Finite Group Databasebeta

← #11798172

Subgroups — #11798172

Order 1152.

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

normal subgroups only

This group has 569 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×D24C211yesmaximal, characteristic
2#10592815C211yesmaximal, characteristic, complemented
3C2×#63548C212yesmaximal, complemented
4C3×#4549C211yesmaximal, characteristic, complemented
5#10589115C212yesmaximal, complemented
6#72036—31nomaximal
7#69110—31nomaximal
8C22×C3×D12—21no—
9C22×C32×C8C2211yescharacteristic
10#64133—21no—
11C2×#3746C2211yescharacteristic, complemented
12C2×C3×D24—21no—
13C2×C3×D24—24no—
14C2×C3×D24C2212yes—
15#63645C2211yescharacteristic, complemented
16#63643—21no—
17C2×C3×C3⋊C16C2212yescomplemented
18#63548—24no—
19C22×C3×D8—31no—
20C22×D24D311yescharacteristic
21C2×#827—32no—
22#5116—31no—
23#5116—31no—
24#5116—61no—
25C2×#800—32no—
26C3×C22⋊C16—31no—
27#4562—32no—
28#4549D311yescharacteristic, complemented
29#4509—32no—
30C23×C3×D3—41no—
31C22×C32×C4D411yescharacteristic
32C2×#634—21no—
33C2×C3×D12—24no—
34C2×C3×D12—22no—
35C2×C3×D12—42no—
36C2×C3×D12—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×D24—24no—
45C3×D24—22no—
46C3×D24D414yes—
47C3×C3⋊C16—22no—
48#2187—91noSylow
49C22×C3×D4—61no—
50C22×D12—21no—
51C2×C3×D8—32no—
52C2×C3×D8—64no—
53C2×C3×D8—61no—
54C22×C3×C8—21no—
55C22×C3×C8C2×D311yescharacteristic
56C22×C3×C8C2×D311yescharacteristic
57#924—62no—
58#924—61no—
59#924—61no—
60C2×Dic12—31no—
61C2×Dic12—31no—
62C2×Dic12—61no—
63C2×D24—21no—
64C2×D24—24no—
65C2×D24C2×D312yes—
66C2×C3×C16—32no—
67#827—64no—
68#818—31no—
69#818—31no—
70#818—61no—
71#816—62no—
72#816—61no—
73#816—61no—
74C2×C3⋊C16C2×D312yescomplemented
75#800—64no—
76C23×C32D811yescharacteristic
77C22×C3×D3—44no—
78C22×C3×D3—42no—
79C22×C3×D3—82no—
80C22×C3×D3—82no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4D811yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C3⋊Dic3—41no—
86C2×C3⋊Dic3—41no—
87#634—41no—
88#634—41no—
89C3×D12—24no—
90C3×D12—44no—
91C3×D12—42no—
92C32×C8—21no—
93C32×C8C2×D412yes—
94C22×D8—31no—
95C2×SD32—92no—
96#467—91no—
97#357—92no—
98C22⋊C16—31no—
99C24×C3—121no—
100C23×D3—41no—
101C2×C3×D4—64no—
102C2×C3×D4—62no—
103C2×C3×D4—122no—
104C2×C3×D4—121no—
105C22×C3×C4—21no—
106C22×C3×C4D1211yescharacteristic
107C22×C3×C4#6711yescharacteristic
108C2×D12—24no—
109C2×D12—22no—
110C2×D12—42no—
111C2×D12—41no—
112C2×Dic6—62no—
113C2×Dic6—61no—
114C2×Dic6—61no—
115C3×D8—34no—
116C3×D8—64no—
117C3×D8—62no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C2×C3×C8—21no—
121C2×C3×C8—21no—
122C2×C3×C8—42no—
123C2×C3×C8D1211yescharacteristic
124C2×C3×C8#6711yescharacteristic
125C2×C3×C8C22×D311yescharacteristic
126C2×C3×C8C22×D311yescharacteristic
127#217—122no—
128#217—121no—
129#217—121no—
130C4⋊Dic3—62no—
131C4⋊Dic3—61no—
132C4⋊Dic3—61no—
133Dic3⋊C4—122no—
134Dic3⋊C4—121no—
135Dic3⋊C4—121no—
136Dic12—62no—
137Dic12—62no—
138Dic12—122no—
139D24—24no—
140D24—22no—
141D24#6714yes—
142C3×C16—62no—
143C3⋊C16—22no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32SD3212yescomplemented
147C22×C32C2×D811yescharacteristic
148C2×C3×D3—44no—
149C2×C3×D3—88no—
150C2×C3×D3—82no—
151C2×C3×D3—82no—
152C32×C4—21no—
153C32×C4#12011yescharacteristic
154C32×C4AGL(1,ℤ/8)11yescharacteristic
155C3⋊Dic3—81no—
156C3⋊Dic3—81no—
157C22×D4—61no—
158C2×Q16—91no—
159C2×D8—32no—
160C2×D8—64no—
161C2×D8—61no—
162C22×C8D3211yescharacteristic
163C22⋊Q8—181no—
164SD32—184no—
165C2×C16—32no—
166#107—91no—
167#103—181no—
168C23×C3—21no—
169C23×C3—124no—
170C23×C3—122no—
171C23×C3—242no—
172C23×C3—242no—
