Finite Group Databasebeta

← #13139461

Subgroups — #13139461

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic
2#11721815C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4#11720332C214yesmaximal, complemented
5#10591231—31nomaximal
6#10588920—31nomaximal
7C22×C3×C4×C9C2211yescharacteristic
8C2×C9×#67C2212yes—
9C9×#235—21no—
10C9×#235—21no—
11C9×#235—21no—
12C9×#235—21no—
13C2×C9×D12C411yescharacteristic, complemented
14C2×C4×D3×C9C412yescomplemented
15C2×C9×Dic6C411yescharacteristic, complemented
16#88551C2212yes—
17C4⋊(C3⋊Dic9)—21no—
18C4×C3⋊Dic9—21no—
19C3×#3676C2212yes—
20C3×C4⋊Dic9—21no—
21C3×C4×Dic9—21no—
22C2×C3×#235D311yescharacteristic
23#64135—31no—
24C3×#926—31no—
25C2×C9×#41—31no—
26#63580—34no—
27#63488—31no—
28#63488—61no—
29#63488D311yescharacteristic, complemented
30#63393—34no—
31C23×C3×C9C2311yescharacteristic
32C22×D3×C9C2×C412yes—
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×C9D411yescharacteristic
39C2×C3⋊Dic9—22no—
40C9×#67—22no—
41C9×#67—22no—
42C2×C9×Dic3C2×C412yes—
43C2×C3×Dic9—22no—
44C9×D12—21no—
45C9×D12—21no—
46C4×D3×C9—22no—
47C4×D3×C9—22no—
48C9×Dic6—21no—
49C9×Dic6—21no—
50#5240—91no—
51#5006—91no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67C2×D312yes—
54C3×#235—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235—21no—
58C2×C3×D12Dic311yescharacteristic
59C2×C3×C4×D3Dic312yes—
60C2×C3×Dic6Dic311yescharacteristic
61#3757—32no—
62C4⋊(C3⋊Dic3)—61no—
63C4×C3⋊Dic3—61no—
64C3×#217—32no—
65C3×C4⋊Dic3—61no—
66C3×C4×Dic3—61no—
67C9×#41—61no—
68C9×#41—61no—
69C9×#41—61no—
70C9×#41—61no—
71C2×Q8×C9—31no—
72C2×D4×C9—32no—
73C2×D4×C9—31no—
74C22×C4×C9—21no—
75C22×C4×C9—32no—
76C22×C4×C9C2×D311yescharacteristic
77#3676—32no—
78#3676—62no—
79#3676C2×D312yes—
80C4⋊Dic9—21no—
81C4⋊Dic9—62no—
82C4⋊Dic9—61no—
83C4×Dic9—21no—
84C4×Dic9—62no—
85C4×Dic9—61no—
86C22×C3×C9—41no—
87C22×C3×C9C22×C411yescharacteristic
88C22×C3×C9C2×D411yescharacteristic
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×C9C22⋊C411yescharacteristic
96C3⋊Dic9—42no—
97C9×Dic3—22no—
98C3×Dic9—42no—
99C2×C3×#41—31no—
100C2×#235D911yescharacteristic
101C3×#117—91no—
102#926—31no—
103#926—91no—
104#926—91no—
105#926—181no—
106#834—94no—
107#806—94no—
108C23×C32C22×D311yescharacteristic
109C22×C3×D3C2×Dic312yes—
110C2×C32×C4C2×Dic311yescharacteristic
111C2×C32×C4C2×Dic311yescharacteristic
112C2×C32×C4#6711yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4#6711yescharacteristic
116C2×C3⋊Dic3—62no—
117C3×#67—22no—
118C3×#67—22no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3C2×Dic312yes—
121C3×D12—21no—
122C3×D12—21no—
123C3×C4×D3—22no—
124C3×C4×D3—22no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C23×C9—21no—
128C23×C9—32no—
129C23×C9C22×D311yescharacteristic
130Q8×C9—61no—
131Q8×C9—61no—
132D4×C9—62no—
133D4×C9—62no—
134D4×C9—61no—
135D4×C9—61no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—32no—
143C2×C4×C9—62no—
144C2×C4×C9—62no—
145C2×C4×C9C4×D311yescharacteristic
146C2×C4×C9C4×D311yescharacteristic
