Finite Group Databasebeta

← #80717

Subgroups — #80717

Order 384.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 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×#912C211yesmaximal, characteristic, complemented
2C2×#901C211yesmaximal, characteristic, complemented
3#5608C212yesmaximal, complemented
4C3×#466C211yesmaximal, characteristic, complemented
5#5157C211yesmaximal, characteristic, complemented
6#5116C211yesmaximal, characteristic, complemented
7#5042C212yesmaximal, complemented
8#4887C211yesmaximal, characteristic, complemented
9D3×#107C211yesmaximal, characteristic, complemented
10#4780C212yesmaximal, complemented
11#4771C212yesmaximal, complemented
12#3061—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×#235C2211yescharacteristic
15C2×C3×D8C2211yescharacteristic
16C22×C3×C8C2211yescharacteristic
17C3×#121C2212yes—
18#939—22no—
19#938—22no—
20#938C2212yes—
21C2×#214C2212yes—
22#924C2212yes—
23C2×#208C2211yescharacteristic
24#912—24no—
25C2×Dic12C2211yescharacteristic
26#901—24no—
27C2×ΓL(1,𝔽25)C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29C2×D3×C8C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#897C2212yes—
32D3×C4⋊C4C2212yes—
33#884—22no—
34#878—22no—
35C3×#107C2211yescharacteristic
36C3×#102C2212yes—
37#832C2212yes—
38#818C2211yescharacteristic
39#816C2212yes—
40#810C2212yes—
41#807C2212yes—
42C2×#135—31no—
43C2×#131—31no—
44#537—32no—
45#468—32no—
46#468—32no—
47#467—31no—
48#467—31no—
49#466—31no—
50#466D311yescharacteristic, complemented
51C2×#107—31no—
52#417—32no—
53C2×C3×D4—22no—
54C2×C3×D4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—22no—
57C2×#67—22no—
58C2×#67D411yescharacteristic
59C2×#67D411yescharacteristic
60C22×Dic3—22no—
61#237—24no—
62#237—24no—
63#235—22no—
64#235—22no—
65C2×D12D411yescharacteristic
66C2×C4×D3—22no—
67C2×C4×D3C2311yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×Dic6D411yescharacteristic
70C2×Dic6C2312yes—
71C3×D8—22no—
72C2×C3×C8—22no—
73C2×C3×C8C2311yescharacteristic
74C2×C3×C8C2311yescharacteristic
75C3×C4⋊C4C2312yes—
76C3×C22⋊C4—22no—
77#217—22no—
78#217—22no—
79#214—24no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—22no—
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85#208—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88Dic12—22no—
89ΓL(1,𝔽25)—22no—
90ΓL(1,𝔽25)—22no—
91D3×C8—22no—
92D3×C8D414yescomplemented
93C2×#41—31no—
94C2×#41—32no—
95#135—64no—
96C2×Q16—31no—
97C2×SD16—32no—
98C2×D8C2×D311yescharacteristic
99#131—64no—
100C2×M16—31no—
101C2×M16—31no—
102C2×M16—31no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8C2×D311yescharacteristic
106C22⋊Q8—32no—
107C22⋊Q8—32no—
108C22⋊Q8—62no—
109#121—32no—
110#121—62no—
111#121—62no—
112#121—62no—
113#121C2×D312yes—
114C2×C4⋊C4—32no—
115#107—31no—
116#107—32no—
117#107C2×D311yescharacteristic
118#103—32no—
119#103—32no—
120#102—32no—
121#102C2×D312yes—
122C23×C3—22no—
123C23×C3C2×D411yescharacteristic
124C22×D3C2×D411yescharacteristic
125C22×D3C2×D411yescharacteristic
126C3×D4—22no—
127C3×D4—22no—
128C3×D4—82no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2411yescharacteristic
134#67—22no—
135#67—22no—
136#67—44no—
137#67—82no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—44no—
141C2×Dic3C2×D411yescharacteristic
142C2×Dic3C2×D411yescharacteristic
143D12—22no—
144C4×D3—21no—
145C4×D3—21no—
146C4×D3—82no—
147C4×D3C2×D412yes—
148C4×D3#4112yes—
149Dic6—22no—
150Dic6—22no—
151Dic6—22no—
152C3×C8—21no—
153C3×C8C2×D412yes—
154C3⋊C8—21no—
155C3⋊C8C2×D412yes—
156#41—62no—
157#41—62no—
158#41—64no—
159#41—64no—
160C2×Q8—31no—
161C2×Q8—32no—
162C2×D4—22no—
163C2×D4—31no—
164C2×D4—31no—
165C2×D4—31no—
166C2×D4—62no—
167C2×D4—62no—
168C2×D4C22×D312yes—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4C22×D311yescharacteristic
174Q16—62no—
175SD16—64no—
176D8—22no—
177M16—62no—
178M16—62no—
179M16—62no—
180C2×C8—22no—
181C2×C8—31no—
182C2×C8—34no—
183C2×C8—31no—
184C2×C8—62no—
185C2×C8C22×D311yescharacteristic
186C2×C8C22×D311yescharacteristic
187C4⋊C4—32no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4C22×D312yes—
192C22⋊C4—22no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3C22×D411yescharacteristic
203C2×D3—21no—
204C2×D3—21no—
205C2×D3—21no—
206C2×D3—21no—
207C2×D3—21no—
208C2×D3#12112yes—
209C3×C4—21no—
210C3×C4—42no—
211C3×C4C22×D411yescharacteristic
212C3×C4C2×#4111yescharacteristic
213Dic3—21no—
214Dic3—42no—
215Dic3—42no—
216Dic3#12112yes—
217C23—22no—
218C23—31no—
219C23—31no—
220C23D3×D411yescharacteristic
221Q8—62no—
222Q8—62no—
223Q8—62no—
224D4—22no—
225D4—22no—
226D4—62no—
227D4—62no—
228D4—62no—
229D4—82no—
230D4—124no—
231D4—242no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—31no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—31no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—124no—
244C2×C4—242no—
245C2×C4D3×D411yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—21no—
248C8—32no—
249C8—61no—
250C8D3×D412yes—
251C2×C3—41no—
252C2×C3—82no—
253C2×C3C2×#12111yescharacteristic
254C2×C3#57611yescharacteristic
255C2×C3#57811yescharacteristic
256D3—22no—
257D3—41no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—32no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—61no—
266C22—61no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22C2×D3×D411yescharacteristic
271C4—21no—
272C4—32no—
273C4—42no—
274C4—61no—
275C4—122no—
276C4—122no—
277C4C2×D3×D411yescharacteristic
278C4D3×#4111yescharacteristic
279C3#306111yescharacteristic, complemented, Sylow
280C2—41no—
281C2—62no—
282C2—82no—
283C2—121no—
284C2D3×#12111yescharacteristic
285C2#575711yescharacteristic
286C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3061
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#912C2semidirect
2. C2×#901C2semidirect
3. #5608C2semidirect
4. C3×#466C2semidirect
5. #5157C2semidirect
6. #5116C2semidirect
7. #5042C2semidirect
8. #4887C2semidirect
9. D3×#107C2semidirect
10. #4780C2semidirect
11. #4771C2semidirect
50. #466D3semidirect
92. D3×C8D4semidirect
279. C3#3061semidirect

Maximal subgroups

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

subgroupindex
C2×#9122
C2×#9012
#56082
C3×#4662
#51572
#51162
#50422
#48872
D3×#1072
#47802
#47712
#30613