Finite Group Databasebeta

← #13155543

Subgroups — #13155543

Order 1728.

566 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 112 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#11723423C211yesmaximal, characteristic
2#11722070C211yesmaximal, characteristic, complemented
3#11722050C212yesmaximal, complemented
4C3×(C2×D3)⋊(C3⋊C8)C212yesmaximal
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591062—31nomaximal
7#10590697—32nomaximal
8C2×C32×C4×D3C2211yescharacteristic
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C2×C32×C3⋊C8C2212yes—
13Dic3⋊(C3⋊Dic3)C412yes—
14(C2×D3)⋊(C3⋊Dic3)C412yes—
15#64135—31no—
16#63855—32no—
17#63855—61no—
18#63641—32no—
19C3×#830—32no—
20C3×(C2×D3)⋊C8—32no—
21C3×#815—32no—
22#63572—32no—
23#63572—32no—
24#63572—62no—
25(C2×D3)⋊(C3⋊C8)D312yes—
26#63555D311yescharacteristic, complemented
27#63553—31no—
28Dic3×C3⋊C8—32no—
29C22×C32×D3C2×C411yescharacteristic
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C412yes—
32C2×C3×C3⋊Dic3C2×C412yes—
33C2×C32×Dic3C2×C411yescharacteristic
34C32×C4×D3—22no—
35C3×C3⋊(C3⋊C8)—22no—
36C32×C3⋊C8—22no—
37#4720—92no—
38#4709—91no—
39C22×C32×C4—31no—
40C2×C3×C4×D3—21no—
41C2×C3×C4×D3C2×D312yes—
42C2×C32×C8—32no—
43#3757—32no—
44C4⋊(C3⋊Dic3)—32no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—32no—
47C4×C3⋊Dic3—62no—
48C4×C3⋊Dic3—61no—
49C4×C3⋊Dic3—121no—
50C4×C3⋊Dic3C2×D311yescharacteristic
51C2×C3⋊(C3⋊C8)C2×D312yes—
52C3×C4×Dic3—32no—
53C3×C4×Dic3—61no—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—62no—
58C2×C3×C3⋊C8C2×D312yes—
59C2×C3×C3⋊C8C2×D312yes—
60Dic3⋊Dic3—32no—
61Dic3⋊Dic3—32no—
62Dic3⋊Dic3—62no—
63(C2×D3)⋊Dic3—32no—
64(C2×D3)⋊Dic3—32no—
65(C2×D3)⋊Dic3—62no—
66C22×C33C22×C411yescharacteristic
67C2×C32×D3—21no—
68C2×C32×D3—22no—
69C33×C4#4112yes—
70C3⋊(C3⋊Dic3)—22no—
71C3×C3⋊Dic3—22no—
72C32×Dic3—21no—
73#926—92no—
74#926—181no—
75#872—91no—
76C3×C22⋊C8—92no—
77C3×#97—91no—
78#830—32no—
79(C2×D3)⋊C8—32no—
80#815—32no—
81Dic3⋊C8—92no—
82Dic3⋊C8—92no—
83Dic3⋊C8—92no—
84Dic3⋊C8—182no—
85C8×Dic3—92no—
86C4×C3⋊C8—91no—
87C23×C32—31no—
88C22×C3×D3—21no—
89C22×C3×D3C4×D312yes—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—31no—
93C2×C32×C4—41no—
94C2×C32×C4—62no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C22×D312yes—
97C2×C3⋊Dic3—32no—
98C2×C3⋊Dic3—32no—
99C2×C3⋊Dic3—32no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—122no—
104C2×C3⋊Dic3C4×D312yes—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3C4×D312yes—
110C3×C4×D3—24no—
111C3×C4×D3—22no—
112C3×C4×D3—22no—
113C32×C8—62no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—22no—
116C3×C3⋊C8—22no—
117C3×C3⋊C8—42no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—122no—
121#433—271noSylow
122C2×C33#11711yescharacteristic
123C2×C33#13112yes—
124C32×D3—41no—
125C22×C3×C4—32no—
126C22×C3×C4—61no—
127C2×C4×D3D3211yescharacteristic
128C2×C3×C8—32no—
129C2×C3×C8—32no—
130C2×C3×C8—62no—
131C2×C3×C8—92no—
132C3×C42—91no—
133#217—92no—
134#217—92no—
135#217—182no—
136(C2×D3)⋊C4—92no—
137C4⋊Dic3—92no—
138C4⋊Dic3—92no—
139C4⋊Dic3—182no—
140Dic3⋊C4—92no—
141C4×Dic3—32no—
142C4×Dic3—61no—
143C4×Dic3—91no—
144C4×Dic3—92no—
145C4×Dic3—182no—
146C4×Dic3—181no—
147C4×Dic3—361no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—62no—
151C2×C3⋊C8D3212yes—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—41no—
155C22×C32—61no—
156C22×C32—62no—
