Finite Group Databasebeta

← #13160567

Subgroups — #13160567

Order 1728.

2252 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×#88804C212yesmaximal
3C3×#63658C211yesmaximal, characteristic, complemented
4C32×C3⋊#95C211yesmaximal, characteristic, complemented
5#11722061C212yesmaximal, complemented
6#10592363—31nomaximal
7#10591219—34nomaximal
8C23×C3⋊(C3⋊D3)C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—24no—
12C2×C4×C3⋊(C3⋊D3)C414yescomplemented
13C22×C3×C3⋊Dic3C2212yes—
14C22×C32×Dic3C2212yes—
15#88804—24no—
16C22×C4×C3⋊D3—31no—
17C22×C4×C3⋊D3—34no—
18C22×C4×C3⋊D3—64no—
19C2×#3752—32no—
20C2×#3722—38no—
21C32×#95—31no—
22#63658—32no—
23#63658D311yescharacteristic, complemented
24C3×C3⋊#95—34no—
25C3×C3⋊#95D314yescomplemented
26#63579—38no—
27C23×C33C2311yescharacteristic
28C22×C3⋊(C3⋊D3)—22no—
29C22×C3⋊(C3⋊D3)—21no—
30C22×C3⋊(C3⋊D3)—21no—
31C22×C3⋊(C3⋊D3)C2×C412yes—
32C22×C3⋊(C3⋊D3)D414yes—
33C2×C33×C4—22no—
34C2×C33×C4C2×C412yes—
35C2×C3⋊(C3⋊Dic3)—22no—
36C2×C3⋊(C3⋊Dic3)C2×C412yes—
37C4×C3⋊(C3⋊D3)—28no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44#4664—94no—
45#4664—91no—
46C23×C3⋊D3—31no—
47C23×C3⋊D3—34no—
48C23×C3⋊D3—64no—
49C22×C32×C4—24no—
50C22×C32×C4—32no—
51C22×C32×C4C2×D314yes—
52C22×C32×C4C2×D311yescharacteristic
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3—34no—
55C22×C3⋊Dic3—64no—
56C22×C3⋊Dic3C2×D312yes—
57C2×C4×C3⋊D3—34no—
58C2×C4×C3⋊D3—316no—
59C2×C4×C3⋊D3—64no—
60C2×C4×C3⋊D3—616no—
61C2×C4×C3⋊D3—616no—
62C2×C4×C3⋊D3—632no—
63C22×C3×Dic3—38no—
64C22×C3×Dic3C2×D318yes—
65#3752—64no—
66#3722—616no—
67C22×C33C22×C411yescharacteristic
68C22×C33C2×D412yes—
69C22×C33#4112yes—
70C22×C33#4111yescharacteristic
71C22×C33#4111yescharacteristic
72C2×C3⋊(C3⋊D3)—22no—
73C2×C3⋊(C3⋊D3)—22no—
74C2×C3⋊(C3⋊D3)—24no—
75C2×C3⋊(C3⋊D3)—24no—
76C2×C3⋊(C3⋊D3)C22⋊C414yes—
77C33×C4—22no—
78C3⋊(C3⋊Dic3)—22no—
79C3×C3⋊Dic3—42no—
80C32×Dic3—42no—
81C22×C4×D3—94no—
82C22×C4×D3—91no—
83C22×C4×D3—184no—
84C2×(C2×D3)⋊C4—98no—
85C2×(C2×D3)⋊C4—92no—
86C3×#95—34no—
87C3×#95—91no—
88C3⋊#95—34no—
89C3⋊#95—98no—
90C3⋊#95—92no—
91C3⋊#95C3⋊D311yescharacteristic, complemented
92C23×C32—24no—
93C23×C32C22×D314yes—
94C23×C32C22×D311yescharacteristic
95C22×C3⋊D3—32no—
96C22×C3⋊D3—34no—
97C22×C3⋊D3—38no—
98C22×C3⋊D3—316no—
99C22×C3⋊D3—68no—
100C22×C3⋊D3—616no—
101C22×C3⋊D3—68no—
102C22×C3⋊D3—68no—
103C22×C3⋊D3—616no—
104C22×C3⋊D3—68no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—64no—
107C22×C3⋊D3—61no—
108C22×C3⋊D3—64no—
109C22×C3⋊D3—61no—
110C2×C32×C4—216no—
111C2×C32×C4—28no—
112C2×C32×C4—22no—
113C2×C32×C4—28no—
114C2×C32×C4—62no—
115C2×C32×C4—62no—
116C2×C32×C4—62no—
117C2×C32×C4C4×D318yes—
118C2×C32×C4C4×D312yes—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—32no—
123C2×C3⋊Dic3—38no—
124C2×C3⋊Dic3—68no—
125C2×C3⋊Dic3—616no—
