Finite Group Databasebeta

← #13158892

Subgroups — #13158892

Order 1728.

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

normal subgroups only

This group has 764 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
1#11723911C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7#11722038C211yesmaximal, characteristic
8#10592721—31nomaximal
9#10590302—34nomaximal
10C2×C32×Dic6C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15#88857—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21C32×C4×Q8—31no—
22#64079—31no—
23#64079—34no—
24#64079—64no—
25C3×C4×Dic6D314yes—
26#63662—31no—
27#63662—31no—
28#63635—31no—
29#63634—31no—
30C4⋊(C3⋊(C3⋊C8))D311yescharacteristic
31C3×C4⋊(C3⋊C8)—34no—
32#63575—34no—
33#63573—34no—
34#63566—34no—
35#63566—34no—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C32×Dic6—21no—
44C32×Dic6Q812yes—
45C3×C3⋊(C3⋊C8)—21no—
46C3×C3⋊(C3⋊C8)—21no—
47#5022—94no—
48#4829—91no—
49C2×C32×Q8—31no—
50C2×C3×Dic6C2×D314yes—
51C32×C4⋊C4—31no—
52C32×C4⋊C4—61no—
53C32×C42—24no—
54C32×C42—61no—
55C32×C42C2×D311yescharacteristic
56C32×C42C2×D314yes—
57C4⋊(C3⋊Dic3)—31no—
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—34no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)—64no—
63#3750—68no—
64#3750—68no—
65#3750—61no—
66#3750—61no—
67#3750—64no—
68#3750—64no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3C2×D314yes—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3—24no—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8—34no—
76C22×C33C2×D411yescharacteristic
77C33×C4—21no—
78C33×C4—21no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3⋊(C3⋊Dic3)—41no—
83C32×Dic3—22no—
84C32×Dic3—41no—
85C3×C4×Q8—34no—
86#870—94no—
87#870—91no—
88#870—184no—
89C4×Dic6C3⋊D311yescharacteristic
90C3×C4⋊C8—91no—
91#835—94no—
92#835—94no—
93#818—91no—
94#817—91no—
95#816—91no—
96#816—91no—
97#808—94no—
98#807—94no—
99C4⋊(C3⋊C8)—34no—
100C32×Q8—32no—
101C32×Q8—61no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4—61no—
106C2×C32×C4—61no—
107C2×C32×C4D1211yescharacteristic
108C2×C32×C4D1211yescharacteristic
109C2×C32×C4#6714yes—
110C2×C32×C4#6714yes—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—68no—
114C2×C3⋊Dic3—68no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—64no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—24no—
121C3×Dic6—24no—
122C3×Dic6Dic618yes—
123C3⋊(C3⋊C8)—21no—
124C3⋊(C3⋊C8)—21no—
125C3×C3⋊C8—64no—
126C3×C3⋊C8—64no—
127#479—271noSylow
128C2×C33C22⋊Q811yescharacteristic
129C2×C33#13511yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—34no—
132C2×Dic6C2×C3⋊D311yescharacteristic
133C2×C3×C8—91no—
134C2×C3×C8—91no—
135C3×C4⋊C4—34no—
136C3×C4⋊C4—64no—
137C3×C42—24no—
138C3×C42—64no—
139C3×C42D3214yes—
140C3×C42C2×C3⋊D311yescharacteristic
141C4⋊Dic3—94no—
142C4⋊Dic3—94no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—184no—
146C4⋊Dic3—184no—
147C4⋊Dic3C2×C3⋊D311yescharacteristic
148Dic3⋊C4—21no—
149Dic3⋊C4—188no—
150Dic3⋊C4—188no—
151Dic3⋊C4—184no—
152Dic3⋊C4—184no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155C4×Dic3—21no—
156C2×C3⋊C8—34no—
157C2×C3⋊C8—34no—
158C22×C32—24no—
159C22×C32C2×D1211yescharacteristic
160C22×C32C2×#6714yes—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—24no—
