Finite Group Databasebeta

← #13159903

Subgroups — #13159903

Order 1728.

728 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 156 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#11723536C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic
3#11722184C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic
5C32×#816C211yesmaximal, characteristic
6#11722056C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592492—31nomaximal
9#10590817—34nomaximal
10C2×C32×Dic6C2211yescharacteristic
11C2×#6293—21no—
12C2×C33×C8C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21Q8×C3⋊Dic3—31no—
22#64097—31no—
23#63919—34no—
24#63843—34no—
25C32×#103—31no—
26#63662—31no—
27#63645D311yescharacteristic
28#63641—31no—
29#63641—34no—
30#63641—64no—
31#63634—31no—
32C3×#818—34no—
33C3×#816D314yes—
34#63576—34no—
35#63565—34no—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C32×Dic6—21no—
44C32×Dic6Q812yescomplemented
45#6293—42no—
46C33×C8—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48#4875—91no—
49#4793—94no—
50C2×C32×Q8—31no—
51C2×#634—61no—
52C2×C3×Dic6C2×D314yes—
53C2×#627—64no—
54C2×C32×C8—24no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4—31no—
58C4⋊(C3⋊Dic3)—61no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—61no—
62C4×C3⋊Dic3—31no—
63C4×C3⋊Dic3—34no—
64C4×C3⋊Dic3—61no—
65C4×C3⋊Dic3—64no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72#3724—64no—
73Dic3⋊Dic3—64no—
74Dic32—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—21no—
79C3⋊(C3⋊Dic3)Q1612yes—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—22no—
83C32×Dic3—41no—
84Q8×Dic3—94no—
85#888—94no—
86#888—91no—
87#887—91no—
88C3×#107—91no—
89C3×#103—34no—
90#835—94no—
91#818—34no—
92#816C3⋊D311yescharacteristic
93Dic3⋊C8—94no—
94Dic3⋊C8—91no—
95Dic3⋊C8—184no—
96#810—91no—
97#807—94no—
98#807—91no—
99C32×Q8—32no—
100C32×Q8—61no—
101C2×C32×C4—24no—
102C2×C32×C4—61no—
103C2×C32×C4—61no—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—34no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—64no—
114#634—122no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119C3×Dic6—24no—
120C3×Dic6Dic618yescomplemented
121#627—128no—
122C32×C8—21no—
123C32×C8—24no—
124C32×C8—44no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—124no—
128#477—271noSylow
129C2×C33C22⋊Q811yescharacteristic
130C2×C33C2×Q1611yescharacteristic
131C2×C33AGL(1,ℤ/8)11yescharacteristic
132C2×C3×Q8—34no—
133C2×Dic6—181no—
134C2×Dic6—184no—
135C2×Dic6C2×C3⋊D311yescharacteristic
136C2×C3×C8—24no—
137C2×C3×C8D3214yes—
138C2×C3×C8C2×C3⋊D311yescharacteristic
139C3×C4⋊C4—34no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—91no—
142C4⋊Dic3—34no—
143C4⋊Dic3—34no—
144C4⋊Dic3—184no—
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—184no—
149C4×Dic3—94no—
150C4×Dic3—91no—
151C4×Dic3—184no—
152C4×Dic3—181no—
153C4×Dic3—184no—
154C2×C3⋊C8—94no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—184no—
157C22×C32—24no—
158C22×C32D3×D414yes—
159C22×C32D3×D411yescharacteristic
160C32×C4—24no—
161C32×C4—24no—
162C32×C4—62no—
163C32×C4—121no—
