Finite Group Databasebeta

← #13147597

Subgroups — #13147597

Order 1728.

970 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88639C212yesmaximal, complemented
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×#63431C211yesmaximal, characteristic, complemented
5#11721038C212yesmaximal, complemented
6#11721020C214yesmaximal, complemented
7D3×C4⋊Dic9C212yesmaximal, complemented
8#11720936C212yesmaximal, complemented
9#10593896—31nomaximal
10#10589657—31nomaximal
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2212yes—
13C2×C9×D12C2211yescharacteristic
14C2×C4×D3×C9C2212yes—
15C2×C3×Dic18C2212yes—
16C2×#6195C2214yes—
17C2×D3×Dic9C2214yes—
18#88639—28no—
19C3×C42×C9C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C9×(C2×D3)⋊C4C2212yes—
23C9×C4⋊Dic3C2211yescharacteristic
24C3×C4⋊Dic9C2212yes—
25C3×C4⋊Dic9C2212yes—
26#88445C2214yes—
27Dic3⋊Dic9C2214yes—
28C2×#3796—32no—
29C4×C4⋊(C3⋊D3)—31no—
30C3×C4×D12D311yescharacteristic
31C3×#869—31no—
32#63909—32no—
33#63890—34no—
34D3×C4⋊Dic3—32no—
35#63871—32no—
36C2×#3699—32no—
37C4×D4×C9—31no—
38#63492—32no—
39C2×C4⋊Dic9—32no—
40#63459—32no—
41#63448—34no—
42C4×D36—31no—
43C4×D36—61no—
44#63431D311yescharacteristic, complemented
45C22×C3⋊D9C2312yes—
46C22×D3×C9C2312yes—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3×C4×C9C2311yescharacteristic
50C2×C3⋊Dic9C2312yes—
51C4⋊(C3⋊D9)—22no—
52C4×C3⋊D9—24no—
53C2×C9×Dic3C2312yes—
54C2×C3×Dic9C2314yes—
55C9×D12D414yescomplemented
56C4×D3×C9—24no—
57C3×Dic18—24no—
58#6195—28no—
59D3×Dic9—28no—
60#5726—91no—
61#5555—91no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4×C3⋊D3—32no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×C4×D3C2×D312yes—
66C2×C3×Dic6—32no—
67C2×#626—34no—
68C2×D3×Dic3—34no—
69#3796—68no—
70C32×C42C2×D311yescharacteristic
71#3752—32no—
72C4⋊(C3⋊Dic3)—31no—
73C3×(C2×D3)⋊C4C2×D312yes—
74C3×C4⋊Dic3—32no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3C2×D311yescharacteristic
77Dic3⋊Dic3—34no—
78#3722—34no—
79C2×D4×C9—31no—
80C22×C4×C9—32no—
81C2×#611—34no—
82C22×Dic9—34no—
83#3699—68no—
84C2×D36—31no—
85C2×D36—61no—
86C2×C4×D9—32no—
87C2×C4×D9—62no—
88C2×Dic18C2×D312yes—
89C9×C4⋊C4—31no—
90C9×C22⋊C4—32no—
91C42×C9—21no—
92C42×C9C2×D311yescharacteristic
93(C2×D9)⋊C4—32no—
94(C2×D9)⋊C4—34no—
95(C2×D9)⋊C4—62no—
96C4⋊Dic9—31no—
97C4⋊Dic9—34no—
98C4⋊Dic9—61no—
99C4⋊Dic9C2×D312yes—
100C4⋊Dic9C2×D312yes—
101C22×C3×C9C2411yescharacteristic
102C2×C3⋊D9—22no—
103C2×C3⋊D9—22no—
104C2×C3⋊D9—22no—
105C2×D3×C9—22no—
106C2×D3×C9—22no—
107C2×D3×C9C2×D414yes—
108C3×C4×C9—21no—
109C3×C4×C9C2×D412yes—
110C3×C4×C9#4112yescomplemented
111C3⋊Dic9—22no—
112C9×Dic3—22no—
113C3×Dic9—24no—
114C2×#239—92no—
115C2×#237—92no—
116C3×C4⋊Q8—91no—
117C3×C4×D4—31no—
118#946—92no—
119#930—92no—
120C2×C4⋊Dic3—92no—
121#897—92no—
122#895—92no—
123#894—94no—
124D3×C4⋊C4—92no—
125#886—94no—
126C4×D12—91no—
127C4×D12—91no—
128C4×D12—181no—
129C4×D12D911yescharacteristic, complemented
130#869—31no—
131C22×C3⋊D3—32no—
132C22×C3×D3C22×D312yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—32no—
137C4⋊(C3⋊D3)—62no—
