Finite Group Databasebeta

← #13155186

Subgroups — #13155186

Order 1728.

514 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 100 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×C8×C3⋊D3C211yesmaximal, characteristic
2#11722066C211yesmaximal, characteristic, complemented
3#11722000C212yesmaximal, complemented
4C3×#63544C211yesmaximal, characteristic, complemented
5C3×#63543C212yesmaximal, complemented
6#10590625—31nomaximal
7#10590226—32nomaximal
8C4×D3×C3⋊D3C411yescharacteristic
9C8×C3⋊(C3⋊D3)C2211yescharacteristic
10C3×C8×C3⋊D3C2211yescharacteristic
11C32×D3×C8C2211yescharacteristic
12#88785C411yescharacteristic
13C3⋊D3×C3⋊C8C411yescharacteristic
14D3×C3⋊(C3⋊C8)C411yescharacteristic
15C3×C3⋊(C3⋊C16)C2212yescomplemented
16C32×C3⋊C16C2212yescomplemented
17C2×C8×C3⋊D3—31no—
18D32×C8—32no—
19D32×C8—61no—
20C16×C3⋊D3—32no—
21C3×#812—32no—
22C3×#798—32no—
23C3×#798—32no—
24#63544—31no—
25#63544D311yescharacteristic, complemented
26#63543—32no—
27#63543D312yescomplemented
28#63542—32no—
29#63542—32no—
30#63542—62no—
31C2×D3×C3⋊D3C811yescharacteristic
32C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C32×C4×D3C2×C411yescharacteristic
35#6289C811yescharacteristic
36Dic3×C3⋊D3C811yescharacteristic
37D3×C3⋊Dic3C811yescharacteristic
38C33×C8C2311yescharacteristic
39C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
40C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
41C32×C3⋊C8C2×C411yescharacteristic
42D3×M32—91no—
43D3×M32—92no—
44C2×C4×C3⋊D3—31no—
45C4×D32—32no—
46C4×D32—61no—
47C2×C32×C8—31no—
48C2×C3⋊(C3⋊C8)—31no—
49C8×C3⋊D3—31no—
50C8×C3⋊D3—32no—
51C8×C3⋊D3—31no—
52C8×C3⋊D3—31no—
53C8×C3⋊D3—61no—
54C8×C3⋊D3—62no—
55C8×C3⋊D3—121no—
56C8×C3⋊D3C2×D311yescharacteristic
57C3×D3×C8—21no—
58C3×D3×C8—32no—
59C3×D3×C8—61no—
60C3×D3×C8C2×D312yes—
61#3710—32no—
62#3710—61no—
63D3×C3⋊C8—32no—
64D3×C3⋊C8—32no—
65D3×C3⋊C8—61no—
66D3×C3⋊C8—61no—
67C32×C16—32no—
68C3⋊(C3⋊C16)C2×D312yescomplemented
69C3×C3⋊C16—22no—
70C3×C3⋊C16—32no—
71C3×C3⋊C16—32no—
72C3×C3⋊C16—62no—
73C3×C3⋊C16C2×D312yescomplemented
74C3×C3⋊C16C2×D312yescomplemented
75C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
76C2×C3×C3⋊D3C2×C811yescharacteristic
77C2×C32×D3C2×C811yescharacteristic
78D3×C3⋊D3—22no—
79C33×C4C22×C411yescharacteristic
80C3⋊(C3⋊Dic3)C2×C811yescharacteristic
81C3×C3⋊Dic3C2×C811yescharacteristic
82C32×Dic3C2×C811yescharacteristic
83C2×D3×C8—92no—
84C2×D3×C8—91no—
85C2×D3×C8—181no—
86C3×M32—91no—
87C3×M32—92no—
88#812—32no—
89#812—91no—
90#798—32no—
91#798—32no—
92#798—92no—
93D3×C16—92no—
94D3×C16—92no—
95D3×C16—92no—
96D3×C16—182no—
97C22×C3⋊D3—31no—
98C2×D32—32no—
99C2×D32—61no—
100C2×C32×C4—31no—
101C2×C3⋊Dic3—31no—
102C4×C3⋊D3—31no—
103C4×C3⋊D3—32no—
