Finite Group Databasebeta

← #10591519

Subgroups — #10591519

Order 576.

358 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#3795C211yesmaximal, characteristic, complemented
2C2×#3754C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4C2×C3×#213C211yesmaximal, characteristic, complemented
5#63567C211yesmaximal, characteristic, complemented
6#63562C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#5158—31nomaximal
9#5037—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×#67—21no—
13C2×C3×D12C2211yescharacteristic
14C2×C3×C4×D3C2211yescharacteristic
15C2×#625—21no—
16C2×D3×Dic3—21no—
17#3795—22no—
18#3795—22no—
19#3754—22no—
20C2×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×#217—21no—
22C3×#213—22no—
23C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C3⋊C8C2211yescharacteristic
25C2×#237—31no—
26C2×#235—31no—
27C2×C3×D8—31no—
28C3×#121—31no—
29#938D311yescharacteristic, complemented
30C2×#214—31no—
31C2×#214—31no—
32C2×#214—61no—
33C2×#213D311yescharacteristic, complemented
34#832—31no—
35#830—31no—
36(C2×D3)⋊C8—31no—
37#816—31no—
38#810—31no—
39#809—31no—
40C23×C32—21no—
41C22×C3×D3—21no—
42C22×C3×D3D411yescharacteristic
43C32×D4—21no—
44C32×D4—21no—
45C2×C32×C4C2311yescharacteristic
46C2×C3⋊Dic3—21no—
47#634—21no—
48#634D412yescomplemented
49C3×#67—42no—
50C2×C3×Dic3—21no—
51C2×C3×Dic3D411yescharacteristic
52C3×D12—21no—
53C3×D12D412yescomplemented
54C3×C4×D3—21no—
55C3×C4×D3—21no—
56#625—24no—
57D3×Dic3—24no—
58C3⋊(C3⋊C8)—21no—
59C3×C3⋊C8—21no—
60#449—91noSylow
61C2×C3×D4—21no—
62C2×C3×D4—31no—
63C2×C3×D4—61no—
64C2×C3×D4C2×D311yescharacteristic
65C2×C3×D4C2×D311yescharacteristic
66C22×C3×C4—31no—
67C2×#67—21no—
68C2×#67—61no—
69C2×#67—61no—
70C22×Dic3—61no—
71#237—62no—
72#237—62no—
73#235—62no—
74#235—62no—
75C2×D12C2×D311yescharacteristic
76C2×C4×D3—61no—
77C2×C4×D3C2×D311yescharacteristic
78C2×Dic6—31no—
79C2×Dic6—31no—
80C2×Dic6—61no—
81C3×D8—62no—
82C2×C3×C8—31no—
83C3×C4⋊C4—31no—
84C3×C22⋊C4—61no—
85#217—21no—
86#214—62no—
87#214—62no—
88#214—64no—
89#213—22no—
90C4⋊Dic3C2×D311yescharacteristic
91C2×C3⋊C8—31no—
92C2×C3⋊C8—31no—
93C2×C3⋊C8—61no—
94C2×C3⋊C8C2×D311yescharacteristic
95C22×C32—41no—
96C22×C32—41no—
97C22×C32—41no—
98C22×C32C2×D411yescharacteristic
99C2×C3×D3—22no—
100C2×C3×D3—21no—
101C2×C3×D3—21no—
102C2×C3×D3—21no—
103C2×C3×D3—41no—
104C2×C3×D3—41no—
105C32×C4C2×D411yescharacteristic
106C32×C4C2×D411yescharacteristic
107C3⋊Dic3—22no—
108C3×Dic3—21no—
109C3×Dic3—41no—
110C2×#41—91no—
111C2×SD16—91no—
112C2×D8—31no—
113#121—31no—
114#103—91no—
115#102—91no—
116C22⋊C8—91no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—31no—
120C23×C3—41no—
121C23×C3—61no—
122C22×D3—21no—
123C22×D3#6711yescharacteristic
124C3×D4—22no—
125C3×D4—21no—
126C3×D4—21no—
127C3×D4—21no—
128C3×D4—21no—
129C3×D4—32no—
130C3×D4—41no—
131C3×D4—61no—
132C3×D4—122no—
133C2×C3×C4—21no—
134C2×C3×C4—31no—
135C2×C3×C4—61no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4C22×D311yescharacteristic
139C2×C3×C4C22×D311yescharacteristic
140#67—42no—
141#67—64no—
142#67—64no—
143C2×Dic3—21no—
144C2×Dic3—64no—
145C2×Dic3—61no—
146C2×Dic3—61no—
