Finite Group Databasebeta

← #13161234

Subgroups — #13161234

Order 1728.

820 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 166 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
1Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722055C211yesmaximal, characteristic
6#11722054C211yesmaximal, characteristic
7#11722050C211yesmaximal, characteristic
8#10592495—31nomaximal
9#10591615—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Q8×C3⋊Dic3—31no—
22Q8×C3⋊Dic3—34no—
23Q8×C3⋊Dic3—64no—
24#64099—31no—
25#63872—34no—
26C32×#103—31no—
27#63662D311yescharacteristic
28#63645—31no—
29#63641—31no—
30#63635—31no—
31C3×#835—34no—
32C3×#835D314yes—
33#63575—34no—
34#63573—34no—
35#63572—34no—
36C33×Q8—21no—
37C33×Q8Q812yes—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3—21no—
45C3×C3⋊(C3⋊C8)—21no—
46C32×C3⋊C8—21no—
47#4799—91no—
48#4796—94no—
49C2×C32×Q8—24no—
50C2×C32×Q8C2×D314yes—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×C8—31no—
53C32×C4⋊C4—31no—
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—34no—
56C4⋊(C3⋊Dic3)—61no—
57C4⋊(C3⋊Dic3)—61no—
58C4⋊(C3⋊Dic3)—64no—
59C4⋊(C3⋊Dic3)—64no—
60C4⋊(C3⋊Dic3)—68no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—61no—
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—34no—
65C4×C3⋊Dic3—61no—
66C4×C3⋊Dic3—64no—
67C4×C3⋊Dic3—64no—
68C4×C3⋊Dic3—68no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C2×C3×C3⋊C8—34no—
73C2×C3×C3⋊C8C2×D314yes—
74Dic3⋊Dic3—64no—
75Dic3⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C33×C4—22no—
78C33×C4C2×Q811yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)—21no—
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—41no—
85Q8×Dic3—94no—
86Q8×Dic3—91no—
87Q8×Dic3—184no—
88#890—94no—
89#890—91no—
90C3×#103—34no—
91C3×#103—91no—
92#835—34no—
93#835C3⋊D311yescharacteristic
94#818—94no—
95#817—91no—
96Dic3⋊C8—94no—
97Dic3⋊C8—91no—
98#808—94no—
99#807—91no—
100C32×Q8—28no—
101C32×Q8—24no—
102C32×Q8—21no—
103C32×Q8—28no—
104C32×Q8Dic618yes—
105C32×Q8Dic612yes—
106C2×C32×C4—28no—
107C2×C32×C4—24no—
108C2×C32×C4—21no—
109C2×C32×C4—24no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—68no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—61no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—64no—
125C32×C8—61no—
126C3⋊(C3⋊C8)—21no—
127C3×C3⋊C8—24no—
128C3×C3⋊C8—64no—
129#479—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33#13711yescharacteristic
133C2×C3×Q8—24no—
134C2×C3×Q8D3214yes—
135C2×C3×Q8C2×C3⋊D311yescharacteristic
136C2×C3×C8—34no—
137C2×C3×C8—91no—
138C3×C4⋊C4—34no—
139C3×C4⋊C4—91no—
140C4⋊Dic3—34no—
141C4⋊Dic3—94no—
142C4⋊Dic3—91no—
143C4⋊Dic3—184no—
144C4⋊Dic3—181no—
145C4⋊Dic3—184no—
146C4⋊Dic3—184no—
147C4⋊Dic3—181no—
148C4⋊Dic3—188no—
149C4⋊Dic3C2×C3⋊D311yescharacteristic
150Dic3⋊C4—181no—
151Dic3⋊C4—184no—
152C4×Dic3—94no—
153C4×Dic3—91no—
154C4×Dic3—184no—
155C4×Dic3—184no—
156C4×Dic3—181no—
157C4×Dic3—188no—
158C2×C3⋊C8—34no—
159C2×C3⋊C8C2×C3⋊D311yescharacteristic
160C22×C32—24no—
161C22×C32D3×D414yes—
162C22×C32D3×D411yescharacteristic
163C32×C4—216no—
164C32×C4—28no—
165C32×C4—22no—
166C32×C4—24no—