157C22×C32C2×C4×D311yescharacteristic
158C22×C32C2×C4×D312yes—
159C2×C3×D3—22no—
160C2×C3×D3—22no—
161C2×C3×D3—22no—
162C2×C3×D3—22no—
163C2×C3×D3—42no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C32×C4—61no—
169C32×C4#23512yes—
170C32×C4#23712yes—
171C32×C4#23712yes—
172C3⋊Dic3—22no—
173C3⋊Dic3—62no—
174C3⋊Dic3—64no—
175C3⋊Dic3—62no—
176C3⋊Dic3—62no—
177C3⋊Dic3—122no—
178C3⋊Dic3—122no—
179C3⋊Dic3—124no—
180C3×Dic3—22no—
181C3×Dic3—22no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—122no—
185#117—271no—
186C4⋊C8—272no—
187C22⋊C8—92no—
188#97—91no—
189C4×C8—271no—
190C33#43311yescharacteristic, complemented, Sylow
191C23×C3—32no—
192C23×C3—61no—
193C22×D3#62411yescharacteristic
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—32no—
197C2×C3×C4—41no—
198C2×C3×C4—61no—
199C2×C3×C4—64no—
200C2×C3×C4—62no—
201C2×C3×C4—62no—
202C2×C3×C4—92no—
203C2×C3×C4C2×D3212yes—
204C2×C3×C4C2×D3211yescharacteristic
205C2×Dic3—32no—
206C2×Dic3—32no—
207C2×Dic3—62no—
208C2×Dic3—92no—
209C2×Dic3—92no—
210C2×Dic3—92no—
211C2×Dic3—182no—
212C2×Dic3—182no—
213C2×Dic3—182no—
214C2×Dic3—362no—
215C2×Dic3#62411yescharacteristic
216C4×D3—22no—
217C3×C8—62no—
218C3×C8—62no—
219C3×C8—122no—
220C3×C8—182no—
221C3⋊C8—22no—
222C3⋊C8—62no—
223C3⋊C8—62no—
224C3⋊C8—122no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32—121no—
233C2×C32#87211yescharacteristic
234C2×C32#87712yes—
235C2×C32#90112yes—
236C2×C32#90712yes—
237C2×C32#90712yes—
238C3×D3—42no—
239C3×D3—42no—
240C22×C4—31no—
241C2×C8—32no—
242C2×C8—92no—
243C4⋊C4—272no—
244C22⋊C4—272no—
245C42—91no—
246C42—271no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—41no—
250C22×C3—62no—
251C22×C3—62no—
252C22×C3—62no—
253C22×C3—62no—
254C22×C3—122no—
255C22×C3C4×D3212yes—
256C22×C3C2×#62411yescharacteristic
257C2×D3—22no—
258C2×D3—21no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—42no—
263C3×C4—62no—
264C3×C4—62no—
265C3×C4—182no—
266C3×C4#379512yes—
267C3×C4#380412yes—
268C3×C4#380512yes—
269Dic3—21no—
270Dic3—62no—
271Dic3—62no—
272Dic3—122no—
273Dic3—184no—
274Dic3—182no—
275Dic3—182no—
276Dic3—182no—
277Dic3—362no—
278Dic3—362no—
279Dic3—364no—
280C32—21no—
281C32—22no—
282C32—41no—
283C32#470911yescharacteristic, complemented
284C32#472012yescomplemented
285C23—31no—
286C2×C4—31no—
287C2×C4—62no—
288C2×C4—92no—
289C2×C4—272no—
290C2×C4D3311yescharacteristic
291C8—62no—
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—122no—
301C2×C3—122no—
302C2×C3#6380412yes—
303C2×C3#6380512yes—
304C2×C3#6381712yes—
305C2×C3#6387712yes—
306C2×C3#6395711yescharacteristic
307D3—41no—
308C22—62no—
309C22—61no—
310C22D3×#62411yescharacteristic
311C4—61no—
312C4—182no—
313C4—542no—
314C4#8896112yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059069712yescomplemented
319C3#1059106211yescharacteristic, complemented
320C2—121no—
321C2#1172282512yes—
322C2#1172291611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C32×D3C2×C4
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722070C2semidirect
3. #11722050C2semidirect
5. C3×#63555C2semidirect
26. #63555D3semidirect
190. C33#433semidirect
283. C32#4709semidirect
284. C32#4720semidirect
318. C3#10590697semidirect
319. C3#10591062semidirect

Maximal subgroups

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

subgroupindex
#117234232
#117220702
#117220502
C3×(C2×D3)⋊(C3⋊C8)2
C3×#635552
#105910623
#105906973