126C2×C3⋊Dic3—68no—
127C2×C3⋊Dic3—62no—
128C4×C3⋊D3—68no—
129C4×C3⋊D3—632no—
130C4×C3⋊D3—664no—
131C2×C3×Dic3—28no—
132C2×C3×Dic3—28no—
133C2×C3×Dic3—28no—
134C2×C3×Dic3—68no—
135C2×C3×Dic3—68no—
136C2×C3×Dic3—68no—
137#381—271noSylow
138C2×C33C2×C22⋊C411yescharacteristic
139C2×C33#11711yescharacteristic
140C2×C33#11711yescharacteristic
141C2×C33#12312yes—
142C2×C33#12312yes—
143C3⋊(C3⋊D3)—24no—
144C23×D3—94no—
145C23×D3—91no—
146C23×D3—184no—
147C22×C3×C4—24no—
148C22×C3×C4—38no—
149C22×C3×C4—92no—
150C22×C3×C4D3214yes—
151C22×C3×C4C2×C3⋊D311yescharacteristic
152C22×Dic3—38no—
153C22×Dic3—94no—
154C22×Dic3—91no—
155C22×Dic3—184no—
156C22×Dic3C2×C3⋊D312yes—
157C2×C4×D3—916no—
158C2×C4×D3—94no—
159C2×C4×D3—1816no—
160C2×C4×D3—1816no—
161C2×C4×D3—184no—
162C2×C4×D3—1832no—
163(C2×D3)⋊C4—184no—
164(C2×D3)⋊C4—1816no—
165C22×C32—24no—
166C22×C32—24no—
167C22×C32—28no—
168C22×C32—24no—
169C22×C32—28no—
170C22×C32C2×C4×D314yes—
171C22×C32C2×C4×D311yescharacteristic
172C22×C32#23518yes—
173C22×C32#23512yes—
174C22×C32D3×D418yes—
175C22×C32D3×D412yes—
176C22×C32#23714yes—
177C22×C32#23711yescharacteristic
178C22×C32#23914yes—
179C22×C32#23911yescharacteristic
180C2×C3⋊D3—34no—
181C2×C3⋊D3—316no—
182C2×C3⋊D3—616no—
183C2×C3⋊D3—616no—
184C2×C3⋊D3—632no—
185C2×C3⋊D3—632no—
186C2×C3⋊D3—616no—
187C2×C3⋊D3—68no—
188C2×C3⋊D3—62no—
189C2×C3⋊D3—616no—
190C2×C3⋊D3—64no—
191C2×C3⋊D3—68no—
192C2×C3⋊D3—62no—
193C2×C3⋊D3—616no—
194C2×C3⋊D3—64no—
195C32×C4—216no—
196C32×C4—28no—
197C32×C4—22no—
198C32×C4—122no—
199C3⋊Dic3—42no—
200C3⋊Dic3—616no—
201C3⋊Dic3—68no—
202C3⋊Dic3—62no—
203C3×Dic3—48no—
204C3×Dic3—128no—
205C23×C4—271no—
206C2×C22⋊C4—272no—
207#95—31no—
208#95—91no—
209#95—272no—
210C33#38111yescharacteristic, complemented, Sylow
211C23×C3—24no—
212C23×C3C2×D3214yes—
213C23×C3C22×C3⋊D311yescharacteristic
214C22×D3—98no—
215C22×D3—916no—
216C22×D3—92no—
217C22×D3—94no—
218C22×D3—188no—
219C22×D3—1816no—
220C22×D3—188no—
221C22×D3—188no—
222C22×D3—1816no—
223C22×D3—184no—
224C22×D3—184no—
225C22×D3—188no—
226C22×D3—182no—
227C22×D3—181no—
228C22×D3—181no—
229C2×C3×C4—216no—
230C2×C3×C4—28no—
231C2×C3×C4—22no—
232C2×C3×C4—28no—
233C2×C3×C4—68no—
234C2×C3×C4—68no—
235C2×C3×C4—68no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4—182no—
239C2×C3×C4#62418yes—
240C2×C3×C4C4×C3⋊D312yes—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—68no—
245C2×Dic3—68no—
246C2×Dic3—68no—
247C2×Dic3—98no—
248C2×Dic3—92no—
249C2×Dic3—188no—
250C2×Dic3—1816no—
251C2×Dic3—188no—
252C2×Dic3—182no—
253C4×D3—188no—
254C4×D3—1832no—
255C4×D3—1864no—
256C2×C32—24no—
257C2×C32—28no—
258C2×C32—28no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32#87214yes—
262C2×C32#87211yescharacteristic
263C2×C32D3×C22⋊C414yes—
264C2×C32D3×C22⋊C411yescharacteristic
265C2×C32#88318yes—
266C2×C32#88312yes—
