Finite Group Databasebeta

← #13167074

Subgroups — #13167074

Order 1728.

394 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 59 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#11723678C211yesmaximal, characteristic
2C3×#63838C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic
4#11722779C212yesmaximal
5#11722775C212yesmaximal
6#10593391—31nomaximal
7C2×C3×#634—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3⋊#622C2212yes—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×Dic3⋊C4—21no—
12#88815—21no—
13#88815C2212yes—
14C3×#3724C2211yescharacteristic
15C3×Dic32—21no—
16C3×#887—61no—
17#63873—31no—
18#63866—61no—
19#63838D311yescharacteristic
20#63779—31no—
21#63743—32no—
22#63740—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×#634—42no—
28C2×C32×Dic3—21no—
29C2×C32×Dic3—21no—
30C3×#627—21no—
31C3×#627Q812yescomplemented
32C3⋊#622—22no—
33#5021—91nomaximal
34C2×#634—21no—
35C2×C3×Dic6—61no—
36C2×C3×Dic6—61no—
37C2×C3×Dic6—61no—
38C2×C3×Dic6—121no—
39C2×#627C2×D311yescharacteristic
40C2×#622—32no—
41C32×C4⋊C4—61no—
42C4×C3⋊Dic3C2×D311yescharacteristic
43C3×Dic3⋊C4—21no—
44C3×Dic3⋊C4—21no—
45C3×Dic3⋊C4—41no—
46C3×Dic3⋊C4—61no—
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—61no—
51#3724—32no—
52#3724—61no—
53#3724C2×D311yescharacteristic
54Dic3⋊Dic3—62no—
55Dic3⋊Dic3—62no—
56Dic32—21no—
57C22×C33C2×D411yescharacteristic
58C33×C4—21no—
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3Q1612yes—
61C3×C3⋊Dic3C2×Q811yescharacteristic
62C3×C3⋊Dic3#4111yescharacteristic
63C32×Dic3—22no—
64C32×Dic3—41no—
65C3×C4×Q8—91no—
66#944—181no—
67#887—61no—
68#869—91no—
69#835—92no—
70#807—92no—
71C4⋊(C3⋊C8)—91no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—61no—
77C2×C32×C4—61no—
78C2×C32×C4#6711yescharacteristic
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3#6711yescharacteristic
81C2×C3⋊Dic3C22×D311yescharacteristic
82#634—42no—
83C2×C3×Dic3—21no—
84C2×C3×Dic3—21no—
85C2×C3×Dic3—21no—
86C2×C3×Dic3—21no—
87C2×C3×Dic3—41no—
88C2×C3×Dic3—41no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—122no—
96C3×Dic6—62no—
97C3×Dic6—62no—
98C3×Dic6—122no—
99C3×Dic6—122no—
100C3×Dic6—124no—
101#627—21no—
102#627Dic612yescomplemented
103#622—62no—
104#477—271noSylow
105C2×C33C22⋊Q811yescharacteristic
106C2×C33C2×Q1611yescharacteristic
107C2×C33AGL(1,ℤ/8)11yescharacteristic
108C2×C3×Q8—91no—
109C2×C3×Q8—181no—
110C2×Dic6—61no—
111C2×Dic6—61no—
112C2×Dic6—61no—
113C2×Dic6—121no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—121no—
118C3×C42—91no—
119C3×C42—181no—
120C4⋊Dic3—92no—
121C4⋊Dic3—181no—
122Dic3⋊C4—21no—
123Dic3⋊C4—61no—
124Dic3⋊C4—182no—
125Dic3⋊C4—182no—
126C4×Dic3—61no—
127C4×Dic3—61no—
128C4×Dic3—61no—
129C4×Dic3—61no—
130C2×C3⋊C8—92no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32C2×#6711yescharacteristic
136C32×C4—22no—
137C32×C4—21no—
138C32×C4—41no—
139C32×C4—42no—
140C32×C4—62no—
141C32×C4—81no—
142C32×C4—121no—
143C3⋊Dic3—42no—
144C3⋊Dic3#21612yes—
145C3⋊Dic3C2×Dic611yescharacteristic
146C3⋊Dic3#23511yescharacteristic
147C3×Dic3—22no—
148C3×Dic3—22no—
149C3×Dic3—42no—
150C3×Dic3—41no—
151C3×Dic3—41no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—81no—
159C3×Dic3—122no—
160C3×Dic3—124no—
161C3×Dic3—122no—
162C4⋊Q8—271no—
163C4×Q8—91no—
164#107—272no—
165C4⋊C8—271no—
166#103—272no—
167C33#47711yescharacteristic, complemented, Sylow
168C3×Q8—92no—
169C3×Q8—181no—
170C3×Q8—362no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4—121no—
182C2×C3×C4—121no—
183C2×C3×C4—182no—
184C2×C3×C4GO+(4,𝔽2)11yescharacteristic
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—122no—
194Dic6—62no—
195Dic6—62no—
196Dic6—122no—
197Dic6—122no—
198Dic6—124no—
199C3⋊C8—182no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32#92411yescharacteristic
213C2×C32C2×#21611yescharacteristic
214C2×C32#94911yescharacteristic
215C2×Q8—91no—
216C2×Q8—181no—
217C2×C8—272no—
218C4⋊C4—61no—
219C4⋊C4—91no—
220C4⋊C4—272no—
221C4⋊C4—541no—
222C42—91no—
223C42—181no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3C2×GO+(4,𝔽2)11yescharacteristic
229C3×C4—22no—
230C3×C4—21no—
231C3×C4—41no—
232C3×C4—42no—
233C3×C4—62no—
234C3×C4—62no—
235C3×C4—81no—
236C3×C4—91no—
237C3×C4—91no—
238C3×C4—92no—
239C3×C4—122no—
240C3×C4—121no—
241C3×C4—121no—
242C3×C4—241no—
243C3×C4—362no—
244Dic3—22no—
245Dic3—41no—
246Dic3—61no—
247Dic3—61no—
248Dic3—62no—
249Dic3—62no—
250Dic3—61no—
251Dic3—61no—
252Dic3—122no—
253Dic3—124no—
254Dic3—122no—
255C32—21no—
256C32—21no—
257C32—41no—
258C32—41no—
259C32#502111yescharacteristic, complemented
260Q8—92no—
261Q8—181no—
262Q8—362no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—182no—
268C2×C4D32⋊D311yescharacteristic
269C8—542no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3D32⋊Q811yescharacteristic
283C2×C3#6422811yescharacteristic
284C2×C3#6423611yescharacteristic
285C22C2×D32⋊D311yescharacteristic
286C4—21no—
287C4—62no—
288C4—91no—
289C4—91no—
290C4—92no—
291C4—121no—
292C4—362no—
293C3—21no—
294C3—21no—
295C3—41no—
296C3—41no—
297C3#1059339111yescharacteristic, complemented
298C2#1172430011yescharacteristic
299C2#1172431111yescharacteristic
300C2#1172432211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#477
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
31. C3×#627Q8semidirect
102. #627Dic6semidirect
167. C33#477semidirect
259. C32#5021semidirect
297. C3#10593391semidirect

Maximal subgroups

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

subgroupindex
#117236782
C3×#638382
#117231042
#117227792
#117227752
#105933913
#50219