Finite Group Databasebeta

← #73887

Subgroups — #73887

Order 384.

788 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#930C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic, complemented
3C2×#895C211yesmaximal, characteristic, complemented
4C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×#884C212yesmaximal, complemented
6C3×#380C211yesmaximal, characteristic, complemented
7#4988C211yesmaximal, characteristic, complemented
8#4972C211yesmaximal, characteristic, complemented
9#4959C212yesmaximal, complemented
10#4676C212yesmaximal, complemented
11#4665C212yesmaximal, complemented
12#2416—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2211yescharacteristic
16C22×D12C2212yes—
17C22×D12C2211yescharacteristic
18C22×C4×D3C2212yes—
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2211yescharacteristic
24#930—24no—
25C4×#67—24no—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36#895—24no—
37D3×C4⋊C4—24no—
38#884—28no—
39C3×#95C2212yes—
40C3⋊#95C2212yes—
41C2×#121—31no—
42C2×#121—31no—
43C2×#121—32no—
44C2×C4×D4—31no—
45C22×C4⋊C4—31no—
46#396—32no—
47#394—32no—
48#390—31no—
49#389—31no—
50#386—32no—
51#380D311yescharacteristic, complemented
52C24×C3C2311yescharacteristic
53C23×D3C2312yes—
54C23×D3C2311yescharacteristic
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—42no—
67C2×#67D414yescomplemented
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C2×D12—24no—
73C2×D12—24no—
74C2×D12—24no—
75C2×D12—22no—
76C2×D12—22no—
77C2×D12—22no—
78C2×C4×D3—22no—
79C2×C4×D3—22no—
80C2×C4×D3—24no—
81C2×C4×D3—24no—
82C2×C4×D3—42no—
83C2×C4×D3D414yescomplemented
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C22⋊C4—24no—
87#217—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—24no—
91(C2×D3)⋊C4—24no—
92(C2×D3)⋊C4—42no—
93C4⋊Dic3D414yescomplemented
94Dic3⋊C4—24no—
95Dic3⋊C4—24no—
96Dic3⋊C4—22no—
97C4×Dic3—22no—
98C22×D4—31no—
99C22×D4—32no—
100C22×D4—32no—
101C22×D4—31no—
102C23×C4—31no—
103C23×C4—32no—
104C23×C4C2×D311yescharacteristic
105#121—68no—
106#121—64no—
107#121—64no—
108C4×D4—64no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—64no—
114C2×C4⋊C4C2×D311yescharacteristic
115C2×C4⋊C4C2×D311yescharacteristic
116C2×C22⋊C4—31no—
117C2×C22⋊C4—31no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4C2×D312yes—
123C2×C42—31no—
124#95—32no—
125#95C2×D312yes—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3C2×D412yes—
133C23×C3C2411yescharacteristic
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—22no—
142C22×D3—22no—
143C22×D3—22no—
144C22×D3—22no—
145C22×D3C2×D412yes—
146C22×D3C2×D412yes—
147C22×D3#4112yes—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—21no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—44no—
158C2×C3×C4C2×D412yes—
159C2×C3×C4C2×D412yes—
160C2×C3×C4#4112yes—
161#67—28no—
162#67—48no—
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—22no—
166C2×Dic3—21no—
167C2×Dic3—21no—
168C2×Dic3—22no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3C2×D412yes—
173C2×Dic3C2×D414yes—
174C2×Dic3C2×D412yes—
175D12—48no—
176D12—44no—
177C4×D3—28no—
178C4×D3—48no—
179C24—31no—
180C24—32no—
181C24C22×D311yescharacteristic
182C2×D4—34no—
183C2×D4—64no—
184C2×D4—64no—
185C2×D4—64no—
186C2×D4—64no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—64no—
190C2×D4—62no—
191C2×D4—62no—
192C2×D4—62no—
193C2×D4—122no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—34no—
201C22×C4—31no—
202C22×C4—64no—
