Finite Group Databasebeta

← #13161232

Subgroups — #13161232

Order 1728.

708 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C32×Q8×Dic3C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722053C211yesmaximal, characteristic
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592721—31nomaximal
9#10591547—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C32×C4×Q8—31no—
22#64079—31no—
23C3×Q8×Dic3D314yes—
24#63869—34no—
25#63662—31no—
26#63662—34no—
27#63662—64no—
28#63662D311yescharacteristic
29#63635—31no—
30#63634—31no—
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#835—34no—
33#63575—34no—
34#63574—34no—
35Dic3⋊(C3⋊C8)—34no—
36C33×Q8—21no—
37C33×Q8Q812yes—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3×C3⋊(C3⋊C8)—21no—
47#5022—94no—
48#4796—91no—
49C2×C32×Q8—24no—
50C2×C32×Q8C2×D314yes—
51C2×C32×Q8C2×D311yescharacteristic
52C32×C4⋊C4—31no—
53C32×C4⋊C4—61no—
54C32×C42—31no—
55C32×C42—61no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—34no—
58C4⋊(C3⋊Dic3)—64no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—61no—
61#3750—61no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×C3⋊(C3⋊C8)—34no—
64C2×C3⋊(C3⋊C8)—64no—
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4⋊Dic3—24no—
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3C2×D314yes—
69C3×C4×Dic3—24no—
70C3×C4×Dic3C2×D314yes—
71C2×C3×C3⋊C8—34no—
72#3724—64no—
73Dic3⋊Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—22no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—21no—
81C32×Dic3—21no—
82C32×Dic3—41no—
83C3×C4×Q8—34no—
84Q8×Dic3C3⋊D311yescharacteristic
85#890—91no—
86#870—94no—
87C3×#103—91no—
88#835—34no—
89#835—94no—
90#835—91no—
91#835—184no—
92#818—91no—
93Dic3⋊C8—91no—
94#808—94no—
95#808—91no—
96#807—94no—
97C4⋊(C3⋊C8)—94no—
98C32×Q8—21no—
99C32×Q8—28no—
100C32×Q8—24no—
101C32×Q8—44no—
102C32×Q8Dic618yes—
103C32×Q8Dic612yes—
104C2×C32×C4—21no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4—44no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3#6714yes—
120C2×C3×Dic3#6714yes—
121C3⋊(C3⋊C8)—21no—
122C3⋊(C3⋊C8)—61no—
123C3⋊(C3⋊C8)—64no—
124C3⋊(C3⋊C8)—124no—
125C3×C3⋊C8—64no—
126#479—271noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C3×Q8—24no—
131C2×C3×Q8D3214yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C2×C3×C8—91no—
134C3×C4⋊C4—34no—
135C3×C4⋊C4—64no—
136C3×C4⋊C4—91no—
137C3×C42—34no—
138C3×C42—64no—
139C4⋊Dic3—21no—
140C4⋊Dic3—34no—
141C4⋊Dic3—94no—
142C4⋊Dic3—91no—
143C4⋊Dic3—184no—
144C4⋊Dic3—181no—
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—184no—
147Dic3⋊C4—184no—
148Dic3⋊C4—181no—
149C4×Dic3—21no—
150C4×Dic3C2×C3⋊D311yescharacteristic
151C2×C3⋊C8—34no—
152C2×C3⋊C8—94no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—184no—
155C22×C32—24no—
156C22×C32D3×D411yescharacteristic
157C22×C32C2×#6714yes—
158C32×C4—22no—
159C32×C4—24no—
160C32×C4—24no—
161C32×C4—28no—
162C32×C4—48no—
163C32×C4—61no—
164C32×C4—61no—
165C32×C4—121no—
166C32×C4C2×Dic614yes—
