Finite Group Databasebeta

← #76317

Subgroups — #76317

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#940C211yesmaximal, characteristic
2C2×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×#886C212yesmaximal, complemented
4C3×#390C211yesmaximal, characteristic, complemented
5#4988C211yesmaximal, characteristic, complemented
6#4984C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4961C212yesmaximal, complemented
9#4958C212yesmaximal, complemented
10#4939C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#2445—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C23×Dic3C2212yes—
17C22×D12C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23#940—24no—
24D4×Dic3—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2212yes—
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35#886—28no—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×#127—31no—
40C2×#123—32no—
41C2×C4×D4—31no—
42#394—32no—
43#394—32no—
44#390—31no—
45#390—31no—
46#390D311yescharacteristic, complemented
47#389—31no—
48#386—32no—
49C4×C4⋊C4—31no—
50C24×C3C2312yes—
51C23×D3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4D414yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—42no—
64C2×#67—42no—
65C22×Dic3—24no—
66C22×Dic3—24no—
67C22×Dic3—42no—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C2×D12—22no—
74C2×D12—22no—
75C2×D12—22no—
76C3×C4⋊C4—22no—
77C3×C22⋊C4—24no—
78C3×C22⋊C4—42no—
79C3×C42—22no—
80#217—24no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—42no—
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—22no—
86C4⋊Dic3—24no—
87C4⋊Dic3—22no—
88C4⋊Dic3D414yescomplemented
89C4×Dic3—22no—
90C4×Dic3—22no—
91C22×D4—32no—
92C22×D4—32no—
93C22×D4—31no—
94C22×D4C2×D311yescharacteristic
95C23×C4—32no—
96#127—64no—
97#123—68no—
98C4×D4—64no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4C2×D311yescharacteristic
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4C2×D312yes—
110C2×C22⋊C4C2×D312yes—
111C2×C42—31no—
112C2×C42—31no—
113C2×C42C2×D311yescharacteristic
114#95—31no—
115#95—31no—
116#95—32no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2×D414yes—
124C23×C3C2411yescharacteristic
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C3×D4—28no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4#4112yes—
144C2×C3×C4#4112yes—
145#67—48no—
146#67—48no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—48no—
160C2×Dic3C2×D412yes—
161C2×Dic3C2×D412yes—
162C2×Dic3#4112yes—
163D12—44no—
164C24—32no—
165C24C22×D312yes—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—64no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—122no—
176C2×D4—122no—
177C2×D4D1214yes—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—64no—
184C22×C4—64no—
185C22×C4—122no—
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D312yes—
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—34no—
192C4⋊C4—64no—
193C4⋊C4—62no—
194C4⋊C4—62no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—122no—
201C22⋊C4—122no—
202C42—22no—
203C42—62no—
204C42—62no—
205C22×C3—24no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3C22×D411yescharacteristic
213C22×C3C22×D411yescharacteristic
214C22×C3C2×#4111yescharacteristic
215C22×C3C2×#4111yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3H(5,𝔽2)11yescharacteristic
218C22×C3H(5,𝔽2)11yescharacteristic
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—42no—
226C3×C4—22no—
227C3×C4—41no—
228C3×C4—42no—
229C3×C4#12314yescomplemented
230Dic3—22no—
231Dic3—22no—
232Dic3—22no—
233Dic3—42no—
234Dic3—41no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23—62no—
246C23—62no—
247C23—62no—
248C23C2×D1214yes—
249C23C23×D311yescharacteristic
250D4—28no—
251D4—128no—
252D4—128no—
253D4—124no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—62no—
277C2×C4—128no—
278C2×C4C2×D1212yes—
279C2×C4D3×D412yes—
280C2×C4#23712yes—
281C2×C4#23712yes—
282C2×C3—44no—
283C2×C3C2×#12311yescharacteristic
284C2×C3#53211yescharacteristic
285C2×C3#54011yescharacteristic
286C2×C3D4211yescharacteristic
287C2×C3#54611yescharacteristic
288C2×C3#55011yescharacteristic
289C2×C3#55911yescharacteristic
290D3—82no—
291C22—24no—
292C22—24no—
293C22—24no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—122no—
299C22—122no—
300C22—122no—
301C22—122no—
302C22—122no—
303C22—122no—
304C22—122no—
305C22C22×D1211yescharacteristic
306C22C2×D3×D411yescharacteristic
307C22C2×#23711yescharacteristic
308C22C2×#23711yescharacteristic
309C22#100411yescharacteristic
310C22D3×#4111yescharacteristic
311C22#100911yescharacteristic
312C4—22no—
313C4—42no—
314C4—41no—
315C4—62no—
316C4—62no—
317C4—62no—
318C4—122no—
319C4—121no—
320C4#88614yescomplemented
321C3#244511yescharacteristic, complemented, Sylow
322C2—44no—
323C2—242no—
324C2C2×#88611yescharacteristic
325C2D4×D1211yescharacteristic
326C2#555411yescharacteristic
327C2#559611yescharacteristic
328C2#561911yescharacteristic
329C2#568111yescharacteristic
330C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#940C2
hypercenter#390D3
hypocenterC3#2445
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic3C2semidirect
3. C2×#886C2semidirect
4. C3×#390C2semidirect
5. #4988C2semidirect
6. #4984C2semidirect
7. #4961C2semidirect
8. #4961C2semidirect
9. #4958C2semidirect
10. #4939C2semidirect
11. C4×C4⋊Dic3C2semidirect
46. #390D3semidirect
88. C4⋊Dic3D4semidirect
229. C3×C4#123semidirect
320. C4#886semidirect
321. C3#2445semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
C2×D4×Dic32
C2×#8862
C3×#3902
#49882
#49842
#49612
#49582
#49392
C4×C4⋊Dic32
#24453