203C22×C4—64no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—122no—
207C22×C4C22×D312yes—
208C22×C4C22×D311yescharacteristic
209C22×C4C22×D312yes—
210C22×C4C22×D311yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—34no—
214C4⋊C4—64no—
215C4⋊C4—62no—
216C4⋊C4—64no—
217C22⋊C4—24no—
218C22⋊C4—64no—
219C22⋊C4—64no—
220C22⋊C4—64no—
221C22⋊C4—62no—
222C22⋊C4—62no—
223C22⋊C4—122no—
224C42—62no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3C22×D411yescharacteristic
233C22×C3C22×D411yescharacteristic
234C22×C3C22×D411yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3H(5,𝔽2)11yescharacteristic
238C22×C32−1+411yescharacteristic
239C2×D3—22no—
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—42no—
246C2×D3—42no—
247C2×D3—42no—
248C2×D3—42no—
249C2×D3—42no—
250C2×D3—42no—
251C2×D3—42no—
252C2×D3#12114yescomplemented
253C3×C4—42no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4#12114yescomplemented
257Dic3—22no—
258Dic3—22no—
259Dic3—24no—
260Dic3—41no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—32no—
268C23—32no—
269C23—32no—
270C23—61no—
271C23—61no—
272C23—61no—
273C23—61no—
274C23—62no—
275C23—62no—
276C23—62no—
277C23—62no—
278C23—62no—
279C23—62no—
280C23—62no—
281C23D3×D412yes—
282C23C23×D311yescharacteristic
283D4—68no—
284D4—128no—
285D4—128no—
286D4—124no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—34no—
299C2×C4—44no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—62no—
304C2×C4—61no—
305C2×C4—62no—
306C2×C4—68no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—128no—
311C2×C4#23512yes—
312C2×C4D3×D412yes—
313C2×C4D3×D412yes—
314C2×C3—42no—
315C2×C3C2×#12111yescharacteristic
316C2×C3C2×#12111yescharacteristic
317C2×C3#53711yescharacteristic
318C2×C3D4211yescharacteristic
319C2×C3#54711yescharacteristic
320C2×C3#54811yescharacteristic
321C2×C3#55311yescharacteristic
322D3—24no—
323D3—82no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—34no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—62no—
333C22—62no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—62no—
338C22—122no—
339C22—122no—
340C22—122no—
341C22—122no—
342C22—122no—
343C22—122no—
344C22—122no—
345C22C2×#23511yescharacteristic
346C22C2×D3×D411yescharacteristic
347C22C2×D3×D411yescharacteristic
348C22C2×D3×D411yescharacteristic
349C22#100711yescharacteristic
350C22D3×#4111yescharacteristic
351C22#100911yescharacteristic
352C4—41no—
353C4—42no—
354C4—42no—
355C4—62no—
356C4—62no—
357C4—64no—
358C4—121no—
359C4#88414yescomplemented
360C3#241611yescharacteristic, complemented, Sylow
361C2—42no—
362C2—64no—
363C2—242no—
364C2C2×#88411yescharacteristic
365C2#550711yescharacteristic
366C2#553011yescharacteristic
367C2D3×#12111yescharacteristic
368C2#560911yescharacteristic
369C2#562911yescharacteristic
370C2#563311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
hypercenter#380D3
hypocenterC3#2416
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#930C2semidirect
2. C2×C4×#67C2semidirect
3. C2×#895C2semidirect
4. C2×D3×C4⋊C4C2semidirect
5. C2×#884C2semidirect
6. C3×#380C2semidirect
7. #4988C2semidirect
8. #4972C2semidirect
9. #4959C2semidirect
10. #4676C2semidirect
11. #4665C2semidirect
51. #380D3semidirect
67. C2×#67D4semidirect
83. C2×C4×D3D4semidirect
93. C4⋊Dic3D4semidirect
252. C2×D3#121semidirect
256. C3×C4#121semidirect
359. C4#884semidirect
360. C3#2416semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×C4×#672
C2×#8952
C2×D3×C4⋊C42
C2×#8842
C3×#3802
#49882
#49722
#49592
#46762
#46652
#24163