147C2×C4×C9D1211yescharacteristic
148C2×C4×C9D1211yescharacteristic
149C2×C4×C9#6711yescharacteristic
150C2×C4×C9#6711yescharacteristic
151C2×Dic9—22no—
152C2×Dic9—64no—
153C2×Dic9—62no—
154#410—271noSylow
155C2×C3×C9—21no—
156C2×C3×C9—21no—
157C2×C3×C9—21no—
158C2×C3×C9C2×C22⋊C411yescharacteristic
159D3×C9—42no—
160C3×#41—61no—
161C3×#41—61no—
162C3×#41—61no—
163C3×#41—61no—
164C2×C3×Q8—31no—
165C2×C3×D4—32no—
166C2×C3×D4—31no—
167C22×C3×C4—21no—
168C22×C3×C4—32no—
169C22×C3×C4C2×D911yescharacteristic
170C22×C3×C4D3211yescharacteristic
171C2×#67C2×D912yes—
172#235—21no—
173#235—21no—
174#235—21no—
175#235—21no—
176C2×D12Dic911yescharacteristic, complemented
177C2×C4×D3Dic912yescomplemented
178C2×Dic6Dic911yescharacteristic, complemented
179C3×C4⋊C4—181no—
180C3×C22⋊C4—92no—
181C3×C42—181no—
182#217—32no—
183#217—92no—
184#217—92no—
185#217—182no—
186C4⋊Dic3—61no—
187C4⋊Dic3—182no—
188C4⋊Dic3—181no—
189C4⋊Dic3—181no—
190C4×Dic3—61no—
191C4×Dic3—182no—
192C4×Dic3—181no—
193C4×Dic3—181no—
194C22×C32—41no—
195C22×C32C22×Dic311yescharacteristic
196C22×C32C2×#6711yescharacteristic
197C22×C32C2×#6711yescharacteristic
198C2×C3×D3—22no—
199C2×C3×D3—42no—
200C32×C4#21711yescharacteristic
201C32×C4#21711yescharacteristic
202C32×C4#21711yescharacteristic
203C32×C4#21711yescharacteristic
204C3⋊Dic3—122no—
205C3×Dic3—22no—
206C3×Dic3—122no—
207C22×C9—21no—
208C22×C9—21no—
209C22×C9—21no—
210C22×C9—41no—
211C22×C9—42no—
212C22×C9—62no—
213C22×C9—122no—
214C22×C9C2×C4×D311yescharacteristic
215C22×C9C2×D1211yescharacteristic
216C22×C9C2×#6711yescharacteristic
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—62no—
222C4×C9(C2×D3)⋊C411yescharacteristic
223C4×C9(C2×D3)⋊C411yescharacteristic
224C4×C9(C2×D3)⋊C411yescharacteristic
225C4×C9(C2×D3)⋊C411yescharacteristic
226Dic9—42no—
227Dic9—124no—
228Dic9—122no—
229C2×#41—31no—
230#117—91no—
231#117—271no—
232#99—274no—
233C3×C9#41011yescharacteristic, complemented, Sylow
234C23×C3—21no—
235C23×C3—32no—
236C23×C3C22×D911yescharacteristic
237C23×C3C2×D3211yescharacteristic
238C22×D3C2×Dic912yes—
239C3×Q8—61no—
240C3×Q8—61no—
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—61no—
244C3×D4—61no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—32no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—182no—
255C2×C3×C4C2×Dic911yescharacteristic
256C2×C3×C4C2×Dic911yescharacteristic
257C2×C3×C4#61111yescharacteristic
258C2×C3×C4#61111yescharacteristic
259C2×C3×C4#61111yescharacteristic
260C2×C3×C4#61111yescharacteristic
261C2×C3×C4D3×Dic311yescharacteristic
262C2×C3×C4D3×Dic311yescharacteristic
263C2×C3×C4#62511yescharacteristic
264C2×C3×C4#62511yescharacteristic
265C2×C3×C4#62611yescharacteristic
266C2×C3×C4#62611yescharacteristic
267#67—22no—
268#67—22no—
269C2×Dic3—62no—
270C2×Dic3—184no—
271C2×Dic3—182no—
272C2×Dic3—182no—
273C2×Dic3C2×Dic912yes—
274D12—21no—
275D12—21no—
276C4×D3—22no—
277C4×D3—22no—
278Dic6—21no—
279Dic6—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32C2×#21711yescharacteristic
284C3×D3—42no—
