Finite Group Databasebeta

← #13173434

Subgroups — #13173434

Order 1728.

760 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic
3C2×C4×C3⋊#157C211yesmaximal, characteristic
4#11723106C214yesmaximal
5#10593837—31nomaximal
6C22×C3⋊#157C2212yes—
7C22×C3⋊#157C2212yes—
8C2×C33×Q8C411yescharacteristic, complemented
9C3×Q8×C3⋊D3—22no—
10C3×Q8×C3⋊D3—22no—
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C411yescharacteristic, complemented
14C2×C3×#634C412yescomplemented
15C4⋊(C3⋊#157)—22no—
16C4×C3⋊#157—22no—
17C2×Q8×C3⋊D3D311yescharacteristic
18C2×C3×D3×Q8—62no—
19C2×C4⋊#157—31no—
20C2×C4×#157—31no—
21#63781—34no—
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×C3⋊#157—22no—
24C2×C3⋊#157—22no—
25C2×C3⋊#157—22no—
26C2×C3⋊#157—24no—
27C2×C3⋊#157Q814yescomplemented
28C33×Q8—22no—
29C2×C33×C4C2×C412yes—
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C412yes—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3D412yes—
36C3×C4×C3⋊D3D412yes—
37C3×#634—22no—
38C3×#634—24no—
39#5229—91nomaximal
40C22×#157—32no—
41C22×#157—32no—
42C2×C32×Q8—22no—
43C2×C32×Q8—42no—
44C2×C32×Q8Dic311yescharacteristic, complemented
45Q8×C3⋊D3—22no—
46Q8×C3⋊D3—22no—
47C2×C4×C3⋊D3C2×D312yes—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×#634Dic311yescharacteristic, complemented
50C2×#634Dic312yescomplemented
51C3×D3×Q8—68no—
52C3×D3×Q8—68no—
53C2×C3×C4×D3—62no—
54C2×C3×C4×D3—64no—
55C2×C3×Dic6—64no—
56C2×C3×Dic6—62no—
57C4⋊#157—62no—
58C4×#157—62no—
59C22×C33C22×C411yescharacteristic
60C2×C3×C3⋊D3C2×D411yescharacteristic
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C2×C3×C3⋊D3C2×Q812yes—
63C2×C3×C3⋊D3#4112yes—
64C3⋊#157—24no—
65C3⋊#157—42no—
66C33×C4—22no—
67C33×C4C22⋊C412yes—
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3C22⋊C412yes—
70C22×C3×Q8—91no—
71C2×D3×Q8—62no—
72C2×C4⋊Dic3—91no—
73C2×C4×Dic3—91no—
74C3⋊#95—94no—
75C22×C3⋊D3C22×D311yescharacteristic
76C22×C3×D3—62no—
77C2×#157—34no—
78C2×#157—62no—
79C2×#157—62no—
80C2×#157—64no—
81C2×#157—62no—
82C32×Q8—22no—
83C32×Q8—28no—
84C32×Q8—48no—
85C2×C32×C4—24no—
86C2×C32×C4—22no—
87C2×C32×C4—44no—
88C2×C32×C4—42no—
89C2×C32×C4C2×Dic312yes—
90C2×C32×C4C2×Dic311yescharacteristic
91C2×C3⋊Dic3C2×Dic312yes—
92C2×C3⋊Dic3C2×Dic311yescharacteristic
93C4×C3⋊D3—22no—
94C4×C3⋊D3—22no—
95C4×C3⋊D3#6712yes—
96C4×C3⋊D3#6712yes—
97#634—22no—
98#634—24no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—64no—
101C3×C4×D3—64no—
102C3×C4×D3—64no—
103C3×C4×D3—68no—
104C3×C4×D3—68no—
105C3×Dic6—616no—
106C3×Dic6—68no—
107#391—271noSylow
108C2×C33C2×C22⋊C411yescharacteristic
109C2×C33C4×Q812yes—
110C3×C3⋊D3C22⋊Q812yes—
111C3×C3⋊D3#12411yescharacteristic
112C3×C3⋊D3C4⋊Q811yescharacteristic
113C2×C3×Q8—22no—
114C2×C3×Q8—42no—
115C2×C3×Q8—92no—
116C2×C3×Q8—91no—
117C2×C3×Q8—182no—
118C2×C3×Q8—182no—
119C2×C3×Q8#15711yescharacteristic, complemented
120C22×C3×C4—91no—
121C22×C3×C4—92no—
122C22×Dic3—92no—
123C22×Dic3—92no—
124D3×Q8—68no—
125D3×Q8—68no—
126C2×C4×D3—62no—
127C2×C4×D3—64no—
128C2×Dic6—64no—
129C2×Dic6—62no—
130C4⋊Dic3—182no—
131C4×Dic3—182no—
132C22×C32—22no—
133C22×C32—42no—
134C22×C32C22×Dic311yescharacteristic
135C2×C3⋊D3D3×Q812yes—
136C2×C3⋊D3#23912yes—
137C2×C3⋊D3C2×#6711yescharacteristic
138C2×C3⋊D3C2×#6711yescharacteristic
139C2×C3×D3—62no—
