Finite Group Databasebeta

← #13138311

Subgroups — #13138311

Order 1728.

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

normal subgroups only

This group has 374 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
1#11720992C211yesmaximal, characteristic
2#11720969C211yesmaximal, characteristic, complemented
3#11720543C211yesmaximal, characteristic
4C9×#817C211yesmaximal, characteristic, complemented
5C3×Dic9⋊C8C211yesmaximal, characteristic, complemented
6#11720267C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic, complemented
8#10590759—31nomaximal
9#10588550—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×(C3⋊D9)⋊C4—21no—
12C2×#6189—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic9⋊Dic3—21no—
21#63918—31no—
22#63894—31no—
23#63648—31no—
24C3×#817D311yescharacteristic
25C3×Dic3⋊C8—31no—
26#63573—31no—
27#63564—31no—
28#63455—31no—
29#63450—31no—
30C9×#106—31no—
31#63380—31no—
32#63380—61no—
33Dic9⋊C8D311yescharacteristic, complemented
34#63369—31no—
35#63367—31no—
36C22×C3⋊D9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C4⋊(C3⋊D9)—21no—
40C4⋊(C3⋊D9)Q812yescomplemented
41C2×C9×Dic3—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45(C3⋊D9)⋊C4—42no—
46#6189—42no—
47C3×C8×C9—21no—
48C3×C9⋊C8—21no—
49#4870—91no—
50#4842—91no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×#627—61no—
53C2×#624—61no—
54C2×C32×C8C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×C4⋊Dic3C2×D311yescharacteristic
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8—31no—
60Dic3⋊Dic3—61no—
61#3722—61no—
62C2×D36—31no—
63C2×D36—61no—
64C2×C4×D9—61no—
65C2×Dic18—61no—
66C2×C8×C9—21no—
67C2×C8×C9C2×D311yescharacteristic
68C9×C4⋊C4—31no—
69C9×C4⋊C4—31no—
70(C2×D9)⋊C4—61no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73Dic9⋊C4—61no—
74C4×Dic9C2×D311yescharacteristic
75C2×C9⋊C8C2×D311yescharacteristic
76C22×C3×C9C2×D411yescharacteristic
77C2×C3⋊D9—22no—
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C3×C4×C9C2×Q811yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—41no—
83C9×Dic3—41no—
84C9×Dic3—41no—
85C3×Dic9—21no—
86C3×Dic9SD1612yes—
87#893—91no—
88#888—91no—
89C4×D12—91no—
90#869—91no—
91C3×#106—31no—
92C3×C4⋊C8—91no—
93#821—91no—
94#821—91no—
95#821—91no—
96#821—181no—
97#817—91no—
98#817D911yescharacteristic, complemented
99Dic3⋊C8—31no—
100#809—91no—
101#808—91no—
102C22×C3⋊D3—61no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—61no—
105C4⋊(C3⋊D3)—32no—
106C4⋊(C3⋊D3)—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111#627—122no—
112#624—122no—
113C22×D9—61no—
114C22×D9—121no—
115C32×C8—21no—
116C3×C3⋊C8—61no—
117C2×C4×C9—21no—
118C2×C4×C9—61no—
119C2×C4×C9—61no—
120C2×C4×C9C22×D311yescharacteristic
121C2×Dic9—61no—
122C2×Dic9—121no—
123C2×Dic9D1211yescharacteristic
124C2×Dic9D1211yescharacteristic
125D36—32no—
126D36—62no—
127D36—61no—
128D36—121no—
129C4×D9—122no—
130Dic18—122no—
131C8×C9—21no—
132C8×C9—41no—
133C9⋊C8—21no—
134#476—271noSylow
135C2×C3×C9C22⋊Q811yescharacteristic
136C2×C3×C9C2×SD1611yescharacteristic
137C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
138C3⋊D9—42no—
139C2×D12—91no—
140C2×D12—91no—
141C2×D12—181no—
142C2×C4×D3—181no—
143C2×C4×D3—181no—
144C2×Dic6—181no—
145C2×Dic6—181no—
146C2×C3×C8—21no—
147C2×C3×C8—91no—
148C2×C3×C8C2×D911yescharacteristic
149C2×C3×C8D3211yescharacteristic
150C3×C4⋊C4—31no—
151C3×C4⋊C4—31no—
152C3×C42—91no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155C4⋊Dic3—91no—
156C4⋊Dic3—91no—
157C4⋊Dic3—181no—
158C4⋊Dic3—181no—
159C4⋊Dic3C2×D911yescharacteristic
160C4⋊Dic3C2×D911yescharacteristic
161Dic3⋊C4—181no—
162C4×Dic3—31no—
163C2×C3⋊C8—31no—
164C22×C32D3×D411yescharacteristic
165C2×C3⋊D3—62no—
166C2×C3⋊D3—121no—
167C2×C3⋊D3—121no—
