Finite Group Databasebeta

← #13161486

Subgroups — #13161486

Order 1728.

674 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 111 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×#64083C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C3×#63630C211yesmaximal, characteristic, complemented
4#11722075C214yesmaximal, complemented
5#10590282—31nomaximal
6#10590279—31nomaximal
7#10590275—31nomaximal
8C2×C3×C4⋊(C3⋊D3)—22no—
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C33×C42C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×#3750—22no—
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14#64083D311yescharacteristic
15#64079D311yescharacteristic, complemented
16C3×#874—31no—
17C3×#874—31no—
18C3×#874—61no—
19C3×#870—31no—
20C3×#870—31no—
21C3×#870—61no—
22#63630D311yescharacteristic, complemented
23C3×#803—31no—
24C3×#803—31no—
25C3×#803—61no—
26#63564—34no—
27#63563—34no—
28#63562—34no—
29C22×C3×C3⋊D3—22no—
30C2×C33×C4D412yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C3×C4⋊(C3⋊D3)—24no—
34C3×C4⋊(C3⋊D3)—44no—
35C3×C3⋊(C3⋊C8)—22no—
36#5071—91no—
37#5067—91no—
38#4713—91no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)C2×D312yes—
41C2×C3×D12—32no—
42C2×C3×D12—32no—
43C2×C3×D12—62no—
44C2×C3×D12—64no—
45C2×C3×D12—62no—
46C2×C3×D12—62no—
47C32×C42—21no—
48C32×C42—21no—
49C32×C42—21no—
50C32×C42—41no—
51C32×C42C2×D311yescharacteristic
52C32×C42C2×D311yescharacteristic
53C32×C42C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)C2×D312yes—
55#3750—22no—
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—62no—
60C3×Dic3⋊C4—64no—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—62no—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—42no—
68C2×C3×C3⋊D3—44no—
69C33×C4—22no—
70C33×C4#4112yes—
71C3×C3⋊Dic3—42no—
72C3×#127—91no—
73C3×#125—91no—
74#874—31no—
75#874—31no—
76#874—61no—
77#870—31no—
78#870—31no—
79#870—61no—
80C3×#97—91no—
81#832—94no—
82#821—94no—
83#809—94no—
84#803—31no—
85#803—31no—
86#803—61no—
87C22×C3⋊D3—22no—
88C22×C3×D3—64no—
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—42no—
98C2×C32×C4—41no—
99C2×C32×C4D1212yes—
100C2×C32×C4#6712yes—
101C2×C32×C4#6712yes—
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—22no—
106C4⋊(C3⋊D3)—24no—
107C4⋊(C3⋊D3)—44no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C3×D12—64no—
112C3×D12—64no—
113C3×D12—124no—
114C3×D12—128no—
115C3×D12—124no—
116C3×D12—124no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—64no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—62no—
121#490—271noSylow
122C2×C33#12411yescharacteristic
123C2×C33AGL(1,ℤ/8)12yes—
124C3×C3⋊D3—82no—
125C2×C3×D4—92no—
126C2×C3×D4—182no—
127C2×D12—32no—
128C2×D12—32no—
129C2×D12—62no—
130C2×D12—64no—
131C2×D12—62no—
132C2×D12—62no—
133C2×C3×C8—92no—
134C3×C4⋊C4—92no—
135C3×C4⋊C4—182no—
136C3×C42—21no—
137C3×C42—21no—
138C3×C42—21no—
139C3×C42—41no—
140C3×C42D3211yescharacteristic
141C3×C42D3211yescharacteristic
142C3×C42D3211yescharacteristic
143C4⋊Dic3—32no—
144C4⋊Dic3—32no—
145C4⋊Dic3—62no—
146Dic3⋊C4—64no—
147Dic3⋊C4—62no—
148Dic3⋊C4—62no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—62no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32C2×D1211yescharacteristic
157C22×C32C2×#6711yescharacteristic
158C22×C32C2×#6711yescharacteristic
159C2×C3⋊D3—44no—
160C2×C3⋊D3—42no—
161C2×C3×D3—128no—
162C2×C3×D3—124no—
163C2×C3×D3—122no—
