Finite Group Databasebeta

← #13167080

Subgroups — #13167080

Order 1728.

394 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 57 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#11723665C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic
4#11722779C212yesmaximal
5#11722774C212yesmaximal
6#10593391—31nomaximal
7C2×C32×Dic6—21no—
8C2×#6296C2211yescharacteristic
9C2×C3⋊#622C2212yes—
10C3×#3750—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#88815C2211yescharacteristic
13#88813—21no—
14C3×#3724—21no—
15C3×#3724C2212yes—
16C3×#944—61no—
17#63873D311yescharacteristic
18#63838—31no—
19#63837—61no—
20#63779—31no—
21#63743—32no—
22#63740—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—22no—
29C32×Dic6—42no—
30#6296—21no—
31#6296Q812yescomplemented
32C3⋊#622—22no—
33#5091—91nomaximal
34C2×C32×Q8—61no—
35C2×C3×Dic6—21no—
36C2×C3×Dic6—21no—
37C2×C3×Dic6—41no—
38C2×#627—31no—
39C2×#627—61no—
40C2×#622—32no—
41#3750—21no—
42C4×C3⋊Dic3C2×D311yescharacteristic
43C3×Dic3⋊C4—62no—
44C3×Dic3⋊C4—62no—
45C3×Dic3⋊C4—61no—
46C3×Dic3⋊C4—61no—
47C3×Dic3⋊C4—121no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50#3724—21no—
51#3724—31no—
52#3724C2×D312yes—
53Dic3⋊Dic3—61no—
54Dic32—61no—
55Dic32—61no—
56Dic32—61no—
57C22×C33C2×D411yescharacteristic
58C33×C4—21no—
59C3×C3⋊Dic3—22no—
60C3×C3⋊Dic3—41no—
61C3×C3⋊Dic3Q1612yes—
62C3×C3⋊Dic3C2×Q811yescharacteristic
63C3×C3⋊Dic3#4111yescharacteristic
64C32×Dic3—42no—
65C3×C4⋊Q8—91no—
66#944—61no—
67#887—181no—
68C4×Dic6—91no—
69#810—92no—
70#807—92no—
71C4⋊(C3⋊C8)—91no—
72C32×Q8—122no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—41no—
76C2×C32×C4—41no—
77C2×C32×C4—62no—
78C2×C32×C4#6711yescharacteristic
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3—21no—
81C2×C3⋊Dic3#6711yescharacteristic
82C2×C3⋊Dic3C22×D311yescharacteristic
83C2×C3×Dic3—22no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—121no—
96C3×Dic6—42no—
97C3×Dic6—42no—
98C3×Dic6—44no—
99#627—32no—
100#627—62no—
101#627—62no—
102#627—61no—
103#622—62no—
104#477—271noSylow
105C2×C33C22⋊Q811yescharacteristic
106C2×C33C2×Q1611yescharacteristic
107C2×C33AGL(1,ℤ/8)11yescharacteristic
108C2×C3×Q8—61no—
109C2×C3×Q8—61no—
110C2×C3×Q8—121no—
111C2×Dic6—21no—
112C2×Dic6—91no—
113C2×Dic6—181no—
114C3×C4⋊C4—92no—
115C3×C4⋊C4—181no—
116C3×C4⋊C4—181no—
117C3×C42—91no—
118C4⋊Dic3—91no—
119Dic3⋊C4—62no—
120Dic3⋊C4—62no—
121Dic3⋊C4—61no—
122Dic3⋊C4—61no—
123Dic3⋊C4—121no—
124Dic3⋊C4—181no—
125C4×Dic3—61no—
126C4×Dic3—61no—
127C4×Dic3—181no—
128C4×Dic3—181no—
129C4×Dic3—181no—
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—81no—
141C32×C4—122no—
142C3⋊Dic3—22no—
143C3⋊Dic3—41no—
144C3⋊Dic3#21612yes—
145C3⋊Dic3D3×Q811yescharacteristic
146C3⋊Dic3#23911yescharacteristic
147C3×Dic3—42no—
148C3×Dic3—44no—
149C3×Dic3—42no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—62no—
157C3×Dic3—62no—
158C3×Dic3—122no—
159C3×Dic3—121no—
160C3×Dic3—121no—
161C3×Dic3—241no—
162C4⋊Q8—91no—
163C4×Q8—271no—
164#107—272no—
165C4⋊C8—271no—
166#103—272no—
167C33#47711yescharacteristic, complemented, Sylow
168C3×Q8—122no—
169C3×Q8—122no—
170C3×Q8—124no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—122no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4GO+(4,𝔽2)11yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3—121no—
193C2×Dic3—121no—
194Dic6—42no—
195Dic6—92no—
196Dic6—182no—
197Dic6—181no—
198Dic6—182no—
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#93211yescharacteristic
213C2×C32C2×#21611yescharacteristic
214C2×C32#94611yescharacteristic
215C2×Q8—61no—
216C2×Q8—271no—
217C2×C8—272no—
218C4⋊C4—92no—
219C4⋊C4—181no—
220C4⋊C4—181no—
221C4⋊C4—271no—
222C42—91no—
223C42—541no—
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—81no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—92no—
237C3×C4—122no—
238C3×C4—124no—
239C3×C4—122no—
240C3×C4—182no—
241C3×C4—361no—
242Dic3—42no—
243Dic3—61no—
244Dic3—61no—
245Dic3—62no—
246Dic3—62no—
247Dic3—61no—
248Dic3—61no—
249Dic3—62no—
250Dic3—62no—
251Dic3—122no—
252Dic3—121no—
253Dic3—121no—
254Dic3—241no—
255C32—21no—
256C32—21no—
257C32—41no—
258C32—41no—
259C32#509111yescharacteristic, complemented
260Q8—122no—
261Q8—272no—
262Q8—541no—
263C2×C4—62no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—181no—
267C2×C4—181no—
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—91no—
288C4—91no—
289C4—92no—
290C4—122no—
291C4—182no—
292C4—361no—
293C3—21no—
294C3—21no—
295C3—41no—
296C3—41no—
297C3#1059339111yescharacteristic, complemented
298C2#1172429811yescharacteristic
299C2#1172431111yescharacteristic
300C2#1172432311yescharacteristic
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. #6296Q8semidirect
167. C33#477semidirect
259. C32#5091semidirect
297. C3#10593391semidirect

Maximal subgroups

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

subgroupindex
#117236652
C3×#638732
#117231042
#117227792
#117227742
#105933913
#50919