168C32×C4#23511yescharacteristic
169C32×C4D3×Q811yescharacteristic
170C3⋊Dic3—121no—
171C3×Dic3—32no—
172C3×Dic3—41no—
173C3×Dic3—41no—
174C3×Dic3—61no—
175C22×C9—21no—
176C22×C9C2×D1211yescharacteristic
177C2×D9—62no—
178C2×D9—122no—
179C2×D9—121no—
180C2×D9—121no—
181C2×D9—241no—
182C2×D9—241no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—121no—
186C4×C9—121no—
187C4×C9#23711yescharacteristic
188C4×C9D3×Q811yescharacteristic
189Dic9—21no—
190Dic9—121no—
191Dic9—241no—
192Dic9#20412yes—
193C4⋊Q8—271no—
194C4×D4—271no—
195#106—31no—
196#106—271no—
197C4⋊C8—91no—
198#102—271no—
199#102—271no—
200C3×C9#47611yescharacteristic, complemented, Sylow
201C22×D3—181no—
202C22×D3—181no—
203C22×D3—361no—
204C2×C3×C4—21no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4C22×D911yescharacteristic
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—31no—
214C2×Dic3—31no—
215C2×Dic3—181no—
216C2×Dic3—181no—
217C2×Dic3—361no—
218D12—92no—
219D12—92no—
220D12—182no—
221D12—181no—
222D12—181no—
223D12—361no—
224C4×D3—362no—
225C4×D3—362no—
226Dic6—362no—
227Dic6—362no—
228C3×C8—21no—
229C3×C8—21no—
230C3×C8—41no—
231C3×C8—181no—
232C3⋊C8—61no—
233C2×C32#89611yescharacteristic
234C2×C32D3×SD1611yescharacteristic
235C2×C32#91411yescharacteristic
236C3⋊D3—122no—
237C2×C9—21no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9#89711yescharacteristic
241C2×C9C2×#20411yescharacteristic
242C2×C9#90811yescharacteristic
243D9—122no—
244D9—242no—
245C2×Q8—541no—
246C2×D4—271no—
247C22×C4—541no—
248C2×C8—91no—
249C2×C8D3×D911yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—271no—
253C4⋊C4—541no—
254C22⋊C4—541no—
255C42—91no—
256C22×C3—21no—
257C22×C3D4×D911yescharacteristic
258C22×C3D3×D1211yescharacteristic
259C2×D3—182no—
260C2×D3—182no—
261C2×D3—362no—
262C2×D3—361no—
263C2×D3—361no—
264C2×D3—361no—
265C2×D3—361no—
266C2×D3—721no—
267C2×D3—721no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—92no—
271C3×C4—121no—
272C3×C4—121no—
273C3×C4—181no—
274C3×C4#369711yescharacteristic
275C3×C4Q8×D911yescharacteristic
276C3×C4#379711yescharacteristic
277C3×C4#380411yescharacteristic
278Dic3—32no—
279Dic3—41no—
280Dic3—41no—
281Dic3—61no—
282Dic3—361no—
283Dic3—361no—
284Dic3—721no—
285C32#487011yescharacteristic
286C9—21no—
287C9#484211yescharacteristic, complemented
288C23—541no—
289Q8—1082no—
290D4—272no—
291D4—541no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—541no—
297C2×C4—1082no—
298C2×C4C2×D3×D911yescharacteristic
299C8—21no—
300C8—181no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3#6345811yescharacteristic
305C2×C3SD16×D911yescharacteristic
306C2×C3#6347611yescharacteristic
307C2×C3D3×#20411yescharacteristic
308C2×C3#6379411yescharacteristic
309C2×C3#6389511yescharacteristic
310D3—362no—
311D3—362no—
312D3—722no—
313C22—542no—
314C22—1081no—
315C22—1081no—
316C22D9×D1211yescharacteristic
317C4—92no—
318C4—121no—
319C4—121no—
320C4—181no—
321C4—1081no—
322C4(C3⋊D9)⋊Q811yescharacteristic
323C4#8865811yescharacteristic
324C3—21no—
325C3#1058855011yescharacteristic, complemented
326C3#1059075911yescharacteristic
327C2—1082no—
328C2D9×#20411yescharacteristic
329C2#1172081511yescharacteristic
330C2#1172099411yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×C8D3×D9
hypocenterC3×C9#476
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
2. #11720969C2semidirect
4. C9×#817C2semidirect
5. C3×Dic9⋊C8C2semidirect
7. #11720264C2semidirect
33. Dic9⋊C8D3semidirect
40. C4⋊(C3⋊D9)Q8semidirect
98. #817D9semidirect
200. C3×C9#476semidirect
287. C9#4842semidirect
325. C3#10588550semidirect

Maximal subgroups

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

subgroupindex
#117209922
#117209692
#117205432
C9×#8172
C3×Dic9⋊C82
#117202672
#117202642
#105907593
#105885503