Finite Group Databasebeta

← #13148776

Subgroups — #13148776

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#11721746C211yesmaximal, characteristic, complemented
2C9×#893C211yesmaximal, characteristic
3C3×#63452C211yesmaximal, characteristic, complemented
4#11721026C212yesmaximal, complemented
5#11721025C211yesmaximal, characteristic
6#11721023C212yesmaximal, complemented
7D12×Dic9C211yesmaximal, characteristic
8D3×C4⋊Dic9C212yesmaximal, complemented
9#11720970C211yesmaximal, characteristic, complemented
10C4×#6189C211yesmaximal, characteristic, complemented
11#11720955C212yesmaximal, complemented
12#10594452—31nomaximal
13#10589645—31nomaximal
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9C2212yes—
16C2×D3×Dic9C2212yes—
17C2×D3×Dic9C2212yes—
18C2×#6189C2211yescharacteristic
19C2×#6189C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2212yes—
23#88534C2212yes—
24C4×C3⋊Dic9C2211yescharacteristic
25C9×(C2×D3)⋊C4C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×C4×Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33Dic3⋊Dic9C2211yescharacteristic
34Dic3⋊Dic9C2212yes—
35#88442C2211yescharacteristic
36Dic9⋊Dic3C2211yescharacteristic
37#64099—31no—
38C3×#893D311yescharacteristic
39C3×#890—31no—
40#63918—31no—
41C4×#627—31no—
42#63899—32no—
43#63898—31no—
44#63893—32no—
45Dic3×D12—31no—
46D3×C4⋊Dic3—32no—
47#63851—32no—
48C4×D4×C9—31no—
49D4×Dic9—31no—
50D4×Dic9—31no—
51C2×C4⋊Dic9—32no—
52#63486—32no—
53#63452—31no—
54#63452—61no—
55#63452D311yescharacteristic, complemented
56#63440—32no—
57#63440—32no—
58#63431—31no—
59C4×Dic18—31no—
60C22×D3×C9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C9×Dic3C2311yescharacteristic
67C2×C9×Dic3C2311yescharacteristic
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C9×D12Q814yescomplemented
72C4×D3×C9—24no—
73D3×Dic9—42no—
74D3×Dic9—42no—
75#6189—22no—
76#6189—42no—
77#5690—91no—
78#5543—91no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×C4×D3C2×D312yes—
81C2×#627—31no—
82C2×#627—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—32no—
87C4⋊(C3⋊Dic3)—31no—
88#3750—32no—
89C4×C3⋊Dic3—31no—
90C3×(C2×D3)⋊C4C2×D312yes—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—31no—
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—31no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—31no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D4×C9—31no—
103C22×C4×C9—32no—
104C22×Dic9—32no—
105C22×Dic9—32no—
106C2×Dic18—31no—
107C2×Dic18—32no—
108C9×C4⋊C4—21no—
109C9×C4⋊C4C2×D311yescharacteristic
110C9×C22⋊C4—32no—
111C42×C9—31no—
112#3676—32no—
113#3676—32no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—31no—
117C4⋊Dic9—32no—
118C4⋊Dic9—62no—
119C4⋊Dic9—61no—
120C4⋊Dic9C2×D311yescharacteristic
121C4⋊Dic9C2×D312yes—
122Dic9⋊C4—32no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—62no—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—31no—
128C4×Dic9—61no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×D3×C9—22no—
132C2×D3×C9—22no—
133C2×D3×C9C2×Q814yes—
134C3×C4×C9—22no—
135C3×C4×C9C2×Q812yes—
136C3⋊Dic9—21no—
137C3⋊Dic9—22no—
138C3⋊Dic9—21no—
139C9×Dic3—21no—
140C9×Dic3#4112yes—
141C3×Dic9—21no—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×#125—91no—
145C3×C4×D4—31no—
146D4×Dic3—91no—
147D4×Dic3—91no—
148C2×C4⋊Dic3—92no—
149#924—92no—