104C4×C3⋊D3—31no—
105C4×C3⋊D3—31no—
106C4×C3⋊D3—61no—
107C4×C3⋊D3—62no—
108C4×C3⋊D3—121no—
109C4×C3⋊D3C4×D311yescharacteristic
110C3×C4×D3—21no—
111C3×C4×D3—32no—
112C3×C4×D3—61no—
113C3×C4×D3C4×D312yes—
114#624—32no—
115#624—61no—
116D3×Dic3—32no—
117D3×Dic3—32no—
118D3×Dic3—61no—
119D3×Dic3—61no—
120C32×C8—22no—
121C32×C8—21no—
122C32×C8—31no—
123C32×C8—41no—
124C32×C8C22×D311yescharacteristic
125C32×C8C22×D312yes—
126C3⋊(C3⋊C8)—32no—
127C3⋊(C3⋊C8)—31no—
128C3⋊(C3⋊C8)—61no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—121no—
131C3⋊(C3⋊C8)C4×D311yescharacteristic
132C3×C3⋊C8—21no—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8C4×D312yes—
136C2×M32—271noSylow
137C2×C33C22×C811yescharacteristic
138C3⋊(C3⋊D3)M3212yescomplemented
139C3×C3⋊D3—21no—
140C32×D3—21no—
141C2×C4×D3—92no—
142C2×C4×D3—91no—
143C2×C4×D3—181no—
144C2×C3×C8—32no—
145C2×C3×C8—61no—
146C2×C3×C8—91no—
147C2×C3⋊C8—92no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—181no—
150D3×C8—32no—
151D3×C8—61no—
152D3×C8—92no—
153D3×C8—92no—
154D3×C8—91no—
155D3×C8—92no—
156D3×C8—91no—
157D3×C8—91no—
158D3×C8—181no—
159D3×C8—182no—
160D3×C8—181no—
161D3×C8—181no—
162D3×C8—361no—
163D3×C8D3211yescharacteristic
164C3×C16—32no—
165C3×C16—32no—
166C3×C16—62no—
167C3×C16—92no—
168C3⋊C16—32no—
169C3⋊C16—32no—
170C3⋊C16—62no—
171C3⋊C16D3212yescomplemented
172C22×C32—31no—
173C2×C3⋊D3—32no—
174C2×C3⋊D3—31no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—62no—
177C2×C3⋊D3—62no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3D3×C811yescharacteristic
180C2×C3×D3—21no—
181C2×C3×D3—32no—
182C2×C3×D3—61no—
183C2×C3×D3D3×C812yes—
184D32—64no—
185D32—64no—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4—31no—
189C32×C4—41no—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4C2×C4×D312yes—
192C3⋊Dic3—32no—
193C3⋊Dic3—31no—
194C3⋊Dic3—61no—
195C3⋊Dic3—62no—
196C3⋊Dic3—121no—
197C3⋊Dic3D3×C811yescharacteristic
198C3×Dic3—21no—
199C3×Dic3—32no—
200C3×Dic3—61no—
201C3×Dic3D3×C812yes—
202C22×C8—271no—
203M32—91no—
204M32—92no—
205M32—271no—
206C2×C16—272no—
207C33C2×M3211yescharacteristic, complemented, Sylow
208C22×D3—92no—
209C22×D3—91no—
210C22×D3—181no—
211C2×C3×C4—32no—
212C2×C3×C4—61no—
213C2×C3×C4—91no—
214C2×Dic3—92no—
215C2×Dic3—91no—
216C2×Dic3—181no—
217C4×D3—32no—
218C4×D3—61no—
219C4×D3—92no—
220C4×D3—92no—
221C4×D3—91no—
222C4×D3—92no—
223C4×D3—91no—
224C4×D3—91no—
225C4×D3—181no—
226C4×D3—182no—
227C4×D3—181no—
228C4×D3—181no—
229C4×D3—361no—
230C4×D3#62411yescharacteristic
231C3×C8—22no—
232C3×C8—21no—
233C3×C8—32no—
234C3×C8—41no—
235C3×C8—61no—
236C3×C8—91no—
237C3×C8C2×D3212yes—
238C3×C8C2×D3211yescharacteristic
239C3⋊C8—32no—
240C3⋊C8—61no—
241C3⋊C8—92no—
