Finite Group Databasebeta

← #13140819

Subgroups — #13140819

Order 1728.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#63508C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3#11721036C211yesmaximal, characteristic
4C2×#88413C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720294C211yesmaximal, characteristic, complemented
7#11720279C211yesmaximal, characteristic, complemented
8#10591587—31nomaximal
9#10588854—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×#6194—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×#215—22no—
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C3×Dic9⋊C4—21no—
20C2×C9×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic9—21no—
22#88413—22no—
23C3×#946—31no—
24C2×C3×#215D311yescharacteristic
25#63908—31no—
26C2×#3711—31no—
27#63662—31no—
28#63575—31no—
29#63562—31no—
30C2×C9×SD16—31no—
31#63508D311yescharacteristic, complemented
32#63500—31no—
33C2×#3663—31no—
34#63395—31no—
35#63395—61no—
36#63392—31no—
37#63378—31no—
38C22×D3×C9—21no—
39C22×C3×D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C9×D12—21no—
48C9×D12—21no—
49C3×C4×D9—21no—
50C3×C4×D9—21no—
51#6194—42no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8D412yescomplemented
54#5174—91no—
55#5049—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C3×D12C2×D311yescharacteristic
58C2×C3×C4×D3—31no—
59C2×#625—61no—
60C4⋊(C3⋊Dic3)—31no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×#215—22no—
63C3×(C2×D3)⋊C4—61no—
64C3×C4⋊Dic3—31no—
65C3×Dic3⋊C4—61no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67(C2×D3)⋊Dic3—61no—
68#3711—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—31no—
72C2×#611—61no—
73C2×C4×D9C2×D311yescharacteristic
74C9×SD16—62no—
75C2×C8×C9—31no—
76#3676—61no—
77(C2×D9)⋊C4—21no—
78C4⋊Dic9—31no—
79C4⋊Dic9—61no—
80C4⋊Dic9C2×D311yescharacteristic
81Dic9⋊C4—21no—
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84#3663—62no—
85C22×C3×C9C2×D411yescharacteristic
86C2×D3×C9—41no—
87C2×D3×C9—41no—
88C2×D3×C9—41no—
89C2×C3×D9—21no—
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3⋊Dic9—41no—
96C3×Dic9—21no—
97C3×Dic9—41no—
98C2×C3×SD16—31no—
99C3×C22⋊Q8—91no—
100#946—31no—
101C2×#215D911yescharacteristic, complemented
102#938—91no—
103#930—91no—
104C2×#208—91no—
105C2×ΓL(1,𝔽25)—91no—
106#835—91no—
107#835—91no—
108#835—181no—
109#832—91no—
110#818—91no—
111#809—91no—
112#807—91no—
113C22×C3×D3—21no—
114C22×C3×D3—31no—
115C32×Q8—21no—
116C32×Q8—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—61no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—61no—
122C3×D12—21no—
123C3×D12—21no—
124C3×C4×D3—61no—
125C3×C4×D3—61no—
126#625—122no—
127C23×C9—61no—
128C22×D9#6711yescharacteristic
129C3⋊(C3⋊C8)—61no—
130C3×C3⋊C8#6712yes—
131Q8×C9—22no—
132Q8×C9—21no—
133Q8×C9—21no—
134Q8×C9—41no—
135D4×C9—61no—
136D4×C9—61no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9C22×D311yescharacteristic
141#611—122no—
142C2×Dic9—21no—
143C2×Dic9—61no—
144C2×Dic9—121no—
145C2×Dic9#6711yescharacteristic
146C4×D9—21no—
147C4×D9—21no—
148C8×C9—32no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#468—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
154C2×C3×C9#13711yescharacteristic
155D3×C9—81no—
156C3×D9—41no—
157C2×C3×Q8—21no—
158C2×C3×Q8C2×D911yescharacteristic
159C2×C3×Q8D3211yescharacteristic
160C2×C3×D4—31no—
161C22×C3×C4—91no—
162C2×#67—181no—
163C2×#67—181no—
164C2×D12C2×D911yescharacteristic
165C2×C4×D3—31no—
166C3×SD16—62no—
167C2×C3×C8—31no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C22⋊C4—181no—
171#217—181no—
172#215—22no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4—181no—
175C4⋊Dic3—31no—
176C4⋊Dic3—91no—
177C4⋊Dic3—91no—
178C4⋊Dic3—181no—
179Dic3⋊C4—61no—
180#208—182no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—181no—
184C2×C3⋊C8C2×D911yescharacteristic
185ΓL(1,𝔽25)—182no—
186C22×C32D3×D411yescharacteristic
187C2×C3×D3—41no—
188C2×C3×D3—41no—
189C2×C3×D3—41no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—41no—
194C32×C4#23711yescharacteristic
