Finite Group Databasebeta

← #13139417

Subgroups — #13139417

Order 1728.

538 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2#11721815C211yesmaximal, characteristic, complemented
3C2×C9×#208C211yesmaximal, characteristic, complemented
4#11720291C214yesmaximal, complemented
5#10591110—31nomaximal
6#10588819—31nomaximal
7C22×C3×C4×C9C2211yescharacteristic
8C2×C3×#611—21no—
9C3×#3697—21no—
10C3×#3697—21no—
11C3×#3697C412yescomplemented
12C3×#3697C2212yes—
13C2×C3×D36C411yescharacteristic, complemented
14C2×C3×C4×D9—21no—
15C2×C3×Dic18C411yescharacteristic, complemented
16#88551—21no—
17C4⋊(C3⋊Dic9)—21no—
18C4×C3⋊Dic9C2212yescomplemented
19C9×#208—21no—
20C9×#208C2212yescomplemented
21C2×C9×C3⋊C8—21no—
22C2×C3×#235—31no—
23#64135—31no—
24C2×C3×#208D311yescharacteristic
25C2×#3697D311yescharacteristic
26#63568—34no—
27C2×C9×M16—31no—
28#63488—31no—
29#63488—61no—
30#63384—34no—
31C23×C3×C9D411yescharacteristic
32C22×C3×D9—21no—
33C2×C3×C4×C9C2×C411yescharacteristic
34C2×C3×C4×C9C2×C411yescharacteristic
35C2×C3×C4×C9D411yescharacteristic
36C2×C3×C4×C9D411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—22no—
40C3×#611—22no—
41C3×#611—22no—
42C2×C3×Dic9—21no—
43C3×D36—21no—
44C3×D36C2×C412yes—
45C3×C4×D9—22no—
46C3×C4×D9—22no—
47C3×Dic18—21no—
48C3×Dic18C2×C412yes—
49C9×C3⋊C8—22no—
50#5236—91no—
51#5139—91no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—61no—
54C3×#235—32no—
55C3×#235—32no—
56C3×#235—61no—
57C3×#235—61no—
58C2×C3×D12—31no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—31no—
61#3757—61no—
62C4⋊(C3⋊Dic3)—61no—
63C4×C3⋊Dic3—32no—
64C3×#208—21no—
65C3×#208C2×D312yes—
66C2×C3×C3⋊C8—21no—
67C22×C4×C9—21no—
68C22×C4×C9C2×D311yescharacteristic
69C2×#611—21no—
70#3697—21no—
71#3697—21no—
72#3697Dic312yescomplemented
73#3697C2×D312yes—
74C2×D36Dic311yescharacteristic, complemented
75C2×C4×D9—21no—
76C2×Dic18Dic311yescharacteristic, complemented
77C9×M16—32no—
78C9×M16—61no—
79C2×C8×C9—61no—
80#3676—62no—
81#3676—61no—
82C4⋊Dic9—62no—
83C4⋊Dic9—61no—
84C4×Dic9—32no—
85C4×Dic9—62no—
86C22×C3×C9—41no—
87C22×C3×C9C22⋊C411yescharacteristic
88C22×C3×C9C22⋊C411yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—22no—
91C2×C3×D9—42no—
92C3×C4×C9C22⋊C411yescharacteristic
93C3×C4×C9C22⋊C411yescharacteristic
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—42no—
97C3×Dic9—22no—
98C2×C3×#41—91no—
99C2×#235—31no—
100C2×C3×M16—31no—
101#926—91no—
102#926—91no—
103#926—181no—
104C2×#208D911yescharacteristic, complemented
105#837—94no—
106#825—94no—
107C23×C32D1211yescharacteristic
108C22×C3×D3—61no—
109C2×C32×C4C4×D311yescharacteristic
110C2×C32×C4C4×D311yescharacteristic
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4#6711yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—62no—
116C3×#67—62no—
117C3×#67—62no—
118C2×C3×Dic3—61no—
119C3×D12—32no—
120C3×D12—61no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—32no—
124C3×Dic6—61no—
125C23×C9—21no—
126C23×C9#6711yescharacteristic
127C22×D9—21no—
128C3×C3⋊C8—22no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9C2×Dic311yescharacteristic
