Finite Group Databasebeta

← #80947

Subgroups — #80947

Order 384.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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#5779C212yesmaximal, complemented
2#5766C211yesmaximal, characteristic, complemented
3#5753C211yesmaximal, characteristic, complemented
4C3×#473C211yesmaximal, characteristic, complemented
5#5205C212yesmaximal, complemented
6#5183C211yesmaximal, characteristic, complemented
7#5144C211yesmaximal, characteristic, complemented
8#4896C211yesmaximal, characteristic, complemented
9#4895C211yesmaximal, characteristic, complemented
10#4755C212yesmaximal, complemented
11#4752C212yesmaximal, complemented
12#3293—31nomaximal, Sylow
13#1010C2212yes—
14#1007C2212yes—
15C3×#137C2212yes—
16C2×C3×Q16C2211yescharacteristic
17C3×#131C2211yescharacteristic
18#950C2211yescharacteristic
19#944C2212yes—
20C2×#216C2212yes—
21#942C2212yes—
22#919—21no—
23#918—22no—
24#918—22no—
25D3×Q16—22no—
26D3×Q16—21no—
27#916—22no—
28#915—22no—
29#909—22no—
30#909C2212yes—
31#909C2212yes—
32#907C2212yes—
33C2×Dic12—21no—
34C2×Dic12C2211yescharacteristic
35#904—21no—
36#904C2211yescharacteristic
37#901C2211yescharacteristic
38C3×#108C2211yescharacteristic
39C3×#101C2212yes—
40#837C2211yescharacteristic
41#825C2211yescharacteristic
42#825C2212yes—
43#824C2212yes—
44#822C2211yescharacteristic
45C8×Dic3C2211yescharacteristic
46#578—32no—
47#578—31no—
48#578—31no—
49#494—31no—
50#473—31no—
51#473—32no—
52#473D311yescharacteristic, complemented
53#456—32no—
54#456—32no—
55#443—31no—
56#443—31no—
57C3×#41C2311yescharacteristic
58C2×C3×Q8C2312yes—
59#239—22no—
60#239—22no—
61D3×Q8—22no—
62D3×Q8—22no—
63D3×Q8—22no—
64#237—22no—
65#237—42no—
66#235—22no—
67#235—42no—
68#235C2311yescharacteristic
69#235C2312yes—
70C2×Dic6—22no—
71C2×Dic6C2312yes—
72C3×Q16—22no—
73C3×Q16—21no—
74C3×Q16—21no—
75C3×SD16—22no—
76C3×M16—21no—
77C3×M16C2311yescharacteristic
78C3×M16C2312yes—
79C2×C3×C8—21no—
80C2×C3×C8C2311yescharacteristic
81#216—22no—
82#216—22no—
83#216—22no—
84#216D411yescharacteristic
85#215—22no—
86#215D411yescharacteristic
87#214D411yescharacteristic
88#213D411yescharacteristic
89Dic3⋊C4—42no—
90C4×Dic3C2311yescharacteristic
91#208C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93Dic12—22no—
94Dic12—22no—
95Dic12—21no—
96Dic12—21no—
97Dic12D411yescharacteristic
98D24D411yescharacteristic
99#204—22no—
100#204—22no—
101#204D412yes—
102ΓL(1,𝔽25)—21no—
103ΓL(1,𝔽25)—22no—
104D3×C8—21no—
105D3×C8—22no—
1062−1+4—32no—
1072−1+4—32no—
108#137—32no—
109#137—32no—
110#137—32no—
111#137—62no—
112#137—62no—
113#137—62no—
114#137—62no—
115#137C2×D312yes—
116#135—31no—
117#135—31no—
118#135—61no—
119#135—62no—
120#135—61no—
121C2×Q16—32no—
122C2×Q16—31no—
123C2×Q16—61no—
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#108C2×D311yescharacteristic
133#104—31no—
134#104—32no—
135#104—31no—
136#101—32no—
137#101C2×D312yes—
138C4×C8—31no—
139C3×Q8—22no—
140C3×Q8—22no—
141C3×Q8C2×D411yescharacteristic
142C3×D4—21no—
143C3×D4C2×D411yescharacteristic
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4C2411yescharacteristic
