Finite Group Databasebeta

← #13159682

Subgroups — #13159682

Order 1728.

812 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 154 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#11723484C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic, complemented
3#11722191C211yesmaximal, characteristic
4C3×#63644C211yesmaximal, characteristic, complemented
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic
8#10592546—31nomaximal
9#10590769—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6289—21no—
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20#88804—21no—
21C4×C4⋊(C3⋊D3)—31no—
22#64079—31no—
23#63894—34no—
24#63869—34no—
25C32×C4⋊C8—31no—
26#63648—31no—
27#63648—31no—
28#63648—34no—
29#63648—64no—
30#63645—31no—
31#63644D311yescharacteristic, complemented
32C3×#817—34no—
33C3×Dic3⋊C8D314yescomplemented
34#63575—34no—
35#63564—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))Q812yes—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45#6289—42no—
46C33×C8—21no—
47C32×C3⋊C8—21no—
48#4871—91no—
49#4847—94no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×C4⋊(C3⋊D3)—34no—
52C2×C4⋊(C3⋊D3)—64no—
53C2×C4×C3⋊D3—61no—
54C2×#624—64no—
55C2×C32×C8—24no—
56C2×C32×C8—31no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59C32×C42—31no—
60#3752—61no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—61no—
67#3750—61no—
68C3×C4⋊Dic3—34no—
69C3×C4⋊Dic3—34no—
70C3×C4×Dic3C2×D314yes—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—64no—
73Dic3⋊Dic3—64no—
74#3722—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3⋊(C3⋊D3)—22no—
77C2×C3⋊(C3⋊D3)—41no—
78C2×C3⋊(C3⋊D3)—41no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—21no—
85C32×Dic3—21no—
86#893—91no—
87#890—91no—
88C4×D12—94no—
89#870—94no—
90C3×#106—91no—
91C3×C4⋊C8—34no—
92#821—94no—
93#821—94no—
94#821—91no—
95#821—184no—
96#818—94no—
97#817—34no—
98Dic3⋊C8C3⋊D311yescharacteristic, complemented
99#809—91no—
100#807—91no—
101C22×C3⋊D3—61no—
102C22×C3⋊D3—64no—
103C22×C3⋊D3—124no—
104C2×C32×C4—24no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—124no—
114C4⋊(C3⋊D3)—32no—
115C4⋊(C3⋊D3)—38no—
116C4⋊(C3⋊D3)—68no—
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—64no—
119C4⋊(C3⋊D3)—124no—
120C4×C3⋊D3—122no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3D1214yes—
124C2×C3×Dic3D1214yes—
125#624—128no—
126C32×C8—21no—
127C32×C8—24no—
128C32×C8—44no—
129C32×C8—61no—
130C3×C3⋊C8—24no—
131#478—271noSylow
132C2×C33C22⋊Q811yescharacteristic
133C2×C33#13511yescharacteristic
134C2×C33AGL(1,ℤ/8)11yescharacteristic
135C3⋊(C3⋊D3)—42no—
136C2×D12—94no—
137C2×D12—91no—
138C2×D12—184no—
139C2×C4×D3—184no—
140C2×C4×D3—181no—
141C2×C3×C8—24no—
142C2×C3×C8—34no—
143C2×C3×C8D3214yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4—91no—
146C3×C4⋊C4—91no—
147C3×C42—34no—
148(C2×D3)⋊C4—184no—
149(C2×D3)⋊C4—181no—
150C4⋊Dic3—34no—
151C4⋊Dic3—34no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—181no—
156Dic3⋊C4—184no—
157Dic3⋊C4—184no—
158Dic3⋊C4—181no—
159C4×Dic3C2×C3⋊D311yescharacteristic
160C2×C3⋊C8C2×C3⋊D311yescharacteristic
161C22×C32—24no—
162C22×C32C2×D1214yes—
163C22×C32D3×D411yescharacteristic
164C2×C3⋊D3—62no—
165C2×C3⋊D3—68no—
166C2×C3⋊D3—128no—
167C2×C3⋊D3—121no—
168C2×C3⋊D3—121no—
169C2×C3⋊D3—124no—
170C2×C3⋊D3—124no—
171C2×C3⋊D3—244no—
172C2×C3⋊D3—244no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—61no—
176C32×C4—61no—
177C32×C4#23511yescharacteristic