167C32×C4C2×Dic611yescharacteristic
168C32×C4#23514yes—
169C32×C4#23711yescharacteristic
170C3⋊Dic3—41no—
171C3⋊Dic3—121no—
172C3⋊Dic3—124no—
173C3⋊Dic3—244no—
174C3×Dic3—24no—
175C3×Dic3—24no—
176C3×Dic3—44no—
177C3×Dic3—124no—
178#125—271no—
179C4×Q8—31no—
180#107—271no—
181#106—271no—
182C4⋊C8—271no—
183#103—91no—
184#103—271no—
185C33#47911yescharacteristic, complemented, Sylow
186C3×Q8—24no—
187C3×Q8—21no—
188C3×Q8—28no—
189C3×Q8—44no—
190C3×Q8#62718yes—
191C3×Q8#63412yes—
192C2×C3×C4—24no—
193C2×C3×C4—21no—
194C2×C3×C4—24no—
195C2×C3×C4—34no—
196C2×C3×C4—34no—
197C2×C3×C4—44no—
198C2×C3×C4—64no—
199C2×C3×C4—181no—
200C2×C3×C4C2×D3214yes—
201C2×C3×C4C22×C3⋊D311yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—64no—
204C2×Dic3—184no—
205C2×Dic3—181no—
206C2×Dic3—364no—
207C2×Dic3#63811yescharacteristic
208C2×Dic3#63811yescharacteristic
209C3×C8—181no—
210C3⋊C8—64no—
211C3⋊C8—184no—
212C3⋊C8—181no—
213C3⋊C8—364no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32#87811yescharacteristic
218C2×C32#91511yescharacteristic
219C2×C32#91911yescharacteristic
220C2×C32#92414yes—
221C2×C32#94214yes—
222C2×C32#95014yes—
223C2×Q8D3×C3⋊D311yescharacteristic
224C2×C8—91no—
225C2×C8—271no—
226C4⋊C4—31no—
227C4⋊C4—61no—
228C4⋊C4—91no—
229C4⋊C4—271no—
230C4⋊C4—541no—
231C4⋊C4—541no—
232C42—31no—
233C42—61no—
234C22×C3—24no—
235C22×C3D3×#6714yes—
236C22×C3C2×#63811yescharacteristic
237C3×C4—22no—
238C3×C4—28no—
239C3×C4—24no—
240C3×C4—24no—
241C3×C4—48no—
242C3×C4—64no—
243C3×C4—64no—
244C3×C4—124no—
245C3×C4—361no—
246C3×C4#380414yes—
247C3×C4C2×#62714yes—
248C3×C4C2×#63411yescharacteristic
249C3×C4#382811yescharacteristic
250Dic3—21no—
251Dic3—21no—
252Dic3—41no—
253Dic3—124no—
254Dic3—361no—
255Dic3—364no—
256Dic3—724no—
257C32—24no—
258C32#479611yescharacteristic, complemented
259C32#502214yescomplemented
260Q8—21no—
261Q8#629312yes—
262C2×C4—21no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—61no—
266C2×C4—181no—
267C2×C4—541no—
268C2×C4C2×D3×C3⋊D311yescharacteristic
269C8—181no—
270C8—541no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3#6394614yes—
275C2×C3#6394914yes—
276C2×C3#6396414yes—
277C2×C3#6413311yescharacteristic
278C2×C3#6415111yescharacteristic
279C2×C3#6415911yescharacteristic
280C22D3×#63811yescharacteristic
281C4—22no—
282C4—61no—
283C4—61no—
284C4—121no—
285C4—361no—
286C4—1081no—
287C4#8903111yescharacteristic
288C4C2×#629311yescharacteristic
289C3—24no—
290C3#1059154714yescomplemented
291C3#1059272111yescharacteristic, complemented
292C2#1172357911yescharacteristic
293C2#1172358211yescharacteristic
294C2#1172364011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#479
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
185. C33#479semidirect
258. C32#4796semidirect
259. C32#5022semidirect
290. C3#10591547semidirect
291. C3#10592721semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×Dic32
#117234542
#117222052
C3×#636622
#117220572
#117220532
Dic3⋊(C3⋊(C3⋊C8))2
#105927213
#105915473