Finite Group Databasebeta

← #81852

Subgroups — #81852

Order 384.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#5716C211yesmaximal, characteristic, complemented
2#5680C211yesmaximal, characteristic, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4#5173C212yesmaximal, complemented
5#5166C212yesmaximal, complemented
6#5158C212yesmaximal, complemented
7#5153C212yesmaximal, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5060C211yesmaximal, characteristic, complemented
10#4705C211yesmaximal, characteristic, complemented
11#4703C211yesmaximal, characteristic, complemented
12#3219—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×C3×SD16C2212yes—
15C2×C3×D8C2212yes—
16C3×#127C2211yescharacteristic
17C3×#124C2211yescharacteristic
18#946C2211yescharacteristic
19Q8×Dic3C2211yescharacteristic
20C2×#216C2212yes—
21#938—21no—
22#938—21no—
23#938C2211yescharacteristic
24#936C2212yes—
25D4×Dic3—21no—
26D4×Dic3—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2212yes—
29#883—21no—
30#67⋊C4—21no—
31#879—21no—
32#877—21no—
33C42×D3C2211yescharacteristic
34#870C2211yescharacteristic
35#869C2211yescharacteristic
36C3×#97C2211yescharacteristic
37#832C2212yes—
38#832C2212yes—
39(C2×D3)⋊C8C2212yes—
40#816C2212yes—
41#816C2212yes—
42Dic3⋊C8C2212yes—
43#803C2211yescharacteristic
44#532—31no—
45#529—31no—
46#497—31no—
47#496D311yescharacteristic, complemented
48#491—31no—
49#490—31no—
50#464—32no—
51#463—32no—
52#452—32no—
53#449—32no—
54#432—31no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C2×C3×D4C2311yescharacteristic
60C2×#67—21no—
61C2×#67—22no—
62C22×Dic3—21no—
63C22×Dic3—22no—
64#237—42no—
65#237—42no—
66C2×C4×D3D411yescharacteristic
67C2×C4×D3D411yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×Dic6C2312yes—
70C3×SD16—42no—
71C3×D8—42no—
72C2×C3×C8C2312yes—
73C3×C22⋊C4—21no—
74C3×C22⋊C4—21no—
75C3×C42C2311yescharacteristic
76#217—21no—
77#217—21no—
78#217—22no—
79#217—22no—
80#216—42no—
81#214—42no—
82(C2×D3)⋊C4—21no—
83C4⋊Dic3—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3C2311yescharacteristic
87C4⋊Dic3C2311yescharacteristic
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91C4×Dic3—21no—
92C4×Dic3D411yescharacteristic
93C4×Dic3D411yescharacteristic
94C4×Dic3C2311yescharacteristic
95C2×C3⋊C8C2312yes—
96C2×#41—32no—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16C2×D312yes—
100C2×D8C2×D312yes—
101C4⋊Q8—31no—
102#127C2×D311yescharacteristic
103#126—61no—
104#125—31no—
105#124—32no—
106#124C2×D311yescharacteristic
107#123—61no—
108C22⋊Q8—31no—
109#121—31no—
110#121—61no—
111#121—61no—
112C4×Q8—31no—
113C4×D4—31no—
114C4×D4—61no—
115C4×D4—61no—
116C4×D4—61no—
117#117—61no—
118C2×C42—31no—
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C22⋊C8—32no—
125#97—31no—
126#97C2×D311yescharacteristic
127C23×C3—21no—
128C23×C3—22no—
129C22×D3C2×D411yescharacteristic
130C3×Q8—22no—
131C3×D4—22no—
132C3×D4—22no—
133C3×D4—22no—
134C3×D4—42no—
135C3×D4—42no—
136C2×C3×C4—21no—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2411yescharacteristic
140#67—44no—
141#67—42no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—44no—
146C2×Dic3—42no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C4×D3—22no—
151C4×D3—41no—
152C4×D3—41no—
153Dic6—22no—
154Dic6—22no—
155C3×C8—22no—
156C3⋊C8—22no—
157#41—122no—
158#41—122no—
159C2×Q8—32no—
160C2×Q8C22×D311yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—61no—
164C2×D4—62no—
165C2×D4C22×D312yes—
166C2×D4C22×D311yescharacteristic
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—61no—
171C22×C4—62no—
172Q16—122no—
173SD16—42no—
174SD16—122no—
175D8—42no—
176C2×C8—32no—
177C2×C8C22×D312yes—
178C4⋊C4—31no—
179C4⋊C4—31no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C42—31no—
194C42—31no—
195C42—31no—
196C42—61no—
197C42C22×D311yescharacteristic
198C22×C3—42no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3C22×D411yescharacteristic
205C2×D3—21no—
206C2×D3—21no—
207C2×D3—21no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—41no—
211C3×C4H(5,𝔽2)12yes—
212Dic3—21no—
213Dic3—21no—
214Dic3—21no—
215Dic3—42no—
216Dic3—41no—
217Dic3—41no—
218C23—21no—
219C23—22no—
220C23—31no—
221Q8—22no—
222Q8—62no—
223Q8—62no—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—42no—
229D4—124no—
230D4—122no—
231C2×C4—21no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—62no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—124no—
242C2×C4—122no—
243C2×C4D3×D411yescharacteristic
244C2×C4D3×D411yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—22no—
247C8—62no—
248C2×C3—82no—
249C2×C3—81no—
250C2×C3#53511yescharacteristic
251C2×C3#57511yescharacteristic
252C2×C3#57511yescharacteristic
253D3—41no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—61no—
261C22—61no—
262C22—61no—
263C22C2×D3×D411yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—61no—
268C4—61no—
269C4—61no—
270C4—122no—
271C4—121no—
272C4—121no—
273C4#100412yes—
274C3#321911yescharacteristic, complemented, Sylow
275C2—82no—
276C2—81no—
277C2—121no—
278C2#571511yescharacteristic
279C2#577311yescharacteristic
280C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC22×C3C22×D4
hypercenter#496D3
hypocenterC3#3219
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. #5680C2semidirect
3. C3×#496C2semidirect
4. #5173C2semidirect
5. #5166C2semidirect
6. #5158C2semidirect
7. #5153C2semidirect
8. #5071C2semidirect
9. #5060C2semidirect
10. #4705C2semidirect
11. #4703C2semidirect
47. #496D3semidirect
274. C3#3219semidirect

Maximal subgroups

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

subgroupindex
#57162
#56802
C3×#4962
#51732
#51662
#51582
#51532
#50712
#50602
#47052
#47032
#32193