136C2×C4×C9C2×Dic311yescharacteristic
137C2×C4×C9#6711yescharacteristic
138C2×C4×C9#6711yescharacteristic
139C2×C4×C9#6711yescharacteristic
140C2×C4×C9C22×D311yescharacteristic
141#611—22no—
142#611—22no—
143C2×Dic9—21no—
144C2×Dic9—64no—
145C2×Dic9—62no—
146D36—21no—
147D36C2×Dic312yes—
148C4×D9—22no—
149C4×D9—22no—
150Dic18—21no—
151Dic18C2×Dic312yes—
152C8×C9—62no—
153#421—271noSylow
154C2×C3×C9—21no—
155C2×C3×C9—21no—
156C2×C3×C9—21no—
157C2×C3×C9C2×C22⋊C411yescharacteristic
158C3×D9—42no—
159C3×#41—92no—
160C3×#41—92no—
161C3×#41—181no—
162C3×#41—181no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C22×C3×C4—21no—
167C22×C3×C4—181no—
168C22×C3×C4C2×D911yescharacteristic
169C22×C3×C4D3211yescharacteristic
170C2×#67—61no—
171#235—32no—
172#235—32no—
173#235—61no—
174#235—61no—
175C2×D12—31no—
176C2×C4×D3—61no—
177C2×Dic6—31no—
178C3×M16—32no—
179C3×M16—61no—
180C2×C3×C8—61no—
181#217—182no—
182#217—181no—
183#217—181no—
184C4⋊Dic3—182no—
185C4⋊Dic3—181no—
186C4⋊Dic3—181no—
187C4×Dic3—92no—
188C4×Dic3—92no—
189C4×Dic3—182no—
190#208—21no—
191#208C2×D912yescomplemented
192C2×C3⋊C8—21no—
193C22×C32—41no—
194C22×C32(C2×D3)⋊C411yescharacteristic
195C22×C32(C2×D3)⋊C411yescharacteristic
196C22×C32C2×D1211yescharacteristic
197C2×C3×D3—62no—
198C2×C3×D3—122no—
199C32×C4(C2×D3)⋊C411yescharacteristic
200C32×C4(C2×D3)⋊C411yescharacteristic
201C32×C4C2×C4×D311yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—122no—
204C3×Dic3—62no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—21no—
208C22×C9—41no—
209C22×C9—42no—
210C22×C9#21711yescharacteristic
211C22×C9#21711yescharacteristic
212C22×C9C2×#6711yescharacteristic
213C2×D9—22no—
214C2×D9—42no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9#21711yescharacteristic
220C4×C9#21711yescharacteristic
221C4×C9C22×Dic311yescharacteristic
222C4×C9C2×#6711yescharacteristic
223Dic9—22no—
224Dic9—124no—
225Dic9—122no—
226C2×#41—91no—
227C2×M16—31no—
228#117—271no—
229#104—274no—
230C3×C9#42111yescharacteristic, complemented, Sylow
231C23×C3—21no—
232C23×C3—181no—
233C23×C3D3611yescharacteristic
234C23×C3#62611yescharacteristic
235C22×D3—61no—
236C3×Q8—92no—
237C3×Q8—181no—
238C3×D4—92no—
239C3×D4—182no—
240C3×D4—182no—
241C3×D4—181no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—181no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4C4×D911yescharacteristic
252C2×C3×C4C4×D911yescharacteristic
253C2×C3×C4D3611yescharacteristic
254C2×C3×C4#61111yescharacteristic
255C2×C3×C4#61111yescharacteristic
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4D3×Dic311yescharacteristic
258C2×C3×C4D3×Dic311yescharacteristic
259C2×C3×C4#62511yescharacteristic
260C2×C3×C4#62511yescharacteristic
261C2×C3×C4#62611yescharacteristic
262C2×C3×C4C2×D3211yescharacteristic
263#67—62no—
264#67—62no—
265C2×Dic3—61no—
266C2×Dic3—184no—
267C2×Dic3—182no—
268C2×Dic3—182no—
269D12—32no—
270D12—61no—
271C4×D3—62no—
272C4×D3—62no—
273Dic6—32no—
274Dic6—61no—
275C3×C8—62no—
276C3⋊C8—22no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32C2×(C2×D3)⋊C411yescharacteristic
