Finite Group Databasebeta

← #13159306

Subgroups — #13159306

Order 1728.

860 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 162 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#11723929C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6#11722052C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic
8#10592554—31nomaximal
9#10590416—34nomaximal
10C2×#6308—21no—
11C2×#6293—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64097—31no—
22#64097—31no—
23#64097—34no—
24#64097—64no—
25#63919—34no—
26C32×#106—31no—
27#63644—31no—
28C8×C3⋊Dic3—31no—
29#63635—31no—
30#63635D311yescharacteristic
31C3×#808—34no—
32C3×#808D314yes—
33#63573—34no—
34#63573—34no—
35#63553—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41#6308—42no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44#6293—42no—
45C3×C3⋊(C3⋊C8)Q812yescomplemented
46C32×C3⋊C8Q812yescomplemented
47#4855—94no—
48#4855—91no—
49C2×#634—61no—
50C2×#634—61no—
51C2×#634—64no—
52C2×#634—68no—
53C2×#627—64no—
54C2×C32×C8—31no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—31no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—61no—
64#3750—61no—
65#3750—64no—
66#3750—68no—
67C4×C3⋊Dic3—31no—
68C4×C3⋊Dic3—34no—
69C4×C3⋊Dic3—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74C2×C3×C3⋊C8C2×D314yes—
75#3724—64no—
76C22×C33C2×D411yescharacteristic
77C33×C4—41no—
78C33×C4C2×Q811yescharacteristic
79C33×C4C2×Q811yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C3⋊(C3⋊Dic3)SD1612yes—
82C3⋊(C3⋊Dic3)SD1612yes—
83C3×C3⋊Dic3—41no—
84C32×Dic3—41no—
85#888—94no—
86#888—94no—
87#888—91no—
88#888—91no—
89#888—184no—
90C3×#106—34no—
91C3×#106—91no—
92#817—94no—
93#817—91no—
94C8×Dic3—94no—
95C8×Dic3—91no—
96#808—34no—
97#808—94no—
98#808—91no—
99#808C3⋊D311yescharacteristic
100C2×C32×C4—28no—
101C2×C32×C4—24no—
102C2×C32×C4—21no—
103C2×C32×C4—24no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
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—
114C2×C3⋊Dic3—68no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—61no—
117#634—122no—
118#634—122no—
119#634—128no—
120#634—1216no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—64no—
123#627—128no—
124C32×C8—32no—
125C3⋊(C3⋊C8)Dic612yescomplemented
126C3×C3⋊C8—38no—
127C3×C3⋊C8Dic618yescomplemented
128#498—271noSylow
129C2×C33C4⋊Q811yescharacteristic
130C2×C33C2×SD1611yescharacteristic
131C2×C33C2×SD1611yescharacteristic
132C2×Dic6—181no—
133C2×Dic6—184no—
134C2×Dic6—184no—
135C2×Dic6—181no—
136C2×Dic6—188no—
137C2×C3×C8—34no—
138C2×C3×C8—91no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—34no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4D3214yes—
143C3×C4⋊C4C2×C3⋊D311yescharacteristic
144C4⋊Dic3—34no—
145C4⋊Dic3—94no—
146C4⋊Dic3—91no—
147C4⋊Dic3—184no—
148C4⋊Dic3C2×C3⋊D311yescharacteristic
149Dic3⋊C4—181no—
150Dic3⋊C4—184no—
151Dic3⋊C4—184no—
152Dic3⋊C4—181no—
153Dic3⋊C4—188no—
154C4×Dic3—94no—
155C4×Dic3—91no—
156C4×Dic3—184no—
157C2×C3⋊C8—34no—
158C2×C3⋊C8C2×C3⋊D311yescharacteristic
159C22×C32—24no—
160C22×C32D3×D414yes—
161C22×C32D3×D411yescharacteristic
162C32×C4—24no—
163C32×C4—24no—
164C32×C4—48no—
165C32×C4—44no—
166C32×C4—41no—
167C32×C4—121no—
168C32×C4C2×Dic614yes—
