Finite Group Databasebeta

← #13156571

Subgroups — #13156571

Order 1728.

550 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723455C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic
3#11722071C211yesmaximal, characteristic
4#11722041C212yesmaximal
5C3×#63565C212yesmaximal
6#10591616—31nomaximal
7#10590562—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C2212yes—
13#88807—22no—
14C3×Dic3⋊Dic3—22no—
15#64153—31no—
16C3×#890—32no—
17#63872D311yescharacteristic
18#63866—32no—
19#63866—61no—
20#63637—32no—
21C3×#835—32no—
22C3×#810—32no—
23#63567—32no—
24#63567—32no—
25#63567—62no—
26#63565D312yes—
27#63555—31no—
28#63554—32no—
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)—21no—
31#6308—22no—
32C2×C3×C3⋊Dic3D412yes—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—22no—
35C32×Dic6—22no—
36C3×C3⋊(C3⋊C8)—22no—
37#5069—91no—
38#4797—92no—
39C2×C32×Q8—31no—
40C2×#634—31no—
41C2×#634—32no—
42C2×#634—62no—
43C2×#634—61no—
44C2×#634—121no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6C2×D312yes—
47C32×C4⋊C4—32no—
48#3750—62no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3—62no—
53C3×C4⋊Dic3C2×D312yes—
54C3×C4⋊Dic3C2×D312yes—
55C3×Dic3⋊C4—62no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—62no—
61Dic3⋊Dic3—22no—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—62no—
64Dic3⋊Dic3—64no—
65C22×C33C2×D411yescharacteristic
66C33×C4#4112yes—
67C3⋊(C3⋊Dic3)—41no—
68C3×C3⋊Dic3—22no—
69C32×Dic3—41no—
70C32×Dic3—42no—
71C3×#125—91no—
72#944—92no—
73#944—181no—
74#890—32no—
75#869—91no—
76C3×#103—92no—
77#835—32no—
78#816—92no—
79#815—92no—
80#810—32no—
81#810—92no—
82#810—92no—
83#810—182no—
84#803—91no—
85C32×Q8—62no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—61no—
90C2×C32×C4—62no—
91C2×C32×C4C22×D312yes—
92C2×C32×C4C22×D311yescharacteristic
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—62no—
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—122no—
97C2×C3⋊Dic3—122no—
98C2×C3⋊Dic3#6712yes—
99#634—62no—
100#634—64no—
101#634—62no—
102#634—62no—
103#634—122no—
104#634—122no—
105#634—124no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—42no—
113C2×C3×Dic3—62no—
114C3×Dic6—22no—
115C3×Dic6—22no—
116C3×Dic6—42no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—122no—
121#491—271noSylow
122C2×C33#12411yescharacteristic
123C2×C33#13712yes—
124C2×C3×Q8—32no—
125C2×C3×Q8—61no—
126C2×Dic6—91no—
127C2×Dic6—92no—
128C2×Dic6—182no—
129C2×Dic6—181no—
130C2×Dic6—361no—
131C2×Dic6D3211yescharacteristic
132C2×C3×C8—92no—
133C3×C4⋊C4—32no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—62no—
136C3×C4⋊C4—182no—
137C3×C42—91no—
138C4⋊Dic3—62no—
139C4⋊Dic3—182no—
140C4⋊Dic3D3212yes—
141Dic3⋊C4—62no—
142Dic3⋊C4—182no—
143Dic3⋊C4—182no—
144Dic3⋊C4—184no—
145C4×Dic3—32no—
146C4×Dic3—61no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—62no—
150C22×C32—22no—
151C22×C32—21no—
152C22×C32—41no—
153C22×C32D3×D412yes—
154C22×C32C2×#6711yescharacteristic
155C32×C4—22no—
