Finite Group Databasebeta

← #13139475

Subgroups — #13139475

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×C9×#235C211yesmaximal, characteristic
2#11721815C211yesmaximal, characteristic, complemented
3C2×C3×#3667C211yesmaximal, characteristic, complemented
4#11720300C214yesmaximal, complemented
5#10591110—31nomaximal
6#10588914—31nomaximal
7C22×C3×C4×C9C2211yescharacteristic
8C2×C9×#67—21no—
9C9×#235—21no—
10C9×#235—21no—
11C9×#235C412yescomplemented
12C9×#235C2212yes—
13C2×C9×D12C411yescharacteristic, complemented
14C2×C4×D3×C9—21no—
15C2×C9×Dic6C411yescharacteristic, complemented
16#88551—21no—
17C4⋊(C3⋊Dic9)—21no—
18C4×C3⋊Dic9C2212yescomplemented
19C3×#3667—21no—
20C3×#3667C2212yescomplemented
21C2×C3×C9⋊C8—21no—
22C2×C3×#235D311yescharacteristic
23#64135—31no—
24C2×C3×#208—31no—
25C2×C9×#41—31no—
26#63568—34no—
27#63488—31no—
28#63488—61no—
29C2×#3667D311yescharacteristic, complemented
30#63396—34no—
31C23×C3×C9D411yescharacteristic
32C22×D3×C9—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—
40C9×#67—22no—
41C9×#67—22no—
42C2×C9×Dic3—21no—
43C9×D12—21no—
44C9×D12C2×C412yes—
45C4×D3×C9—22no—
46C4×D3×C9—22no—
47C9×Dic6—21no—
48C9×Dic6C2×C412yes—
49C3×C9⋊C8—22no—
50#5236—91no—
51#5139—91no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—21no—
54C3×#235—21no—
55C3×#235—21no—
56C3×#235Dic312yes—
57C3×#235C2×D312yes—
58C2×C3×D12Dic311yescharacteristic
59C2×C3×C4×D3—21no—
60C2×C3×Dic6Dic311yescharacteristic
61#3757—61no—
62C4⋊(C3⋊Dic3)—61no—
63C4×C3⋊Dic3—32no—
64C3×#208—32no—
65C3×#208—61no—
66C2×C3×C3⋊C8—61no—
67C9×#41—32no—
68C9×#41—32no—
69C9×#41—61no—
70C9×#41—61no—
71C2×Q8×C9—31no—
72C2×D4×C9—31no—
73C2×D4×C9—61no—
74C22×C4×C9—21no—
75C22×C4×C9—61no—
76C22×C4×C9C2×D311yescharacteristic
77#3676—62no—
78#3676—61no—
79C4⋊Dic9—62no—
80C4⋊Dic9—61no—
81C4×Dic9—32no—
82C4×Dic9—62no—
83#3667—21no—
84#3667C2×D312yescomplemented
85C2×C9⋊C8—21no—
86C22×C3×C9—41no—
87C22×C3×C9C22⋊C411yescharacteristic
88C22×C3×C9C22⋊C411yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×D3×C9—22no—
91C2×D3×C9—42no—
92C3×C4×C9C22⋊C411yescharacteristic
93C3×C4×C9C22⋊C411yescharacteristic
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—42no—
97C9×Dic3—22no—
98C2×C3×#41—31no—
99C2×#235D911yescharacteristic
100C2×C3×M16—91no—
101#926—91no—
102#926—91no—
103#926—181no—
104C2×#208—31no—
105#837—94no—
106#825—94no—
107C23×C32#6711yescharacteristic
108C22×C3×D3—21no—
109C2×C32×C4C2×Dic311yescharacteristic
110C2×C32×C4C2×Dic311yescharacteristic
111C2×C32×C4#6711yescharacteristic
112C2×C32×C4#6711yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—62no—
116C3×#67—22no—
117C3×#67—22no—
118C2×C3×Dic3—21no—
119C3×D12—21no—
120C3×D12C2×Dic312yes—
121C3×C4×D3—22no—
122C3×C4×D3—22no—
123C3×Dic6—21no—
124C3×Dic6C2×Dic312yes—
125C23×C9—21no—
126C23×C9—61no—
127C23×C9D1211yescharacteristic
128C3×C3⋊C8—62no—
129Q8×C9—32no—
