Finite Group Databasebeta

← #13167101

Subgroups — #13167101

Order 1728.

364 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 47 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×#88812C211yesmaximal, characteristic, complemented
2C3×#63822C211yesmaximal, characteristic
3#11723098C211yesmaximal, characteristic, complemented
4#11722768C212yesmaximal, complemented
5#11722767C212yesmaximal, complemented
6#10593336—31nomaximal
7C2×C3×#634—21no—
8C32×#235—21no—
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10#88812—21no—
11#88812C2212yes—
12C3×#3718C2211yescharacteristic
13C3×#3715—21no—
14C3×#3713C2211yescharacteristic
15C3⋊#3708C2212yes—
16C3×#950—61no—
17C2×#3719—61no—
18C2×#3718—31no—
19#63822D311yescharacteristic
20#63773—31no—
21#63736—32no—
22#63735—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—41no—
25C3×#634—22no—
26C3×#634—41no—
27C32×#67—41no—
28C32×D12—21no—
29C32×C4×D3—41no—
30C32×Dic6—21no—
31C3×C3⋊(C3⋊C8)D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33#5195—91nomaximal
34C32×#41—61no—
35C2×#634—21no—
36C3×#235—21no—
37C3×#235—21no—
38C3×#235—41no—
39C2×C3×Dic6—61no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—121no—
42C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
43C3×#216—61no—
44C3×#215—61no—
45C3×#214—61no—
46C3×#213—61no—
47C2×C3×C3⋊C8—61no—
48C2×C3×C3⋊C8—61no—
49#3719—62no—
50#3719—62no—
51#3718—32no—
52#3718—61no—
53#3718C2×D311yescharacteristic
54#3715—21no—
55#3713C2×D311yescharacteristic
56#3708—32no—
57C22×C33D811yescharacteristic
58C2×C32×D3—41no—
59C33×C4D811yescharacteristic
60C33×C4C2×D411yescharacteristic
61C3×C3⋊Dic3—42no—
62C32×Dic3—41no—
63C3×#135—91no—
64#950—61no—
65C2×#216—181no—
66C2×Dic12—91no—
67#829—92no—
68#827—92no—
69#812—91no—
70C32×Q8—61no—
71C32×D4—61no—
72C32×D4—121no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—41no—
76C2×C32×C4—41no—
77C2×C32×C4—121no—
78C2×C32×C4#6711yescharacteristic
79C2×C3⋊Dic3—41no—
80#634—22no—
81#634—41no—
82C3×#67—41no—
83C3×#67—42no—
84C3×#67—41no—
85C2×C3×Dic3—121no—
86C2×C3×Dic3—122no—
87C2×C3×Dic3—121no—
88C3×D12—21no—
89C3×D12—21no—
90C3×D12—41no—
91C3×C4×D3—41no—
92C3×C4×D3—42no—
93C3×C4×D3—41no—
94C3×Dic6—21no—
95C3×Dic6—21no—
96C3×Dic6—41no—
97C3×Dic6—62no—
98C3×Dic6—62no—
99C3×Dic6—122no—
100C3×Dic6—121no—
101C3×Dic6—122no—
102C3×Dic6—121no—
103C3⋊(C3⋊C8)#6711yescharacteristic
104C3⋊(C3⋊C8)C22×D311yescharacteristic
105C3×C3⋊C8—61no—
106C3×C3⋊C8—61no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109#510—271noSylow
110C2×C33—21no—
111C2×C33C2×D811yescharacteristic
112C32×D3—81no—
113C3×#41—61no—
114C3×#41—61no—
115C3×#41—121no—
116C2×C3×Q8—181no—
117#235—21no—
118C2×Dic6—61no—
119C2×Dic6—61no—
120C2×Dic6—121no—
121C3×Q16—91no—
122C3×SD16—181no—
123C3×D8—91no—
124C2×C3×C8—91no—
125#216—61no—
126#216—182no—
127#216—182no—
128#215—61no—
129#214—61no—
130#213—61no—
131C2×C3⋊C8—61no—
132C2×C3⋊C8—61no—
133Dic12—92no—
134Dic12—181no—
135C3⋊C16—92no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—121no—
141C22×C32#21311yescharacteristic
