Finite Group Databasebeta

← #13159564

Subgroups — #13159564

Order 1728.

728 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 162 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#11723437C211yesmaximal, characteristic
2#11722184C211yesmaximal, characteristic, complemented
3C3×#63647C211yesmaximal, characteristic
4C32×#815C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592408—31nomaximal
9#10590697—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17#88807C411yescharacteristic
18#88807C411yescharacteristic
19#88803C411yescharacteristic
20#88803C411yescharacteristic
21#64081—31no—
22#63855—34no—
23C32×#97—31no—
24#63647—31no—
25#63647D311yescharacteristic
26#63641—31no—
27#63641—31no—
28#63641—34no—
29#63641—64no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×(C2×D3)⋊C8—34no—
32C3×#815D314yescomplemented
33#63572—34no—
34(C2×D3)⋊(C3⋊C8)—34no—
35Dic3×C3⋊C8—34no—
36C22×C3×C3⋊D3C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
40C2×C3×C3⋊Dic3C2×C411yescharacteristic
41C3×C4×C3⋊D3—21no—
42C3×C4×C3⋊D3—21no—
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C33×C8—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4720—91no—
50#4709—94no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×C3×C4×D3—34no—
53C2×C32×C8—24no—
54C2×C32×C8—31no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C2×D311yescharacteristic
57C32×C42—31no—
58#3752—31no—
59#3752—31no—
60#3750—31no—
61#3750—31no—
62C4×C3⋊Dic3—31no—
63C4×C3⋊Dic3—34no—
64C4×C3⋊Dic3—64no—
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4×Dic3C2×D314yes—
70C2×C3×C3⋊C8—34no—
71C2×C3×C3⋊C8C2×D314yes—
72Dic3⋊Dic3—34no—
73Dic3⋊Dic3—34no—
74(C2×D3)⋊Dic3—34no—
75(C2×D3)⋊Dic3—34no—
76C22×C33C22×C411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3—21no—
85C32×Dic3—21no—
86C32×Dic3—21no—
87#926—91no—
88#872—94no—
89C3×C22⋊C8—91no—
90C3×#97—34no—
91#830—91no—
92(C2×D3)⋊C8—34no—
93(C2×D3)⋊C8—94no—
94#815C3⋊D311yescharacteristic, complemented
95Dic3⋊C8—94no—
96Dic3⋊C8—94no—
97Dic3⋊C8—91no—
98Dic3⋊C8—91no—
99Dic3⋊C8—184no—
100C8×Dic3—91no—
101C4×C3⋊C8—94no—
102C22×C3⋊D3C4×D311yescharacteristic
103C22×C3×D3—34no—
104C2×C32×C4—24no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3C4×D311yescharacteristic
116C4×C3⋊D3—21no—
117C4×C3⋊D3—21no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3C4×D314yes—
120C2×C3×Dic3C4×D314yes—
121C3×C4×D3—64no—
122C3×C4×D3—64no—
123C32×C8—21no—
124C32×C8—24no—
125C32×C8—44no—
126C32×C8—61no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—61no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—124no—
131C3×C3⋊C8—24no—
132C3×C3⋊C8—64no—
133#433—271noSylow
134C2×C33#11711yescharacteristic
135C2×C33#13111yescharacteristic
136C2×C33#13111yescharacteristic
137C3×C3⋊D3—41no—
138C22×C3×C4—91no—
139C2×C4×D3—34no—
140C2×C3×C8—24no—
141C2×C3×C8—34no—
142C2×C3×C8—91no—
143C2×C3×C8D3214yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C42—34no—
146#217—91no—
147#217—91no—
148(C2×D3)⋊C4—94no—
149(C2×D3)⋊C4—94no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152Dic3⋊C4—94no—
153Dic3⋊C4—94no—
154C4×Dic3—94no—
155C4×Dic3—91no—
156C4×Dic3—184no—
157C4×Dic3C2×C3⋊D311yescharacteristic
158C2×C3⋊C8—34no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C2×C3⋊C8C2×C3⋊D311yescharacteristic
163C22×C32—24no—
164C22×C32C2×C4×D314yes—
165C22×C32C2×C4×D311yescharacteristic
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—64no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—61no—
