Finite Group Databasebeta

← #82352

Subgroups — #82352

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×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#939C212yesmaximal, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#5212C211yesmaximal, characteristic, complemented
7#4994C212yesmaximal, complemented
8#4961C212yesmaximal, complemented
9#4958C212yesmaximal, complemented
10#4939C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2416—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2212yes—
17C22×#67C2212yes—
18C23×Dic3C2212yes—
19C22×D12C2211yescharacteristic
20C3×C4×D4—24no—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C42C2211yescharacteristic
25C2×#217C2212yes—
26#939—28no—
27#930—24no—
28#930—24no—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×(C2×D3)⋊C4C2212yes—
33C2×C4⋊Dic3—24no—
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2212yes—
37C2×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2212yes—
39C3⋊#95C2212yes—
40C3⋊#95C2212yes—
41C2×#121—32no—
42C2×#121—31no—
43C2×#121—31no—
44C2×C4×D4D311yescharacteristic, complemented
45C22×C4⋊C4—31no—
46#396—32no—
47#394—32no—
48#390—31no—
49#389—31no—
50#386—32no—
51#380—31no—
52C24×C3C2311yescharacteristic
53C24×C3C2311yescharacteristic
54C23×D3C2312yes—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4D414yescomplemented
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—22no—
63C22×C3×C4D414yescomplemented
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C2×#67—24no—
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—24no—
73C2×#67—24no—
74C2×#67—42no—
75C22×Dic3—24no—
76C22×Dic3—24no—
77C22×Dic3—42no—
78C22×Dic3C2312yes—
79C22×Dic3C2312yes—
80C22×Dic3C2312yes—
81C2×D12—22no—
82C2×D12—22no—
83C2×D12—22no—
84C3×C4⋊C4—22no—
85C3×C22⋊C4—22no—
86C3×C22⋊C4—22no—
87C3×C42—22no—
88#217—24no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—24no—
91(C2×D3)⋊C4—24no—
92(C2×D3)⋊C4—42no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—24no—
96C4⋊Dic3D414yescomplemented
97Dic3⋊C4—24no—
98C22×D4—32no—
99C22×D4—32no—
100C22×D4—31no—
101C22×D4C2×D311yescharacteristic
102C23×C4—32no—
103C23×C4C2×D311yescharacteristic
104C23×C4C2×D311yescharacteristic
105#121—64no—
106#121—68no—
107#121—64no—
108C4×D4—24no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—64no—
115C2×C4⋊C4C2×D311yescharacteristic
116C2×C22⋊C4—32no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4C2×D311yescharacteristic
122C2×C22⋊C4C2×D311yescharacteristic
123C2×C42C2×D311yescharacteristic
124#95—32no—
125#95—32no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3C2×D412yes—
137C23×C3C2×D412yes—
138C23×C3C2×D412yes—
139C23×C3#4112yes—
140C23×C3C2411yescharacteristic
141C22×D3—22no—
142C22×D3—22no—
143C22×D3—22no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—22no—
148C3×D4—28no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—28no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—44no—
161C2×C3×C4C2×D412yes—
162C2×C3×C4C2×D412yes—
163C2×C3×C4C2×D412yes—
164C2×C3×C4C2×D412yes—
165C2×C3×C4#4112yes—
166#67—48no—
167#67—48no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3—22no—
173C2×Dic3—22no—
174C2×Dic3—22no—
175C2×Dic3—22no—
176C2×Dic3—48no—
177C2×Dic3C2×D414yes—
178D12—44no—
179C24—32no—
180C24C22×D311yescharacteristic
181C24C22×D311yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—64no—
185C2×D4—64no—
186C2×D4—64no—
187C2×D4—64no—
188C2×D4—64no—
189C2×D4—62no—
190C2×D4—62no—
191C2×D4—62no—