164C32×C4—24no—
165C32×C4—24no—
166C32×C4—24no—
167C32×C4—44no—
168C32×C4—44no—
169C32×C4—62no—
170C32×C4—121no—
171C32×C4C2×Dic614yes—
172C32×C4#23514yes—
173C32×C4#23711yescharacteristic
174C32×C4D3×Q811yescharacteristic
175C3⋊Dic3—128no—
176C3⋊Dic3—128no—
177C3⋊Dic3—121no—
178C3⋊Dic3—121no—
179C3⋊Dic3—124no—
180C3⋊Dic3—124no—
181C3×Dic3—28no—
182C3×Dic3—44no—
183#125—271no—
184C4×Q8—31no—
185#107—271no—
186#106—271no—
187C4⋊C8—91no—
188#103—271no—
189#103—271no—
190C33#47911yescharacteristic, complemented, Sylow
191C3×Q8—38no—
192C3×Q8—64no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4—64no—
197C2×C3×C4—64no—
198C2×C3×C4#62614yes—
199C2×C3×C4#62614yes—
200C2×C3×C4#63811yescharacteristic
201C2×C3×C4#63811yescharacteristic
202C2×C3×C4C2×D3214yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—188no—
207C2×Dic3—188no—
208C2×Dic3—184no—
209C2×Dic3—184no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212Dic6—21no—
213Dic6#63412yes—
214C3×C8—181no—
215C3×C8—181no—
216C3⋊C8—64no—
217C3⋊C8—64no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32#89711yescharacteristic
222C2×C32#90411yescharacteristic
223C2×C32#90911yescharacteristic
224C2×C32#92414yes—
225C2×C32#94214yes—
226C2×C32#95014yes—
227C2×Q8—31no—
228C2×C8—91no—
229C2×C8—91no—
230C4⋊C4—31no—
231C4⋊C4—61no—
232C4⋊C4—271no—
233C4⋊C4—271no—
234C4⋊C4—541no—
235C4⋊C4—541no—
236C42—61no—
237C42D3×C3⋊D311yescharacteristic
238C22×C3—24no—
239C22×C3C2×#62614yes—
240C22×C3C2×#63811yescharacteristic
241C3×C4—24no—
242C3×C4—24no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—24no—
246C3×C4—24no—
247C3×C4—44no—
248C3×C4—44no—
249C3×C4—68no—
250C3×C4—124no—
251C3×C4D3×Dic614yes—
252C3×C4#379514yes—
253C3×C4C2×#63411yescharacteristic
254C3×C4#382811yescharacteristic
255Dic3—22no—
256Dic3—41no—
257Dic3—368no—
258Dic3—368no—
259Dic3—364no—
260Dic3—364no—
261Dic3—361no—
262Dic3—361no—
263C32—24no—
264C32#482911yescharacteristic, complemented
265C32#502214yescomplemented
266Q8—32no—
267Q8—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—541no—
271C2×C4—541no—
272C2×C4#629111yescharacteristic
273C2×C4#629111yescharacteristic
274C2×C4C2×D3×C3⋊D311yescharacteristic
275C8—181no—
276C8—181no—
277C2×C3—24no—
278C2×C3—24no—
279C2×C3—24no—
280C2×C3#6382914yes—
281C2×C3#6383314yes—
282C2×C3#6385714yes—
283C2×C3#6413311yescharacteristic
284C2×C3#6415111yescharacteristic
285C2×C3#6415911yescharacteristic
286C22C2×#629111yescharacteristic
287C4—21no—
288C4—21no—
289C4—62no—
290C4—121no—
291C4—1081no—
292C4—1081no—
293C4D3×#63411yescharacteristic
294C4#8902111yescharacteristic
295C3—24no—
296C3#1059030214yescomplemented
297C3#1059272111yescharacteristic, complemented
298C2#1172337711yescharacteristic
299C2#1172338111yescharacteristic
300C2#1172342611yescharacteristic
subgroupG/H
centerC22C2×#6291
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#479
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
190. C33#479semidirect
264. C32#4829semidirect
265. C32#5022semidirect
296. C3#10590302semidirect
297. C3#10592721semidirect

Maximal subgroups

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

subgroupindex
#117239112
C32×C4×Dic62
C3×C4⋊(C3⋊(C3⋊C8))2
#117220572
#117220542
#117220382
#105927213
#105903023