173C23×C3D2411yescharacteristic
174C23×C3#21311yescharacteristic
175C22×D3—44no—
176C22×D3—42no—
177C22×D3—82no—
178C22×D3—82no—
179C3×D4—64no—
180C3×D4—124no—
181C3×D4—122no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4D2411yescharacteristic
191C2×C3×C4#21311yescharacteristic
192C2×C3×C4C2×D1211yescharacteristic
193C2×C3×C4C2×#6711yescharacteristic
194C2×Dic3—122no—
195C2×Dic3—122no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198C2×Dic3—121no—
199C2×Dic3—121no—
200D12—24no—
201D12—44no—
202D12—42no—
203Dic6—122no—
204Dic6—122no—
205Dic6—121no—
206Dic6—121no—
207Dic6—121no—
208Dic6—121no—
209C3×C8—21no—
210C3×C8—21no—
211C3×C8—22no—
212C3×C8—41no—
213C3×C8D3×D412yes—
214C3×C8C2×#6712yes—
215C2×C32—22no—
216C2×C32#44711yescharacteristic
217C2×C32C2×SD3211yescharacteristic
218C2×C32#50911yescharacteristic
219C3×D3—84no—
220C24—121no—
221C2×Q8—181no—
222C2×D4—64no—
223C2×D4—62no—
224C2×D4—122no—
225C2×D4—121no—
226C22×C4#62611yescharacteristic
227Q16—182no—
228D8—34no—
229D8—64no—
230D8—62no—
231C2×C8—22no—
232C2×C8#62611yescharacteristic
233C2×C8C2×D3211yescharacteristic
234C4⋊C4—181no—
235C4⋊C4—361no—
236C22⋊C4—361no—
237C16—62no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—124no—
247C22×C3—248no—
248C22×C3—242no—
249C22×C3—242no—
250C22×C3#80012yescomplemented
251C22×C3#82712yescomplemented
252C22×C3C2×D2411yescharacteristic
253C22×C3C2×#21311yescharacteristic
254C2×D3—44no—
255C2×D3—88no—
256C2×D3—82no—
257C2×D3—82no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4#88211yescharacteristic
264C3×C4#90811yescharacteristic
265C3×C4#93211yescharacteristic
266C3×C4#95311yescharacteristic
267Dic3—242no—
268Dic3—242no—
269Dic3—241no—
270Dic3—241no—
271Dic3—241no—
272Dic3—241no—
273C32#218711yescharacteristic, complemented, Sylow
274C23—124no—
275C23—122no—
276C23—242no—
277C23—242no—
278C23#371411yescharacteristic
279Q8—361no—
280Q8—361no—
281D4—64no—
282D4—124no—
283D4—122no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—361no—
287C2×C4—361no—
288C2×C4#371411yescharacteristic
289C2×C4C2×#62611yescharacteristic
290C8—21no—
291C8D3×#6712yes—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—42no—
298C2×C3—244no—
299C2×C3#473211yescharacteristic
300C2×C3C2×#80011yescharacteristic
301C2×C3#490811yescharacteristic
302C2×C3#514711yescharacteristic
303C2×C3C2×#82711yescharacteristic
304C2×C3#521711yescharacteristic
305D3—84no—
306C22—21no—
307C22—21no—
308C22—124no—
309C22—248no—
310C22—242no—
311C22—242no—
312C22#6354812yescomplemented
313C22C2×#371411yescharacteristic
314C4—21no—
315C4—721no—
316C4—721no—
317C4#6383011yescharacteristic
318C4#6397711yescharacteristic
319C3—21no—
320C3#6911011yescharacteristic, complemented
321C3#7203611yescharacteristic, complemented
322C2—22no—
323C2—244no—
324C2#1059063311yescharacteristic
325C2C2×#6354811yescharacteristic
326C2#1059116211yescharacteristic
subgroupG/H
centerC22C2×#3714
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC22×C32×C8C22
socleC22×C32C2×D8
prime radicalC22×C3×D24C2
hypercenterC22×C8D32
hypocenterC32#2187
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
2. #10592815C2semidirect
3. C2×#63548C2semidirect
4. C3×#4549C2semidirect
5. #10589115C2semidirect
11. C2×#3746C22semidirect
15. #63645C22semidirect
17. C2×C3×C3⋊C16C22semidirect
28. #4549D3semidirect
74. C2×C3⋊C16C2×D3semidirect
146. C22×C32SD32semidirect
250. C22×C3#800semidirect
251. C22×C3#827semidirect
273. C32#2187semidirect
312. C22#63548semidirect
320. C3#69110semidirect
321. C3#72036semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D242
#105928152
C2×#635482
C3×#45492
#105891152
#720363
#691103