Finite Group Databasebeta

← #13153912

Subgroups — #13153912

Order 1728.

1070 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D4×C9C211yesmaximal, characteristic
2#11721835C212yesmaximal
3#11721815C211yesmaximal, characteristic, complemented
4C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
5#11721813C211yesmaximal, characteristic, complemented
6#11721813C211yesmaximal, characteristic, complemented
7#11721745C212yesmaximal, complemented
8#11721725C212yesmaximal, complemented
9#11721723C212yesmaximal, complemented
10#11721707C211yesmaximal, characteristic, complemented
11C4×#6226C211yesmaximal, characteristic, complemented
12#10596012—31nomaximal
13#10589643—33nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic
17C22×C3⋊Dic9C2212yes—
18C2×#6226C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C22⋊C4C2211yescharacteristic
21C3×C9×C22⋊C4C2211yescharacteristic
22C3×C42×C9C2211yescharacteristic
23#88551C2212yes—
24#88551C2212yes—
25#88551C2212yes—
26C4⋊(C3⋊Dic9)C2212yes—
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C4⋊(C3⋊Dic9)C2211yescharacteristic
30#88534C2212yes—
31#88534C2212yes—
32#88534C2212yes—
33#88534C2212yes—
34#88534C2212yes—
35C4×C3⋊Dic9C2212yes—
36C4×C3⋊Dic9C2212yes—
37C32×C4×D4D311yescharacteristic
38#64144—32no—
39#64135—31no—
40C2×C4⋊(C3⋊Dic3)—31no—
41#64133—31no—
42#64133—31no—
43#64098—32no—
44#64088—32no—
45#64086—32no—
46#64079—31no—
47C4×#634—31no—
48C4×D4×C9D313yes—
49#63497—36no—
50#63488—33no—
51C2×C4⋊Dic9—33no—
52#63486—33no—
53#63486—33no—
54#63451—36no—
55#63441—36no—
56#63439—36no—
57#63432—33no—
58C4×Dic18—33no—
59C23×C3×C9C2311yescharacteristic
60C23×C3×C9C2311yescharacteristic
61C3×D4×C9—22no—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3⋊Dic9—24no—
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2312yes—
74#6226—22no—
75#5541—93no—
76#5541—91no—
77C2×C32×D4C2×D311yescharacteristic
78C22×C32×C4C2×D311yescharacteristic
79C22×C32×C4C2×D311yescharacteristic
80C22×C3⋊Dic3—32no—
81C2×#634—31no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C22⋊C4C2×D311yescharacteristic
84C32×C22⋊C4C2×D311yescharacteristic
85C32×C42C2×D311yescharacteristic
86#3757—32no—
87#3757—32no—
88#3757—32no—
89C4⋊(C3⋊Dic3)—31no—
90C4⋊(C3⋊Dic3)—31no—
91C4⋊(C3⋊Dic3)—31no—
92C4⋊(C3⋊Dic3)—32no—
93#3750—32no—
94#3750—32no—
95#3750—32no—
96#3750—32no—
97#3750—32no—
98C4×C3⋊Dic3—32no—
99C4×C3⋊Dic3—32no—
100C2×D4×C9C2×D313yes—
101C22×C4×C9C2×D313yes—
102C22×C4×C9C2×D313yes—
103C22×Dic9—36no—
104C2×Dic18—33no—
105C9×C4⋊C4C2×D313yes—
106C9×C22⋊C4C2×D313yes—
107C9×C22⋊C4C2×D313yes—
108C42×C9C2×D313yes—
109#3676—36no—
110#3676—36no—
111#3676—36no—
112C4⋊Dic9—33no—
113C4⋊Dic9—33no—
114C4⋊Dic9—33no—
115C4⋊Dic9—36no—
116Dic9⋊C4—36no—
117Dic9⋊C4—36no—
118Dic9⋊C4—36no—
119Dic9⋊C4—36no—
120Dic9⋊C4—36no—
121C4×Dic9—36no—
122C4×Dic9—36no—
123C22×C3×C9—21no—
124C22×C3×C9—21no—
125C22×C3×C9—21no—
126C22×C3×C9—21no—
127C22×C3×C9—21no—
128C22×C3×C9#4112yes—
129C22×C3×C9C2411yescharacteristic
130C3×C4×C9—21no—
131C3×C4×C9—21no—
132C3×C4×C9—21no—
133C3×C4×C9—21no—
134C3×C4×C9#4112yes—
135C3⋊Dic9—22no—
136C3⋊Dic9—22no—
137C3⋊Dic9—22no—
138C3⋊Dic9—22no—
139C3×C4×D4D913yes—
140C3×C4×D4C3⋊D311yescharacteristic