142C2×C3×D3—41no—
143C2×C3×D3—42no—
144C2×C3×D3—41no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4—41no—
153C32×C4—121no—
154C32×C4#21311yescharacteristic
155C32×C4C2×#6711yescharacteristic
156C3⋊Dic3—42no—
157C3×Dic3—41no—
158C3×Dic3—42no—
159C3×Dic3—41no—
160C3×Dic3—122no—
161C3×Dic3—124no—
162C3×Dic3—122no—
163#135—91no—
164C2×Q16—271no—
165Q32—272no—
166SD32—272no—
167M32—271no—
168C33#51011yescharacteristic, complemented, Sylow
169C3×Q8—61no—
170C3×Q8—61no—
171C3×Q8—121no—
172C3×Q8—182no—
173C3×Q8—361no—
174C3×D4—61no—
175C3×D4—61no—
176C3×D4—121no—
177C3×D4—121no—
178C3×D4—122no—
179C3×D4—121no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—121no—
185C2×C3×C4—122no—
186C2×C3×C4—121no—
187C2×C3×C4—361no—
188C2×C3×C4GO+(4,𝔽2)11yescharacteristic
189#67—41no—
190C2×Dic3—121no—
191C2×Dic3—122no—
192C2×Dic3—121no—
193D12—21no—
194C4×D3—41no—
195Dic6—21no—
196Dic6—62no—
197Dic6—62no—
198Dic6—122no—
199Dic6—121no—
200Dic6—122no—
201Dic6—121no—
202C3×C8—91no—
203C3×C8—91no—
204C3⋊C8—61no—
205C3⋊C8—61no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C2×C32—22no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—42no—
216C2×C32—81no—
217C2×C32—241no—
218C2×C32C2×#21311yescharacteristic
219C3×D3—81no—
220C3×D3—82no—
221C3×D3—81no—
222#41—61no—
223C2×Q8—181no—
224Q16—91no—
225Q16—272no—
226Q16—541no—
227SD16—181no—
228D8—91no—
229C2×C8—91no—
230C16—272no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—121no—
236C22×C3—122no—
237C22×C3—121no—
238C22×C3#377411yescharacteristic
239C2×D3—41no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—121no—
249C3×C4—122no—
250C3×C4—121no—
251C3×C4—362no—
252C3×C4#377411yescharacteristic
253C3×C4C2×GO+(4,𝔽2)11yescharacteristic
254Dic3—41no—
255Dic3—122no—
256Dic3—124no—
257Dic3—122no—
258C32—21no—
259C32—21no—
260C32—41no—
261C32—41no—
262C32#519511yescharacteristic, complemented
263Q8—61no—
264Q8—182no—
265Q8—361no—
266D4—61no—
267D4—121no—
268C2×C4—121no—
269C2×C4—361no—
270C2×C4D32⋊D311yescharacteristic
271C8—91no—
272C8—91no—
273C2×C3—22no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—41no—
278C2×C3—42no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—81no—
282C2×C3—241no—
283C2×C3—242no—
284C2×C3—241no—
285C2×C3C2×#377411yescharacteristic
286D3—81no—
287C22—121no—
288C22#8893511yescharacteristic
289C4—121no—
290C4—362no—
291C4#8893511yescharacteristic
292C4C2×D32⋊D311yescharacteristic
293C3—21no—
294C3—21no—
295C3—41no—
296C3—41no—
297C3#1059333611yescharacteristic, complemented
298C2—21no—
299C2—241no—
300C2C2×#8893511yescharacteristic
subgroupG/H
centerC2C2×#88935
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×D8
prime radicalC3×#63822C2
hypercenterC2×C4D32⋊D3
hypocenterC33#510
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88812C2semidirect
3. #11723098C2semidirect
4. #11722768C2semidirect
5. #11722767C2semidirect
168. C33#510semidirect
262. C32#5195semidirect
297. C3#10593336semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
C3×#638222
#117230982
#117227682
#117227672
#105933363
#51959