138C4×C3⋊D3—64no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3C22×D312yes—
141C3×D12D1214yes—
142C3×C4×D3—24no—
143C3×Dic6—64no—
144#626—68no—
145D3×Dic3—68no—
146C23×C9—32no—
147C22×D9—32no—
148C22×D9—62no—
149D4×C9—34no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9—32no—
154C2×C4×C9—64no—
155C2×C4×C9C22×D311yescharacteristic
156C2×C4×C9C22×D311yescharacteristic
157C2×C4×C9C22×D311yescharacteristic
158#611—68no—
159C2×Dic9—32no—
160C2×Dic9—62no—
161C2×Dic9—68no—
162C2×Dic9C22×D314yes—
163D36—64no—
164D36—62no—
165C4×D9—64no—
166C4×D9—124no—
167Dic18—24no—
168#547—271noSylow
169C2×C3×C9C22×D411yescharacteristic
170C2×C3×C9C2×#4111yescharacteristic
171C2×C3×C92−1+411yescharacteristic
172C3⋊D9—42no—
173D3×C9—24no—
174C2×C3×Q8—92no—
175C2×C3×D4—31no—
176C22×C3×C4—32no—
177C2×#67—94no—
178C22×Dic3—94no—
179#239—188no—
180#237—188no—
181C2×D12—94no—
182C2×D12—91no—
183C2×D12—91no—
184C2×D12—181no—
185C2×D12C2×D911yescharacteristic
186C2×C4×D3—94no—
187C2×C4×D3—92no—
188C2×C4×D3—92no—
189C2×C4×D3—182no—
190C2×C4×D3C2×D912yes—
191C2×Dic6—32no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—92no—
195C3×C22⋊C4—32no—
196C3×C42—21no—
197C3×C42C2×D911yescharacteristic
198C3×C42D3211yescharacteristic
199(C2×D3)⋊C4—94no—
200(C2×D3)⋊C4—92no—
201(C2×D3)⋊C4—92no—
202(C2×D3)⋊C4—94no—
203(C2×D3)⋊C4—182no—
204(C2×D3)⋊C4C2×D912yes—
205C4⋊Dic3—32no—
206C4⋊Dic3—32no—
207C4⋊Dic3—94no—
208C4⋊Dic3—91no—
209C4⋊Dic3—91no—
210C4⋊Dic3—181no—
211C4⋊Dic3C2×D911yescharacteristic
212Dic3⋊C4—94no—
213C22×C32C23×D311yescharacteristic
214C2×C3⋊D3—62no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—62no—
217C2×C3×D3—22no—
218C2×C3×D3—22no—
219C2×C3×D3C2×D1214yes—
220C32×C4—21no—
221C32×C4C2×D1212yes—
222C32×C4#23712yes—
223C3⋊Dic3—62no—
224C3×Dic3—22no—
225C3×Dic3—64no—
226C22×C9—21no—
227C22×C9—34no—
228C22×C9—62no—
229C22×C9—62no—
230C22×C9C23×D311yescharacteristic
231C2×D9—64no—
232C2×D9—62no—
233C2×D9—62no—
234C2×D9—62no—
235C2×D9—122no—
236C2×D9—122no—
237C4×C9—21no—
238C4×C9—22no—
239C4×C9—22no—
240C4×C9—41no—
241C4×C9—62no—
242C4×C9D3×D412yes—
243C4×C9#23912yescomplemented
244Dic9—24no—
245Dic9—62no—
246Dic9—122no—
247C2×#41—272no—
248C4⋊Q8—91no—
249#123—274no—
250C22⋊Q8—272no—
251#121—272no—
252C4×D4—31no—
253C4×D4—271no—
254C2×C4⋊C4—272no—
255C3×C9#54711yescharacteristic, complemented, Sylow
256C23×C3—32no—
257C22×D3—92no—
258C22×D3—92no—
259C22×D3—182no—
260C22×D3C22×D912yes—
261C3×Q8—184no—
262C3×D4—34no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—32no—
267C2×C3×C4—64no—
268C2×C3×C4—94no—
269C2×C3×C4C22×D911yescharacteristic
270C2×C3×C4C22×D911yescharacteristic
271C2×C3×C4C22×D911yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273C2×C3×C4C2×D3211yescharacteristic
274C2×C3×C4C2×D3211yescharacteristic
275#67—188no—
276C2×Dic3—34no—
277C2×Dic3—92no—
278C2×Dic3—92no—
279C2×Dic3—182no—
280C2×Dic3—188no—
281C2×Dic3C22×D912yes—
282D12—184no—
283D12—188no—
284D12—182no—
285D12—182no—
286D12D3614yescomplemented
287C4×D3—24no—
288C4×D3—188no—
289C4×D3—184no—
290C4×D3—184no—
291C4×D3—364no—
