Finite Group Databasebeta

← #13147486

Subgroups — #13147486

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×#88634C212yesmaximal, complemented
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C3×C4×D36C211yesmaximal, characteristic, complemented
4C9×#869C211yesmaximal, characteristic, complemented
5#11721004C212yesmaximal, complemented
6#11720988C214yesmaximal, complemented
7D9×C4⋊Dic3C212yesmaximal, complemented
8#11720929C212yesmaximal, complemented
9#10593896—31nomaximal
10#10589828—31nomaximal
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2212yes—
13C2×C3×D36C2211yescharacteristic
14C2×C3×C4×D9C2212yes—
15C2×C9×Dic6C2212yes—
16C2×#6192C2214yes—
17C2×Dic3×D9C2214yes—
18#88634—28no—
19C3×C42×C9C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C3×(C2×D9)⋊C4C2212yes—
23C9×C4⋊Dic3C2212yes—
24C9×C4⋊Dic3C2212yes—
25C3×C4⋊Dic9C2211yescharacteristic
26#88445C2214yes—
27Dic9⋊Dic3C2214yes—
28C2×#3796—32no—
29C4×C4⋊(C3⋊D3)—31no—
30C3×C4×D12—31no—
31C3×#869D311yescharacteristic
32#63909—32no—
33#63890—34no—
34D3×C4⋊Dic3—32no—
35#63871—32no—
36C2×#3701—32no—
37C9×C4⋊Q8—31no—
38#63508—32no—
39#63457—32no—
40#63456—34no—
41C4⋊C4×D9—32no—
42C4×D36—31no—
43C4×D36—61no—
44C4×D36D311yescharacteristic, complemented
45C22×C3⋊D9C2312yes—
46C22×C3×D9C2312yes—
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×Dic3C2314yes—
54C2×C3×Dic9C2312yes—
55C3×D36D414yescomplemented
56C3×C4×D9—24no—
57C9×Dic6—24no—
58#6192—28no—
59Dic3×D9—28no—
60#5726—91no—
61#5555—91no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4×C3⋊D3—32no—
64C2×C3×D12—31no—
65C2×C3×C4×D3—32no—
66C2×C3×Dic6C2×D312yes—
67C2×#626—34no—
68C2×D3×Dic3—34no—
69#3796—68no—
70C32×C42C2×D311yescharacteristic
71#3752—32no—
72C4⋊(C3⋊Dic3)—31no—
73C3×(C2×D3)⋊C4—32no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3C2×D312yes—
76C3×C4⋊Dic3C2×D312yes—
77Dic3⋊Dic3—34no—
78#3722—34no—
79C2×Q8×C9—32no—
80#3701—68no—
81C2×D36—34no—
82C2×D36—31no—
83C2×D36—61no—
84C2×D36C2×D311yescharacteristic
85C2×C4×D9—34no—
86C2×C4×D9—32no—
87C2×C4×D9—62no—
88C2×C4×D9C2×D312yes—
89C9×C4⋊C4—32no—
90C9×C4⋊C4—32no—
91C42×C9—21no—
92C42×C9C2×D311yescharacteristic
93(C2×D9)⋊C4—32no—
94(C2×D9)⋊C4—34no—
95(C2×D9)⋊C4—62no—
96(C2×D9)⋊C4C2×D312yes—
97C4⋊Dic9—31no—
98C4⋊Dic9—61no—
99C4⋊Dic9C2×D311yescharacteristic
100Dic9⋊C4—34no—
101C22×C3×C9C2411yescharacteristic
102C2×C3⋊D9—22no—
103C2×C3⋊D9—22no—
104C2×C3⋊D9—22no—
105C2×C3×D9—22no—
106C2×C3×D9—22no—
107C2×C3×D9C2×D414yes—
108C3×C4×C9—21no—
109C3×C4×C9C2×D412yes—
110C3×C4×C9#4112yescomplemented
111C3⋊Dic9—22no—
112C9×Dic3—24no—
113C3×Dic9—22no—
114C2×#239—92no—
115C2×#237—92no—
116C3×C4⋊Q8—31no—
117C3×C4×D4—91no—
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—31no—
127C4×D12—91no—
128C4×D12—91no—
129C4×D12—181no—
130#869D911yescharacteristic, complemented
131C22×C3⋊D3—32no—
132C22×C3×D3—32no—
