Finite Group Databasebeta

← #84018

Subgroups — #84018

Order 384.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#578C211yesmaximal, characteristic, complemented
2#5838C212yesmaximal, complemented
3#5771C211yesmaximal, characteristic, complemented
4#5246C212yesmaximal, complemented
5#5205C212yesmaximal, complemented
6#5204C212yesmaximal, complemented
7#5196C211yesmaximal, characteristic, complemented
8#5192C211yesmaximal, characteristic, complemented
9#5183C211yesmaximal, characteristic, complemented
10#5144C211yesmaximal, characteristic, complemented
11#5143C211yesmaximal, characteristic, complemented
12#3293—31nomaximal, Sylow
13C3×2−1+4C2212yes—
14#1010C2212yes—
15C3×#137—22no—
16C3×#137C2212yes—
17C3×#135—21no—
18C3×#135C2211yescharacteristic
19C2×C3×Q16—21no—
20C2×C3×Q16C2211yescharacteristic
21C3×#131C2211yescharacteristic
22#951—22no—
23#951C2212yes—
24#951C2212yes—
25#950—22no—
26#950C2211yescharacteristic
27#948C2212yes—
28#944C2212yes—
29C2×#216—22no—
30C2×#216C2212yes—
31#942—22no—
32#942C2212yes—
33D3×Q16—21no—
34#915—22no—
35#912—21no—
36C2×Dic12C2211yescharacteristic
37#901C2211yescharacteristic
38#837C2211yescharacteristic
39#837C2212yes—
40#836C2212yes—
41#825C2211yescharacteristic
42#824C2212yes—
43#822C2211yescharacteristic
44#819C2211yescharacteristic
45C8×Dic3C2211yescharacteristic
46#578—32no—
47#578—31no—
48#578D311yescharacteristic, complemented
49#494—31no—
50#473—32no—
51#473—31no—
52#473—31no—
53#456—32no—
54#456—32no—
55#443—31no—
56#443—31no—
57C3×#41—22no—
58C3×#41—22no—
59C3×#41—42no—
60C3×#41C2312yes—
61C3×#41C2311yescharacteristic
62C2×C3×Q8—22no—
63C2×C3×Q8—22no—
64C2×C3×Q8C2312yes—
65D3×Q8—22no—
66#237—22no—
67#237—42no—
68#235—22no—
69#235C2311yescharacteristic
70C2×Dic6—22no—
71C2×Dic6C2312yes—
72C3×Q16—22no—
73C3×Q16—21no—
74C3×Q16—21no—
75C3×Q16D411yescharacteristic
76C3×SD16—22no—
77C3×SD16D412yes—
78C3×D8D411yescharacteristic
79C3×M16—21no—
80C3×M16C2311yescharacteristic
81C2×C3×C8—21no—
82C2×C3×C8C2311yescharacteristic
83#216—22no—
84#216—22no—
85#216—22no—
86#216—22no—
87#216—22no—
88#216D411yescharacteristic
89#215—22no—
90#215D411yescharacteristic
91#214—22no—
92#214—22no—
93#214D411yescharacteristic
94#213D411yescharacteristic
95Dic3⋊C4—42no—
96C4×Dic3C2311yescharacteristic
97#208—22no—
98#208C2312yes—
99#208C2311yescharacteristic
100C2×C3⋊C8—22no—
101C2×C3⋊C8C2311yescharacteristic
102Dic12—21no—
103Dic12—21no—
104ΓL(1,𝔽25)—21no—
105D3×C8—21no—
1062−1+4—32no—
1072−1+4C2×D312yes—
108#137—22no—
109#137—32no—
110#137—32no—
111#137—32no—
112#137—62no—
113#137—62no—
114#137—62no—
115#137C2×D312yes—
116#135—21no—
117#135—31no—
118#135—62no—
119#135—61no—
120#135C2×D311yescharacteristic
121C2×Q16—21no—
122C2×Q16—32no—
123C2×Q16—31no—
124C2×Q16—62no—
125C2×Q16—61no—
126C2×Q16C2×D311yescharacteristic
127#131—32no—
128#131—31no—
129#131C2×D311yescharacteristic
130C4⋊Q8—32no—
131#108—31no—
132#108—31no—
133#104—31no—
134#104—32no—
135#104—31no—
136#101—32no—
137#101—32no—
138C4×C8—31no—
139C3×Q8—22no—
140C3×Q8—22no—
141C3×Q8—42no—
142C3×Q8C2×D411yescharacteristic
143C3×Q8C2×D412yes—
144C3×D4—22no—