169C32×C4C2×Dic611yescharacteristic
170C32×C4D3×Q814yes—
171C32×C4D3×Q811yescharacteristic
172C3⋊Dic3—32no—
173C3⋊Dic3—32no—
174C3⋊Dic3—38no—
175C3⋊Dic3—38no—
176C3⋊Dic3—41no—
177C3⋊Dic3—68no—
178C3⋊Dic3—68no—
179C3⋊Dic3—128no—
180C3⋊Dic3—124no—
181C3⋊Dic3—121no—
182C3×Dic3—44no—
183C3×Dic3—124no—
184C4⋊Q8—271no—
185C4⋊Q8—271no—
186#106—31no—
187#106—91no—
188#106—271no—
189#106—271no—
190C4×C8—271no—
191C33#49811yescharacteristic, complemented, Sylow
192C2×C3×C4—28no—
193C2×C3×C4—24no—
194C2×C3×C4—21no—
195C2×C3×C4—24no—
196C2×C3×C4—64no—
197C2×C3×C4—181no—
198C2×C3×C4C2×D3214yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—64no—
202C2×Dic3—94no—
203C2×Dic3—94no—
204C2×Dic3—91no—
205C2×Dic3—91no—
206C2×Dic3—184no—
207C2×Dic3—184no—
208C2×Dic3—188no—
209C2×Dic3—184no—
210C2×Dic3—181no—
211Dic6—362no—
212Dic6—368no—
213Dic6—362no—
214Dic6—368no—
215Dic6—3616no—
216C3×C8—38no—
217C3×C8—92no—
218C3⋊C8—38no—
219C3⋊C8#63412yescomplemented
220C2×C32—24no—
221C2×C32—24no—
222C2×C32—24no—
223C2×C32#88814yes—
224C2×C32#88811yescharacteristic
225C2×C32D3×SD1614yes—
226C2×C32D3×SD1614yes—
227C2×C32D3×SD1611yescharacteristic
228C2×C32D3×SD1611yescharacteristic
229C2×Q8—541no—
230C2×Q8—541no—
231C2×C8—31no—
232C2×C8—91no—
233C4⋊C4—31no—
234C4⋊C4—91no—
235C4⋊C4—271no—
236C4⋊C4—541no—
237C4⋊C4—541no—
238C4⋊C4D3×C3⋊D311yescharacteristic
239C42—271no—
240C22×C3—24no—
241C22×C3#381114yes—
242C22×C3D4×C3⋊D311yescharacteristic
243C3×C4—24no—
244C3×C4—24no—
245C3×C4—48no—
246C3×C4—41no—
247C3×C4—44no—
248C3×C4—124no—
249C3×C4—361no—
250C3×C4D3×Dic614yes—
251C3×C4D3×Dic614yes—
252C3×C4C2×#63411yescharacteristic
253C3×C4Q8×C3⋊D311yescharacteristic
254Dic3—41no—
255Dic3—92no—
256Dic3—92no—
257Dic3—98no—
258Dic3—98no—
259Dic3—124no—
260Dic3—188no—
261Dic3—188no—
262Dic3—368no—
263Dic3—361no—
264Dic3—364no—
265C32—24no—
266C32#485514yescomplemented
267C32#485511yescharacteristic, complemented
268Q8—1082no—
269Q8—1082no—
270C2×C4—21no—
271C2×C4—61no—
272C2×C4—181no—
273C2×C4—271no—
274C2×C4—271no—
275C2×C4—541no—
276C2×C4C2×D3×C3⋊D311yescharacteristic
277C8—32no—
278C8—92no—
279C2×C3—24no—
280C2×C3—24no—
281C2×C3—24no—
282C2×C3#6384014yes—
283C2×C3#6393014yes—
284C2×C3#6394014yes—
285C2×C3#6409711yescharacteristic
286C2×C3SD16×C3⋊D311yescharacteristic
287C2×C3SD16×C3⋊D311yescharacteristic
288C22#8903911yescharacteristic
289C4—41no—
290C4—121no—
291C4—272no—
292C4—272no—
293C4—361no—
294C4—1081no—
295C4D3×#63411yescharacteristic
296C4C3⋊D3×Dic611yescharacteristic
297C3—24no—
298C3#1059041614yescomplemented
299C3#1059255411yescharacteristic, complemented
300C2#1172339511yescharacteristic
301C2#1172355311yescharacteristic
302C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#498
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
45. C3×C3⋊(C3⋊C8)Q8semidirect
46. C32×C3⋊C8Q8semidirect
125. C3⋊(C3⋊C8)Dic6semidirect
127. C3×C3⋊C8Dic6semidirect
191. C33#498semidirect
219. C3⋊C8#634semidirect
266. C32#4855semidirect
267. C32#4855semidirect
298. C3#10590416semidirect
299. C3#10592554semidirect

Maximal subgroups

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

subgroupindex
#117239292
#117235362
C3×#636352
C32×#8082
#117220542
#117220522
#117220182
#105925543
#105904163