Finite Group Databasebeta

← #13137697

Subgroups — #13137697

Order 1728.

378 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 89 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
1D3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2#11721037C211yesmaximal, characteristic
3#11720526C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
6#11720266C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10590500—31nomaximal
9#10588651—31nomaximal
10C2×C4×D3×C9—21no—
11C2×C4×D3×C9C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6194—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21C3×D3×C4⋊C4D311yescharacteristic
22#63908—31no—
23#63635—31no—
24C3×#809—31no—
25#63573—31no—
26#63562—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28C2×C9×C4⋊C4—31no—
29#63492—31no—
30#63389—31no—
31#63369—31no—
32#63369D311yescharacteristic, complemented
33#63367—31no—
34#63367—31no—
35#63367—61no—
36C22×D3×C9D411yescharacteristic
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3—21no—
42C2×C9×Dic3D411yescharacteristic
43C4×D3×C9—21no—
44C4×D3×C9—21no—
45C4×D3×C9—42no—
46C3×D36—21no—
47C3×D36—21no—
48#6194—42no—
49C3⋊(C9⋊C8)—21no—
50C3×C9⋊C8—21no—
51#4968—91no—
52#4777—91no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—21no—
55C2×C3×C4×D3C2×D311yescharacteristic
56C2×#625—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C2×C3⋊(C3⋊C8)—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×Dic3⋊C4—21no—
62C2×C3×C3⋊C8—31no—
63(C2×D3)⋊Dic3—61no—
64C22×C4×C9—31no—
65C22×C4×C9—61no—
66C2×#611—61no—
67C2×D36C2×D311yescharacteristic
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4—61no—
71C9×C4⋊C4C2×D311yescharacteristic
72(C2×D9)⋊C4—61no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×D3×C9—21no—
80C2×D3×C9—21no—
81C2×D3×C9SD1612yes—
82C2×C3×D9—41no—
83C2×C3×D9—41no—
84C2×C3×D9—41no—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—41no—
89C9×Dic3—21no—
90C9×Dic3—41no—
91C2×C3×C4⋊C4—31no—
92#938—91no—
93#930—91no—
94D3×C4⋊C4D911yescharacteristic, complemented
95C3×#102—91no—
96#832—91no—
97#830—91no—
98(C2×D3)⋊C8—91no—
99#817—91no—
100#809—31no—
101#809—91no—
102#808—91no—
103#808—91no—
104#808—91no—
105#808—181no—
106C22×C3×D3—61no—
107C22×C3×D3#6711yescharacteristic
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3#6711yescharacteristic
113C3×D12—61no—
114C3×D12—61no—
115C3×C4×D3—21no—
116C3×C4×D3—21no—
117C3×C4×D3—42no—
118#625—122no—
119C23×C9—31no—
120C22×D9—21no—
121C3⋊(C3⋊C8)—61no—
122C3×C3⋊C8—61no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—31no—
126C2×C4×C9—41no—
127C2×C4×C9—61no—
128C2×C4×C9—61no—
129C2×C4×C9—61no—
130C2×C4×C9—122no—
131C2×C4×C9C22×D311yescharacteristic
132#611—122no—
133C2×Dic9—61no—
134C2×Dic9—121no—
135D36—21no—
136D36—21no—
137C9⋊C8—21no—
138C9⋊C8—62no—
139C9⋊C8—61no—
140#481—271noSylow
141C2×C3×C9#12311yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144D3×C9—22no—
145C3×D9—81no—
146C2×C3×D4—91no—
147C22×C3×C4—31no—
148C22×C3×C4—61no—
149C2×#67—181no—
150C2×#67—181no—
151C2×D12—31no—
152C2×C4×D3—21no—
153C2×C4×D3C2×D911yescharacteristic
154C2×C3×C8—91no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4C2×D911yescharacteristic
159C3×C4⋊C4D3211yescharacteristic
160#217—181no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—181no—
165C4⋊Dic3C2×D911yescharacteristic
166Dic3⋊C4—21no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—181no—
171C22×C32C2×#6711yescharacteristic
172C2×C3×D3—21no—
173C2×C3×D3—21no—
174C2×C3×D3—121no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3#21512yes—
178C32×C4—41no—
179C32×C4#23511yescharacteristic
180C32×C4#23511yescharacteristic
181C3⋊Dic3—121no—
182C3×Dic3—21no—
183C3×Dic3—41no—
184C22×C9—21no—
185C22×C9—32no—
