Finite Group Databasebeta

← #76353

Subgroups — #76353

Order 384.

676 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#940C211yesmaximal, characteristic
2C2×#885C212yesmaximal, complemented
3C2×C4×Dic6C211yesmaximal, characteristic, complemented
4C3×#390C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
8#4961C212yesmaximal, complemented
9#4959C212yesmaximal, complemented
10#4957C212yesmaximal, complemented
11#4939C211yesmaximal, characteristic, complemented
12#2548—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C22×D12C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24#940—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36#885—28no—
37C4×Dic6—24no—
38C3⋊#95C2212yes—
39C2×#127—31no—
40C2×#124—32no—
41C2×C4×Q8—31no—
42#394—32no—
43#394—32no—
44#390—31no—
45#390—31no—
46#390—32no—
47#390—31no—
48#390D311yescharacteristic, complemented
49C4×C4⋊C4—31no—
50C24×C3C2312yes—
51C23×D3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
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×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×D12—22no—
71C2×D12—22no—
72C2×D12—22no—
73C2×Dic6—22no—
74C2×Dic6—22no—
75C2×Dic6D414yescomplemented
76C3×C4⋊C4—22no—
77C3×C22⋊C4—24no—
78C3×C22⋊C4—42no—
79C3×C42—22no—
80#217—24no—
81#217—42no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—42no—
85(C2×D3)⋊C4—42no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88Dic3⋊C4—22no—
89Dic3⋊C4—22no—
90C4×Dic3—22no—
91C4×Dic3—24no—
92C4×Dic3D414yes—
93C22×Q8—31no—
94C22×D4—32no—
95C22×D4—32no—
96C22×D4—31no—
97C22×D4C2×D311yescharacteristic
98#127—64no—
99#124—68no—
100C4×Q8—64no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4C2×D311yescharacteristic
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4C2×D312yes—
113C2×C22⋊C4C2×D312yes—
114C2×C42—31no—
115C2×C42—31no—
116C2×C42—32no—
117C2×C42C2×D311yescharacteristic
118#95—32no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3C2411yescharacteristic
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C3×D4—44no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D412yes—
145C2×C3×C4#4112yes—
146C2×C3×C4#4112yes—
147#67—48no—
148#67—48no—
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—22no—
158C2×Dic3—21no—
159C2×Dic3C2×D412yes—
160C2×Dic3C2×D412yes—
161C2×Dic3C2×D412yes—
162C2×Dic3C2×D412yes—
163C2×Dic3#4112yes—
164D12—44no—
165Dic6—28no—
166C24—32no—
167C24C22×D312yes—
168C2×Q8—34no—
169C2×Q8—62no—
170C2×Q8—62no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4—64no—
178C2×D4—62no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—122no—
182C2×D4—122no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—32no—
187C22×C4—31no—
188C22×C4C22×D312yes—
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D311yescharacteristic
191C22×C4C22×D311yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—62no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C22⋊C4—64no—
202C22⋊C4—122no—
203C22⋊C4—122no—
204C22⋊C4—122no—
205C42—22no—
206C42—34no—
207C42—64no—
208C42—62no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3C22×D411yescharacteristic
217C22×C3C22×D411yescharacteristic
218C22×C3C2×#4111yescharacteristic
219C22×C3C2×#4111yescharacteristic
220C22×C3C2×#4111yescharacteristic
221C22×C3H(5,𝔽2)11yescharacteristic
222C22×C3H(5,𝔽2)11yescharacteristic
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—42no—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C3×C4—22no—
231C3×C4—41no—
232C3×C4—42no—
233C3×C4#12414yescomplemented
234Dic3—22no—
235Dic3—22no—
236Dic3—41no—
237Dic3—42no—
238Dic3#12114yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—62no—
247C23—62no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23C23×D311yescharacteristic
254Q8—68no—
255D4—44no—
256D4—128no—
257D4—128no—
258D4—124no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4#23512yes—
283C2×C4D3×D412yes—
284C2×C4D3×D412yes—
285C2×C4#23712yes—
286C2×C3—82no—
287C2×C3C2×#12111yescharacteristic
288C2×C3C2×#12411yescharacteristic
289C2×C3#53511yescharacteristic
290C2×C3#55011yescharacteristic
291C2×C3#55011yescharacteristic
292C2×C3#55911yescharacteristic
293C2×C3#55911yescharacteristic
294D3—82no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—122no—
303C22—122no—
304C22—122no—
305C22—122no—
306C22—122no—
307C22—122no—
308C22—122no—
309C22C2×#23511yescharacteristic
310C22C2×D3×D411yescharacteristic
311C22C2×D3×D411yescharacteristic
312C22C2×#23711yescharacteristic
313C22#100411yescharacteristic
314C22D3×#4111yescharacteristic
315C22#100911yescharacteristic
316C4—22no—
317C4—34no—
318C4—42no—
319C4—41no—
320C4—62no—
321C4—62no—
322C4—122no—
323C4—121no—
324C4#88514yescomplemented
325C3#254811yescharacteristic, complemented, Sylow
326C2—82no—
327C2—242no—
328C2C2×#88511yescharacteristic
329C2#555311yescharacteristic
330C2#555911yescharacteristic
331C2D3×#12111yescharacteristic
332C2#560711yescharacteristic
333C2#568111yescharacteristic
334C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#940C2
hypercenter#390D3
hypocenterC3#2548
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#885C2semidirect
3. C2×C4×Dic6C2semidirect
4. C3×#390C2semidirect
5. #5221C2semidirect
6. #4988C2semidirect
7. Dic3×C4⋊C4C2semidirect
8. #4961C2semidirect
9. #4959C2semidirect
10. #4957C2semidirect
11. #4939C2semidirect
48. #390D3semidirect
75. C2×Dic6D4semidirect
233. C3×C4#124semidirect
324. C4#885semidirect
325. C3#2548semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
C2×#8852
C2×C4×Dic62
C3×#3902
#52212
#49882
Dic3×C4⋊C42
#49612
#49592
#49572
#49392
#25483