175C32×C4—61no—
176C32×C4#23514yes—
177C32×C4#23514yes—
178C32×C4#23711yescharacteristic
179C32×C4#23711yescharacteristic
180C3⋊Dic3—21no—
181C3⋊Dic3—68no—
182C3⋊Dic3—61no—
183C3⋊Dic3—64no—
184C3⋊Dic3—61no—
185C3⋊Dic3—64no—
186C3⋊Dic3—124no—
187C3×Dic3—24no—
188C3×Dic3—24no—
189C3×Dic3—64no—
190#117—271no—
191C4⋊C8—271no—
192C4⋊C8—271no—
193C22⋊C8—91no—
194C22⋊C8—271no—
195#97—31no—
196C4×C8—271no—
197C33#43311yescharacteristic, complemented, Sylow
198C23×C3—91no—
199C22×D3—34no—
200C2×C3×C4—24no—
201C2×C3×C4—34no—
202C2×C3×C4—34no—
203C2×C3×C4—91no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—34no—
209C2×Dic3—94no—
210C2×Dic3—94no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—184no—
214C2×Dic3—184no—
215C2×Dic3C4×C3⋊D311yescharacteristic
216C2×Dic3C4×C3⋊D311yescharacteristic
217C4×D3—64no—
218C4×D3—64no—
219C3×C8—24no—
220C3×C8—21no—
221C3×C8—44no—
222C3×C8—64no—
223C3×C8—181no—
224C3⋊C8—21no—
225C3⋊C8—64no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#87214yes—
233C2×C32#87711yescharacteristic
234C2×C32#90111yescharacteristic
235C2×C32#90714yes—
236C2×C32#90714yes—
237C2×C32#90711yescharacteristic
238C3⋊D3—41no—
239C3×D3—124no—
240C22×C4—91no—
241C2×C8—31no—
242C2×C8—91no—
243C2×C8—271no—
244C2×C8D3×C3⋊D311yescharacteristic
245C4⋊C4—271no—
246C4⋊C4—271no—
247C22⋊C4—271no—
248C22⋊C4—271no—
249C42—31no—
250C42—271no—
251C22×C3—24no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3—181no—
255C22×C3C4×D3214yes—
256C22×C3C2×C4×C3⋊D311yescharacteristic
257C2×D3—64no—
258C2×D3—64no—
259C2×D3—64no—
260C3×C4—24no—
261C3×C4—24no—
262C3×C4—64no—
263C3×C4—64no—
264C3×C4—181no—
265C3×C4#379514yes—
266C3×C4#380414yes—
267C3×C4#382811yescharacteristic
268C3×C4#382811yescharacteristic
269Dic3—21no—
270Dic3—21no—
271Dic3—64no—
272Dic3—188no—
273Dic3—184no—
274Dic3—184no—
275Dic3—181no—
276Dic3—181no—
277Dic3—364no—
278C32—24no—
279C32#470914yescomplemented
280C32#472011yescharacteristic, complemented
281C23—91no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—91no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—271no—
288C2×C4—271no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—21no—
291C8—61no—
292C8—181no—
293C8—541no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—361no—
298C2×C3#6380414yes—
299C2×C3#6380514yes—
300C2×C3#6387714yes—
301C2×C3#6408111yescharacteristic
302C2×C3#6411611yescharacteristic
303C2×C3#6411611yescharacteristic
304D3—124no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22C4×D3×C3⋊D311yescharacteristic
309C4—61no—
310C4—61no—
311C4—181no—
312C4—541no—
313C4—541no—
314C4#8902111yescharacteristic
315C4#8903111yescharacteristic
316C3—24no—
317C3#1059069714yescomplemented
318C3#1059240811yescharacteristic, complemented
319C2—361no—
320C2#1172333311yescharacteristic
321C2#1172333411yescharacteristic
322C2#1172346411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11722184C2semidirect
4. C32×#815C2semidirect
5. #11722050C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
32. C3×#815D3semidirect
94. #815C3⋊D3semidirect
197. C33#433semidirect
279. C32#4709semidirect
280. C32#4720semidirect
317. C3#10590697semidirect
318. C3#10592408semidirect

Maximal subgroups

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

subgroupindex
#117234372
#117221842
C3×#636472
C32×#8152
#117220502
#117220232
Dic3×C3⋊(C3⋊C8)2
#105924083
#105906973