Finite Group Databasebeta

← #13167060

Subgroups — #13167060

Order 1728.

414 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 75 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#11724514C211yesmaximal, characteristic
2C2×#88808C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4#11722763C214yesmaximal, complemented
5#10593327—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7#88994C412yes—
8C4⋊(C3⋊#157)C412yes—
9C4×C3⋊#157C2212yes—
10C2×C3×C3⋊(C3⋊C8)—21no—
11C32×#208—21no—
12#88808—21no—
13#88808C2212yes—
14C3×#3712—21no—
15C3×#3710C2212yes—
16#64286—31no—
17C3×D3×M16—61no—
18#63818D311yescharacteristic
19C2×#3710—31no—
20C2×D3×C3⋊C8—61no—
21#63732—34no—
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C2×C3⋊#157C2×C414yes—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3Q811yescharacteristic
26C3×C4×C3⋊D3C2×C411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3×C4×C3⋊D3Q811yescharacteristic
29C3×C4×C3⋊D3C2311yescharacteristic
30C3×C3⋊(C3⋊C8)—22no—
31C32×C3⋊C8—22no—
32#5119—91nomaximal
33C2×C4×C3⋊D3C2×D311yescharacteristic
34C2×C3×C4×D3—61no—
35C2×C3×C4×D3—61no—
36C22⋊#157—32no—
37C4⋊#157—32no—
38C4×#157—32no—
39C32×M16—61no—
40C2×C3⋊(C3⋊C8)—21no—
41C3×#208—21no—
42C3×#208—21no—
43C3×#208—41no—
44C2×C3×C3⋊C8—61no—
45C2×C3×C3⋊C8—61no—
46C2×C3×C3⋊C8—121no—
47C3×ΓL(1,𝔽25)—62no—
48C3×D3×C8—62no—
49#3712—21no—
50#3710—32no—
51#3710—61no—
52#3710C2×D312yes—
53D3×C3⋊C8—62no—
54D3×C3⋊C8—62no—
55C22×C33C4⋊C411yescharacteristic
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3C4⋊C411yescharacteristic
58C2×C3×C3⋊D3C22×C411yescharacteristic
59C3⋊#157—24no—
60C33×C4C4⋊C411yescharacteristic
61C33×C4C2×D411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C3×C3⋊Dic3C2×Q811yescharacteristic
64C2×C3×M16—91no—
65#926—91no—
66C22×C3⋊C8—91no—
67D3×M16—61no—
68C2×D3×C8—181no—
69Dic3⋊C8—94no—
70C22×C3⋊D3C4×D311yescharacteristic
71C22×C3×D3—61no—
72C22×C3×D3—61no—
73C2×#157—34no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—41no—
77C2×C32×C4—41no—
78C2×C32×C4#6711yescharacteristic
79C2×C3⋊Dic3Dic611yescharacteristic
80C4×C3⋊D3Dic611yescharacteristic
81C4×C3⋊D3C4×D311yescharacteristic
82C4×C3⋊D3#6711yescharacteristic
83C4×C3⋊D3C22×D311yescharacteristic
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94C32×C8—62no—
95C3⋊(C3⋊C8)—22no—
96C3×C3⋊C8—22no—
97C3×C3⋊C8—22no—
98C3×C3⋊C8—42no—
99C3×C3⋊C8—62no—
100C3×C3⋊C8—62no—
101C3×C3⋊C8—122no—
102#424—271noSylow
103C2×C33—21no—
104C2×C33C2×C4⋊C411yescharacteristic
105C3×C3⋊D3—21no—
106C3×C3⋊D3#13112yes—
107C22×C3×C4—91no—
108C2×C4×D3—61no—
109C2×C4×D3—61no—
110C3×M16—61no—
111C3×M16—61no—
112C3×M16—121no—
113C3×M16—181no—
114C2×C3×C8—92no—
115C2×C3×C8—181no—
116#217—92no—
117C4⋊Dic3—92no—
118C4×Dic3—92no—
119#208—21no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—61no—
122C2×C3⋊C8—92no—
123C2×C3⋊C8—121no—
124C2×C3⋊C8—181no—
125ΓL(1,𝔽25)—62no—
126D3×C8—62no—
127D3×C8—182no—
128D3×C8—182no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32Dic3⋊C411yescharacteristic
134C2×C3⋊D3—22no—
135C2×C3⋊D3Dic3⋊C411yescharacteristic
136C2×C3⋊D3C2×C4×D311yescharacteristic
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—61no—
141C2×C3×D3—64no—
142C2×C3×D3—122no—
143#157—64no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—41no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4Dic3⋊C411yescharacteristic
153C32×C4C2×#6711yescharacteristic
154C3⋊Dic3Dic3⋊C411yescharacteristic
155C3⋊Dic3C2×Dic611yescharacteristic
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C2×M16—91no—
161C22×C8—271no—
162#117—271no—
163C4⋊C8—274no—
164C33#42411yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C22×D3—61no—
167C22×D3—61no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4GO+(4,𝔽2)11yescharacteristic
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—94no—
181C4×D3—61no—
182C4×D3—61no—
183C4×D3—61no—
184C4×D3—61no—
185C4×D3—61no—
186C4×D3—61no—
187C4×D3—61no—
188C4×D3—61no—
189C3×C8—62no—
190C3×C8—62no—
191C3×C8—122no—
192C3×C8—182no—
193C3⋊C8—22no—
194C3⋊C8—62no—
195C3⋊C8—62no—
196C3⋊C8—122no—
197C2×C32—22no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—42no—
205C2×C32—81no—
206C2×C32C2×Dic3⋊C411yescharacteristic
207C3⋊D3—21no—
208C3⋊D3#90712yes—
209C3×D3—62no—
210C3×D3—62no—
211C3×D3—62no—
212C3×D3—121no—
213C22×C4—91no—
214M16—61no—
215M16—181no—
216C2×C8—92no—
217C2×C8—181no—
218C2×C8—272no—
219C2×C8—541no—
220C4⋊C4—272no—
221C22⋊C4—272no—
222C42—272no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—91no—
228C22×C3—91no—
229C22×C3—182no—
230C22×C3#377311yescharacteristic
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—64no—
236C2×D3—122no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—91no—
246C3×C4—91no—
247C3×C4#377311yescharacteristic
248C3×C4C2×GO+(4,𝔽2)11yescharacteristic
249Dic3—61no—
250Dic3—61no—
251Dic3—61no—
252Dic3—61no—
253Dic3—184no—
254C32—21no—
255C32—21no—
256C32—41no—
257C32—41no—
258C32#511911yescharacteristic, complemented
259C23—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—274no—
266C2×C4D32⋊D311yescharacteristic
267C8—62no—
268C8—182no—
269C2×C3—22no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—41no—
274C2×C3—42no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—81no—
278C2×C3—92no—
279C2×C3—181no—
280C2×C3C2×#377311yescharacteristic
281D3—62no—
282D3—62no—
283D3—62no—
284D3—121no—
285C22—91no—
286C22—91no—
287C22—182no—
288C22#8893411yescharacteristic
289C4—91no—
290C4—91no—
291C4—544no—
292C4#8893411yescharacteristic
293C4C2×D32⋊D311yescharacteristic
294C3—21no—
295C3—21no—
296C3—41no—
297C3—41no—
298C3#1059332711yescharacteristic, complemented
299C2—21no—
300C2—92no—
301C2—181no—
302C2C2×#8893411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D32⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#424
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722763C2semidirect
164. C33#424semidirect
258. C32#5119semidirect
298. C3#10593327semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×#888082
C3×#638182
#117227632
#105933273
#51199