267C2×C32#89214yes—
268C2×C32#89211yescharacteristic
269C2×C32#89418yes—
270C2×C32#89412yes—
271C3⋊D3—632no—
272C3⋊D3—616no—
273C3⋊D3—64no—
274C24—271no—
275C22×C4—32no—
276C22×C4—92no—
277C22×C4—271no—
278C22×C4—274no—
279C22×C4—544no—
280C22×C4D3×C3⋊D311yescharacteristic
281C22⋊C4—544no—
282C22×C3—24no—
283C22×C3—24no—
284C22×C3—28no—
285C22×C3—24no—
286C22×C3—28no—
287C22×C3#379614yes—
288C22×C3#379614yes—
289C22×C3#379818yes—
290C22×C3C2×#62414yes—
291C22×C3#381118yes—
292C22×C3C2×C4×C3⋊D311yescharacteristic
293C22×C3#382812yes—
294C22×C3D4×C3⋊D312yes—
295C22×C3#383011yescharacteristic
296C22×C3#383211yescharacteristic
297C2×D3—94no—
298C2×D3—916no—
299C2×D3—1816no—
300C2×D3—1816no—
301C2×D3—1832no—
302C2×D3—1832no—
303C2×D3—1816no—
304C2×D3—182no—
305C2×D3—184no—
306C2×D3—182no—
307C2×D3—184no—
308C2×D3—188no—
309C2×D3—1816no—
310C2×D3—1816no—
311C2×D3—188no—
312C3×C4—216no—
313C3×C4—22no—
314C3×C4—28no—
315C3×C4—128no—
316C3×C4—362no—
317Dic3—42no—
318Dic3—128no—
319Dic3—1816no—
320Dic3—182no—
321Dic3—188no—
322C32—24no—
323C32#466414yescomplemented
324C32#466411yescharacteristic, complemented
325C23—272no—
326C23—274no—
327C23—541no—
328C23—541no—
329C23—542no—
330C23C2×D3×C3⋊D311yescharacteristic
331C2×C4—22no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—272no—
339C2×C4—548no—
340C2×C4—542no—
341C2×C4#628912yes—
342C2×C3—24no—
343C2×C3—24no—
344C2×C3—28no—
345C2×C3—28no—
346C2×C3—24no—
347C2×C3#6384914yes—
348C2×C3#6385318yes—
349C2×C3#6385318yes—
350C2×C3#6387414yes—
351C2×C3#6397414yes—
352C2×C3#6408111yescharacteristic
353C2×C3C22⋊C4×C3⋊D311yescharacteristic
354C2×C3#6409212yes—
355C2×C3#6410111yescharacteristic
356C2×C3#6410312yes—
357D3—1832no—
358D3—184no—
359D3—1816no—
360C22—274no—
361C22—542no—
362C22—544no—
363C22—544no—
364C22—542no—
365C22#8901511yescharacteristic
366C22#8901711yescharacteristic
367C22#8902012yes—
368C22C2×#628911yescharacteristic
369C22#8903912yes—
370C4—22no—
371C4—122no—
372C4—362no—
373C4—542no—
374C3—24no—
375C3#1059121914yescomplemented
376C3#1059236311yescharacteristic, complemented
377C2—544no—
378C2#1172341311yescharacteristic
379C2#1172342012yes—
380C2#1172342112yes—
381C2#1172345211yescharacteristic
382C2#1172362711yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, cosocleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#381
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63658C2semidirect
4. C32×C3⋊#95C2semidirect
5. #11722061C2semidirect
12. C2×C4×C3⋊(C3⋊D3)C4semidirect
23. #63658D3semidirect
25. C3×C3⋊#95D3semidirect
91. C3⋊#95C3⋊D3semidirect
210. C33#381semidirect
323. C32#4664semidirect
324. C32#4664semidirect
375. C3#10591219semidirect
376. C3#10592363semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊(C3⋊D3)2
C2×#888042
C3×#636582
C32×C3⋊#952
#117220612
#105923633
#105912193