Finite Group Databasebeta

← #13147563

Subgroups — #13147563

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721706C211yesmaximal, characteristic, complemented
2C3×#63434C211yesmaximal, characteristic
3C9×#870C211yesmaximal, characteristic, complemented
4#11720996C212yesmaximal
5#11720991C212yesmaximal
6D9×C4⋊Dic3C211yesmaximal, characteristic
7#11720969C211yesmaximal, characteristic, complemented
8#11720935C211yesmaximal, characteristic, complemented
9#11720915C212yesmaximal, complemented
10#11720907C211yesmaximal, characteristic
11#11720905C212yesmaximal, complemented
12#10593959—31nomaximal
13#10589791—31nomaximal
14C2×#6226C2212yes—
15C2×C3×C4×D9C2211yescharacteristic
16C2×Dic3×D9C2212yes—
17C2×#6189C2212yes—
18C3×C42×C9C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2212yes—
22C3×(C2×D9)⋊C4C2211yescharacteristic
23C3×(C2×D9)⋊C4C2211yescharacteristic
24C9×C4⋊Dic3C2211yescharacteristic
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C9×Dic3⋊C4C2212yes—
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64078—31no—
38C3×#872—31no—
39C3×#870D311yescharacteristic
40#63918—31no—
41#63896—32no—
42#63893—32no—
43D3×C4⋊Dic3—31no—
44#63869—31no—
45#63845—32no—
46#63841—31no—
47#63840—32no—
48C9×#125—31no—
49#63459—32no—
50#63458—32no—
51#63454—31no—
52C4⋊C4×D9—31no—
53#63452—31no—
54#63451—32no—
55#63450—32no—
56#63450—31no—
57#63434D311yescharacteristic
58#63431—31no—
59#63431—61no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66#6226—42no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9Q814yescomplemented
73Dic3×D9—42no—
74#6189—42no—
75#5689—91no—
76#5519—91no—
77C2×#634—32no—
78C2×C3×C4×D3—31no—
79C2×#627—32no—
80C2×D3×Dic3—32no—
81C32×C42C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—32no—
84C4⋊(C3⋊Dic3)—31no—
85C3×(C2×D3)⋊C4—31no—
86C3×(C2×D3)⋊C4—31no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—31no—
90C3×Dic3⋊C4—31no—
91C3×Dic3⋊C4C2×D312yes—
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3—31no—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99Dic32—32no—
100C2×C4×D9—32no—
101C2×C4×D9C2×D311yescharacteristic
102C2×Dic18—32no—
103C2×Dic18—32no—
104C2×Dic18—62no—
105C9×C4⋊C4—32no—
106C9×C4⋊C4—32no—
107C9×C4⋊C4—31no—
108C9×C4⋊C4—31no—
109C42×C9—21no—
110C42×C9C2×D311yescharacteristic
111(C2×D9)⋊C4—32no—
112(C2×D9)⋊C4C2×D311yescharacteristic
113(C2×D9)⋊C4C2×D311yescharacteristic
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—31no—
117C4⋊Dic9—32no—
118C4⋊Dic9—61no—
119C4⋊Dic9—62no—
120C4⋊Dic9—61no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4C2×D311yescharacteristic
125Dic9⋊C4C2×D311yescharacteristic
126C4×Dic9—32no—
127C4×Dic9C2×D311yescharacteristic
128C22×C3×C9C2411yescharacteristic
129C2×C3×D9—21no—
130C2×C3×D9—21no—
131C2×C3×D9C2×Q812yes—
132C3×C4×C9—21no—
133C3×C4×C9—21no—
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—22no—
136C3⋊Dic9—22no—
137C9×Dic3—22no—
138C9×Dic3—22no—
139C3×Dic9—21no—
140C3×Dic9—21no—
141C3×Dic9C2×Q812yes—
142C3×#125—31no—
143C3×#117—91no—
144#926—91no—
145C2×C4⋊Dic3—91no—
146#897—92no—
147#896—92no—
148#892—91no—
149D3×C4⋊C4—91no—
150#890—91no—
151#890—92no—
152#889—92no—
153#888—91no—
154#888—92no—
155#888—92no—