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—32no—
140C2×C3×Dic3C22×D314yes—
141C3×D12—34no—
142C3×C4×D3—64no—
143C3×Dic6—24no—
144#626—68no—
145D3×Dic3—68no—
146C22×D9—32no—
147C22×D9—62no—
148C22×D9C22×D312yes—
149Q8×C9—64no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9—34no—
154C2×C4×C9C22×D311yescharacteristic
155C2×C4×C9C22×D311yescharacteristic
156C2×C4×C9C22×D311yescharacteristic
157C2×Dic9—32no—
158C2×Dic9—62no—
159C2×Dic9C22×D312yes—
160D36—64no—
161D36—68no—
162D36—62no—
163D36D1214yescomplemented
164C4×D9—24no—
165C4×D9—68no—
166C4×D9—64no—
167C4×D9—124no—
168#547—271noSylow
169C2×C3×C9C22×D411yescharacteristic
170C2×C3×C9C2×#4111yescharacteristic
171C2×C3×C92−1+411yescharacteristic
172C3⋊D9—42no—
173C3×D9—24no—
174C2×C3×Q8—32no—
175C2×C3×D4—91no—
176C22×C3×C4—92no—
177C2×#67—94no—
178C22×Dic3—94no—
179#239—188no—
180#237—188no—
181C2×D12—31no—
182C2×D12—94no—
183C2×D12—91no—
184C2×D12—91no—
185C2×D12—181no—
186C2×C4×D3—32no—
187C2×C4×D3—94no—
188C2×C4×D3—92no—
189C2×C4×D3—92no—
190C2×C4×D3—182no—
191C2×Dic6C2×D912yes—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—32no—
194C3×C4⋊C4—91no—
195C3×C22⋊C4—92no—
196C3×C42—21no—
197C3×C42C2×D911yescharacteristic
198C3×C42D3211yescharacteristic
199(C2×D3)⋊C4—32no—
200(C2×D3)⋊C4—94no—
201(C2×D3)⋊C4—92no—
202(C2×D3)⋊C4—92no—
203(C2×D3)⋊C4—94no—
204(C2×D3)⋊C4—182no—
205C4⋊Dic3—31no—
206C4⋊Dic3—91no—
207C4⋊Dic3—91no—
208C4⋊Dic3—94no—
209C4⋊Dic3—181no—
210C4⋊Dic3C2×D912yes—
211C4⋊Dic3C2×D912yes—
212Dic3⋊C4—94no—
213C22×C32C23×D311yescharacteristic
214C2×C3⋊D3—62no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—62no—
217C2×C3×D3—34no—
218C2×C3×D3—62no—
219C2×C3×D3—62no—
220C32×C4—21no—
221C32×C4D3×D412yes—
222C32×C4#23912yes—
223C3⋊Dic3—62no—
224C3×Dic3—24no—
225C3×Dic3—62no—
226C22×C9—21no—
227C22×C9C23×D311yescharacteristic
228C2×D9—22no—
229C2×D9—22no—
230C2×D9—64no—
231C2×D9—62no—
232C2×D9—62no—
233C2×D9—62no—
234C2×D9—122no—
235C2×D9—122no—
236C2×D9C2×D1214yes—
237C4×C9—21no—
238C4×C9—22no—
239C4×C9—22no—
240C4×C9—41no—
241C4×C9—64no—
242C4×C9C2×D1212yes—
243C4×C9#23712yescomplemented
244Dic9—22no—
245Dic9—62no—
246Dic9—122no—
247C2×#41—272no—
248C4⋊Q8—31no—
249#123—274no—
250C22⋊Q8—272no—
251#121—272no—
252C4×D4—91no—
253C4×D4—271no—
254C2×C4⋊C4—272no—
255C3×C9#54711yescharacteristic, complemented, Sylow
256C23×C3—92no—
257C22×D3—32no—
258C22×D3—92no—
259C22×D3—92no—
260C22×D3—182no—
261C3×Q8—64no—
262C3×D4—94no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—34no—
267C2×C3×C4—92no—
268C2×C3×C4—184no—
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—32no—
277C2×Dic3—92no—
278C2×Dic3—92no—
279C2×Dic3—182no—
280C2×Dic3—188no—
281C2×Dic3C22×D914yes—
282D12—34no—
283D12—184no—
284D12—188no—
285D12—182no—
286D12—182no—
287C4×D3—64no—
288C4×D3—188no—
289C4×D3—184no—
290C4×D3—184no—
291C4×D3—364no—