130Q8×C9—61no—
131D4×C9—32no—
132D4×C9—62no—
133D4×C9—62no—
134D4×C9—61no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—61no—
142C2×C4×C9—62no—
143C2×C4×C9—62no—
144C2×C4×C9C4×D311yescharacteristic
145C2×C4×C9C4×D311yescharacteristic
146C2×C4×C9D1211yescharacteristic
147C2×C4×C9#6711yescharacteristic
148C2×C4×C9#6711yescharacteristic
149C2×C4×C9C22×D311yescharacteristic
150C2×Dic9—64no—
151C2×Dic9—62no—
152C9⋊C8—22no—
153#421—271noSylow
154C2×C3×C9—21no—
155C2×C3×C9—21no—
156C2×C3×C9—21no—
157C2×C3×C9C2×C22⋊C411yescharacteristic
158D3×C9—42no—
159C3×#41—32no—
160C3×#41—32no—
161C3×#41—61no—
162C3×#41—61no—
163C2×C3×Q8—31no—
164C2×C3×D4—31no—
165C2×C3×D4—61no—
166C22×C3×C4—21no—
167C22×C3×C4—61no—
168C22×C3×C4C2×D911yescharacteristic
169C22×C3×C4D3211yescharacteristic
170C2×#67—21no—
171#235—21no—
172#235—21no—
173#235Dic912yescomplemented
174#235C2×D912yes—
175C2×D12Dic911yescharacteristic, complemented
176C2×C4×D3—21no—
177C2×Dic6Dic911yescharacteristic, complemented
178C3×M16—92no—
179C3×M16—181no—
180C2×C3×C8—181no—
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—32no—
191#208—61no—
192C2×C3⋊C8—61no—
193C22×C32—41no—
194C22×C32#21711yescharacteristic
195C22×C32#21711yescharacteristic
196C22×C32C2×#6711yescharacteristic
197C2×C3×D3—22no—
198C2×C3×D3—42no—
199C32×C4#21711yescharacteristic
200C32×C4#21711yescharacteristic
201C32×C4C22×Dic311yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—122no—
204C3×Dic3—22no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—21no—
208C22×C9—41no—
209C22×C9—42no—
210C22×C9—62no—
211C22×C9—122no—
212C22×C9(C2×D3)⋊C411yescharacteristic
213C22×C9(C2×D3)⋊C411yescharacteristic
214C22×C9C2×D1211yescharacteristic
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—62no—
220C4×C9(C2×D3)⋊C411yescharacteristic
221C4×C9(C2×D3)⋊C411yescharacteristic
222C4×C9C2×C4×D311yescharacteristic
223C4×C9C2×#6711yescharacteristic
224Dic9—124no—
225Dic9—122no—
226C2×#41—31no—
227C2×M16—91no—
228#117—271no—
229#104—274no—
230C3×C9#42111yescharacteristic, complemented, Sylow
231C23×C3—21no—
232C23×C3—61no—
233C23×C3#61111yescharacteristic
234C23×C3#62611yescharacteristic
235C22×D3—21no—
236C3×Q8—32no—
237C3×Q8—61no—
238C3×D4—32no—
239C3×D4—62no—
240C3×D4—62no—
241C3×D4—61no—
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—61no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4C2×Dic911yescharacteristic
252C2×C3×C4C2×Dic911yescharacteristic
253C2×C3×C4#61111yescharacteristic
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—22no—
264#67—22no—
265C2×Dic3—21no—
266C2×Dic3—184no—
267C2×Dic3—182no—
268C2×Dic3—182no—
269D12—21no—
270D12C2×Dic912yes—
271C4×D3—22no—
272C4×D3—22no—
273Dic6—21no—
274Dic6C2×Dic912yes—
275C3×C8—182no—
276C3⋊C8—62no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32C2×#21711yescharacteristic
281C3×D3—42no—