186C22×C9—61no—
187C22×C9—61no—
188C22×C9D3×D411yescharacteristic
189C2×D9—41no—
190C2×D9—41no—
191C2×D9—41no—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—41no—
195C4×C9—61no—
196C4×C9—81no—
197C4×C9—121no—
198C4×C9#23511yescharacteristic
199C4×C9#23711yescharacteristic
200Dic9—121no—
201Dic9—241no—
202#121—271no—
203C2×C4⋊C4—31no—
204#106—271no—
205#106—271no—
206#102—91no—
207#102—271no—
208C22⋊C8—271no—
209C3×C9#48111yescharacteristic, complemented, Sylow
210C23×C3—31no—
211C23×C3—181no—
212C22×D3—61no—
213C22×D3#61111yescharacteristic
214C3×D4—181no—
215C3×D4—181no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—41no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—122no—
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227#67—362no—
228#67—362no—
229C2×Dic3—21no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232C2×Dic3—361no—
233C2×Dic3#61111yescharacteristic
234D12—61no—
235D12—61no—
236C4×D3—21no—
237C4×D3—21no—
238C4×D3—42no—
239C3×C8—181no—
240C3⋊C8—61no—
241C3⋊C8—182no—
242C3⋊C8—181no—
243C3⋊C8—181no—
244C2×C32#92911yescharacteristic
245C2×C32#93211yescharacteristic
246C2×C32C2×#21511yescharacteristic
247C3×D3—22no—
248C3×D3—241no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—62no—
253C2×C9#88311yescharacteristic
254C2×C9#91111yescharacteristic
255C2×C9D3×SD1611yescharacteristic
256D9—81no—
257C2×D4—91no—
258C2×D4—541no—
259C22×C4—31no—
260C22×C4—61no—
261C2×C8—91no—
262C2×C8—271no—
263C4⋊C4—31no—
264C4⋊C4—61no—
265C4⋊C4—271no—
266C4⋊C4D3×D911yescharacteristic
267C22⋊C4—541no—
268C22×C3—21no—
269C22×C3—32no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3C2×#61111yescharacteristic
276C22×C3D3×#6711yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3#367512yes—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—61no—
288C3×C4—81no—
289C3×C4—121no—
290C3×C4#369711yescharacteristic
291C3×C4#369711yescharacteristic
292C3×C4#379511yescharacteristic
293C3×C4#379811yescharacteristic
294Dic3—21no—
295Dic3—41no—
296Dic3—361no—
297Dic3—361no—
298Dic3—721no—
299C32#496811yescharacteristic
300C9—21no—
301C9#477711yescharacteristic, complemented
302C23—31no—
303C23—181no—
304D4—181no—
305D4—181no—
306D4—1082no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—541no—
314C2×C4C2×D3×D911yescharacteristic
315C8—181no—
316C8—541no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—62no—
321C2×C3—721no—
322C2×C3#6349111yescharacteristic
323C2×C3#6349511yescharacteristic
324C2×C3C2×#367511yescharacteristic
325C2×C3#6390511yescharacteristic
326C2×C3#6393211yescharacteristic
327C2×C3D3×#21511yescharacteristic
328D3—22no—
329D3—241no—
330C22—32no—
331C22—61no—
332C22—61no—
333C22—361no—
334C22—361no—
335C22—361no—
336C22D3×#61111yescharacteristic
337C4—41no—
338C4—61no—
339C4—121no—
340C4—1081no—
341C4#8864011yescharacteristic
342C4#8864111yescharacteristic
343C3—21no—
344C3#1058865111yescharacteristic, complemented
345C3#1059050011yescharacteristic
346C2—62no—
347C2—721no—
348C2#1172103211yescharacteristic
349C2#1172116811yescharacteristic
350C2D3×#367511yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radical#11721037C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#481
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. D3×C9×C4⋊C4C2semidirect
3. #11720526C2semidirect
4. C3×#63369C2semidirect
5. (C2×D3)⋊(C9⋊C8)C2semidirect
6. #11720266C2semidirect
7. #11720264C2semidirect
32. #63369D3semidirect
94. D3×C4⋊C4D9semidirect
209. C3×C9#481semidirect
301. C9#4777semidirect
344. C3#10588651semidirect

Maximal subgroups

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

subgroupindex
D3×C9×C4⋊C42
#117210372
#117205262
C3×#633692
(C2×D3)⋊(C9⋊C8)2
#117202662
#117202642
#105905003
#105886513