Finite Group Databasebeta

← #13138705

Subgroups — #13138705

Order 1728.

536 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88635C211yesmaximal, characteristic
2C2×C3×#3663C211yesmaximal, characteristic, complemented
3D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic, complemented
5C9×#816C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590671—31nomaximal
9#10588483—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×#6192—21no—
14C2×(C3⋊D9)⋊C4—21no—
15C2×Dic3×D9—21no—
16#88635—22no—
17#88635C414yescomplemented
18C2×C3×C8×C9C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C3×#3663—22no—
23Dic9⋊Dic3—21no—
24C2×#3799—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26D3×C4⋊Dic3—31no—
27C2×#3701—31no—
28#63648—31no—
29C3×#816D311yescharacteristic
30#63566—31no—
31#63564—31no—
32C2×#3663D311yescharacteristic, complemented
33C4⋊C4×D9—31no—
34C9×#103—31no—
35#63395—31no—
36#63380—31no—
37#63380—61no—
38#63369—31no—
39C22×C3⋊D9—21no—
40C22×C3×D9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)C2×C412yescomplemented
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×C4×D9C2×C412yes—
49C3×C4×D9D412yes—
50C9×Dic6—21no—
51C9×Dic6C2×C412yescomplemented
52#6192—24no—
53(C3⋊D9)⋊C4—24no—
54Dic3×D9—42no—
55C3×C8×C9—21no—
56C3×C9⋊C8—21no—
57#5133—91no—
58#4803—91no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×C4×D3—31no—
61C2×C3×Dic6C2×D311yescharacteristic
62C2×#626—61no—
63C2×#624—61no—
64C2×D3×Dic3—61no—
65#3799—34no—
66#3799—62no—
67C2×C32×C8C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)—31no—
69C3×C4⋊Dic3C2×D311yescharacteristic
70C2×C3×C3⋊C8—31no—
71C3×ΓL(1,𝔽25)—62no—
72Dic3⋊Dic3—61no—
73C2×Q8×C9—31no—
74#3701—34no—
75#3701—62no—
76C2×D36—31no—
77C2×D36—61no—
78C2×D36—61no—
79C2×C4×D9—61no—
80C2×C4×D9—61no—
81C2×C4×D9C2×D311yescharacteristic
82C2×C8×C9—21no—
83C2×C8×C9C2×D311yescharacteristic
84C9×C4⋊C4—31no—
85C4⋊Dic9—31no—
86C4⋊Dic9—61no—
87Dic9⋊C4—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89#3663—22no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—22no—
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C2×C3×D9—21no—
95C2×C3×D9—21no—
96C2×C3×D9C22⋊C412yes—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—41no—
100C9×Dic3—22no—
101C9×Dic3—41no—
102C3×Dic9C22⋊C412yes—
103C2×#239—91no—
104C2×#235—91no—
105C2×C3×M16—91no—
106C2×C4⋊Dic3—91no—
107C2×ΓL(1,𝔽25)—31no—
108D3×C4⋊C4—91no—
109C3×#103—31no—
110#835—91no—
111#821—91no—
112#821—91no—
113#821—91no—
114#821—181no—
115#816—91no—
116#816D911yescharacteristic, complemented
117#809—91no—
118C22×C3⋊D3—61no—
119C22×C3×D3—31no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—61no—
122C4⋊(C3⋊D3)—32no—
123C4⋊(C3⋊D3)—61no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—31no—
127C3×C4×D3—32no—
128C3×C4×D3—32no—
129C3×Dic6—21no—
130C3×Dic6C4×D312yes—
131#626—64no—
132#624—64no—
133D3×Dic3—122no—
134C22×D9—61no—
135C22×D9—121no—
136C22×D9D1211yescharacteristic
137C32×C8—21no—
138C3×C3⋊C8—61no—
139Q8×C9—32no—
140Q8×C9—61no—
141C2×C4×C9—21no—
142C2×C4×C9—61no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145C2×Dic9—61no—
146C2×Dic9—121no—
147C2×Dic9D1211yescharacteristic
148D36—32no—
149D36—62no—
150D36—64no—
151D36—61no—
152D36—121no—
153C4×D9—64no—
154C4×D9—122no—
155C4×D9C4×D312yes—
156C4×D9#6712yes—
157C8×C9—21no—
158C8×C9—41no—
159C9⋊C8—21no—
160#417—271noSylow
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
163C2×C3×C9#13711yescharacteristic
164C3⋊D9—42no—
165C3×D9—22no—
166C2×C3×Q8—31no—
167C22×C3×C4—91no—
168C2×#67—181no—
169C22×Dic3—181no—
170#239—94no—
171#239—182no—
172#235—94no—
173#235—182no—
174C2×D12—91no—
175C2×D12—91no—
176C2×D12—181no—
177C2×D12—181no—
178C2×C4×D3—31no—
179C2×C4×D3—181no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C2×Dic6C2×D911yescharacteristic
183C3×M16—182no—
184C2×C3×C8—21no—
185C2×C3×C8—91no—
186C2×C3×C8C2×D911yescharacteristic
187C2×C3×C8D3211yescharacteristic
188C3×C4⋊C4—31no—
189C4⋊Dic3—91no—
190C4⋊Dic3—91no—
191C4⋊Dic3—181no—
192C4⋊Dic3—181no—
193C4⋊Dic3C2×D911yescharacteristic
194Dic3⋊C4—181no—
195C2×C3⋊C8—31no—
196ΓL(1,𝔽25)—62no—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—62no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C2×C3×D3—32no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4D3×D411yescharacteristic
