Finite Group Databasebeta

← #13161229

Subgroups — #13161229

Order 1728.

744 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64156C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic, complemented
3#11722205C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic
6#11722033C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592499—31nomaximal
9#10591614—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3752—21no—
15C3×#3752—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64156D311yescharacteristic
22#64099—31no—
23C3×#947—34no—
24#63869—34no—
25C32×#103—31no—
26#63662—31no—
27#63662—34no—
28#63662—64no—
29#63648—31no—
30C8×C3⋊Dic3—31no—
31#63636—31no—
32C3×#835D314yescomplemented
33#63563—34no—
34#63562—34no—
35Dic3×C3⋊C8—34no—
36C22×C3×C3⋊D3—21no—
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5056—91no—
50#4800—94no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C3×D12—34no—
56C2×C32×C8—31no—
57C32×C4⋊C4—31no—
58#3752—21no—
59#3752—21no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—61no—
63C4⋊(C3⋊Dic3)—64no—
64#3750—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×(C2×D3)⋊C4—64no—
70C3×(C2×D3)⋊C4—64no—
71C3×C4⋊Dic3C2×D314yes—
72C3×C4×Dic3—34no—
73C2×C3×C3⋊C8C2×D314yes—
74#3724—64no—
75Dic3⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C32×Dic3—41no—
87C3×#124—91no—
88#947—34no—
89#890—94no—
90#870—91no—
91C3×#103—34no—
92#835—94no—
93#835—91no—
94#835—184no—
95#835C3⋊D311yescharacteristic, complemented
96#832—91no—
97#832—91no—
98#821—94no—
99C8×Dic3—94no—
100#809—94no—
101C4×C3⋊C8—91no—
102C22×C3⋊D3—21no—
103C22×C3×D3—64no—
104C32×Q8—28no—
105C32×Q8—24no—
106C32×Q8—21no—
107C32×Q8—21no—
108C32×Q8—24no—
109C32×Q8—44no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—44no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3#6711yescharacteristic
121C2×C3⋊Dic3#6711yescharacteristic
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)—21no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C3×D12—64no—
128C3×D12—64no—
129C32×C8—61no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—24no—
134#488—271noSylow
135C2×C33#12411yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C33#13511yescharacteristic
138C3×C3⋊D3—81no—
139C2×C3×Q8—24no—
140C2×C3×Q8D3214yes—
141C2×C3×Q8C2×C3⋊D311yescharacteristic
142C2×C3×D4—91no—
143C2×D12—34no—
144C2×C3×C8—34no—
145C3×C4⋊C4—34no—
146C3×C22⋊C4—181no—
147C3×C22⋊C4—181no—
148C3×C42—91no—
149(C2×D3)⋊C4—64no—
150(C2×D3)⋊C4—64no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—184no—
154C4⋊Dic3—184no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—184no—
159C4×Dic3—34no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32C2×#6711yescharacteristic
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C2×C3×D3—124no—
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—48no—
176C32×C4—44no—
177C32×C4—41no—
178C32×C4—121no—
179C32×C4#23514yes—
180C32×C4#23714yes—
181C32×C4#23711yescharacteristic
182C32×C4#23711yescharacteristic
183C3⋊Dic3—21no—
184C3⋊Dic3—21no—
185C3⋊Dic3—121no—
186C3⋊Dic3—124no—
187C3⋊Dic3—244no—
188C3×Dic3—44no—
189C3×Dic3—64no—
190C3×Dic3—64no—
191#125—271no—
192#124—91no—
193#103—31no—
194#103—271no—
195#102—271no—
196#102—271no—
197C4×C8—271no—
198C33#48811yescharacteristic, complemented, Sylow
199C23×C3—181no—
200C22×D3—64no—
201C3×Q8—28no—
202C3×Q8—24no—
203C3×Q8—24no—
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8—44no—
207C3×D4—181no—
208C3×D4—181no—
209C2×C3×C4—24no—
210C2×C3×C4—21no—
211C2×C3×C4—24no—
212C2×C3×C4—44no—
213C2×C3×C4—64no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—34no—
220C2×Dic3—34no—
221C2×Dic3—184no—
222C2×Dic3—181no—
223C2×Dic3—364no—
224D12—64no—
225D12—64no—
226C3×C8—64no—
227C3⋊C8—21no—
228C3⋊C8—184no—
229C3⋊C8—181no—
230C3⋊C8—364no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32#88514yes—
235C2×C32#91614yes—
236C2×C32#91914yes—
237C2×C32#93611yescharacteristic
238C2×C32#95011yescharacteristic
239C2×C32#95011yescharacteristic
240C3⋊D3—81no—
241C3×D3—244no—
242C2×Q8D3×C3⋊D311yescharacteristic
243C2×D4—91no—
244C2×C8—31no—
245C2×C8—271no—
246C4⋊C4—31no—
247C4⋊C4—271no—
248C4⋊C4—541no—
249C4⋊C4—541no—
250C22⋊C4—181no—
251C22⋊C4—181no—
252C42—91no—
253C22×C3—24no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3—361no—
257C22×C3D3×#6714yes—
258C22×C3D4×C3⋊D311yescharacteristic
259C2×D3—124no—
260C2×D3—124no—
261C2×D3—124no—
262C3×C4—24no—
263C3×C4—24no—
264C3×C4—48no—
265C3×C4—41no—
266C3×C4—44no—
267C3×C4—124no—
268C3×C4—181no—
269C3×C4—181no—
270C3×C4#380414yes—
271C3×C4#380514yes—
272C3×C4#382811yescharacteristic
273C3×C4#383011yescharacteristic
274Dic3—41no—
275Dic3—64no—
276Dic3—64no—
277Dic3—361no—
278Dic3—364no—
279Dic3—724no—
280C32—24no—
281C32#480014yescomplemented
282C32#505611yescharacteristic, complemented
283C23—181no—
284Q8—21no—
285Q8—21no—
286D4—181no—
287D4—181no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—541no—
293C2×C4C2×D3×C3⋊D311yescharacteristic
294C8—61no—
295C8—541no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—721no—
300C2×C3#6394714yes—
301C2×C3#6394914yes—
302C2×C3#6395914yes—
303C2×C3#6409411yescharacteristic
304C2×C3#6412511yescharacteristic
305C2×C3#6412811yescharacteristic
306D3—244no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22C3⋊D3×#6711yescharacteristic
311C4—41no—
312C4—121no—
313C4—181no—
314C4—181no—
315C4—1081no—
316C4#8902911yescharacteristic
317C4#8903111yescharacteristic
318C3—24no—
319C3#1059161414yescomplemented
320C3#1059249911yescharacteristic, complemented
321C2—721no—
322C2#1172358311yescharacteristic
323C2#1172358511yescharacteristic
324C2#1172359111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#488
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
2. #11723447C2semidirect
3. #11722205C2semidirect
4. C32×#835C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
32. C3×#835D3semidirect
95. #835C3⋊D3semidirect
198. C33#488semidirect
281. C32#4800semidirect
282. C32#5056semidirect
319. C3#10591614semidirect
320. C3#10592499semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117234472
#117222052
C32×#8352
#117220342
#117220332
C3⋊C8×C3⋊Dic32
#105924993
#105916143