292Dic6—24no—
293C2×C32C2×D3×D411yescharacteristic
294C2×C32C2×#23911yescharacteristic
295C2×C32#100711yescharacteristic
296C3⋊D3—122no—
297C3×D3—64no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9C22×D1211yescharacteristic
302C2×C9C2×#23711yescharacteristic
303C2×C9#101011yescharacteristic
304D9—24no—
305D9—124no—
306D9—122no—
307#41—548no—
308C2×Q8—32no—
309C2×D4—91no—
310C2×D4—274no—
311C2×D4—271no—
312C22×C4—92no—
313C22×C4—274no—
314C22×C4—272no—
315C4⋊C4—32no—
316C4⋊C4—32no—
317C4⋊C4—91no—
318C4⋊C4—274no—
319C4⋊C4—271no—
320C22⋊C4—92no—
321C22⋊C4—274no—
322C22⋊C4—272no—
323C42D3×D911yescharacteristic
324C22×C3—21no—
325C22×C3—94no—
326C22×C3—182no—
327C22×C3—182no—
328C22×C3C23×D911yescharacteristic
329C22×C3C22×D3211yescharacteristic
330C2×D3—34no—
331C2×D3—62no—
332C2×D3—62no—
333C2×D3—184no—
334C2×D3—182no—
335C2×D3—182no—
336C2×D3—182no—
337C2×D3—182no—
338C2×D3—182no—
339C2×D3—182no—
340C2×D3—362no—
341C2×D3—362no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—41no—
347C3×C4—64no—
348C3×C4—182no—
349C3×C4D4×D912yes—
350C3×C4#370112yescomplemented
351C3×C4#379612yes—
352C3×C4D3×D1212yes—
353Dic3—24no—
354Dic3—62no—
355Dic3—182no—
356Dic3—182no—
357Dic3—362no—
358C32#572611yescharacteristic
359C9—21no—
360C9#555511yescharacteristic, complemented
361C23—92no—
362C23—272no—
363Q8—64no—
364D4—94no—
365D4—548no—
366D4—542no—
367C2×C4—34no—
368C2×C4—92no—
369C2×C4—184no—
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—184no—
380C2×C3C2×D4×D911yescharacteristic
381C2×C3C2×#370111yescharacteristic
382C2×C3#6371311yescharacteristic
383C2×C3C2×#379611yescharacteristic
384C2×C3#6429711yescharacteristic
385C2×C3C2×D3×D1211yescharacteristic
386D3—64no—
387D3—364no—
388D3—362no—
389D3—362no—
390C22—94no—
391C22—182no—
392C22—182no—
393C22—542no—
394C22—542no—
395C22—542no—
396C22C22×D3×D911yescharacteristic
397C4—21no—
398C4—64no—
399C4—182no—
400C4—542no—
401C4#8863412yescomplemented
402C4D9×D1212yes—
403C3—21no—
404C3#1058982811yescharacteristic, complemented
405C3#1059389611yescharacteristic
406C2—184no—
407C2—1082no—
408C2C2×#8863411yescharacteristic
409C2#1172245811yescharacteristic
410C2C2×D9×D1211yescharacteristic
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×#88634C2semidirect
2. C4×C4⋊(C3⋊D9)C2semidirect
3. C3×C4×D36C2semidirect
4. C9×#869C2semidirect
5. #11721004C2semidirect
6. #11720988C2semidirect
7. D9×C4⋊Dic3C2semidirect
8. #11720929C2semidirect
44. C4×D36D3semidirect
55. C3×D36D4semidirect
110. C3×C4×C9#41semidirect
130. #869D9semidirect
163. D36D12semidirect
243. C4×C9#237semidirect
255. C3×C9#547semidirect
350. C3×C4#3701semidirect
360. C9#5555semidirect
401. C4#88634semidirect
404. C3#10589828semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C4×C4⋊(C3⋊D9)2
C3×C4×D362
C9×#8692
#117210042
#117209882
D9×C4⋊Dic32
#117209292
#105938963
#105898283