195C32×C4C2×#6711yescharacteristic
196C3⋊Dic3—121no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C22×C9—21no—
200C22×C9—121no—
201C22×C9—121no—
202C22×C9—121no—
203C22×C9C2×#6711yescharacteristic
204C2×D9—21no—
205C2×D9—21no—
206C2×D9—21no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—42no—
210C4×C9—41no—
211C4×C9D3×D411yescharacteristic
212C4×C9#23711yescharacteristic
213Dic9—21no—
214Dic9—41no—
215Dic9—121no—
216Dic9—241no—
217C2×SD16—31no—
218C2×M16—271no—
219C22⋊Q8—91no—
220#121—271no—
221#107—271no—
222#103—271no—
223#102—271no—
224C3×C9#46811yescharacteristic, complemented, Sylow
225C23×C3—61no—
226C23×C3—91no—
227C22×D3—21no—
228C22×D3—31no—
229C3×Q8—22no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×D4—61no—
236C3×D4—61no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4C22×D911yescharacteristic
246C2×C3×C4C2×D3211yescharacteristic
247#67—362no—
248#67—362no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253C2×Dic3—361no—
254D12—21no—
255D12—21no—
256C4×D3—61no—
257C4×D3—61no—
258C3×C8—32no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C3⋊C8—361no—
262C3⋊C8#61112yescomplemented
263C2×C32#91411yescharacteristic
264C2×C32#91511yescharacteristic
265C2×C32#93811yescharacteristic
266C3×D3—81no—
267C3×D3—121no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—241no—
272C2×C9#93911yescharacteristic
273C2×C9#94211yescharacteristic
274C2×C9#94911yescharacteristic
275D9—41no—
276C2×Q8D3×D911yescharacteristic
277C2×D4—31no—
278C2×D4—541no—
279C22×C4—91no—
280SD16—62no—
281M16—542no—
282C2×C8—31no—
283C2×C8—271no—
284C4⋊C4—91no—
285C4⋊C4—181no—
286C4⋊C4—271no—
287C22⋊C4—181no—
288C22⋊C4—541no—
289C22×C3—21no—
290C22×C3—121no—
291C22×C3—121no—
292C22×C3—121no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3D4×D911yescharacteristic
297C22×C3D3×#6711yescharacteristic
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—61no—
302C2×D3—61no—
303C2×D3—61no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—42no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—181no—
310C3×C4—361no—
311C3×C4#369911yescharacteristic
312C3×C4C2×#61111yescharacteristic
313C3×C4#380511yescharacteristic
314C3×C4D3×#6711yescharacteristic
315Dic3—61no—
316Dic3—121no—
317Dic3—361no—
318Dic3—361no—
319Dic3—721no—
320C32#517411yescharacteristic
321C9—21no—
322C9#504911yescharacteristic, complemented
323C23—61no—
324C23—91no—
325Q8—21no—
326Q8—21no—
327D4—61no—
328D4—61no—
329D4—1082no—
330C2×C4—21no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—541no—
336C2×C4C2×D3×D911yescharacteristic
337C8—32no—
338C8—541no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—241no—
343C2×C3—361no—
344C2×C3#6347611yescharacteristic
345C2×C3#6347711yescharacteristic
346C2×C3#6350011yescharacteristic
347C2×C3#6393911yescharacteristic
348C2×C3#6394611yescharacteristic
349C2×C3#6397011yescharacteristic
350D3—81no—
351D3—121no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—181no—
356C22—181no—
357C22—181no—
358C22D9×#6711yescharacteristic
359C4—41no—
360C4—181no—
361C4—361no—
362C4—1081no—
363C4#8866311yescharacteristic
364C4D3×#61111yescharacteristic
365C3—21no—
366C3#1058885411yescharacteristic, complemented
367C3#1059158711yescharacteristic
368C2—241no—
369C2—361no—
370C2#1172121111yescharacteristic
371C2#1172121411yescharacteristic
372C2#1172127011yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radical#11721036C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#468
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#63508C2semidirect
2. C2×C9×#215C2semidirect
4. C2×#88413C2semidirect
5. #11720582C2semidirect
6. #11720294C2semidirect
7. #11720279C2semidirect
31. #63508D3semidirect
53. C9×C3⋊C8D4semidirect
101. C2×#215D9semidirect
224. C3×C9#468semidirect
262. C3⋊C8#611semidirect
322. C9#5049semidirect
366. C3#10588854semidirect

Maximal subgroups

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

subgroupindex
C3×#635082
C2×C9×#2152
#117210362
C2×#884132
#117205822
#117202942
#117202792
#105915873
#105888543