140C2×C3×D3—62no—
141C2×C3×D3—64no—
142C2×C3×D3—64no—
143#157—64no—
144#157—122no—
145C32×C4—22no—
146C32×C4—28no—
147C32×C4—24no—
148C32×C4—48no—
149C32×C4—44no—
150C32×C4#21712yes—
151C3⋊Dic3—22no—
152C3⋊Dic3#21712yes—
153C3×Dic3—64no—
154C3×Dic3—68no—
155C22×Q8—91no—
156C2×C4⋊C4—271no—
157C2×C42—271no—
158#95—274no—
159C33#39111yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—62no—
162C3×Q8—28no—
163C3×Q8—22no—
164C3×Q8—48no—
165C3×Q8—184no—
166C3×Q8—182no—
167C2×C3×C4—24no—
168C2×C3×C4—22no—
169C2×C3×C4—44no—
170C2×C3×C4—42no—
171C2×C3×C4—91no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—92no—
175C2×C3×C4—182no—
176C2×C3×C4—182no—
177C2×C3×C4C2×#15712yes—
178C2×C3×C4C2×#15711yescharacteristic
179C2×Dic3—62no—
180C2×Dic3—64no—
181C2×Dic3—94no—
182C2×Dic3—182no—
183C2×Dic3—182no—
184C2×Dic3—182no—
185C2×Dic3—184no—
186C4×D3—64no—
187C4×D3—64no—
188C4×D3—68no—
189C4×D3—68no—
190Dic6—616no—
191Dic6—68no—
192C2×C32—24no—
193C2×C32—22no—
194C2×C32—42no—
195C2×C32—44no—
196C2×C32Q8×Dic312yes—
197C2×C32C2×#21711yescharacteristic
198C3⋊D3#94411yescharacteristic
199C3⋊D3#94612yes—
200C3⋊D3#94711yescharacteristic
201C3×D3—64no—
202C3×D3—62no—
203C3×D3—62no—
204C2×Q8—92no—
205C2×Q8—91no—
206C2×Q8—182no—
207C2×Q8—182no—
208C2×Q8C3⋊#15711yescharacteristic, complemented
209C22×C4—91no—
210C22×C4—92no—
211C22×C4—272no—
212C22×C4—272no—
213C4⋊C4—542no—
214C42—542no—
215C22×C3—22no—
216C22×C3—42no—
217C22×C3—92no—
218C22×C3—92no—
219C22×C3—91no—
220C22×C3—91no—
221C22×C3C22×#15711yescharacteristic
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—64no—
225C2×D3—64no—
226C3×C4—28no—
227C3×C4—22no—
228C3×C4—24no—
229C3×C4—48no—
230C3×C4—44no—
231C3×C4—92no—
232C3×C4—182no—
233C3×C4C22⋊#15712yes—
234Dic3—64no—
235Dic3—68no—
236Dic3—184no—
237Dic3—362no—
238C32—22no—
239C32—42no—
240C32#522911yescharacteristic, complemented
241C23—91no—
242Q8—22no—
243Q8—184no—
244Q8—182no—
245C2×C4—91no—
246C2×C4—92no—
247C2×C4—92no—
248C2×C4—92no—
249C2×C4—182no—
250C2×C4—182no—
251C2×C4—274no—
252C2×C4—542no—
253C2×C4—542no—
254C2×C4—542no—
255C2×C4—544no—
256C2×C4C2×C3⋊#15712yes—
257C2×C4C2×C3⋊#15711yescharacteristic
258C2×C3—24no—
259C2×C3—22no—
260C2×C3—44no—
261C2×C3—42no—
262C2×C3—92no—
263C2×C3—91no—
264C2×C3—91no—
265C2×C3Q8×#15712yes—
266C2×C3C2×C22⋊#15711yescharacteristic
267D3—62no—
268D3—64no—
269D3—62no—
270C22—92no—
271C22—92no—
272C22—91no—
273C22—91no—
274C22C22×C3⋊#15711yescharacteristic
275C4—22no—
276C4—92no—
277C4—182no—
278C4—544no—
279C4—1082no—
280C4#8899412yes—
281C3—22no—
282C3—42no—
283C3#1059383711yescharacteristic, complemented
284C2—92no—
285C2—91no—
286C2—91no—
287C2Q8×C3⋊#15712yes—
288C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC2×C33×Q8C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33#391
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
8. C2×C33×Q8C4semidirect
13. C2×C3×#634C4semidirect
14. C2×C3×#634C4semidirect
27. C2×C3⋊#157Q8semidirect
44. C2×C32×Q8Dic3semidirect
49. C2×#634Dic3semidirect
50. C2×#634Dic3semidirect
119. C2×C3×Q8#157semidirect
159. C33#391semidirect
208. C2×Q8C3⋊#157semidirect
240. C32#5229semidirect
283. C3#10593837semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C4⋊(C3⋊#157)2
C2×C4×C3⋊#1572
#117231062
#105938373
#52299