Finite Group Databasebeta

← #11798203

Subgroups — #11798203

Order 1152.

489 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C22×C3×Dic12C211yesmaximal, characteristic
2#10592815C211yesmaximal, characteristic
3C2×#63551C212yesmaximal, complemented
4C3×#4549C211yesmaximal, characteristic
5#10589118C212yesmaximal, complemented
6#72064—31nomaximal
7#69113—31nomaximal
8C22×C3×Dic6—21no—
9C22×C32×C8C2211yescharacteristic
10#64133—21no—
11C2×#3746C2211yescharacteristic
12C2×C3×Dic12—21no—
13C2×C3×Dic12—24no—
14C2×C3×Dic12C2212yes—
15#63645C2211yescharacteristic
16#63643—21no—
17C2×C3×C3⋊C16C2212yes—
18#63551—24no—
19C22×C3×Q16—31no—
20C22×Dic12D311yescharacteristic
21C2×#829—32no—
22#5116—31no—
23#5116—31no—
24#5116—61no—
25C2×Dic24—32no—
26C3×C22⋊C16—31no—
27#4564—32no—
28#4549D311yescharacteristic
29#4503—32no—
30C22×C32×C4D411yescharacteristic
31C2×#634—21no—
32C22×C3×Dic3—41no—
33C2×C3×Dic6—24no—
34C2×C3×Dic6—22no—
35C2×C3×Dic6—42no—
36C2×C3×Dic6—41no—
37C2×C32×C8—22no—
38C2×C32×C8D411yescharacteristic
39C2×C32×C8C2311yescharacteristic
40#3757—41no—
41C4⋊(C3⋊Dic3)—21no—
42#3750—41no—
43#3746—22no—
44C3×Dic12—24no—
45C3×Dic12—22no—
46C3×Dic12D414yes—
47C3×C3⋊C16—22no—
48#2188—91noSylow
49C22×C3×Q8—61no—
50C22×Dic6—21no—
51C2×C3×Q16—32no—
52C2×C3×Q16—64no—
53C2×C3×Q16—61no—
54C22×C3×C8—21no—
55C22×C3×C8C2×D311yescharacteristic
56C22×C3×C8C2×D311yescharacteristic
57#924—62no—
58#924—61no—
59#924—61no—
60C2×Dic12—21no—
61C2×Dic12—24no—
62C2×Dic12—31no—
63C2×Dic12—31no—
64C2×Dic12—61no—
65C2×Dic12C2×D312yes—
66C2×C3×C16—32no—
67#829—64no—
68#818—31no—
69#818—31no—
70#818—61no—
71#816—62no—
72#816—61no—
73#816—61no—
74C2×C3⋊C16C2×D312yes—
75Dic24—64no—
76C23×C32D811yescharacteristic
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4D811yescharacteristic
80C2×C32×C4C2×D411yescharacteristic
81C2×C3⋊Dic3—41no—
82C2×C3⋊Dic3—41no—
83#634—41no—
84#634—41no—
85C2×C3×Dic3—44no—
86C2×C3×Dic3—42no—
87C3×Dic6—24no—
88C3×Dic6—44no—
89C3×Dic6—42no—
90C32×C8—21no—
91C32×C8C2×D412yes—
92C22×Q16—31no—
93C2×Q32—92no—
94#467—91no—
95#358—92no—
96C22⋊C16—31no—
97C2×C3×Q8—64no—
98C2×C3×Q8—62no—
99C2×C3×Q8—122no—
100C2×C3×Q8—121no—
101C22×C3×C4—21no—
102C22×C3×C4—121no—
103C22×C3×C4D1211yescharacteristic
104C22×C3×C4#6711yescharacteristic
105C22×Dic3—41no—
106C2×Dic6—24no—
107C2×Dic6—22no—
108C2×Dic6—42no—
109C2×Dic6—41no—
110C2×Dic6—62no—
111C2×Dic6—61no—
112C2×Dic6—61no—
113C3×Q16—34no—
114C3×Q16—64no—
115C3×Q16—62no—
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—42no—
121C2×C3×C8D1211yescharacteristic
122C2×C3×C8#6711yescharacteristic
123C2×C3×C8C22×D311yescharacteristic
124C2×C3×C8C22×D311yescharacteristic
125#217—122no—
126#217—121no—
127#217—121no—
128C4⋊Dic3—62no—
129C4⋊Dic3—61no—
130C4⋊Dic3—61no—
131Dic3⋊C4—122no—
132Dic3⋊C4—121no—
133Dic3⋊C4—121no—
134Dic12—24no—
135Dic12—22no—
136Dic12—62no—
137Dic12—62no—
138Dic12—122no—
139Dic12#6714yes—
140C3×C16—62no—
141C3⋊C16—22no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32Q3212yescomplemented
145C22×C32C2×D811yescharacteristic
146C32×C4—21no—
147C32×C4#12011yescharacteristic
148C32×C4AGL(1,ℤ/8)11yescharacteristic
149C3⋊Dic3—81no—
150C3⋊Dic3—81no—
151C3×Dic3—44no—
152C22×Q8—61no—
153C2×Q16—32no—
154C2×Q16—64no—
155C2×Q16—61no—
156C2×Q16—91no—
157C22×C8D3211yescharacteristic