282C2×C9—22no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—41no—
288C2×C9—41no—
289C2×C9—122no—
290C2×C9C2×(C2×D3)⋊C411yescharacteristic
291#41—32no—
292#41—32no—
293#41—61no—
294#41—61no—
295C2×Q8—31no—
296C2×D4—31no—
297C2×D4—61no—
298C22×C4—61no—
299C22×C4D3×D911yescharacteristic
300M16—92no—
301M16—181no—
302C2×C8—181no—
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—62no—
313C22×C3—122no—
314C22×C3#367611yescharacteristic
315C22×C3#367611yescharacteristic
316C22×C3C2×#61111yescharacteristic
317C22×C3(C2×D3)⋊Dic311yescharacteristic
318C22×C3(C2×D3)⋊Dic311yescharacteristic
319C22×C3C2×#62611yescharacteristic
320C2×D3—22no—
321C2×D3—42no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—62no—
327C3×C4#367611yescharacteristic
328C3×C4#367611yescharacteristic
329C3×C4C22×Dic911yescharacteristic
330C3×C4C2×#61111yescharacteristic
331C3×C4(C2×D3)⋊Dic311yescharacteristic
332C3×C4(C2×D3)⋊Dic311yescharacteristic
333C3×C4C2×D3×Dic311yescharacteristic
334C3×C4C2×#62511yescharacteristic
335Dic3—22no—
336Dic3—362no—
337Dic3—364no—
338Dic3—362no—
339C32#523611yescharacteristic
340C9—21no—
341C9#513911yescharacteristic, complemented
342C23—61no—
343C23#619511yescharacteristic
344Q8—32no—
345Q8—61no—
346D4—32no—
347D4—62no—
348D4—62no—
349D4—61no—
350C2×C4—61no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—542no—
354C2×C4D3×Dic911yescharacteristic
355C2×C4D3×Dic911yescharacteristic
356C2×C4#619411yescharacteristic
357C2×C4#619411yescharacteristic
358C2×C4#619511yescharacteristic
359C2×C4C2×D3×D911yescharacteristic
360C8—182no—
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—122no—
372C2×C3C2×#367611yescharacteristic
373C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
374D3—42no—
375C22—41no—
376C22—62no—
377C22—122no—
378C22(C2×D3)⋊Dic911yescharacteristic
379C22(C2×D3)⋊Dic911yescharacteristic
380C22C2×#619511yescharacteristic
381C4—62no—
382C4—1082no—
383C4(C2×D3)⋊Dic911yescharacteristic
384C4(C2×D3)⋊Dic911yescharacteristic
385C4C2×D3×Dic911yescharacteristic
386C4C2×#619411yescharacteristic
387C3—21no—
388C3#1058891411yescharacteristic, complemented
389C3#1059111011yescharacteristic
390C2—21no—
391C2—21no—
392C2—21no—
393C2—122no—
394C2C2×(C2×D3)⋊Dic911yescharacteristic
subgroupG/H
centerC4C2×#6194
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32C2×#217
prime radicalC2×C3×#235D3
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×C3×#3667C2semidirect
4. #11720300C2semidirect
11. C9×#235C4semidirect
13. C2×C9×D12C4semidirect
15. C2×C9×Dic6C4semidirect
18. C4×C3⋊Dic9C22semidirect
20. C3×#3667C22semidirect
29. C2×#3667D3semidirect
84. #3667C2×D3semidirect
173. #235Dic9semidirect
175. C2×D12Dic9semidirect
177. C2×Dic6Dic9semidirect
230. C3×C9#421semidirect
341. C9#5139semidirect
388. C3#10588914semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
#117218152
C2×C3×#36672
#117203002
#105911103
#105889143