292Dic6—64no—
293C2×C32C22×D1211yescharacteristic
294C2×C32C2×#23711yescharacteristic
295C2×C32#101011yescharacteristic
296C3⋊D3—122no—
297C3×D3—24no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—64no—
302C2×C9C2×D3×D411yescharacteristic
303C2×C9C2×#23911yescharacteristic
304C2×C9#100711yescharacteristic
305D9—124no—
306D9—122no—
307#41—548no—
308C2×Q8—92no—
309C2×D4—31no—
310C2×D4—274no—
311C2×D4—271no—
312C22×C4—32no—
313C22×C4—274no—
314C22×C4—272no—
315C4⋊C4—31no—
316C4⋊C4—92no—
317C4⋊C4—92no—
318C4⋊C4—274no—
319C4⋊C4—271no—
320C22⋊C4—32no—
321C22⋊C4—274no—
322C22⋊C4—272no—
323C42D3×D911yescharacteristic
324C22×C3—21no—
325C22×C3—34no—
326C22×C3—62no—
327C22×C3—62no—
328C22×C3C23×D911yescharacteristic
329C22×C3C22×D3211yescharacteristic
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—184no—
333C2×D3—182no—
334C2×D3—182no—
335C2×D3—182no—
336C2×D3—182no—
337C2×D3—182no—
338C2×D3—182no—
339C2×D3—362no—
340C2×D3—362no—
341C2×D3C2×D3614yes—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—41no—
347C3×C4—62no—
348C3×C4—184no—
349C3×C4C2×D3612yes—
350C3×C4#369912yescomplemented
351C3×C4#379612yes—
352C3×C4D3×D1212yes—
353Dic3—22no—
354Dic3—64no—
355Dic3—182no—
356Dic3—182no—
357Dic3—362no—
358C32#555511yescharacteristic
359C9—21no—
360C9#572611yescharacteristic, complemented
361C23—32no—
362C23—272no—
363Q8—184no—
364D4—34no—
365D4—548no—
366D4—542no—
367C2×C4—32no—
368C2×C4—64no—
369C2×C4—94no—
370C2×C4—272no—
371C2×C4—548no—
372C2×C4—544no—
373C2×C4C2×D3×D911yescharacteristic
374C2×C4C2×D3×D911yescharacteristic
375C2×C4C2×D3×D911yescharacteristic
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—64no—
380C2×C3C22×D3611yescharacteristic
381C2×C3C2×#369911yescharacteristic
382C2×C3#6371611yescharacteristic
383C2×C3C2×#379611yescharacteristic
384C2×C3#6429711yescharacteristic
385C2×C3C2×D3×D1211yescharacteristic
386D3—24no—
387D3—364no—
388D3—362no—
389D3—362no—
390C22—34no—
391C22—62no—
392C22—62no—
393C22—542no—
394C22—542no—
395C22—542no—
396C22C22×D3×D911yescharacteristic
397C4—21no—
398C4—62no—
399C4—184no—
400C4—542no—
401C4#8863912yescomplemented
402C4D3×D3612yes—
403C3—21no—
404C3#1058965711yescharacteristic, complemented
405C3#1059389611yescharacteristic
406C2—64no—
407C2—1082no—
408C2C2×#8863911yescharacteristic
409C2#1172246411yescharacteristic
410C2C2×D3×D3611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#547
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88639C2semidirect
2. C4×C4⋊(C3⋊D9)C2semidirect
3. C4×C9×D12C2semidirect
4. C3×#63431C2semidirect
5. #11721038C2semidirect
6. #11721020C2semidirect
7. D3×C4⋊Dic9C2semidirect
8. #11720936C2semidirect
44. #63431D3semidirect
55. C9×D12D4semidirect
110. C3×C4×C9#41semidirect
129. C4×D12D9semidirect
243. C4×C9#239semidirect
255. C3×C9#547semidirect
286. D12D36semidirect
350. C3×C4#3699semidirect
360. C9#5726semidirect
401. C4#88639semidirect
404. C3#10589657semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
C4×C4⋊(C3⋊D9)2
C4×C9×D122
C3×#634312
#117210382
#117210202
D3×C4⋊Dic92
#117209362
#105938963
#105896573