147#67—21no—
148#67—22no—
149#67—42no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—42no—
153D12—42no—
154D12C2×D411yescharacteristic
155D12C2×D412yes—
156C4×D3—22no—
157C4×D3—21no—
158C4×D3—42no—
159C4×D3—42no—
160C4×D3—42no—
161Dic6—22no—
162Dic6—22no—
163Dic6—42no—
164Dic6—42no—
165Dic6C2×D412yes—
166Dic6C2×D411yescharacteristic
167C3×C8—22no—
168C3×C8—21no—
169C3×C8C2×D411yescharacteristic
170C3×C8C2×D411yescharacteristic
171C3⋊C8—21no—
172C3⋊C8C2×D411yescharacteristic
173C3⋊C8C2×D411yescharacteristic
174#41—32no—
175#41—31no—
176#41—62no—
177#41—62no—
178#41—62no—
179#41—62no—
180#41—122no—
181#41—122no—
182#41C22×D311yescharacteristic
183C2×Q8—32no—
184C2×Q8—62no—
185C2×Q8—62no—
186C2×Q8—62no—
187C2×Q8—62no—
188C2×Q8C22×D312yes—
189Q16—22no—
190Q16—21no—
191Q16—21no—
192Q16—31no—
193Q16—31no—
194Q16—62no—
195Q16—62no—
196Q16—62no—
197Q16—62no—
198Q16—62no—
199Q16—61no—
200Q16—61no—
201SD16—22no—
202SD16—31no—
203SD16—31no—
204SD16—32no—
205SD16—62no—
206SD16—62no—
207SD16—62no—
208D8—31no—
209D8—31no—
210M16—21no—
211M16—31no—
212M16—62no—
213M16—61no—
214M16C22×D312yes—
215M16C22×D311yescharacteristic
216C2×C8—21no—
217C2×C8—31no—
218C2×C8—62no—
219C2×C8—61no—
220C2×C8C22×D311yescharacteristic
221C4⋊C4—122no—
222C42—31no—
223C22×C3—21no—
224C22×C3H(5,𝔽2)11yescharacteristic
225C2×D3—21no—
226C2×D3—22no—
227C3×C4—21no—
228C3×C4—42no—
229C3×C4C22×D411yescharacteristic
230C3×C4C22×D411yescharacteristic
231Dic3—22no—
232Dic3—21no—
233Dic3—22no—
234Dic3—42no—
235Q8—22no—
236Q8—22no—
237Q8—31no—
238Q8—32no—
239Q8—62no—
240Q8—62no—
241Q8—122no—
242Q8—122no—
243Q8D3×D411yescharacteristic
244D4—21no—
245D4—31no—
246D4—32no—
247D4—61no—
248D4—62no—
249D4—122no—
250D4—122no—
251D4D3×D411yescharacteristic
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—62no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4C23×D311yescharacteristic
263C8—21no—
264C8—22no—
265C8—31no—
266C8—31no—
267C8—61no—
268C8D3×D411yescharacteristic
269C8D3×D411yescharacteristic
270C2×C3—21no—
271C2×C3—41no—
272C2×C3D4211yescharacteristic
273D3—41no—
274D3—42no—
275C22—21no—
276C22—62no—
277C22—61no—
278C22#100911yescharacteristic
279C4—21no—
280C4—42no—
281C4—62no—
282C4—61no—
283C4—62no—
284C4—122no—
285C4C2×D3×D411yescharacteristic
286C4C2×D3×D411yescharacteristic
287C3#329311yescharacteristic, complemented, Sylow
288C2—21no—
289C2—41no—
290C2—122no—
291C2—121no—
292C2#560911yescharacteristic
subgroupG/H
centerC2#5609
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#473C2
socleC2×C3D42
hypercenter#473D3
hypocenterC3#3293
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #5779C2semidirect
2. #5766C2semidirect
3. #5753C2semidirect
4. C3×#473C2semidirect
5. #5205C2semidirect
6. #5183C2semidirect
7. #5144C2semidirect
8. #4896C2semidirect
9. #4895C2semidirect
10. #4755C2semidirect
11. #4752C2semidirect
52. #473D3semidirect
287. C3#3293semidirect

Maximal subgroups

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

subgroupindex
#57792
#57662
#57532
C3×#4732
#52052
#51832
#51442
#48962
#48952
#47552
#47522
#32933