164C2×C3×D3—124no—
165C2×C3×D3—122no—
166C2×C3×D3—124no—
167C32×C4—22no—
168C32×C4—24no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—22no—
174C32×C4—42no—
175C32×C4—42no—
176C32×C4—42no—
177C32×C4—44no—
178C32×C4#23512yes—
179C32×C4#23712yes—
180C32×C4#23912yes—
181C3⋊Dic3—42no—
182C3×Dic3—124no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185#127—91no—
186#125—91no—
187#102—274no—
188#97—91no—
189C33#49011yescharacteristic, complemented, Sylow
190C23×C3—182no—
191C22×D3—64no—
192C22×D3—62no—
193C22×D3—62no—
194C3×D4—184no—
195C3×D4—364no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—182no—
205C2×C3×C4#62512yes—
206C2×C3×C4#62612yes—
207C2×C3×C4#62612yes—
208C2×C3×C4C2×D3211yescharacteristic
209C2×C3×C4C2×D3211yescharacteristic
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—64no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214D12—64no—
215D12—64no—
216D12—124no—
217D12—128no—
218D12—124no—
219D12—124no—
220C3×C8—182no—
221C3⋊C8—64no—
222C3⋊C8—62no—
223C3⋊C8—62no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32#87511yescharacteristic
233C2×C32#90812yes—
234C2×C32#93212yes—
235C2×C32#93611yescharacteristic
236C2×C32#94711yescharacteristic
237C2×C32#94912yes—
238C3⋊D3—82no—
239C3×D3—244no—
240C3×D3—242no—
241C3×D3—242no—
242C2×D4—92no—
243C2×D4—182no—
244C2×C8—92no—
245C4⋊C4—92no—
246C4⋊C4—182no—
247C42#111311yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—41no—
252C22×C3—364no—
253C22×C3—362no—
254C22×C3C2×#62511yescharacteristic
255C22×C3C2×#62611yescharacteristic
256C22×C3C2×#62611yescharacteristic
257C2×D3—128no—
258C2×D3—124no—
259C2×D3—124no—
260C2×D3—124no—
261C2×D3—122no—
262C2×D3—122no—
263C3×C4—24no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—42no—
271C3×C4—42no—
272C3×C4—44no—
273C3×C4—42no—
274C3×C4—362no—
275C3×C4#379512yes—
276C3×C4#379612yes—
277C3×C4#379912yes—
278Dic3—124no—
279Dic3—122no—
280Dic3—122no—
281C32—21no—
282C32—21no—
283C32—21no—
284C32—41no—
285C32#471311yescharacteristic, complemented
286C32#506711yescharacteristic, complemented
287C32#507111yescharacteristic, complemented
288C23—182no—
289D4—184no—
290D4—364no—
291C2×C4—182no—
292C2×C4#629512yes—
293C2×C4C2×#111311yescharacteristic
294C8—182no—
295C2×C3—22no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—42no—
302C2×C3—41no—
303C2×C3—722no—
304C2×C3#6382312yes—
305C2×C3#6382512yes—
306C2×C3#6383012yes—
307C2×C3#6386011yescharacteristic
308C2×C3#6386311yescharacteristic
309C2×C3#6386411yescharacteristic
310D3—244no—
311D3—242no—
312D3—242no—
313C22—364no—
314C22—362no—
315C22C2×#629511yescharacteristic
316C4—22no—
317C4—362no—
318C4#8904112yes—
319C3—21no—
320C3—21no—
321C3—21no—
322C3—41no—
323C3#1059027511yescharacteristic, complemented
324C3#1059027911yescharacteristic, complemented
325C3#1059028211yescharacteristic, complemented
326C2—722no—
327C2#1172366112yes—
328C2#1172367711yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC3×#64083C2
hypercenterC42#1113
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C3×#63630C2semidirect
4. #11722075C2semidirect
15. #64079D3semidirect
22. #63630D3semidirect
189. C33#490semidirect
285. C32#4713semidirect
286. C32#5067semidirect
287. C32#5071semidirect
323. C3#10590275semidirect
324. C3#10590279semidirect
325. C3#10590282semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C3×#640792
C3×#636302
#117220752
#105902823
#105902793
#105902753