Finite Group Databasebeta

← #13138560

Subgroups — #13138560

Order 1728.

508 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 97 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×#88525C211yesmaximal, characteristic, complemented
2C2×C9×#204C211yesmaximal, characteristic, complemented
3#11721040C211yesmaximal, characteristic, complemented
4#11720953C211yesmaximal, characteristic, complemented
5C2×#88426C211yesmaximal, characteristic, complemented
6C2×#88422C211yesmaximal, characteristic, complemented
7C3×#63375C211yesmaximal, characteristic, complemented
8#10590881—31nomaximal
9#10588856—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×#6195—22no—
15C2×C3×C8×C9C2211yescharacteristic
16#88525—22no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19C9×#204—22no—
20#88445—22no—
21#88426—22no—
22#88422—22no—
23C2×#3745—31no—
24C2×C3×#204D311yescharacteristic
25#63911—31no—
26#63864—31no—
27C2×#3717—31no—
28C2×#3714—31no—
29C3×#815—31no—
30C2×C9×SD16—31no—
31#63509—31no—
32C2×#3675—31no—
33#63502—31no—
34C2×#3673—31no—
35C2×D72—31no—
36C2×D72—61no—
37#63375D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×D3×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D412yes—
48C9×D12—21no—
49C9×D12—21no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52#6195—44no—
53C3×C8×C9D412yescomplemented
54C3×C9⋊C8D412yescomplemented
55#5176—91no—
56#4712—91no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C3×D12C2×D311yescharacteristic
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#626—62no—
62C2×C32×C8C2×D311yescharacteristic
63#3745—62no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C3×#204—22no—
67#3722—62no—
68#3717—62no—
69#3714—62no—
70C2×Q8×C9—31no—
71C2×D4×C9—31no—
72C2×#611—62no—
73C2×D36—31no—
74C2×D36—31no—
75C2×D36—61no—
76C2×D36—61no—
77C9×SD16—62no—
78C2×C8×C9—21no—
79C2×C8×C9C2×D311yescharacteristic
80#3675—62no—
81#3673—62no—
82(C2×D9)⋊C4—62no—
83C4×Dic9C2×D311yescharacteristic
84C2×C9⋊C8C2×D311yescharacteristic
85D72—64no—
86D72—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—42no—
91C2×C3⋊D9—42no—
92C2×D3×C9—41no—
93C2×D3×C9—42no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C9×Dic3—41no—
97C3×Dic9—22no—
98C2×C3×SD16—31no—
99#947—91no—
100C2×#215—91no—
101#940—91no—
102C2×#213—91no—
103C2×D24—91no—
104C2×D24—91no—
105C2×D24—91no—
106C2×D24—181no—
107C2×#204—91no—
108C2×#204D911yescharacteristic, complemented
109#875—91no—
110#874—91no—
111C3×#97—91no—
112#815—31no—
113C22×C3⋊D3—61no—
114C22×C3⋊D3—61no—
115C22×C3×D3—21no—
116C2×C32×C4C22×D311yescharacteristic
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—61no—
120C4⋊(C3⋊D3)—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—32no—
123C3×D12—21no—
124C3×D12—21no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127#626—124no—
128C23×C9—61no—
129C22×D9—61no—
130C22×D9—61no—
131C22×D9—121no—
132C22×D9—121no—
133C32×C8#6712yes—
134C3×C3⋊C8—32no—
135Q8×C9—61no—
136Q8×C9—61no—
137D4×C9—61no—
138D4×C9—61no—
139C2×C4×C9—21no—
140C2×C4×C9—61no—
141C2×C4×C9C22×D311yescharacteristic
142#611—124no—
143C2×Dic9D1212yes—
144D36—62no—
145D36—62no—
146D36—61no—
147D36—61no—
148D36—61no—
149D36—61no—
150D36—121no—
151D36—121no—
152C8×C9—22no—
153C8×C9D1212yescomplemented
154C9⋊C8D1212yescomplemented
155#496—271noSylow
156C2×C3×C9#12711yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)12yes—
158C3⋊D9—81no—
159C3⋊D9—81no—
160D3×C9—81no—
161C2×C3×Q8—31no—
162C2×C3×D4—31no—
163C2×#67—182no—
164C2×D12—91no—
165C2×D12—91no—
166C2×D12—91no—
167C2×D12—91no—
168C2×D12—181no—
169C2×D12—181no—
170C2×D12—182no—
171C2×D12C2×D911yescharacteristic
172C2×Dic6C2×D911yescharacteristic
173C3×SD16—62no—
174C2×C3×C8—21no—
175C2×C3×C8—91no—
176C2×C3×C8C2×D911yescharacteristic
177C2×C3×C8D3211yescharacteristic
178C3×C42—91no—
179#215—182no—
180#213—182no—
181(C2×D3)⋊C4—182no—
182(C2×D3)⋊C4—182no—
183C4×Dic3—31no—
184C2×C3⋊C8—31no—
185D24—184no—
186D24—182no—
187D24—182no—
188D24—182no—
189#204—22no—
190#204—182no—
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—121no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—122no—
195C2×C3⋊D3—122no—
196C2×C3×D3—42no—