281C3×D3—122no—
282C2×C9—22no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—41no—
288C2×C9—41no—
289C2×C9C2×#21711yescharacteristic
290D9—42no—
291#41—92no—
292#41—92no—
293#41—181no—
294#41—181no—
295C2×Q8—91no—
296C2×D4—91no—
297C2×D4—181no—
298C22×C4—181no—
299C22×C4D3×D911yescharacteristic
300M16—32no—
301M16—61no—
302C2×C8—61no—
303C4⋊C4—541no—
304C22⋊C4—541no—
305C42—272no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—42no—
312C22×C3—182no—
313C22×C3—362no—
314C22×C3(C2×D9)⋊C411yescharacteristic
315C22×C3(C2×D9)⋊C411yescharacteristic
316C22×C3C2×D3611yescharacteristic
317C22×C3(C2×D3)⋊Dic311yescharacteristic
318C22×C3(C2×D3)⋊Dic311yescharacteristic
319C22×C3C2×#62611yescharacteristic
320C2×D3—62no—
321C2×D3—122no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—182no—
327C3×C4(C2×D9)⋊C411yescharacteristic
328C3×C4(C2×D9)⋊C411yescharacteristic
329C3×C4C2×C4×D911yescharacteristic
330C3×C4C2×#61111yescharacteristic
331C3×C4(C2×D3)⋊Dic311yescharacteristic
332C3×C4(C2×D3)⋊Dic311yescharacteristic
333C3×C4C2×D3×Dic311yescharacteristic
334C3×C4C2×#62511yescharacteristic
335Dic3—62no—
336Dic3—362no—
337Dic3—364no—
338Dic3—362no—
339C32#513911yescharacteristic
340C9—21no—
341C9#523611yescharacteristic, complemented
342C23—181no—
343C23#619211yescharacteristic
344Q8—92no—
345Q8—181no—
346D4—92no—
347D4—182no—
348D4—182no—
349D4—181no—
350C2×C4—181no—
351C2×C4—182no—
352C2×C4—182no—
353C2×C4—542no—
354C2×C4Dic3×D911yescharacteristic
355C2×C4Dic3×D911yescharacteristic
356C2×C4#619211yescharacteristic
357C2×C4#619411yescharacteristic
358C2×C4#619411yescharacteristic
359C2×C4C2×D3×D911yescharacteristic
360C8—62no—
361C2×C3—22no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—362no—
372C2×C3C2×(C2×D9)⋊C411yescharacteristic
373C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
374D3—122no—
375C22—41no—
376C22—182no—
377C22—362no—
378C22(C2×D9)⋊Dic311yescharacteristic
379C22(C2×D9)⋊Dic311yescharacteristic
380C22C2×#619211yescharacteristic
381C4—182no—
382C4—1082no—
383C4(C2×D9)⋊Dic311yescharacteristic
384C4(C2×D9)⋊Dic311yescharacteristic
385C4C2×Dic3×D911yescharacteristic
386C4C2×#619411yescharacteristic
387C3—21no—
388C3#1058881911yescharacteristic, complemented
389C3#1059111011yescharacteristic
390C2—21no—
391C2—21no—
392C2—21no—
393C2—362no—
394C2C2×(C2×D9)⋊Dic311yescharacteristic
subgroupG/H
centerC4C2×#6194
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32C2×(C2×D3)⋊C4
prime radicalC2×C3×#3697C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11721815C2semidirect
3. C2×C9×#208C2semidirect
4. #11720291C2semidirect
11. C3×#3697C4semidirect
13. C2×C3×D36C4semidirect
15. C2×C3×Dic18C4semidirect
18. C4×C3⋊Dic9C22semidirect
20. C9×#208C22semidirect
72. #3697Dic3semidirect
74. C2×D36Dic3semidirect
76. C2×Dic18Dic3semidirect
104. C2×#208D9semidirect
191. #208C2×D9semidirect
230. C3×C9#421semidirect
341. C9#5236semidirect
388. C3#10588819semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
#117218152
C2×C9×#2082
#117202912
#105911103
#105888193