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—42no—
159C32×C4—121no—
160C32×C4—122no—
161C32×C4#23512yes—
162C32×C4#23712yes—
163C32×C4#23912yes—
164C3⋊Dic3—22no—
165C3⋊Dic3—124no—
166C3⋊Dic3—122no—
167C3⋊Dic3—124no—
168C3⋊Dic3—122no—
169C3⋊Dic3—121no—
170C3×Dic3—42no—
171C3×Dic3—42no—
172C3×Dic3—42no—
173C3×Dic3—42no—
174C3×Dic3—64no—
175C3×Dic3—62no—
176C3×Dic3—62no—
177C3×Dic3—82no—
178C4⋊Q8—271no—
179#125—91no—
180#103—92no—
181#103—272no—
182#97—271no—
183C33#49111yescharacteristic, complemented, Sylow
184C3×Q8—62no—
185C3×Q8—62no—
186C3×Q8—122no—
187C2×C3×C4—22no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—92no—
195C2×C3×C4—122no—
196C2×C3×C4C2×D3212yes—
197C2×C3×C4C2×D3211yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—21no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—62no—
203C2×Dic3—182no—
204C2×Dic3—182no—
205C2×Dic3—181no—
206C2×Dic3—362no—
207C2×Dic3—362no—
208Dic6—22no—
209Dic6—184no—
210Dic6—182no—
211Dic6—182no—
212Dic6—182no—
213Dic6—362no—
214Dic6—362no—
215Dic6—364no—
216C3×C8—182no—
217C3⋊C8—62no—
218C3⋊C8—62no—
219C3⋊C8—122no—
220C2×C32—22no—
221C2×C32—22no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—41no—
226C2×C32—42no—
227C2×C32#88512yes—
228C2×C32#91512yes—
229C2×C32#91812yes—
230C2×C32#94711yescharacteristic
231C2×C32#95112yes—
232C2×Q8—31no—
233C2×Q8—271no—
234C2×C8—92no—
235C4⋊C4—32no—
236C4⋊C4—182no—
237C4⋊C4—542no—
238C42—91no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—41no—
242C22×C3D3×#6712yes—
243C22×C3#381111yescharacteristic
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—42no—
248C3×C4—122no—
249C3×C4—122no—
250C3×C4—122no—
251C3×C4—122no—
252C3×C4—182no—
253C3×C4—242no—
254C3×C4#379612yes—
255C3×C4#379912yes—
256C3×C4#380412yes—
257Dic3—42no—
258Dic3—41no—
259Dic3—64no—
260Dic3—62no—
261Dic3—62no—
262Dic3—364no—
263Dic3—362no—
264Dic3—364no—
265Dic3—362no—
266Dic3—361no—
267C32—22no—
268C32—21no—
269C32—41no—
270C32#479712yescomplemented
271C32#506911yescharacteristic, complemented
272Q8—62no—
273Q8—542no—
274C2×C4—62no—
275C2×C4—61no—
276C2×C4—92no—
277C2×C4—541no—
278C2×C4D3311yescharacteristic
279C8—182no—
280C2×C3—22no—
281C2×C3—22no—
282C2×C3—22no—
283C2×C3—21no—
284C2×C3—22no—
285C2×C3—42no—
286C2×C3—41no—
287C2×C3#6390712yes—
288C2×C3#6392912yes—
289C2×C3#6394312yes—
290C2×C3#6395112yes—
291C2×C3#6395311yescharacteristic
292C22#8895211yescharacteristic
293C4—122no—
294C4—121no—
295C4—182no—
296C4—1081no—
297C4#8896212yes—
298C3—22no—
299C3—21no—
300C3—41no—
301C3#1059056212yescomplemented
302C3#1059161611yescharacteristic, complemented
303C2#1172288812yes—
304C2#1172300511yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
183. C33#491semidirect
270. C32#4797semidirect
271. C32#5069semidirect
301. C3#10590562semidirect
302. C3#10591616semidirect

Maximal subgroups

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

subgroupindex
#117234552
C3×#638722
#117220712
#117220412
C3×#635652
#105916163
#105905623