192C2×D4—122no—
193C2×D4D1214yescomplemented
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—32no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—64no—
203C22×C4—64no—
204C22×C4—122no—
205C22×C4#6714yescomplemented
206C22×C4C22×D311yescharacteristic
207C22×C4C22×D311yescharacteristic
208C22×C4C22×D311yescharacteristic
209C22×C4C22×D311yescharacteristic
210C22×C4C22×D311yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—34no—
213C4⋊C4—64no—
214C4⋊C4—64no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—64no—
220C22⋊C4—64no—
221C22⋊C4—64no—
222C22⋊C4—64no—
223C22⋊C4—122no—
224C42—22no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3#12114yescomplemented
239C22×C3C22×D411yescharacteristic
240C22×C3C22×D411yescharacteristic
241C22×C3C22×D411yescharacteristic
242C22×C3C2×#4111yescharacteristic
243C22×C3C2×#4111yescharacteristic
244C22×C3H(5,𝔽2)11yescharacteristic
245C22×C32−1+411yescharacteristic
246C2×D3—42no—
247C2×D3—42no—
248C2×D3—42no—
249C2×D3—42no—
250C2×D3—42no—
251C2×D3—42no—
252C2×D3—42no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4#12114yescomplemented
258Dic3—24no—
259Dic3—42no—
260Dic3—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—62no—
272C23—62no—
273C23—62no—
274C23—62no—
275C23—62no—
276C23—62no—
277C23—62no—
278C23C2×D1212yes—
279C23C2×D1212yes—
280C23#23512yes—
281C23C2×#6712yes—
282C23C23×D311yescharacteristic
283D4—28no—
284D4—128no—
285D4—128no—
286D4—124no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—34no—
299C2×C4—44no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—128no—
309C2×C4C2×D1212yes—
310C2×C4D3×D412yes—
311C2×C4#23712yes—
312C2×C4C2×#6712yes—
313C2×C4C2×#6712yes—
314C2×C3—24no—
315C2×C3—42no—
316C2×C3C2×#12111yescharacteristic
317C2×C3C2×#12111yescharacteristic
318C2×C3#53711yescharacteristic
319C2×C3D4211yescharacteristic
320C2×C3#54711yescharacteristic
321C2×C3#54811yescharacteristic
322C2×C3#55311yescharacteristic
323D3—82no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—122no—
338C22—122no—
339C22—122no—
340C22—122no—
341C22—122no—
342C22—122no—
343C22—122no—
344C22#93014yescomplemented
345C22C22×D1211yescharacteristic
346C22C2×#23511yescharacteristic
347C22C2×D3×D411yescharacteristic
348C22C2×#23711yescharacteristic
349C22#100911yescharacteristic
350C22#101011yescharacteristic
351C22C22×#6711yescharacteristic
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—64no—
357C4—122no—
358C4—122no—
359C4#93914yescomplemented
360C3#241611yescharacteristic, complemented, Sylow
361C2—24no—
362C2—42no—
363C2—242no—
364C2D4×D1211yescharacteristic
365C2#555211yescharacteristic
366C2#555511yescharacteristic
367C2#556511yescharacteristic
368C2C2×#93011yescharacteristic
369C2C2×#93911yescharacteristic
370C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#2416
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×#939C2semidirect
3. C2×#930C2semidirect
4. C2×#930C2semidirect
5. C22×C4⋊Dic3C2semidirect
6. #5212C2semidirect
7. #4994C2semidirect
8. #4961C2semidirect
9. #4958C2semidirect
10. #4939C2semidirect
11. #4928C2semidirect
44. C2×C4×D4D3semidirect
57. C2×C3×D4D4semidirect
63. C22×C3×C4D4semidirect
96. C4⋊Dic3D4semidirect
193. C2×D4D12semidirect
205. C22×C4#67semidirect
238. C22×C3#121semidirect
257. C3×C4#121semidirect
344. C22#930semidirect
359. C4#939semidirect
360. C3#2416semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9392
C2×#9302
C22×C4⋊Dic32
#52122
#49942
#49612
#49582
#49392
#49282
#24163