156#878—92no—
157#878—92no—
158#872—31no—
159#870—91no—
160#870D911yescharacteristic, complemented
161#869—91no—
162#869—91no—
163#869—91no—
164#869—181no—
165C22×C3×D3—31no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171#634—122no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3C22×D312yes—
176C2×C3×Dic3C22×D312yes—
177C3×C4×D3—34no—
178#627—122no—
179D3×Dic3—122no—
180C22×D9C22×D311yescharacteristic
181C2×C4×C9—21no—
182C2×C4×C9—21no—
183C2×C4×C9—21no—
184C2×C4×C9—32no—
185C2×C4×C9—32no—
186C2×C4×C9C22×D311yescharacteristic
187C2×C4×C9C22×D311yescharacteristic
188C2×C4×C9C22×D311yescharacteristic
189C2×Dic9—32no—
190C2×Dic9—32no—
191C2×Dic9—62no—
192C2×Dic9—62no—
193C2×Dic9C22×D311yescharacteristic
194C2×Dic9C22×D311yescharacteristic
195C2×Dic9C22×D311yescharacteristic
196C4×D9—122no—
197C4×D9Dic614yescomplemented
198Dic18—122no—
199Dic18—122no—
200Dic18—124no—
201#544—271noSylow
202C2×C3×C9C22×Q811yescharacteristic
203C2×C3×C9H(5,𝔽2)11yescharacteristic
204C2×C3×C92−1+411yescharacteristic
205C3×D9—22no—
206C22×C3×C4—91no—
207C22×Dic3—92no—
208C2×C4×D3—31no—
209C2×C4×D3—92no—
210C2×Dic6—92no—
211C2×Dic6—92no—
212C2×Dic6—92no—
213C2×Dic6—92no—
214C2×Dic6—182no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—32no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—91no—
221C3×C22⋊C4—91no—
222C3×C22⋊C4—91no—
223C3×C42—21no—
224C3×C42—91no—
225C3×C42C2×D911yescharacteristic
226C3×C42D3211yescharacteristic
227#217—92no—
228(C2×D3)⋊C4—31no—
229(C2×D3)⋊C4—31no—
230(C2×D3)⋊C4—92no—
231C4⋊Dic3—92no—
232C4⋊Dic3—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—92no—
235C4⋊Dic3—91no—
236C4⋊Dic3—92no—
237C4⋊Dic3—91no—
238C4⋊Dic3—92no—
239C4⋊Dic3—92no—
240C4⋊Dic3—181no—
241C4⋊Dic3—181no—
242C4⋊Dic3—182no—
243C4⋊Dic3C2×D911yescharacteristic
244C4⋊Dic3C2×D911yescharacteristic
245Dic3⋊C4—31no—
246Dic3⋊C4—31no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4C2×D912yes—
253Dic3⋊C4C2×D912yes—
254C4×Dic3—31no—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C22×C32C23×D311yescharacteristic
258C2×C3×D3—32no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4D3×Q812yes—
264C3⋊Dic3—62no—
265C3⋊Dic3—62no—
266C3×Dic3—22no—
267C3×Dic3—22no—
268C3×Dic3—32no—
269C3×Dic3—61no—
270C3×Dic3—61no—
271C22×C9—21no—
272C22×C9C23×D311yescharacteristic
273C2×D9—21no—
274C2×D9—21no—
275C2×D9C2×Dic612yes—
276C4×C9—22no—
277C4×C9—22no—
278C4×C9—21no—
279C4×C9—21no—
280C4×C9—22no—
281C4×C9—62no—
282C4×C9—62no—
283C4×C9C2×Dic612yes—
284Dic9—21no—
285Dic9—21no—
286Dic9—62no—
287Dic9—62no—
288Dic9—122no—
289Dic9—122no—
290Dic9C2×Dic612yes—
291C4⋊Q8—271no—
292C4⋊Q8—272no—
293C4⋊Q8—271no—
294#125—31no—
295#125—271no—
296#125—272no—
297C22⋊Q8—272no—
298C22⋊Q8—272no—
299#117—91no—
300#117—271no—
301C2×C4⋊C4—271no—
302C3×C9#54411yescharacteristic, complemented, Sylow
303C23×C3—91no—
304C22×D3—31no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—91no—
311C2×C3×C4—94no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4C22×D911yescharacteristic
315C2×C3×C4C22×D911yescharacteristic
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×C3×C4C2×D3211yescharacteristic
319C2×C3×C4C2×D3211yescharacteristic
320C2×Dic3—31no—
321C2×Dic3—31no—