167C32×C4—24no—
168C32×C4—121no—
169C32×C4C2×Dic614yes—
170C32×C4C2×Dic611yescharacteristic
171C32×C4#23714yes—
172C32×C4#23711yescharacteristic
173C3⋊Dic3—41no—
174C3⋊Dic3—68no—
175C3⋊Dic3—64no—
176C3⋊Dic3—61no—
177C3⋊Dic3—61no—
178C3⋊Dic3—64no—
179C3⋊Dic3—124no—
180C3⋊Dic3—128no—
181C3⋊Dic3—121no—
182C3⋊Dic3—124no—
183C3×Dic3—44no—
184C3×Dic3—124no—
185#125—271no—
186C4×Q8—271no—
187#107—271no—
188#106—271no—
189C4⋊C8—271no—
190#103—31no—
191#103—91no—
192C33#47911yescharacteristic, complemented, Sylow
193C3×Q8—28no—
194C3×Q8—24no—
195C3×Q8—21no—
196C3×Q8—28no—
197C3×Q8#62718yes—
198C3×Q8#63412yes—
199C2×C3×C4—28no—
200C2×C3×C4—24no—
201C2×C3×C4—21no—
202C2×C3×C4—24no—
203C2×C3×C4—64no—
204C2×C3×C4—181no—
205C2×C3×C4C2×D3214yes—
206C2×C3×C4C22×C3⋊D311yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—64no—
209C2×Dic3—94no—
210C2×Dic3—94no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—184no—
214C2×Dic3—184no—
215C2×Dic3—188no—
216C2×Dic3—184no—
217C2×Dic3—181no—
218C3×C8—64no—
219C3×C8—181no—
220C3⋊C8—21no—
221C3⋊C8—64no—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32#87814yes—
226C2×C32#87811yescharacteristic
227C2×C32#91514yes—
228C2×C32#91611yescharacteristic
229C2×C32#91811yescharacteristic
230C2×C32#91914yes—
231C2×Q8D3×C3⋊D311yescharacteristic
232C2×C8—31no—
233C2×C8—91no—
234C4⋊C4—31no—
235C4⋊C4—91no—
236C4⋊C4—271no—
237C4⋊C4—541no—
238C4⋊C4—541no—
239C4⋊C4—541no—
240C42—271no—
241C42—541no—
242C22×C3—24no—
243C22×C3#381114yes—
244C22×C3D4×C3⋊D311yescharacteristic
245C3×C4—216no—
246C3×C4—22no—
247C3×C4—28no—
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—124no—
251C3×C4—361no—
252C3×C4#380514yes—
253C3×C4C2×#62714yes—
254C3×C4C2×#63411yescharacteristic
255C3×C4#383011yescharacteristic
256Dic3—41no—
257Dic3—124no—
258Dic3—188no—
259Dic3—181no—
260Dic3—181no—
261Dic3—184no—
262Dic3—184no—
263Dic3—364no—
264Dic3—368no—
265Dic3—361no—
266Dic3—364no—
267C32—24no—
268C32#479614yescomplemented
269C32#479911yescharacteristic, complemented
270Q8—21no—
271Q8#629312yes—
272C2×C4—21no—
273C2×C4—61no—
274C2×C4—181no—
275C2×C4—271no—
276C2×C4—271no—
277C2×C4—541no—
278C2×C4C2×D3×C3⋊D311yescharacteristic
279C8—61no—
280C8—181no—
281C2×C3—24no—
282C2×C3—24no—
283C2×C3—24no—
284C2×C3#6395014yes—
285C2×C3#6395114yes—
286C2×C3#6398214yes—
287C2×C3#6408711yescharacteristic
288C2×C3#6412411yescharacteristic
289C2×C3#6412811yescharacteristic
290C22#8903911yescharacteristic
291C4—22no—
292C4—121no—
293C4—361no—
294C4—541no—
295C4—541no—
296C4—1081no—
297C4#8902911yescharacteristic
298C4C2×#629311yescharacteristic
299C3—24no—
300C3#1059161514yescomplemented
301C3#1059249511yescharacteristic, complemented
302C2#1172358811yescharacteristic
303C2#1172358911yescharacteristic
304C2#1172364211yescharacteristic
subgroupG/H
centerC22#89039
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
192. C33#479semidirect
268. C32#4796semidirect
269. C32#4799semidirect
300. C3#10591615semidirect
301. C3#10592495semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊Dic3)2
#117234502
C3×#636622
C32×#8352
#117220552
#117220542
#117220502
#105924953
#105916153