463C4—182no—
464C4—272no—
465C4—541no—
466C4—541no—
467C4(C3⋊D9)⋊Q812yes—
468C3—21no—
469C3#1058983111yescharacteristic, complemented
470C3#1059385511yescharacteristic
471C2—542no—
472C2#1172245811yescharacteristic
473C2C2×(C3⋊D9)⋊Q811yescharacteristic
474C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC42D3×D9
hypocenterC3×C9#544
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C9×#869C2semidirect
3. C3×#63432C2semidirect
7. #11720971C2semidirect
8. #11720937C2semidirect
10. #11720918C2semidirect
11. #11720903C2semidirect
59. #63432D3semidirect
67. C4×C3⋊D9Q8semidirect
166. #869D9semidirect
312. C3×C9#544semidirect
426. C9#5689semidirect
469. C3#10589831semidirect

Maximal subgroups

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

subgroupindex
#117217092
C9×#8692
C3×#634322
#117210002
#117209942
#117209862
#117209712
#117209372
#117209222
#117209182
#117209032
#105938553
#105898313