Finite Group Databasebeta

← #13167065

Subgroups — #13167065

Order 1728.

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

normal subgroups only

This group has 382 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
1C4⋊#6296C211yesmaximal, characteristic
2C3×#63838C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic
4#11722778C212yesmaximal
5#11722775C212yesmaximal
6#10593385—31nomaximal
7C2×#6296—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3⋊#622C2212yes—
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×Dic3⋊C4—21no—
13#88815C2212yes—
14C3×#3724C2211yescharacteristic
15C3×Dic32—21no—
16C3×#887—61no—
17#63919—31no—
18#63918—61no—
19#63838D311yescharacteristic
20#63779—31no—
21#63742—32no—
22#63740—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—21no—
28C2×C32×Dic3—21no—
29#6296—42no—
30C3×#627—21no—
31C3×#627Q812yescomplemented
32C3⋊#622—22no—
33#5020—91nomaximal
34C2×C3×Dic6—61no—
35C2×#627—62no—
36C2×#627—61no—
37C2×#627C2×D311yescharacteristic
38C2×#622—32no—
39C32×C4⋊C4—61no—
40C4⋊(C3⋊Dic3)—21no—
41C4×C3⋊Dic3C2×D311yescharacteristic
42C3×C4⋊Dic3—61no—
43C3×C4⋊Dic3—61no—
44C3×C4⋊Dic3—121no—
45C3×Dic3⋊C4—21no—
46C3×Dic3⋊C4—21no—
47C3×Dic3⋊C4—41no—
48C3×Dic3⋊C4—61no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—61no—
52C3×C4×Dic3—61no—
53#3724—32no—
54#3724C2×D311yescharacteristic
55Dic3⋊Dic3—62no—
56Dic32—21no—
57C22×C33C2×D411yescharacteristic
58C33×C4—21no—
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3SD1612yes—
61C3×C3⋊Dic3C2×Q811yescharacteristic
62C3×C3⋊Dic3#4111yescharacteristic
63C32×Dic3—22no—
64C32×Dic3—41no—
65C3×C4×Q8—91no—
66#888—181no—
67#887—61no—
68#869—91no—
69#835—92no—
70#808—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
82C2×C3×Dic3—21no—
83C2×C3×Dic3—21no—
84C2×C3×Dic3—21no—
85C2×C3×Dic3—21no—
86C2×C3×Dic3—41no—
87C2×C3×Dic3—41no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—122no—
95C3×Dic6—62no—
96C3×Dic6—62no—
97#627—21no—
98#627—124no—
99#627—122no—
100#627Dic612yescomplemented
101#622—62no—
102#475—271noSylow
103C2×C33C22⋊Q811yescharacteristic
104C2×C33C2×SD1611yescharacteristic
105C2×C33#13711yescharacteristic
106C2×C3×Q8—91no—
107C2×Dic6—61no—
108C2×Dic6—181no—
109C2×Dic6—182no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—121no—
114C3×C4⋊C4—181no—
115C3×C42—91no—
116C3×C42—181no—
117C4⋊Dic3—61no—
118C4⋊Dic3—61no—
119C4⋊Dic3—92no—
120C4⋊Dic3—121no—
121Dic3⋊C4—21no—
122Dic3⋊C4—61no—
123Dic3⋊C4—182no—
124C4×Dic3—61no—
125C4×Dic3—61no—
126C4×Dic3—61no—
127C4×Dic3—61no—
128C2×C3⋊C8—92no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32C2×#6711yescharacteristic
134C32×C4—22no—
135C32×C4—21no—
136C32×C4—41no—
137C32×C4—42no—
138C32×C4—62no—
139C32×C4—81no—
140C32×C4—121no—
141C3⋊Dic3—42no—
142C3⋊Dic3#21512yes—
143C3⋊Dic3C2×Dic611yescharacteristic
144C3⋊Dic3#23511yescharacteristic
145C3×Dic3—22no—
146C3×Dic3—22no—
147C3×Dic3—42no—
148C3×Dic3—41no—
149C3×Dic3—41no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—81no—
157C3×Dic3—122no—
158C3×Dic3—122no—
159C3×Dic3—242no—
160C4⋊Q8—271no—
161C4×Q8—91no—
162#106—272no—
163C4⋊C8—271no—
164#103—272no—
165C33#47511yescharacteristic, complemented, Sylow
166C3×Q8—92no—
167C3×Q8—181no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—121no—
179C2×C3×C4—121no—
180C2×C3×C4—182no—
181C2×C3×C4GO+(4,𝔽2)11yescharacteristic
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—122no—
191Dic6—62no—
192Dic6—62no—
193Dic6—364no—
194Dic6—362no—
195C3⋊C8—182no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32#92411yescharacteristic
209C2×C32C2×#21511yescharacteristic
210C2×C32#95111yescharacteristic
211C2×Q8—91no—
212C2×Q8—541no—
213C2×C8—272no—
214C4⋊C4—61no—
215C4⋊C4—91no—
216C4⋊C4—181no—
217C4⋊C4—272no—
218C42—91no—
219C42—181no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4—42no—
229C3×C4—62no—
230C3×C4—62no—
231C3×C4—81no—
232C3×C4—91no—
233C3×C4—91no—
234C3×C4—92no—
235C3×C4—122no—
236C3×C4—121no—
237C3×C4—121no—
238C3×C4—241no—
239C3×C4—362no—
240Dic3—22no—
241Dic3—41no—
242Dic3—61no—
243Dic3—61no—
244Dic3—62no—
245Dic3—62no—
246Dic3—61no—
247Dic3—61no—
248Dic3—122no—
249Dic3—122no—
250Dic3—242no—
251C32—21no—
252C32—21no—
253C32—41no—
254C32—41no—
255C32#502011yescharacteristic, complemented
256Q8—92no—
257Q8—181no—
258Q8—1082no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—182no—
264C2×C4D32⋊D311yescharacteristic
265C8—542no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3D32⋊Q811yescharacteristic
279C2×C3#6422711yescharacteristic
280C2×C3#6423911yescharacteristic
281C22C2×D32⋊D311yescharacteristic
282C4—21no—
283C4—62no—
284C4—91no—
285C4—91no—
286C4—92no—
287C4—121no—
288C4—362no—
289C3—21no—
290C3—21no—
291C3—41no—
292C3—41no—
293C3#1059338511yescharacteristic, complemented
294C2#1172430011yescharacteristic
295C2#1172430811yescharacteristic
296C2#1172432811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#475
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
100. #627Dic6semidirect
165. C33#475semidirect
255. C32#5020semidirect
293. C3#10593385semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#638382
#117231042
#117227782
#117227752
#105933853
#50209