Finite Group Databasebeta

← #13162012

Subgroups — #13162012

Order 1728.

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

normal subgroups only

This group has 582 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×#64156C211yesmaximal, characteristic
2#11723680C211yesmaximal, characteristic, complemented
3C3×#63662C212yesmaximal, complemented
4#11722075C212yesmaximal
5#11722070C211yesmaximal, characteristic, complemented
6#10591614—32nomaximal
7#10591539—31nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C3×#3752—22no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14#88815—22no—
15#64156D311yescharacteristic
16C3×#947—32no—
17C3×#947—61no—
18#63872—31no—
19#63869—32no—
20#63662D312yescomplemented
21C3×#835—32no—
22C3×#835—32no—
23C3×#835—62no—
24#63564—32no—
25#63563—32no—
26#63562—32no—
27#63553—31no—
28Dic3×C3⋊C8—32no—
29C22×C3×C3⋊D3—21no—
30C33×Q8—22no—
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D412yes—
35C3×C4⋊(C3⋊D3)—22no—
36C3×C3⋊(C3⋊C8)—22no—
37#5056—91no—
38#4800—92no—
39C2×C32×Q8—21no—
40C2×C32×Q8—22no—
41C2×C32×Q8—41no—
42C2×C32×Q8C2×D312yes—
43C2×C32×Q8C2×D311yescharacteristic
44C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
45C2×C3×D12—32no—
46C2×C3×D12—61no—
47#3752—22no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×(C2×D3)⋊C4—62no—
52C3×(C2×D3)⋊C4—62no—
53C3×(C2×D3)⋊C4—64no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—62no—
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
66C2×C3×C3⋊D3—41no—
67C2×C3×C3⋊D3—42no—
68C33×C4—41no—
69C33×C4#4112yes—
70C3×C3⋊Dic3—22no—
71C3×C3⋊Dic3—42no—
72C3×#124—91no—
73#947—32no—
74#947—61no—
75#890—92no—
76#870—91no—
77C3×#103—92no—
78#835—32no—
79#835—32no—
80#835—62no—
81#832—92no—
82#821—92no—
83C8×Dic3—92no—
84#809—92no—
85C4×C3⋊C8—91no—
86C22×C3⋊D3—21no—
87C22×C3×D3—62no—
88C22×C3×D3—62no—
89C32×Q8—22no—
90C32×Q8—24no—
91C32×Q8—22no—
92C32×Q8—22no—
93C32×Q8—42no—
94C32×Q8—42no—
95C32×Q8—44no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—41no—
102C2×C32×C4—42no—
103C2×C32×C4—42no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3#6712yes—
108C4⋊(C3⋊D3)—22no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—122no—
115C3×D12—62no—
116C3×D12—62no—
117C3×D12—122no—
118C3⋊(C3⋊C8)—22no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—62no—
121C3×C3⋊C8—122no—
122#488—271noSylow
123C2×C33#12411yescharacteristic
124C2×C33#13512yes—
125C3×C3⋊D3—81no—
126C2×C3×Q8—21no—
127C2×C3×Q8—22no—
128C2×C3×Q8—41no—
129C2×C3×Q8D3211yescharacteristic
130C2×C3×Q8D3212yes—
131C2×C3×D4—91no—
132C2×D12—32no—
133C2×D12—61no—
134C2×C3×C8—92no—
135C3×C4⋊C4—92no—
136C3×C22⋊C4—182no—
137C3×C42—91no—
138(C2×D3)⋊C4—62no—
139(C2×D3)⋊C4—62no—
140(C2×D3)⋊C4—64no—
141C4⋊Dic3—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3—62no—
144C4⋊Dic3—182no—
145Dic3⋊C4—182no—
146Dic3⋊C4—182no—
147C4×Dic3—32no—
148C4×Dic3—61no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—62no—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—41no—
155C22×C32D3×D412yes—
156C22×C32C2×#6711yescharacteristic
157C2×C3⋊D3—41no—
158C2×C3⋊D3—42no—
159C2×C3×D3—122no—