150#897—92no—
151#896—92no—
152#896—92no—
153#893—91no—
154#893D911yescharacteristic
155D3×C4⋊C4—92no—
156#890—31no—
157#890—91no—
158#890—91no—
159#890—181no—
160#888—91no—
161#878—92no—
162#878—92no—
163C4×D12—91no—
164#869—91no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C22×C3×D3C22×D312yes—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C3×D12Dic614yes—
179C3×C4×D3—24no—
180#627—62no—
181#627—122no—
182D3×Dic3—122no—
183D3×Dic3—122no—
184C23×C9—32no—
185D4×C9—34no—
186C2×C4×C9—21no—
187C2×C4×C9—22no—
188C2×C4×C9—31no—
189C2×C4×C9—31no—
190C2×C4×C9—64no—
191C2×C4×C9C22×D312yes—
192C2×C4×C9C22×D311yescharacteristic
193C2×Dic9—31no—
194C2×Dic9—31no—
195C2×Dic9—32no—
196C2×Dic9—61no—
197C2×Dic9—61no—
198C2×Dic9—62no—
199C2×Dic9—122no—
200C2×Dic9—122no—
201C2×Dic9C22×D311yescharacteristic
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D312yes—
204Dic18—62no—
205Dic18—122no—
206#554—271noSylow
207C2×C3×C9C22×Q811yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210D3×C9—24no—
211C2×C3×D4—31no—
212C22×C3×C4—32no—
213C22×Dic3—92no—
214C22×Dic3—92no—
215C2×D12C2×D911yescharacteristic
216C2×C4×D3—92no—
217C2×C4×D3—92no—
218C2×C4×D3C2×D912yes—
219C2×Dic6—91no—
220C2×Dic6—91no—
221C2×Dic6—92no—
222C2×Dic6—92no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4C2×D911yescharacteristic
228C3×C4⋊C4D3211yescharacteristic
229C3×C22⋊C4—32no—
230C3×C42—31no—
231C3×C42—91no—
232#217—92no—
233#217—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4C2×D912yes—
237C4⋊Dic3—31no—
238C4⋊Dic3—32no—
239C4⋊Dic3—91no—
240C4⋊Dic3—92no—
241C4⋊Dic3—91no—
242C4⋊Dic3—92no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—91no—
246C4⋊Dic3—182no—
247C4⋊Dic3—181no—
248Dic3⋊C4—32no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—91no—
254Dic3⋊C4—91no—
255Dic3⋊C4—92no—
256Dic3⋊C4—182no—
257C4×Dic3—31no—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3—181no—
261C4×Dic3C2×D911yescharacteristic
262C22×C32C23×D311yescharacteristic
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3C2×Dic614yes—
266C32×C4—22no—
267C32×C4C2×Dic612yes—
268C3⋊Dic3—61no—
269C3⋊Dic3—61no—
270C3⋊Dic3—62no—
271C3×Dic3—21no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3—62no—
275C3×Dic3#23712yes—
276C22×C9—21no—
277C22×C9—34no—
278C22×C9—62no—
279C22×C9—62no—
280C22×C9C23×D311yescharacteristic
281C4×C9—22no—
282C4×C9—22no—
283C4×C9—32no—
284C4×C9—42no—
285C4×C9—61no—
286C4×C9D3×Q812yes—
287Dic9—21no—
288Dic9—22no—
289Dic9—21no—
290Dic9—61no—
291Dic9—62no—
292Dic9—61no—
293Dic9—121no—
294Dic9—122no—
295Dic9—121no—
296C4⋊Q8—271no—
297#125—91no—
298#125—271no—
299C22⋊Q8—272no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C4×Q8—271no—
303C4×D4—31no—
304C4×D4—271no—
305C4×D4—271no—
306C2×C4⋊C4—272no—
307C3×C9#55411yescharacteristic, complemented, Sylow
308C23×C3—32no—
309C22×D3C22×D912yes—
310C3×D4—34no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—64no—
316C2×C3×C4—91no—
317C2×C3×C4—92no—
318C2×C3×C4—91no—
319C2×C3×C4C22×D912yes—
320C2×C3×C4C22×D911yescharacteristic
321C2×C3×C4C2×D3212yes—
322C2×C3×C4C2×D3211yescharacteristic
323C2×Dic3—31no—
324C2×Dic3—31no—
325C2×Dic3—32no—
326C2×Dic3—91no—
327C2×Dic3—91no—
328C2×Dic3—92no—
329C2×Dic3—91no—
330C2×Dic3—91no—
331C2×Dic3—92no—
332C2×Dic3—181no—
333C2×Dic3—181no—
334C2×Dic3—182no—
335C2×Dic3—362no—