141#935—96no—
142#935—92no—
143#926—93no—
144#926—91no—
145C2×C4⋊Dic3—93no—
146C2×C4⋊Dic3—91no—
147#924—93no—
148#924—93no—
149#924—91no—
150#924—91no—
151#889—96no—
152#889—92no—
153#879—96no—
154#879—92no—
155#877—96no—
156#877—92no—
157#870—93no—
158#870—91no—
159C4×Dic6—93no—
160C4×Dic6—91no—
161C23×C32C22×D311yescharacteristic
162C23×C32C22×D311yescharacteristic
163C32×D4—22no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—64no—
176#634—62no—
177C23×C9C22×D313yes—
178C23×C9C22×D313yes—
179D4×C9—26no—
180C2×C4×C9—26no—
181C2×C4×C9—26no—
182C2×C4×C9C22×D313yes—
183C2×C4×C9C22×D313yes—
184C2×C4×C9C22×D313yes—
185C2×C4×C9C22×D313yes—
186C2×C4×C9C22×D313yes—
187C2×Dic9—36no—
188C2×Dic9—36no—
189C2×Dic9—36no—
190C2×Dic9—36no—
191C2×Dic9—612no—
192Dic18—66no—
193#552—271noSylow
194C2×C3×C9—22no—
195C2×C3×C9—41no—
196C2×C3×C9C2×#4111yescharacteristic
197C2×C3×C9C2×#4111yescharacteristic
198C2×C3×C92−1+411yescharacteristic
199C2×C3×D4C2×D913yes—
200C2×C3×D4C2×C3⋊D311yescharacteristic
201C22×C3×C4C2×D913yes—
202C22×C3×C4C2×D913yes—
203C22×C3×C4C2×C3⋊D311yescharacteristic
204C22×C3×C4C2×C3⋊D311yescharacteristic
205C22×Dic3—96no—
206C22×Dic3—92no—
207C2×Dic6—93no—
208C2×Dic6—91no—
209C3×C4⋊C4C2×D913yes—
210C3×C4⋊C4C2×C3⋊D311yescharacteristic
211C3×C22⋊C4C2×D913yes—
212C3×C22⋊C4C2×D913yes—
213C3×C22⋊C4C2×C3⋊D311yescharacteristic
214C3×C22⋊C4C2×C3⋊D311yescharacteristic
215C3×C42C2×D913yes—
216C3×C42C2×C3⋊D311yescharacteristic
217#217—96no—
218#217—96no—
219#217—96no—
220#217—92no—
221#217—92no—
222#217—92no—
223C4⋊Dic3—93no—
224C4⋊Dic3—93no—
225C4⋊Dic3—93no—
226C4⋊Dic3—96no—
227C4⋊Dic3—91no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—92no—
231Dic3⋊C4—96no—
232Dic3⋊C4—96no—
233Dic3⋊C4—96no—
234Dic3⋊C4—96no—
235Dic3⋊C4—96no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241C4×Dic3—96no—
242C4×Dic3—96no—
243C4×Dic3—92no—
244C4×Dic3—92no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32#23512yes—
251C22×C32C23×D311yescharacteristic
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4#23712yes—
257C3⋊Dic3—62no—
258C3⋊Dic3—62no—
259C3⋊Dic3—62no—
260C3⋊Dic3—62no—
261C22×C9—23no—
262C22×C9—23no—
263C22×C9—23no—
264C22×C9—23no—
265C22×C9—23no—
266C22×C9#23516yes—
267C22×C9C23×D313yes—
268C4×C9—23no—
269C4×C9—23no—
270C4×C9—23no—
271C4×C9—23no—
272C4×C9#23716yes—
273Dic9—66no—
274Dic9—66no—
275Dic9—66no—
276Dic9—66no—
277#126—272no—
278#125—272no—
279#125—271no—
280#123—272no—
281C22⋊Q8—271no—
282C22⋊Q8—271no—
283C4×Q8—271no—
284C4×D4C3⋊D911yescharacteristic
285#117—271no—
286#117—272no—
287C2×C4⋊C4—271no—
288C3×C9#55211yescharacteristic, complemented, Sylow
289C23×C3C22×D913yes—
290C23×C3C22×D913yes—
291C23×C3C22×C3⋊D311yescharacteristic
292C23×C3C22×C3⋊D311yescharacteristic
293C3×D4—26no—
294C3×D4—22no—
295C2×C3×C4—26no—
296C2×C3×C4—26no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4C22×D913yes—
300C2×C3×C4C22×D913yes—
301C2×C3×C4C22×D913yes—
302C2×C3×C4C22×D913yes—
303C2×C3×C4C22×D913yes—
304C2×C3×C4C22×C3⋊D311yescharacteristic
305C2×C3×C4C22×C3⋊D311yescharacteristic
306C2×C3×C4C22×C3⋊D311yescharacteristic
307C2×C3×C4C22×C3⋊D311yescharacteristic
308C2×C3×C4C22×C3⋊D311yescharacteristic