197C2×C3×D3—41no—
198C32×C4D3×D411yescharacteristic
199C32×C4C2×#6711yescharacteristic
200C3×Dic3—41no—
201C3×Dic3—62no—
202C22×C9—21no—
203C22×C9—121no—
204C22×C9—122no—
205C22×C9C2×D1211yescharacteristic
206C2×D9—122no—
207C2×D9—122no—
208C2×D9—121no—
209C2×D9—121no—
210C2×D9—122no—
211C2×D9—122no—
212C2×D9—242no—
213C2×D9—242no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—121no—
217C4×C9C2×D1211yescharacteristic
218C4×C9C2×D1211yescharacteristic
219Dic9—22no—
220C2×SD16—31no—
221C2×SD16—271no—
222C2×D8—271no—
223C2×D8—271no—
224#127—271no—
225#124—271no—
226#97—91no—
227C3×C9#49611yescharacteristic, complemented, Sylow
228C23×C3—61no—
229C22×D3—21no—
230C22×D3—181no—
231C22×D3—181no—
232C22×D3—181no—
233C22×D3—181no—
234C22×D3—361no—
235C22×D3—361no—
236C3×Q8—61no—
237C3×Q8—61no—
238C3×D4—61no—
239C3×D4—61no—
240C2×C3×C4—21no—
241C2×C3×C4—61no—
242C2×C3×C4—92no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245#67—364no—
246C2×Dic3—21no—
247C2×Dic3—32no—
248D12—21no—
249D12—21no—
250D12—182no—
251D12—182no—
252D12—181no—
253D12—181no—
254D12—181no—
255D12—181no—
256D12—181no—
257D12—181no—
258D12—181no—
259D12—181no—
260D12—361no—
261D12—361no—
262D12—364no—
263Dic6—21no—
264Dic6—21no—
265C3×C8—22no—
266C3×C8—92no—
267C3×C8#61112yescomplemented
268C3×C8#62612yes—
269C3⋊C8—32no—
270C2×C32#91412yes—
271C2×C32#94011yescharacteristic
272C3⋊D3—241no—
273C3⋊D3—241no—
274C3×D3—81no—
275C2×C9—22no—
276C2×C9—21no—
277C2×C9—241no—
278C2×C9#87411yescharacteristic
279C2×C9#90812yes—
280D9—242no—
281D9—242no—
282D9—241no—
283D9—241no—
284C2×Q8—31no—
285C2×D4—31no—
286C2×D4—271no—
287C2×D4—271no—
288C2×D4—542no—
289SD16—62no—
290SD16—542no—
291D8—542no—
292D8—542no—
293C2×C8—91no—
294C2×C8D3×D911yescharacteristic
295C22⋊C4—542no—
296C42—91no—
297C22×C3—21no—
298C22×C3—122no—
299C22×C3—121no—
300C22×C3D4×D911yescharacteristic
301C22×C3D3×D1211yescharacteristic
302C2×D3—41no—
303C2×D3—42no—
304C2×D3—362no—
305C2×D3—362no—
306C2×D3—361no—
307C2×D3—361no—
308C2×D3—362no—
309C2×D3—362no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—362no—
313C2×D3—362no—
314C2×D3—722no—
315C2×D3—722no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—121no—
319C3×C4—182no—
320C3×C4D4×D911yescharacteristic
321C3×C4C2×#61111yescharacteristic
322C3×C4D3×D1211yescharacteristic
323C3×C4C2×#62611yescharacteristic
324Dic3—41no—
325Dic3—62no—
326C32#517611yescharacteristic
327C9—21no—
328C9#471211yescharacteristic, complemented
329C23—61no—
330C23—541no—
331C23—541no—
332Q8—61no—
333Q8—61no—
334D4—61no—
335D4—61no—
336D4—541no—
337D4—541no—
338D4—541no—
339D4—541no—
340D4—1084no—
341C2×C4—61no—
342C2×C4—92no—
343C2×C4C2×D3×D911yescharacteristic
344C8—92no—
345C8#619512yescomplemented
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—241no—
349C2×C3#6347612yes—
350C2×C3#6350211yescharacteristic
351C2×C3#6379412yes—
352C2×C3#6391111yescharacteristic
353D3—81no—
354D3—722no—
355D3—722no—
356D3—721no—
357D3—721no—
358D3—721no—
359D3—721no—
360C22—121no—
361C22—122no—
362C22—1081no—
363C22—1081no—
364C22—1082no—
365C22—1082no—
366C22D9×D1211yescharacteristic
367C4—121no—
368C4—182no—
369C4D9×D1211yescharacteristic
370C4C2×#619511yescharacteristic
371C3—21no—
372C3#1058885611yescharacteristic, complemented
373C3#1059088111yescharacteristic
374C2—241no—
375C2—2161no—
376C2—2161no—
377C2#1172081512yes—
378C2#1172104011yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#496
LCSG ▷ C2×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
1. C2×#88525C2semidirect
2. C2×C9×#204C2semidirect
3. #11721040C2semidirect
4. #11720953C2semidirect
5. C2×#88426C2semidirect
6. C2×#88422C2semidirect
7. C3×#63375C2semidirect
37. #63375D3semidirect
53. C3×C8×C9D4semidirect
54. C3×C9⋊C8D4semidirect
108. C2×#204D9semidirect
153. C8×C9D12semidirect
154. C9⋊C8D12semidirect
227. C3×C9#496semidirect
267. C3×C8#611semidirect
328. C9#4712semidirect
345. C8#6195semidirect
372. C3#10588856semidirect

Maximal subgroups

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

subgroupindex
C2×#885252
C2×C9×#2042
#117210402
#117209532
C2×#884262
C2×#884222
C3×#633752
#105908813
#105888563