178C32×C4#23714yes—
179C32×C4D3×Q814yes—
180C32×C4D3×Q811yescharacteristic
181C3⋊Dic3—41no—
182C3⋊Dic3—41no—
183C3⋊Dic3—121no—
184C3⋊Dic3—124no—
185C3⋊Dic3—244no—
186C3×Dic3—24no—
187C3×Dic3—24no—
188C3×Dic3—124no—
189C3×Dic3—124no—
190#125—271no—
191C4×D4—271no—
192#107—271no—
193#106—91no—
194C4⋊C8—31no—
195#102—271no—
196#102—271no—
197C33#47811yescharacteristic, complemented, Sylow
198C22×D3—184no—
199C22×D3—181no—
200C22×D3—364no—
201C2×C3×C4—24no—
202C2×C3×C4—34no—
203C2×C3×C4—34no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—64no—
209C2×Dic3—64no—
210C2×Dic3—184no—
211C2×Dic3—181no—
212C2×Dic3—364no—
213C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
214C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
215D12—98no—
216D12—92no—
217D12—188no—
218D12—184no—
219D12—181no—
220D12—364no—
221C4×D3—362no—
222C4×D3—368no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—44no—
226C3×C8—64no—
227C3⋊C8—21no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32#89611yescharacteristic
232C2×C32#89714yes—
233C2×C32#90414yes—
234C2×C32#90814yes—
235C2×C32#91411yescharacteristic
236C2×C32#91611yescharacteristic
237C3⋊D3—122no—
238C3⋊D3—128no—
239C3⋊D3—248no—
240C2×D4—271no—
241C22×C4—541no—
242C2×C8—31no—
243C2×C8D3×C3⋊D311yescharacteristic
244C4⋊C4—91no—
245C4⋊C4—91no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C4⋊C4—541no—
249C22⋊C4—541no—
250C42—31no—
251C22×C3—24no—
252C22×C3D3×D1214yes—
253C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
254C2×D3—182no—
255C2×D3—188no—
256C2×D3—368no—
257C2×D3—364no—
258C2×D3—364no—
259C2×D3—361no—
260C2×D3—361no—
261C2×D3—724no—
262C2×D3—724no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—64no—
266C3×C4—64no—
267C3×C4—361no—
268C3×C4—361no—
269C3×C4#379714yes—
270C3×C4#380414yes—
271C3×C4#383011yescharacteristic
272C3×C4Q8×C3⋊D311yescharacteristic
273Dic3—21no—
274Dic3—21no—
275Dic3—124no—
276Dic3—124no—
277Dic3—364no—
278Dic3—361no—
279Dic3—724no—
280C32—24no—
281C32#484714yescomplemented
282C32#487111yescharacteristic, complemented
283C23—541no—
284D4—272no—
285D4—541no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4—541no—
291C2×C4—1082no—
292C2×C4C2×D3×C3⋊D311yescharacteristic
293C8—21no—
294C8—61no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3#6379414yes—
299C2×C3#6380614yes—
300C2×C3#6389514yes—
301C2×C3#6410611yescharacteristic
302C2×C3#6411311yescharacteristic
303C2×C3#6411711yescharacteristic
304D3—362no—
305D3—368no—
306D3—728no—
307C22—542no—
308C22—1081no—
309C22—1081no—
310C22D3×C4⋊(C3⋊D3)11yescharacteristic
311C4—61no—
312C4—61no—
313C4—361no—
314C4—361no—
315C4—1081no—
316C4#8901911yescharacteristic
317C4#8902811yescharacteristic
318C3—24no—
319C3#1059076914yescomplemented
320C3#1059254611yescharacteristic, complemented
321C2—1082no—
322C2#1172331511yescharacteristic
323C2#1172333511yescharacteristic
324C2#1172348511yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723454C2semidirect
4. C3×#63644C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722055C2semidirect
31. #63644D3semidirect
33. C3×Dic3⋊C8D3semidirect
98. Dic3⋊C8C3⋊D3semidirect
197. C33#478semidirect
281. C32#4847semidirect
282. C32#4871semidirect
319. C3#10590769semidirect
320. C3#10592546semidirect

Maximal subgroups

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

subgroupindex
#117234842
#117234542
#117221912
C3×#636442
C32×Dic3⋊C82
#117220552
#117220362
#105925463
#105907693