336C2×Dic3—362no—
337C2×Dic3C22×D911yescharacteristic
338C2×Dic3C22×D911yescharacteristic
339D12Dic1814yescomplemented
340C4×D3—24no—
341C4×D3—362no—
342C4×D3—362no—
343Dic6—182no—
344Dic6—182no—
345Dic6—362no—
346Dic6—362no—
347C2×C32C22×Dic611yescharacteristic
348C2×C32C2×#23711yescharacteristic
349C2×C32#100911yescharacteristic
350C3×D3—24no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—64no—
355C2×C9C2×D3×Q811yescharacteristic
356C2×C9D3×#4111yescharacteristic
357C2×C9#100911yescharacteristic
358C2×Q8—271no—
359C2×Q8—272no—
360C2×D4—31no—
361C22×C4—32no—
362C22×C4—272no—
363C22×C4—272no—
364C4⋊C4—92no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—271no—
368C4⋊C4—272no—
369C4⋊C4—271no—
370C4⋊C4—271no—
371C4⋊C4—272no—
372C4⋊C4—271no—
373C4⋊C4—272no—
374C4⋊C4D3×D911yescharacteristic
375C22⋊C4—32no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—31no—
379C42—91no—
380C42—271no—
381C22×C3—21no—
382C22×C3—34no—
383C22×C3—62no—
384C22×C3—62no—
385C22×C3C23×D911yescharacteristic
386C22×C3C22×D3211yescharacteristic
387C2×D3—22no—
388C2×D3—22no—
389C2×D3C2×Dic1814yes—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—32no—
394C3×C4—42no—
395C3×C4—61no—
396C3×C4—181no—
397C3×C4—182no—
398C3×C4—181no—
399C3×C4C2×Dic1812yes—
400C3×C4D3×Dic612yes—
401Dic3—21no—
402Dic3—61no—
403Dic3—62no—
404Dic3—61no—
405Dic3—181no—
406Dic3—182no—
407Dic3—181no—
408Dic3—181no—
409Dic3—182no—
410Dic3—181no—
411Dic3—361no—
412Dic3—361no—
413Dic3—362no—
414Dic3#369912yes—
415C32#554311yescharacteristic
416C9—21no—
417C9#569011yescharacteristic, complemented
418C23—32no—
419Q8—542no—
420Q8—1082no—
421D4—34no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—64no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—92no—
428C2×C4—271no—
429C2×C4—271no—
430C2×C4—272no—
431C2×C4—1082no—
432C2×C4—1082no—
433C2×C4C2×D3×D912yes—
434C2×C4C2×D3×D911yescharacteristic
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—64no—
439C2×C3C22×Dic1811yescharacteristic
440C2×C3C2×#369911yescharacteristic
441C2×C3#6371511yescharacteristic
442C2×C3C2×D3×Dic611yescharacteristic
443C2×C3D3×#23711yescharacteristic
444C2×C3#6432011yescharacteristic
445D3—24no—
446C22—34no—
447C22—62no—
448C22—62no—
449C22C22×D3×D911yescharacteristic
450C4—22no—
451C4—32no—
452C4—61no—
453C4—181no—
454C4—181no—
455C4—182no—
456C4—541no—
457C4—541no—
458C4—542no—
459C4D3×Dic1812yes—
460C3—21no—
461C3#1058964511yescharacteristic, complemented
462C3#1059445211yescharacteristic
463C2—64no—
464C2C2×D3×Dic1811yescharacteristic
465C2D3×#369911yescharacteristic
466C2#1172251511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×D12C2×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#554
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. C3×#63452C2semidirect
4. #11721026C2semidirect
6. #11721023C2semidirect
8. D3×C4⋊Dic9C2semidirect
9. #11720970C2semidirect
10. C4×#6189C2semidirect
11. #11720955C2semidirect
55. #63452D3semidirect
71. C9×D12Q8semidirect
307. C3×C9#554semidirect
339. D12Dic18semidirect
417. C9#5690semidirect
461. C3#10589645semidirect

Maximal subgroups

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

subgroupindex
#117217462
C9×#8932
C3×#634522
#117210262
#117210252
#117210232
D12×Dic92
D3×C4⋊Dic92
#117209702
C4×#61892
#117209552
#105944523
#105896453