147C2×Dic3—121no—
148C2×Dic3#6711yescharacteristic
149D12—21no—
150D12#6712yescomplemented
151C4×D3—21no—
152C4×D3—21no—
153C4×D3—64no—
154Dic6—32no—
155Dic6—32no—
156Dic6—62no—
157Dic6—61no—
158Dic6—61no—
159Dic6—121no—
160C3×C8—61no—
161C3⋊C8—21no—
162C3⋊C8—62no—
163C3⋊C8—61no—
164C3⋊C8—61no—
165C2×C32—81no—
166C2×C32#12011yescharacteristic
167C2×C32#13511yescharacteristic
168C2×C32AGL(1,ℤ/8)11yescharacteristic
169C3×D3—41no—
170C3×D3—42no—
171#41—182no—
172#41—182no—
173C2×Q8—91no—
174C2×D4—31no—
175C2×D4—61no—
176C2×D4—181no—
177C2×D4D3211yescharacteristic
178C22×C4—31no—
179C22×C4—181no—
180SD16—182no—
181D8—62no—
182C2×C8—31no—
183C2×C8—91no—
184C4⋊C4—31no—
185C22⋊C4—61no—
186C22×C3—21no—
187C22×C3—42no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—61no—
195C22×C3—61no—
196C22×C3—62no—
197C22×C3—61no—
198C22×C3—81no—
199C22×C3—81no—
200C22×C3—121no—
201C22×C3—121no—
202C22×C3D3×D411yescharacteristic
203C22×C3C2×#6711yescharacteristic
204C2×D3—21no—
205C2×D3—21no—
206C2×D3—22no—
207C2×D3—21no—
208C2×D3—41no—
209C2×D3—41no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—61no—
213C3×C4—121no—
214C3×C4D3×D411yescharacteristic
215C3×C4D3×D411yescharacteristic
216C3×C4D3×D411yescharacteristic
217C3×C4C2×#6711yescharacteristic
218Dic3—21no—
219Dic3—41no—
220Dic3—62no—
221Dic3—62no—
222Dic3—122no—
223C32#44911yescharacteristic, complemented, Sylow
224C23—21no—
225C23—31no—
226C23—61no—
227Q8—92no—
228Q8—181no—
229D4—21no—
230D4—21no—
231D4—32no—
232D4—61no—
233D4—122no—
234D4—184no—
235C2×C4—31no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—184no—
240C2×C4—181no—
241C2×C4C2×D3211yescharacteristic
242C8—61no—
243C8—181no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—82no—
248C2×C3—81no—
249C2×C3—81no—
250C2×C3—121no—
251C2×C3—122no—
252C2×C3#91111yescharacteristic
253C2×C3#91211yescharacteristic
254C2×C3#93211yescharacteristic
255C2×C3#93711yescharacteristic
256C2×C3#93711yescharacteristic
257C2×C3#95011yescharacteristic
258D3—42no—
259D3—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—61no—
264C22—61no—
265C22—61no—
266C22—62no—
267C22—121no—
268C22—121no—
269C22D3×#6711yescharacteristic
270C4—61no—
271C4—121no—
272C4—182no—
273C4D3×#6711yescharacteristic
274C4#381111yescharacteristic
275C3—21no—
276C3#503711yescharacteristic, complemented
277C3#515811yescharacteristic, complemented
278C2—81no—
279C2—121no—
280C2—122no—
281C2#6393211yescharacteristic
282C2#6393311yescharacteristic
283C2#6398311yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
hypercenterC2×D4D32
hypocenterC32#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#3795C2semidirect
2. C2×#3754C2semidirect
3. C3×#938C2semidirect
4. C2×C3×#213C2semidirect
5. #63567C2semidirect
6. #63562C2semidirect
7. (C2×D3)⋊(C3⋊C8)C2semidirect
29. #938D3semidirect
33. C2×#213D3semidirect
48. #634D4semidirect
53. C3×D12D4semidirect
150. D12#67semidirect
223. C32#449semidirect
276. C3#5037semidirect
277. C3#5158semidirect

Maximal subgroups

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

subgroupindex
C2×#37952
C2×#37542
C3×#9382
C2×C3×#2132
#635672
#635622
(C2×D3)⋊(C3⋊C8)2
#51583
#50373