145C3×D4—21no—
146C3×D4—42no—
147C3×D4C2×D411yescharacteristic
148C3×D4C2×D412yes—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4C2411yescharacteristic
155#67—21no—
156#67—42no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—42no—
160D12C2×D411yescharacteristic
161C4×D3—21no—
162C4×D3—42no—
163Dic6—22no—
164Dic6—22no—
165Dic6—42no—
166Dic6C2×D411yescharacteristic
167C3×C8—21no—
168C3×C8C2×D411yescharacteristic
169C3×C8C2×D411yescharacteristic
170C3⋊C8—21no—
171C3⋊C8—22no—
172C3⋊C8C2×D411yescharacteristic
173C3⋊C8C2×D411yescharacteristic
174#41—22no—
175#41—22no—
176#41—31no—
177#41—42no—
178#41—62no—
179#41—62no—
180#41—122no—
181#41C22×D311yescharacteristic
182#41C22×D312yes—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—32no—
186C2×Q8—62no—
187C2×Q8—62no—
188C2×Q8C22×D312yes—
189Q16—22no—
190Q16—21no—
191Q16—21no—
192Q16—31no—
193Q16—62no—
194Q16—62no—
195Q16—62no—
196Q16—62no—
197Q16—62no—
198Q16—61no—
199Q16—61no—
200Q16#6711yescharacteristic
201SD16—22no—
202SD16—31no—
203SD16—31no—
204SD16—62no—
205SD16—62no—
206SD16—62no—
207SD16#6712yes—
208D8—31no—
209D8#6711yescharacteristic
210M16—21no—
211M16—32no—
212M16—31no—
213M16—62no—
214M16—61no—
215M16C22×D311yescharacteristic
216C2×C8—21no—
217C2×C8—31no—
218C2×C8—62no—
219C2×C8—61no—
220C2×C8C22×D311yescharacteristic
221C4⋊C4—122no—
222C42—31no—
223C22×C3—22no—
224C22×C3—21no—
225C22×C3H(5,𝔽2)11yescharacteristic
226C2×D3—21no—
227C3×C4—22no—
228C3×C4—21no—
229C3×C4—42no—
230C3×C4C22×D411yescharacteristic
231C3×C4C22×D411yescharacteristic
232Dic3—22no—
233Dic3—21no—
234Dic3—42no—
235Q8—22no—
236Q8—22no—
237Q8—31no—
238Q8—42no—
239Q8—62no—
240Q8—62no—
241Q8—122no—
242Q8D3×D411yescharacteristic
243Q8C2×#6712yes—
244D4—21no—
245D4—22no—
246D4—31no—
247D4—42no—
248D4—61no—
249D4—122no—
250D4D3×D411yescharacteristic
251D4C2×#6712yes—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4C23×D311yescharacteristic
263C8—21no—
264C8—31no—
265C8—31no—
266C8—62no—
267C8—61no—
268C8C2×#6711yescharacteristic
269C8C2×#6711yescharacteristic
270C2×C3—21no—
271C2×C3—42no—
272C2×C3—41no—
273C2×C3D4211yescharacteristic
274D3—41no—
275C22—21no—
276C22—22no—
277C22—61no—
278C22#100411yescharacteristic
279C4—21no—
280C4—22no—
281C4—42no—
282C4—62no—
283C4—61no—
284C4—122no—
285C4C2×D3×D411yescharacteristic
286C4C22×#6711yescharacteristic
287C3#329311yescharacteristic, complemented, Sylow
288C2—21no—
289C2—41no—
290C2—42no—
291C2—121no—
292C2D4×#6711yescharacteristic
subgroupG/H
centerC2D4×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#578C2
socleC2×C3D42
hypercenter#578D3
hypocenterC3#3293
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. C3×#578C2semidirect
2. #5838C2semidirect
3. #5771C2semidirect
4. #5246C2semidirect
5. #5205C2semidirect
6. #5204C2semidirect
7. #5196C2semidirect
8. #5192C2semidirect
9. #5183C2semidirect
10. #5144C2semidirect
11. #5143C2semidirect
48. #578D3semidirect
287. C3#3293semidirect

Maximal subgroups

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

subgroupindex
C3×#5782
#58382
#57712
#52462
#52052
#52042
#51962
#51922
#51832
#51442
#51432
#32933