Finite Group Databasebeta

← #13160220

Subgroups — #13160220

Order 1728.

652 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 116 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×D3×D8C211yesmaximal, characteristic, complemented
2#11723337C211yesmaximal, characteristic, complemented
3#11722196C211yesmaximal, characteristic, complemented
4C3×#63654C211yesmaximal, characteristic, complemented
5#11722007C211yesmaximal, characteristic, complemented
6#11722005C211yesmaximal, characteristic, complemented
7#11722001C211yesmaximal, characteristic, complemented
8#10592846—31nomaximal
9#10591033—34nomaximal
10C32×D3×D4—21no—
11#89022—21no—
12C33×D8C2211yescharacteristic
13#88852C2211yescharacteristic, complemented
14C3×#3746C2211yescharacteristic, complemented
15C32×#213—21no—
16C32×D24C2211yescharacteristic
17C32×D3×C8C2211yescharacteristic
18#88797—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic, complemented
20C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
21C2×C32×D8—31no—
22#64113—31no—
23C3×D3×D8D314yescomplemented
24#63808—34no—
25#63654—31no—
26#63654D311yescharacteristic, complemented
27#63653—31no—
28#63653—31no—
29#63653—34no—
30#63653—64no—
31#63639—31no—
32C3×#827—34no—
33#63547—34no—
34#63546—34no—
35#63543—34no—
36C22×C32×D3—41no—
37C33×D4—21no—
38C4⋊(C3⋊(C3⋊D3))—21no—
39C3×#634—21no—
40C32×#67—41no—
41C32×D12—21no—
42C32×C4×D3D411yescharacteristic
43#6291—41no—
44#6289—41no—
45C33×C8C2311yescharacteristic
46C32×C3⋊C8D411yescharacteristic
47#5147—94no—
48#4914—91no—
49C2×C32×D4—61no—
50#3828—61no—
51C3×D3×D4—24no—
52#3799—64no—
53C32×D8—24no—
54C32×D8—31no—
55C32×D8—61no—
56C32×D8C2×D314yes—
57C32×D8C2×D311yescharacteristic
58C2×C32×C8—31no—
59#3746C2×D311yescharacteristic, complemented
60#3745—31no—
61#3745—34no—
62#3745—64no—
63#3744—61no—
64C3×#213—24no—
65C3×Dic12—34no—
66C3×D24C2×D314yes—
67C3×D3×C8C2×D314yes—
68#3717—64no—
69C3⋊(C3⋊C16)—31no—
70C3⋊(C3⋊C16)—34no—
71C3⋊(C3⋊C16)—64no—
72C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
73C3×C3⋊C16—34no—
74C22×C33—41no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C32×D3—41no—
77C2×C32×D3—82no—
78C2×C32×D3D811yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3×C3⋊Dic3—41no—
81C32×Dic3D811yescharacteristic
82C2×C3×D8—34no—
83#919—91no—
84D3×D8C3⋊D311yescharacteristic, complemented
85#904—94no—
86C3×SD32—91no—
87#828—91no—
88#827—34no—
89#827—94no—
90#826—94no—
91#826—94no—
92#826—91no—
93#826—184no—
94#812—94no—
95D48—91no—
96#798—91no—
97C23×C32—121no—
98C22×C3×D3—44no—
99C32×D4—21no—
100C32×D4—24no—
101C32×D4—44no—
102C32×D4—61no—
103C32×D4—121no—
104C2×C32×C4—31no—
105#638—121no—
106C4⋊(C3⋊D3)—61no—
107C4⋊(C3⋊D3)—64no—
108C4⋊(C3⋊D3)—124no—
109C4×C3⋊D3—121no—
110#634—21no—
111C3×#67—44no—
112C3×D12—24no—
113C3×C4×D3#6714yes—
114C3×Dic6—64no—
115#626—124no—
116#624—124no—
117C32×C8—24no—
118C32×C8—31no—
119C32×C8C22×D314yes—
120C32×C8C22×D311yescharacteristic
121C3×C3⋊C8#6714yes—
122#509—271noSylow
123C2×C33—81no—
124C2×C33C2×D811yescharacteristic
125C3⋊(C3⋊D3)—81no—
126C32×D3—21no—
127C32×D3—81no—
128C2×C3×D4—64no—
129#239—181no—
130D3×D4—21no—
131#235—184no—
132C3×Q16—91no—
133C3×D8—24no—
134C3×D8—34no—
135C3×D8—64no—
136C3×D8D3214yes—
137C3×D8C2×C3⋊D311yescharacteristic
138C2×C3×C8—34no—
139#215—181no—
140#213—21no—
141Dic12—34no—
142D24—94no—
143D24—91no—
144D24—184no—
145D24C2×C3⋊D311yescharacteristic
146#204—184no—
147D3×C8C2×C3⋊D311yescharacteristic
148C3×C16—91no—
149C3⋊C16—34no—