158C22⋊Q8—181no—
159Q32—184no—
160C2×C16—32no—
161#107—91no—
162#103—181no—
163C23×C3—21no—
164C23×C3D2411yescharacteristic
165C23×C3#21311yescharacteristic
166C3×Q8—64no—
167C3×Q8—124no—
168C3×Q8—122no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—124no—
178C2×C3×C4—122no—
179C2×C3×C4D2411yescharacteristic
180C2×C3×C4#21311yescharacteristic
181C2×C3×C4C2×D1211yescharacteristic
182C2×C3×C4C2×#6711yescharacteristic
183C2×Dic3—44no—
184C2×Dic3—42no—
185C2×Dic3—122no—
186C2×Dic3—122no—
187C2×Dic3—121no—
188C2×Dic3—121no—
189C2×Dic3—121no—
190C2×Dic3—121no—
191Dic6—24no—
192Dic6—44no—
193Dic6—42no—
194Dic6—122no—
195Dic6—122no—
196Dic6—121no—
197Dic6—121no—
198Dic6—121no—
199Dic6—121no—
200C3×C8—21no—
201C3×C8—21no—
202C3×C8—22no—
203C3×C8—41no—
204C3×C8D3×D412yes—
205C3×C8C2×#6712yes—
206C2×C32—22no—
207C2×C32#44711yescharacteristic
208C2×C32C2×Q3211yescharacteristic
209C2×C32#51011yescharacteristic
210C2×Q8—64no—
211C2×Q8—62no—
212C2×Q8—122no—
213C2×Q8—121no—
214C2×Q8—181no—
215C22×C4—121no—
216C22×C4#62611yescharacteristic
217Q16—34no—
218Q16—64no—
219Q16—62no—
220Q16—182no—
221C2×C8—22no—
222C2×C8#62611yescharacteristic
223C2×C8C2×D3211yescharacteristic
224C4⋊C4—181no—
225C4⋊C4—361no—
226C22⋊C4—361no—
227C16—62no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3Dic2412yescomplemented
237C22×C3#82912yescomplemented
238C22×C3C2×D2411yescharacteristic
239C22×C3C2×#21311yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—124no—
246C3×C4#88211yescharacteristic
247C3×C4#90811yescharacteristic
248C3×C4#93211yescharacteristic
249C3×C4#95311yescharacteristic
250Dic3—44no—
251Dic3—242no—
252Dic3—242no—
253Dic3—241no—
254Dic3—241no—
255Dic3—241no—
256Dic3—241no—
257C32#218811yescharacteristic, complemented, Sylow
258C23#371411yescharacteristic
259Q8—64no—
260Q8—124no—
261Q8—122no—
262Q8—361no—
263Q8—361no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—124no—
267C2×C4—122no—
268C2×C4—361no—
269C2×C4—361no—
270C2×C4#371411yescharacteristic
271C2×C4C2×#62611yescharacteristic
272C8—21no—
273C8D3×#6712yes—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—42no—
280C2×C3#473211yescharacteristic
281C2×C3C2×Dic2411yescharacteristic
282C2×C3#490911yescharacteristic
283C2×C3#517911yescharacteristic
284C2×C3C2×#82911yescharacteristic
285C2×C3#521711yescharacteristic
286C22—21no—
287C22—21no—
288C22#6355112yescomplemented
289C22C2×#371411yescharacteristic
290C4—21no—
291C4—124no—
292C4—721no—
293C4—721no—
294C4#6383011yescharacteristic
295C4#6397711yescharacteristic
296C3—21no—
297C3#6911311yescharacteristic, complemented
298C3#7206411yescharacteristic, complemented
299C2—22no—
300C2C2×#6355111yescharacteristic
301C2#1059064211yescharacteristic
302C2#1059116211yescharacteristic
subgroupG/H
centerC22C2×#3714
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC22×C32×C8C22
socleC22×C32C2×D8
prime radicalC23×C32D8
hypercenterC22×C8D32
hypocenterC32#2188
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×#63551C2semidirect
5. #10589118C2semidirect
144. C22×C32Q32semidirect
236. C22×C3Dic24semidirect
237. C22×C3#829semidirect
257. C32#2188semidirect
288. C22#63551semidirect
297. C3#69113semidirect
298. C3#72064semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic122
#105928152
C2×#635512
C3×#45492
#105891182
#720643
#691133