309C2×Dic3—92no—
310C2×Dic3—92no—
311C2×Dic3—92no—
312C2×Dic3—92no—
313C2×Dic3—96no—
314C2×Dic3—96no—
315C2×Dic3—96no—
316C2×Dic3—96no—
317C2×Dic3—1812no—
318C2×Dic3—184no—
319Dic6—186no—
320Dic6—182no—
321C2×C32—22no—
322C2×C32—41no—
323C2×C32C2×#23511yescharacteristic
324C2×C32C2×#23711yescharacteristic
325C2×C32#101011yescharacteristic
326C2×C9—26no—
327C2×C9—43no—
328C2×C9C2×#23513yes—
329C2×C9C2×#23713yes—
330C2×C9#101013yes—
331C2×Q8—271no—
332C2×D4C2×C3⋊D911yescharacteristic
333C22×C4—272no—
334C22×C4C2×C3⋊D911yescharacteristic
335C22×C4C2×C3⋊D911yescharacteristic
336C4⋊C4—271no—
337C4⋊C4—271no—
338C4⋊C4—272no—
339C4⋊C4—272no—
340C4⋊C4—271no—
341C4⋊C4—272no—
342C4⋊C4—272no—
343C4⋊C4—272no—
344C4⋊C4—272no—
345C4⋊C4C2×C3⋊D911yescharacteristic
346C22⋊C4—272no—
347C22⋊C4—272no—
348C22⋊C4—272no—
349C22⋊C4C2×C3⋊D911yescharacteristic
350C22⋊C4C2×C3⋊D911yescharacteristic
351C42—272no—
352C42—272no—
353C42C2×C3⋊D911yescharacteristic
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—23no—
357C22×C3—23no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—23no—
362C22×C3—23no—
363C22×C3—23no—
364C22×C3#369716yes—
365C22×C3C23×D913yes—
366C22×C3#382812yes—
367C22×C3C23×C3⋊D311yescharacteristic
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—23no—
373C3×C4—23no—
374C3×C4—23no—
375C3×C4—23no—
376C3×C4#369916yes—
377C3×C4#383012yes—
378Dic3—182no—
379Dic3—182no—
380Dic3—182no—
381Dic3—182no—
382Dic3—186no—
383Dic3—186no—
384Dic3—186no—
385Dic3—186no—
386C32#554111yescharacteristic
387C9#554113yescomplemented
388C23C22×C3⋊D911yescharacteristic
389C23C22×C3⋊D911yescharacteristic
390Q8—542no—
391D4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—272no—
395C2×C4—272no—
396C2×C4—272no—
397C2×C4—272no—
398C2×C4—544no—
399C2×C4C22×C3⋊D911yescharacteristic
400C2×C4C22×C3⋊D911yescharacteristic
401C2×C4C22×C3⋊D911yescharacteristic
402C2×C4C22×C3⋊D911yescharacteristic
403C2×C4C22×C3⋊D911yescharacteristic
404C2×C3—22no—
405C2×C3—26no—
406C2×C3—41no—
407C2×C3—43no—
408C2×C3C2×#369713yes—
409C2×C3C2×#369913yes—
410C2×C3#6371613yes—
411C2×C3C2×#382811yescharacteristic
412C2×C3C2×#383011yescharacteristic
413C2×C3#6436711yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22#8873612yes—
420C22C23×C3⋊D911yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—542no—
426C4—542no—
427C4—542no—
428C4—542no—
429C4#8874212yes—
430C3#1058964313yescomplemented
431C3#1059601211yescharacteristic
432C2—22no—
433C2—41no—
434C2C2×#8873611yescharacteristic
435C2C2×#8874211yescharacteristic
436C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
prime radicalC2×C32×D4C2×D3
hypercenterC4×D4C3⋊D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. #11721815C2semidirect
4. C2×C4⋊(C3⋊Dic9)C2semidirect
5. #11721813C2semidirect
6. #11721813C2semidirect
7. #11721745C2semidirect
8. #11721725C2semidirect
9. #11721723C2semidirect
10. #11721707C2semidirect
11. C4×#6226C2semidirect
288. C3×C9#552semidirect
387. C9#5541semidirect
430. C3#10589643semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D4×C92
#117218352
#117218152
C2×C4⋊(C3⋊Dic9)2
#117218132
#117217452
#117217252
#117217232
#117217072
C4×#62262
#105960123
#105896433