206C3⋊Dic3—121no—
207C3×Dic3—22no—
208C3×Dic3—32no—
209C3×Dic3—41no—
210C22×C9—21no—
211C22×C9C2×D1211yescharacteristic
212C2×D9—21no—
213C2×D9—21no—
214C2×D9—62no—
215C2×D9—122no—
216C2×D9—121no—
217C2×D9—121no—
218C2×D9—241no—
219C2×D9—241no—
220C2×D9(C2×D3)⋊C412yes—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—62no—
224C4×C9—121no—
225C4×C9C2×C4×D311yescharacteristic
226C4×C9C2×#6711yescharacteristic
227Dic9—121no—
228Dic9—241no—
229Dic9(C2×D3)⋊C412yes—
230C2×#41—271no—
231C2×M16—91no—
232C2×C4⋊C4—271no—
233#103—31no—
234#103—271no—
235#102—271no—
236#102—271no—
237C3×C9#41711yescharacteristic, complemented, Sylow
238C23×C3—91no—
239C22×D3—31no—
240C22×D3—181no—
241C22×D3—181no—
242C22×D3—361no—
243C3×Q8—32no—
244C3×Q8—61no—
245C2×C3×C4—21no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—91no—
249C2×C3×C4—92no—
250C2×C3×C4—92no—
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253#67—184no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—31no—
257C2×Dic3—181no—
258C2×Dic3—181no—
259C2×Dic3—361no—
260C2×Dic3—362no—
261D12—92no—
262D12—92no—
263D12—182no—
264D12—184no—
265D12—181no—
266D12—181no—
267D12—361no—
268C4×D3—32no—
269C4×D3—32no—
270C4×D3—184no—
271C4×D3—184no—
272C4×D3—362no—
273Dic6—21no—
274Dic6C4×D912yescomplemented
275C3×C8—21no—
276C3×C8—21no—
277C3×C8—41no—
278C3×C8—181no—
279C3⋊C8—61no—
280C2×C32D3×C22⋊C411yescharacteristic
281C2×C32#91411yescharacteristic
282C2×C32#91811yescharacteristic
283C3⋊D3—122no—
284C3×D3—62no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9#90811yescharacteristic
289C2×C9#90911yescharacteristic
290C2×C9C2×(C2×D3)⋊C411yescharacteristic
291D9—22no—
292D9—122no—
293D9—242no—
294#41—274no—
295#41—542no—
296C2×Q8—31no—
297C2×D4—271no—
298C2×D4—541no—
299C22×C4—91no—
300C22×C4—541no—
301C22×C4—541no—
302M16—182no—
303C2×C8—91no—
304C2×C8D3×D911yescharacteristic
305C4⋊C4—31no—
306C4⋊C4—271no—
307C4⋊C4—541no—
308C22×C3—21no—
309C22×C3—92no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3D4×D911yescharacteristic
313C22×C3D3×D1211yescharacteristic
314C2×D3—32no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—182no—
318C2×D3—182no—
319C2×D3—362no—
320C2×D3—361no—
321C2×D3—361no—
322C2×D3—361no—
323C2×D3—361no—
324C2×D3—721no—
325C2×D3—721no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—62no—
329C3×C4—92no—
330C3×C4—121no—
331C3×C4C2×C4×D911yescharacteristic
332C3×C4D4×D911yescharacteristic
333C3×C4C4×D3211yescharacteristic
334C3×C4D3×#6711yescharacteristic
335Dic3—22no—
336Dic3—32no—
337Dic3—41no—
338Dic3—361no—
339Dic3—361no—
340Dic3—721no—
341C32#480311yescharacteristic
342C9—21no—
343C9#513311yescharacteristic, complemented
344C23—91no—
345C23—541no—
346Q8—32no—
347Q8—61no—
348D4—272no—
349D4—544no—
350D4—541no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—544no—
357C2×C4—541no—
358C2×C4—1082no—
359C2×C4C2×D3×D911yescharacteristic
360C8—21no—
361C8—181no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—182no—
366C2×C3C22⋊C4×D911yescharacteristic
367C2×C3#6347611yescharacteristic
368C2×C3#6348011yescharacteristic
369C2×C3#6379411yescharacteristic
370C2×C3#6380111yescharacteristic
371C2×C3D3×(C2×D3)⋊C411yescharacteristic
372D3—62no—
373D3—362no—
374D3—362no—
375D3—722no—
376C22—92no—
377C22—181no—
378C22—181no—
379C22—542no—
380C22—1081no—
381C22—1081no—
382C22D9×D1211yescharacteristic
383C4—62no—
384C4—92no—
385C4—121no—
386C4—1081no—
387C4C4×D3×D911yescharacteristic
388C4D9×#6711yescharacteristic
389C3—21no—
390C3#1058848311yescharacteristic, complemented
391C3#1059067111yescharacteristic
392C2—182no—
393C2—1082no—
394C2#1172081511yescharacteristic
395C2#1172082811yescharacteristic
396C2D9×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88635C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3663C2semidirect
3. D9×C4⋊Dic3C2semidirect
4. #11720543C2semidirect
5. C9×#816C2semidirect
6. #11720298C2semidirect
7. #11720267C2semidirect
17. #88635C4semidirect
32. C2×#3663D3semidirect
44. C4⋊(C3⋊D9)C2×C4semidirect
51. C9×Dic6C2×C4semidirect
116. #816D9semidirect
237. C3×C9#417semidirect
274. Dic6C4×D9semidirect
343. C9#5133semidirect
390. C3#10588483semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C2×C3×#36632
D9×C4⋊Dic32
#117205432
C9×#8162
#117202982
#117202672
#105906713
#105884833