322C2×Dic3—31no—
323C2×Dic3—92no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—182no—
328C2×Dic3—182no—
329C2×Dic3—362no—
330C2×Dic3C22×D912yes—
331C2×Dic3C22×D912yes—
332C4×D3—34no—
333C4×D3—362no—
334Dic6—362no—
335Dic6—362no—
336Dic6—362no—
337Dic6—362no—
338Dic6—364no—
339C2×C32C2×D3×Q811yescharacteristic
340C2×C32#100911yescharacteristic
341C2×C32#101011yescharacteristic
342C3×D3—62no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9—21no—
346C2×C9C22×Dic611yescharacteristic
347C2×C9#100911yescharacteristic
348C2×C9#101011yescharacteristic
349D9—22no—
350C2×Q8—272no—
351C2×Q8—272no—
352C22×C4—91no—
353C22×C4—272no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—32no—
357C4⋊C4—32no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—272no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4—271no—
365C4⋊C4—271no—
366C4⋊C4—272no—
367C22⋊C4—91no—
368C22⋊C4—91no—
369C22⋊C4—272no—
370C42—91no—
371C42—272no—
372C42D3×D911yescharacteristic
373C22×C3—21no—
374C22×C3—92no—
375C22×C3—181no—
376C22×C3—181no—
377C22×C3C23×D911yescharacteristic
378C22×C3C22×D3211yescharacteristic
379C2×D3—32no—
380C2×D3—61no—
381C2×D3—61no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—22no—
389C3×C4—62no—
390C3×C4—62no—
391C3×C4—92no—
392C3×C4—181no—
393C3×C4—181no—
394C3×C4Q8×D912yes—
395C3×C4D3×Dic612yes—
396Dic3—22no—
397Dic3—22no—
398Dic3—32no—
399Dic3—61no—
400Dic3—61no—
401Dic3—182no—
402Dic3—182no—
403Dic3—182no—
404Dic3—182no—
405Dic3—362no—
406Dic3—362no—
407C32#568911yescharacteristic
408C9—21no—
409C9#551911yescharacteristic, complemented
410C23—91no—
411Q8—1082no—
412Q8—1082no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—91no—
416C2×C4—94no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—272no—
420C2×C4—272no—
421C2×C4—1082no—
422C2×C4C2×D3×D911yescharacteristic
423C2×C4C2×D3×D911yescharacteristic
424C2×C4C2×D3×D911yescharacteristic
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—182no—
429C2×C3C2×Q8×D911yescharacteristic
430C2×C3#6371511yescharacteristic
431C2×C3#6371611yescharacteristic
432C2×C3C2×D3×Dic611yescharacteristic
433C2×C3#6429811yescharacteristic
434C2×C3#6430811yescharacteristic
435D3—62no—
436C22—92no—
437C22—181no—
438C22—181no—
439C22C22×D3×D911yescharacteristic
440C4—21no—
441C4—21no—
442C4—62no—
443C4—62no—
444C4—92no—
445C4—181no—
446C4—181no—
447C4—542no—
448C4—542no—
449C4D9×Dic612yes—
450C3—21no—
451C3#1058979111yescharacteristic, complemented
452C3#1059395911yescharacteristic
453C2—182no—
454C2C2×D9×Dic611yescharacteristic
455C2#1172246511yescharacteristic
456C2#1172247811yescharacteristic
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
1. #11721706C2semidirect
3. C9×#870C2semidirect
7. #11720969C2semidirect
8. #11720935C2semidirect
9. #11720915C2semidirect
11. #11720905C2semidirect
72. C3×C4×D9Q8semidirect
160. #870D9semidirect
197. C4×D9Dic6semidirect
302. C3×C9#544semidirect
409. C9#5519semidirect
451. C3#10589791semidirect

Maximal subgroups

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

subgroupindex
#117217062
C3×#634342
C9×#8702
#117209962
#117209912
D9×C4⋊Dic32
#117209692
#117209352
#117209152
#117209072
#117209052
#105939593
#105897913