285C2×C9—22no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—41no—
291C2×C9—41no—
292C2×C9—122no—
293C2×C9C2×(C2×D3)⋊C411yescharacteristic
294#41—61no—
295#41—61no—
296#41—61no—
297#41—61no—
298C2×Q8—31no—
299C2×D4—32no—
300C2×D4—31no—
301C22×C4—32no—
302C22×C4D3×D911yescharacteristic
303C4⋊C4—181no—
304C4⋊C4—541no—
305C22⋊C4—92no—
306C22⋊C4—272no—
307C42—181no—
308C42—541no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—42no—
315C22×C3—62no—
316C22×C3—122no—
317C22×C3C22×Dic911yescharacteristic
318C22×C3C2×#61111yescharacteristic
319C22×C3C2×#61111yescharacteristic
320C22×C3C2×D3×Dic311yescharacteristic
321C22×C3C2×#62511yescharacteristic
322C22×C3C2×#62611yescharacteristic
323C2×D3—22no—
324C2×D3—42no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—62no—
330C3×C4—362no—
331C3×C4#367611yescharacteristic
332C3×C4#367611yescharacteristic
333C3×C4#367611yescharacteristic
334C3×C4#367611yescharacteristic
335C3×C4(C2×D3)⋊Dic311yescharacteristic
336C3×C4(C2×D3)⋊Dic311yescharacteristic
337C3×C4(C2×D3)⋊Dic311yescharacteristic
338C3×C4(C2×D3)⋊Dic311yescharacteristic
339Dic3—22no—
340Dic3—122no—
341Dic3—362no—
342Dic3—364no—
343Dic3—362no—
344C32#524011yescharacteristic
345C9—21no—
346C9#500611yescharacteristic, complemented
347C23—32no—
348C23C2×D3×D911yescharacteristic
349Q8—61no—
350Q8—61no—
351D4—62no—
352D4—62no—
353D4—61no—
354D4—61no—
355C2×C4—32no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—182no—
359C2×C4—542no—
360C2×C4D3×Dic911yescharacteristic
361C2×C4D3×Dic911yescharacteristic
362C2×C4#619411yescharacteristic
363C2×C4#619411yescharacteristic
364C2×C4#619511yescharacteristic
365C2×C4#619511yescharacteristic
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—122no—
377C2×C3C2×#367611yescharacteristic
378C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
379D3—42no—
380C22—41no—
381C22—62no—
382C22—122no—
383C22C2×D3×Dic911yescharacteristic
384C22C2×#619411yescharacteristic
385C22C2×#619511yescharacteristic
386C4—62no—
387C4—362no—
388C4—1082no—
389C4(C2×D3)⋊Dic911yescharacteristic
390C4(C2×D3)⋊Dic911yescharacteristic
391C4(C2×D3)⋊Dic911yescharacteristic
392C4(C2×D3)⋊Dic911yescharacteristic
393C3—21no—
394C3#1058892011yescharacteristic, complemented
395C3#1059123111yescharacteristic
396C2—21no—
397C2—21no—
398C2—21no—
399C2—122no—
400C2C2×(C2×D3)⋊Dic911yescharacteristic
subgroupG/H
centerC4(C2×D3)⋊Dic9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32C2×#217
prime radicalC2×C3×#235D3
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#410
LCSG ▷ C22×C3×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. C3×#63488C2semidirect
4. #11720332C2semidirect
13. C2×C9×D12C4semidirect
14. C2×C4×D3×C9C4semidirect
15. C2×C9×Dic6C4semidirect
29. #63488D3semidirect
176. C2×D12Dic9semidirect
177. C2×C4×D3Dic9semidirect
178. C2×Dic6Dic9semidirect
233. C3×C9#410semidirect
346. C9#5006semidirect
394. C3#10588920semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
#117218152
C3×#634882
#117203322
#105912313
#105889203