160C2×C3×D3—124no—
161C2×C3×D3—122no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—44no—
170C32×C4—44no—
171C32×C4—42no—
172C32×C4—41no—
173C32×C4#23512yes—
174C32×C4#23712yes—
175C32×C4#23712yes—
176C3⋊Dic3—22no—
177C3⋊Dic3—42no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C3×Dic3—122no—
181C3×Dic3—122no—
182C3×Dic3—122no—
183C3×Dic3—242no—
184#125—271no—
185#124—91no—
186#103—92no—
187#102—272no—
188C4×C8—271no—
189C33#48811yescharacteristic, complemented, Sylow
190C23×C3—181no—
191C22×D3—62no—
192C22×D3—62no—
193C3×Q8—24no—
194C3×Q8—22no—
195C3×Q8—22no—
196C3×Q8—22no—
197C3×Q8—42no—
198C3×Q8—42no—
199C3×Q8—44no—
200C3×D4—182no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—92no—
210C2×C3×C4—182no—
211C2×C3×C4C2×D3211yescharacteristic
212C2×C3×C4C2×D3212yes—
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—122no—
219D12—62no—
220D12—62no—
221D12—122no—
222C3×C8—182no—
223C3⋊C8—62no—
224C3⋊C8—62no—
225C3⋊C8—122no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—22no—
231C2×C32—41no—
232C2×C32—42no—
233C2×C32#88512yes—
234C2×C32#91612yes—
235C2×C32#91912yes—
236C2×C32#93611yescharacteristic
237C2×C32#95012yes—
238C3⋊D3—81no—
239C3×D3—242no—
240C3×D3—242no—
241C2×Q8#111311yescharacteristic
242C2×D4—91no—
243C2×C8—92no—
244C4⋊C4—92no—
245C4⋊C4—542no—
246C22⋊C4—182no—
247C42—91no—
248C22×C3—21no—
249C22×C3—22no—
250C22×C3—41no—
251C22×C3—362no—
252C22×C3—361no—
253C22×C3D3×#6712yes—
254C22×C3#381111yescharacteristic
255C2×D3—122no—
256C2×D3—124no—
257C2×D3—122no—
258C2×D3—122no—
259C2×D3—122no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—44no—
265C3×C4—44no—
266C3×C4—41no—
267C3×C4—42no—
268C3×C4—42no—
269C3×C4—182no—
270C3×C4—362no—
271C3×C4#380412yes—
272C3×C4#380412yes—
273C3×C4#380512yes—
274Dic3—62no—
275Dic3—62no—
276Dic3—122no—
277Dic3—122no—
278Dic3—122no—
279Dic3—242no—
280C32—21no—
281C32—22no—
282C32—41no—
283C32#480012yescomplemented
284C32#505611yescharacteristic, complemented
285C23—181no—
286Q8—22no—
287D4—182no—
288C2×C4—21no—
289C2×C4—92no—
290C2×C4—182no—
291C2×C4C2×#111311yescharacteristic
292C8—182no—
293C2×C3—22no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—22no—
298C2×C3—42no—
299C2×C3—41no—
300C2×C3—721no—
301C2×C3#6394712yes—
302C2×C3#6394912yes—
303C2×C3#6395012yes—
304C2×C3#6395311yescharacteristic
305C2×C3#6395912yes—
306D3—242no—
307D3—242no—
308C22—362no—
309C22—361no—
310C22#8904911yescharacteristic
311C4—41no—
312C4—182no—
313C4—362no—
314C4#8904612yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059153911yescharacteristic, complemented
319C3#1059161412yescomplemented
320C2—721no—
321C2#1172371312yes—
322C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8#1113
hypocenterC33#488
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
2. #11723680C2semidirect
3. C3×#63662C2semidirect
5. #11722070C2semidirect
20. #63662D3semidirect
189. C33#488semidirect
283. C32#4800semidirect
284. C32#5056semidirect
318. C3#10591539semidirect
319. C3#10591614semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117236802
C3×#636622
#117220752
#117220702
#105916143
#105915393