150C3⋊C16—94no—
151C3⋊C16—91no—
152C3⋊C16—184no—
153C22×C32—31no—
154C22×C32—41no—
155C22×C32—44no—
156C22×C32—84no—
157C22×C32—121no—
158C22×C32—242no—
159C2×C3⋊D3—121no—
160C2×C3⋊D3—124no—
161C2×C3⋊D3—244no—
162C2×C3×D3—44no—
163C2×C3×D3—88no—
164C2×C3×D3#21314yes—
165C32×C4—24no—
166C32×C4—31no—
167C32×C4D3×D411yescharacteristic
168C32×C4C2×#6714yes—
169C3⋊Dic3—41no—
170C3×Dic3—124no—
171C3×Dic3#21314yes—
172#135—271no—
173C2×D8—31no—
174SD32—91no—
175SD32—271no—
176D16—271no—
177D16—271no—
178M32—271no—
179C33#50911yescharacteristic, complemented, Sylow
180C23×C3—124no—
181C22×D3—41no—
182C3×Q8—181no—
183C3×D4—24no—
184C3×D4—21no—
185C3×D4—44no—
186C3×D4—64no—
187C3×D4—124no—
188C2×C3×C4—34no—
189#67—41no—
190#67—364no—
191D12—21no—
192D12—184no—
193D12—181no—
194D12—364no—
195D12—361no—
196C4×D3—364no—
197C4×D3—361no—
198C4×D3#63811yescharacteristic
199Dic6—64no—
200C3×C8—24no—
201C3×C8—34no—
202C3×C8C2×D3214yes—
203C3×C8C22×C3⋊D311yescharacteristic
204C3⋊C8#63811yescharacteristic
205C2×C32—24no—
206C2×C32—61no—
207C2×C32—81no—
208C2×C32—84no—
209C2×C32—164no—
210C2×C32—241no—
211C2×C32D3×D811yescharacteristic
212C2×C32C2×#21314yes—
213C3⋊D3—241no—
214C3⋊D3—244no—
215C3⋊D3—484no—
216C3×D3—24no—
217C3×D3—84no—
218#41—541no—
219C2×D4—61no—
220Q16—91no—
221SD16—541no—
222D8—31no—
223D8—61no—
224D8—271no—
225D8D3×C3⋊D311yescharacteristic
226C2×C8—31no—
227C16—91no—
228C16—271no—
229C22×C3—34no—
230C22×C3—41no—
231C22×C3—44no—
232C22×C3—84no—
233C22×C3—124no—
234C22×C3—248no—
235C2×D3—41no—
236C2×D3—82no—
237C2×D3—361no—
238C2×D3—364no—
239C2×D3—724no—
240C2×D3#375311yescharacteristic
241C3×C4—24no—
242C3×C4—34no—
243C3×C4—361no—
244C3×C4D3×#6714yes—
245C3×C4C2×#63811yescharacteristic
246Dic3—124no—
247Dic3#375311yescharacteristic
248C32—24no—
249C32#491411yescharacteristic, complemented
250C32#514714yescomplemented
251C23—121no—
252Q8—181no—
253D4—21no—
254D4—61no—
255D4—121no—
256D4—541no—
257D4—1081no—
258C2×C4—31no—
259C2×C4—1081no—
260C8—31no—
261C8C2×D3×C3⋊D311yescharacteristic
262C2×C3—24no—
263C2×C3—64no—
264C2×C3—84no—
265C2×C3—81no—
266C2×C3—164no—
267C2×C3—244no—
268C2×C3D3×#21314yes—
269C2×C3C2×#375311yescharacteristic
270D3—21no—
271D3—81no—
272D3—724no—
273D3—721no—
274D3—1444no—
275C22—31no—
276C22—41no—
277C22—121no—
278C22—242no—
279C22—1081no—
280C4—31no—
281C4—361no—
282C4D3×#63811yescharacteristic
283C3—24no—
284C3#1059103314yescomplemented
285C3#1059284611yescharacteristic, complemented
286C2—61no—
287C2—81no—
288C2—241no—
289C2—2161no—
290C2D3×#375311yescharacteristic
subgroupG/H
centerC2D3×#3753
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×D8C22
socleC2×C33C2×D8
hypercenterD8D3×C3⋊D3
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
1. C32×D3×D8C2semidirect
2. #11723337C2semidirect
3. #11722196C2semidirect
4. C3×#63654C2semidirect
5. #11722007C2semidirect
6. #11722005C2semidirect
7. #11722001C2semidirect
13. #88852C22semidirect
14. C3×#3746C22semidirect
19. C3⋊(C3⋊(C3⋊C16))C22semidirect
20. C3×C3⋊(C3⋊C16)C22semidirect
23. C3×D3×D8D3semidirect
26. #63654D3semidirect
59. #3746C2×D3semidirect
72. C3⋊(C3⋊C16)C2×D3semidirect
84. D3×D8C3⋊D3semidirect
179. C33#509semidirect
249. C32#4914semidirect
250. C32#5147semidirect
284. C3#10591033semidirect
285. C3#10592846semidirect

Maximal subgroups

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

subgroupindex
C32×D3×D82
#117233372
#117221962
C3×#636542
#117220072
#117220052
#117220012
#105928463
#105910333