Finite Group Databasebeta

← #13162011

Subgroups — #13162011

Order 1728.

550 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#64153C211yesmaximal, characteristic
2#11723680C211yesmaximal, characteristic
3C3×#63662C212yesmaximal
4#11722076C212yesmaximal
5#11722071C211yesmaximal, characteristic
6#10591616—31nomaximal
7#10591613—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×#3750—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14#88815—22no—
15#64153D311yescharacteristic
16C3×#944—32no—
17C3×#944—61no—
18#63872—31no—
19#63869—32no—
20#63662D312yes—
21C3×#835—32no—
22C3×#835—32no—
23C3×#835—62no—
24#63567—32no—
25#63566—32no—
26#63565—32no—
27#63555—31no—
28#63554—32no—
29C33×Q8—22no—
30C2×C33×C4—21no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D412yes—
35C3×#634—22no—
36C3×C3⋊(C3⋊C8)—22no—
37#5082—91no—
38#4797—92no—
39C2×C32×Q8—21no—
40C2×C32×Q8—22no—
41C2×C32×Q8—41no—
42C2×C32×Q8C2×D312yes—
43C2×C32×Q8C2×D311yescharacteristic
44C2×#634C2×D311yescharacteristic
45C2×C3×Dic6—32no—
46C2×C3×Dic6—61no—
47C4⋊(C3⋊Dic3)C2×D312yes—
48#3750—22no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C3×Dic3⋊C4—62no—
55C3×Dic3⋊C4—62no—
56C3×Dic3⋊C4—64no—
57C3×C4×Dic3—32no—
58C3×C4×Dic3—61no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—62no—
62#3724—62no—
63Dic3⋊Dic3—62no—
64Dic3⋊Dic3—62no—
65C22×C33C2×D411yescharacteristic
66C33×C4—41no—
67C33×C4#4112yes—
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3—42no—
71C3×C4⋊Q8—91no—
72#944—32no—
73#944—61no—
74#890—92no—
75#870—91no—
76C3×#103—92no—
77#835—32no—
78#835—32no—
79#835—62no—
80#835—92no—
81#816—92no—
82#815—92no—
83#810—92no—
84#803—91no—
85C32×Q8—22no—
86C32×Q8—24no—
87C32×Q8—22no—
88C32×Q8—22no—
89C32×Q8—42no—
90C32×Q8—42no—
91C32×Q8—44no—
92C2×C32×C4—21no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—41no—
98C2×C32×C4—42no—
99C2×C32×C4—42no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3#6712yes—
105#634—22no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C3×Dic6—62no—
115C3×Dic6—62no—
116C3×Dic6—122no—
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—21no—
125C2×C3×Q8—22no—
126C2×C3×Q8—41no—
127C2×C3×Q8—91no—
128C2×C3×Q8D3211yescharacteristic
129C2×C3×Q8D3212yes—
130C2×Dic6—32no—
131C2×Dic6—61no—
132C2×C3×C8—92no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—182no—
135C3×C42—91no—
136C4⋊Dic3—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3—62no—
139C4⋊Dic3—182no—
140Dic3⋊C4—62no—
141Dic3⋊C4—62no—
142Dic3⋊C4—64no—
143Dic3⋊C4—182no—
144Dic3⋊C4—182no—
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—42no—
160C32×C4—44no—
161C32×C4—44no—
162C32×C4—42no—
163C32×C4—41no—
164C32×C4#23512yes—
165C32×C4#23712yes—
166C32×C4#23712yes—
167C3⋊Dic3—22no—
168C3⋊Dic3—41no—
169C3⋊Dic3—42no—
170C3×Dic3—64no—
171C3×Dic3—62no—
172C3×Dic3—62no—
173C3×Dic3—122no—
174C3×Dic3—122no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C3×Dic3—242no—
178C4⋊Q8—91no—
179#125—271no—
180#103—92no—
181#103—272no—
182#97—271no—
183C33#49111yescharacteristic, complemented, Sylow
184C3×Q8—24no—
185C3×Q8—22no—
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—42no—
189C3×Q8—42no—
190C3×Q8—44no—
191C3×Q8—182no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—92no—
201C2×C3×C4—182no—
202C2×C3×C4—181no—
203C2×C3×C4C2×D3211yescharacteristic
204C2×C3×C4C2×D3212yes—
205C2×Dic3—32no—
206C2×Dic3—32no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—122no—
213Dic6—62no—
214Dic6—62no—
215Dic6—122no—
216C3×C8—182no—
217C3⋊C8—62no—
218C3⋊C8—62no—
219C3⋊C8—122no—
220C2×C32—22no—
221C2×C32—22no—
222C2×C32—22no—
223C2×C32—21no—
224C2×C32—22no—
225C2×C32—41no—
226C2×C32—42no—
227C2×C32#88512yes—
228C2×C32#91512yes—
229C2×C32#91812yes—
230C2×C32#93611yescharacteristic
231C2×C32#94212yes—
232C2×Q8—91no—
233C2×Q8#111311yescharacteristic
234C2×C8—92no—
235C4⋊C4—92no—
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—44no—
249C3×C4—44no—
250C3×C4—41no—
251C3×C4—42no—
252C3×C4—42no—
253C3×C4—182no—
254C3×C4—362no—
255C3×C4—361no—
256C3×C4#380412yes—
257C3×C4#380412yes—
258C3×C4#380512yes—
259Dic3—64no—
260Dic3—62no—
261Dic3—62no—
262Dic3—122no—
263Dic3—122no—
264Dic3—122no—
265Dic3—122no—
266Dic3—242no—
267C32—22no—
268C32—21no—
269C32—41no—
270C32#479712yescomplemented
271C32#508211yescharacteristic, complemented
272Q8—22no—
273Q8—182no—
274C2×C4—21no—
275C2×C4—92no—
276C2×C4—182no—
277C2×C4—181no—
278C2×C4C2×#111311yescharacteristic
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#6394612yes—
288C2×C3#6394812yes—
289C2×C3#6395112yes—
290C2×C3#6395311yescharacteristic
291C2×C3#6395912yes—
292C22#8904911yescharacteristic
293C4—41no—
294C4—182no—
295C4—362no—
296C4—361no—
297C4#8904612yes—
298C3—22no—
299C3—21no—
300C3—41no—
301C3#1059161312yescomplemented
302C3#1059161611yescharacteristic, complemented
303C2#1172371212yes—
304C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
183. C33#491semidirect
270. C32#4797semidirect
271. C32#5082semidirect
301. C3#10591613semidirect
302. C3#10591616semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
#117236802
C3×#636622
#117220762
#117220712
#105916163
#105916133