242C3⋊C8—91no—
243C3⋊C8—182no—
244C3⋊C8—181no—
245C3⋊C8—361no—
246C3⋊C8#62411yescharacteristic
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—41no—
250C2×C32—61no—
251C2×C32C2×D3×C811yescharacteristic
252C2×C32C2×D3×C812yes—
253C3⋊D3—21no—
254C3⋊D3—34no—
255C3⋊D3—32no—
256C3⋊D3—62no—
257C3⋊D3—64no—
258C3⋊D3—122no—
259C3×D3—22no—
260C3×D3—22no—
261C3×D3—62no—
262C3×D3—62no—
263C22×C4—271no—
264C2×C8—31no—
265C2×C8—91no—
266C2×C8—271no—
267C2×C8—271no—
268C2×C8—271no—
269C2×C8—271no—
270C16—32no—
271C16—92no—
272C22×C3—32no—
273C22×C3—61no—
274C22×C3—91no—
275C2×D3—32no—
276C2×D3—61no—
277C2×D3—92no—
278C2×D3—91no—
279C2×D3—181no—
280C2×D3—182no—
281C2×D3—182no—
282C2×D3—184no—
283C2×D3—184no—
284C2×D3—361no—
285C2×D3#371011yescharacteristic
286C3×C4—21no—
287C3×C4—22no—
288C3×C4—32no—
289C3×C4—41no—
290C3×C4—61no—
291C3×C4—91no—
292C3×C4C4×D3212yes—
293C3×C4C2×#62411yescharacteristic
294Dic3—32no—
295Dic3—61no—
296Dic3—92no—
297Dic3—91no—
298Dic3—181no—
299Dic3—182no—
300Dic3—361no—
301Dic3#371011yescharacteristic
302C32—22no—
303C32—21no—
304C32—41no—
305C32D3×M3211yescharacteristic, complemented
306C32D3×M3212yescomplemented
307C23—271no—
308C2×C4—31no—
309C2×C4—91no—
310C2×C4—271no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—271no—
314C8—31no—
315C8—91no—
316C8—271no—
317C8D3311yescharacteristic
318C2×C3—21no—
319C2×C3—22no—
320C2×C3—41no—
321C2×C3—62no—
322C2×C3—62no—
323C2×C3—181no—
324C2×C3D32×C812yes—
325C2×C3C2×#371011yescharacteristic
326D3—21no—
327D3—62no—
328D3—62no—
329D3—94no—
330D3—92no—
331D3—182no—
332D3—184no—
333D3—362no—
334C22—31no—
335C22—91no—
336C22—271no—
337C22—542no—
338C4—31no—
339C4—91no—
340C4—271no—
341C4D3×#62411yescharacteristic
342C3—21no—
343C3—22no—
344C3—41no—
345C3#1059022612yescomplemented
346C3#1059062511yescharacteristic, complemented
347C2—61no—
348C2—181no—
349C2—272no—
350C2D3×#371011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC8D33
commutator, socleC2×C33C22×C8
Fitting, cosocleC33×C8C23
prime radicalC2×D3×C3⋊D3C8
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11722066C2semidirect
3. #11722000C2semidirect
4. C3×#63544C2semidirect
5. C3×#63543C2semidirect
15. C3×C3⋊(C3⋊C16)C22semidirect
16. C32×C3⋊C16C22semidirect
25. #63544D3semidirect
27. #63543D3semidirect
68. C3⋊(C3⋊C16)C2×D3semidirect
73. C3×C3⋊C16C2×D3semidirect
74. C3×C3⋊C16C2×D3semidirect
138. C3⋊(C3⋊D3)M32semidirect
171. C3⋊C16D32semidirect
207. C33C2×M32semidirect
305. C32D3×M32semidirect
306. C32D3×M32semidirect
345. C3#10590226semidirect
346. C3#10590625semidirect

Maximal subgroups

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

subgroupindex
D3×C8×C3⋊D32
#117220662
#117220002
C3×#635442
C3×#635432
#105906253
#105902263