164C32×C4C2×Dic614yes—
165C32×C4#23511yescharacteristic
166C32×C4#23714yes—
167C32×C4D3×Q811yescharacteristic
168C3⋊Dic3—32no—
169C3⋊Dic3—38no—
170C3⋊Dic3—41no—
171C3⋊Dic3—41no—
172C3⋊Dic3—68no—
173C3⋊Dic3—61no—
174C3⋊Dic3—64no—
175C3⋊Dic3—124no—
176C3×Dic3—28no—
177C3×Dic3—44no—
178C3×Dic3—124no—
179C3×Dic3—124no—
180C4⋊Q8—271no—
181C4×Q8—271no—
182#107—91no—
183#107—271no—
184C4⋊C8—271no—
185#103—31no—
186#103—271no—
187C33#47711yescharacteristic, complemented, Sylow
188C3×Q8—38no—
189C3×Q8—64no—
190C2×C3×C4—24no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C2×D3214yes—
196C2×C3×C4C22×C3⋊D311yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—64no—
200C2×Dic3—64no—
201C2×Dic3—94no—
202C2×Dic3—91no—
203C2×Dic3—94no—
204C2×Dic3—91no—
205C2×Dic3—184no—
206C2×Dic3—184no—
207Dic6—21no—
208Dic6—362no—
209Dic6—368no—
210Dic6#63412yescomplemented
211C3×C8—24no—
212C3×C8—21no—
213C3×C8—44no—
214C3⋊C8—184no—
215C3⋊C8—181no—
216C3⋊C8—364no—
217C2×C32—24no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32#87814yes—
221C2×C32#89611yescharacteristic
222C2×C32#91111yescharacteristic
223C2×C32#91414yes—
224C2×C32D3×Q1614yes—
225C2×C32D3×Q1611yescharacteristic
226C2×Q8—31no—
227C2×Q8—541no—
228C2×C8—271no—
229C2×C8D3×C3⋊D311yescharacteristic
230C4⋊C4—31no—
231C4⋊C4—91no—
232C4⋊C4—91no—
233C4⋊C4—541no—
234C4⋊C4—541no—
235C42—271no—
236C42—541no—
237C22×C3—24no—
238C22×C3#380214yes—
239C22×C3D4×C3⋊D311yescharacteristic
240C3×C4—24no—
241C3×C4—24no—
242C3×C4—68no—
243C3×C4—124no—
244C3×C4—361no—
245C3×C4—361no—
246C3×C4D3×Dic614yes—
247C3×C4#380414yes—
248C3×C4C2×#63411yescharacteristic
249C3×C4#383011yescharacteristic
250Dic3—22no—
251Dic3—41no—
252Dic3—98no—
253Dic3—92no—
254Dic3—124no—
255Dic3—124no—
256Dic3—188no—
257Dic3—184no—
258Dic3—181no—
259Dic3—364no—
260C32—24no—
261C32#479314yescomplemented
262C32#487511yescharacteristic, complemented
263Q8—32no—
264Q8—61no—
265Q8—1082no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—271no—
271C2×C4—271no—
272C2×C4C2×D3×C3⋊D311yescharacteristic
273C8—21no—
274C8—541no—
275C2×C3—24no—
276C2×C3—24no—
277C2×C3—24no—
278C2×C3#6379814yes—
279C2×C3#6380214yes—
280C2×C3#6389914yes—
281C2×C3#6408711yescharacteristic
282C2×C3#6412311yescharacteristic
283C2×C3Q16×C3⋊D311yescharacteristic
284C22#8902611yescharacteristic
285C4—62no—
286C4—121no—
287C4—272no—
288C4—361no—
289C4—361no—
290C4—541no—
291C4D3×#63411yescharacteristic
292C4#8902811yescharacteristic
293C3—24no—
294C3#1059081714yescomplemented
295C3#1059249211yescharacteristic, complemented
296C2#1172332311yescharacteristic
297C2#1172333011yescharacteristic
298C2#1172350011yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#477
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
44. C32×Dic6Q8semidirect
120. C3×Dic6Dic6semidirect
187. C33#477semidirect
210. Dic6#634semidirect
261. C32#4793semidirect
262. C32#4875semidirect
294. C3#10590817semidirect
295. C3#10592492semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117233972
#117221842
C3×#636452
C32×#8162
#117220562
#117220372
#105924923
#105908173