Finite Group Databasebeta

← #13161125

Subgroups — #13161125

Order 1728.

792 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 148 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×#64155C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722052C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic
8#10592504—31nomaximal
9#10591549—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64155D311yescharacteristic
22C4⋊C4×C3⋊D3—31no—
23C3×#946—34no—
24D3×C4⋊Dic3—34no—
25C32×#103—31no—
26#63662—31no—
27#63662—34no—
28#63662—64no—
29#63647—31no—
30#63644—31no—
31#63635—31no—
32C3×#835D314yescomplemented
33#63574—34no—
34#63573—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3—21no—
45C3×C4×C3⋊D3—21no—
46C2×C32×Dic3—21no—
47Dic3×C3⋊D3—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5043—91no—
51#4805—94no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×C4×D3—34no—
58C2×D3×Dic3—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61#3752—21no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67#3750—61no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×Dic3⋊C4—64no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3SD1612yes—
81C33×C4—41no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88C3×C22⋊Q8—91no—
89#946—34no—
90C2×C4⋊Dic3—91no—
91D3×C4⋊C4—94no—
92C3×#103—34no—
93#835—94no—
94#835—91no—
95#835—184no—
96#835C3⋊D311yescharacteristic, complemented
97#830—91no—
98(C2×D3)⋊C8—94no—
99#817—94no—
100#808—94no—
101#808—91no—
102#808—91no—
103C22×C3⋊D3#6711yescharacteristic
104C22×C3×D3—34no—
105C32×Q8—28no—
106C32×Q8—24no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—44no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122C2×C3⋊Dic3#6711yescharacteristic
123C4×C3⋊D3—21no—
124C4×C3⋊D3—21no—
125C4×C3⋊D3—122no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131D3×Dic3—128no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—124no—
136C3×C3⋊C8—24no—
137#483—271noSylow
138C2×C33#12311yescharacteristic
139C2×C33C2×SD1611yescharacteristic
140C2×C33#13711yescharacteristic
141C3×C3⋊D3—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8D3214yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C22×C3×C4—91no—
146C22×Dic3—181no—
147C2×C4×D3—34no—
148C2×C4×D3—184no—
149C2×C3×C8—34no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C22⋊C4—181no—
154(C2×D3)⋊C4—64no—
155C4⋊Dic3—34no—
156C4⋊Dic3—94no—
157C4⋊Dic3—91no—
158C4⋊Dic3—184no—
159C4⋊Dic3—181no—
160C4⋊Dic3C2×C3⋊D311yescharacteristic
161Dic3⋊C4—64no—
162Dic3⋊C4—184no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—184no—
166C2×C3⋊C8C2×C3⋊D311yescharacteristic
167C22×C32—24no—
168C22×C32D3×D414yes—
169C22×C32C2×#6711yescharacteristic
170C2×C3⋊D3—21no—
171C2×C3⋊D3—21no—
172C2×C3⋊D3#21512yes—
173C2×C3×D3—38no—
174C2×C3×D3—64no—
175C2×C3×D3—64no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—48no—
179C32×C4—44no—
180C32×C4—41no—
181C32×C4—121no—
182C32×C4#23514yes—
183C32×C4#23714yes—
184C32×C4#23711yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—41no—
188C3⋊Dic3—121no—
189C3⋊Dic3—124no—
190C3⋊Dic3—244no—
191C3×Dic3—44no—
192C3×Dic3—64no—
193C3×Dic3—124no—
194C22⋊Q8—91no—
195C2×C4⋊C4—271no—
196#106—271no—
197#106—271no—
198#103—31no—
199#103—271no—
200C22⋊C8—271no—
201C33#48311yescharacteristic, complemented, Sylow
202C23×C3—91no—
203C22×D3—34no—
204C3×Q8—28no—
205C3×Q8—24no—
206C3×Q8—24no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8—44no—
210C2×C3×C4—24no—
211C2×C3×C4—21no—
212C2×C3×C4—24no—
213C2×C3×C4—44no—
214C2×C3×C4—64no—
215C2×C3×C4—91no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—34no—
223C2×Dic3—64no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226C2×Dic3—364no—
227C2×Dic3—362no—
228C4×D3—64no—
229C4×D3—64no—
230C4×D3—368no—
231C3×C8—64no—
232C3⋊C8—21no—
233C3⋊C8—184no—
234C3⋊C8—181no—
235C3⋊C8—364no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32#88314yes—
240C2×C32D3×SD1614yes—
241C2×C32#91814yes—
242C2×C32#93511yescharacteristic
243C2×C32C2×#21511yescharacteristic
244C2×C32#95111yescharacteristic
245C3⋊D3—22no—
246C3×D3—68no—
247C2×Q8D3×C3⋊D311yescharacteristic
248C22×C4—91no—
249C22×C4—541no—
250C2×C8—31no—
251C2×C8—271no—
252C4⋊C4—31no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C4⋊C4—271no—
256C4⋊C4—541no—
257C22⋊C4—181no—
258C22×C3—24no—
259C22×C3—92no—
260C22×C3—181no—
261C22×C3—181no—
262C22×C3D3×#6714yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—38no—
265C2×D3—64no—
266C2×D3—64no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—48no—
270C3×C4—41no—
271C3×C4—44no—
272C3×C4—124no—
273C3×C4—181no—
274C3×C4—361no—
275C3×C4#380414yes—
276C3×C4#380514yes—
277C3×C4#382811yescharacteristic
278C3×C4#383011yescharacteristic
279Dic3—41no—
280Dic3—64no—
281Dic3—124no—
282Dic3—361no—
283Dic3—364no—
284Dic3—724no—
285C32—24no—
286C32#480514yescomplemented
287C32#504311yescharacteristic, complemented
288C23—91no—
289Q8—21no—
290Q8—21no—
291C2×C4—21no—
292C2×C4—61no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—541no—
298C2×C4—1082no—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—61no—
301C8—541no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—182no—
306C2×C3D3×#21514yes—
307C2×C3#6394314yes—
308C2×C3#6396914yes—
309C2×C3#6409211yescharacteristic
310C2×C3SD16×C3⋊D311yescharacteristic
311C2×C3#6412711yescharacteristic
312D3—68no—
313C22—92no—
314C22—181no—
315C22—181no—
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—121no—
319C4—181no—
320C4—361no—
321C4—1081no—
322C4#8902911yescharacteristic
323C4#8903111yescharacteristic
324C3—24no—
325C3#1059154914yescomplemented
326C3#1059250411yescharacteristic, complemented
327C2—182no—
328C2C3⋊D3×#21511yescharacteristic
329C2#1172357611yescharacteristic
330C2#1172362111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11722205C2semidirect
4. C32×#835C2semidirect
5. #11722053C2semidirect
6. #11722052C2semidirect
32. C3×#835D3semidirect
96. #835C3⋊D3semidirect
201. C33#483semidirect
286. C32#4805semidirect
287. C32#5043semidirect
325. C3#10591549semidirect
326. C3#10592504semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C3⋊D3×C4⋊Dic32
#117222052
C32×#8352
#117220532
#117220522
#117220232
#105925043
#105915493