Finite Group Databasebeta

← #82350

Subgroups — #82350

Order 384.

704 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4C2×#926C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5212C211yesmaximal, characteristic, complemented
7#4990C212yesmaximal, complemented
8#4959C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4939C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2414—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2212yes—
17C22×#67C2212yes—
18C22×D12C2211yescharacteristic
19C3×C4×D4—24no—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C42C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2212yes—
26#930—24no—
27#930—24no—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32#926—24no—
33C2×C4⋊Dic3C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2212yes—
37C2×C4×Dic3C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#121—31no—
41C2×#121—31no—
42C2×C4×D4D311yescharacteristic, complemented
43C2×#117—31no—
44#394—32no—
45#392—32no—
46#390—32no—
47#390—31no—
48#389—31no—
49#388—32no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C23×D3C2312yes—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4D414yescomplemented
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C2×#67—24no—
69C2×#67—24no—
70C2×#67—24no—
71C2×#67—42no—
72C2×#67—42no—
73C2×#67—42no—
74C22×Dic3C2312yes—
75C22×Dic3C2312yes—
76C22×Dic3C2312yes—
77C2×D12—22no—
78C2×D12—22no—
79C2×D12—22no—
80C3×C4⋊C4—22no—
81C3×C22⋊C4—22no—
82C3×C22⋊C4—22no—
83C3×C42—22no—
84#217—24no—
85#217—42no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3D414yescomplemented
93Dic3⋊C4—42no—
94C4×Dic3—24no—
95C22×D4—32no—
96C22×D4—32no—
97C22×D4—31no—
98C22×D4C2×D311yescharacteristic
99C23×C4C2×D311yescharacteristic
100C23×C4C2×D311yescharacteristic
101#121—64no—
102#121—64no—
103C4×D4—24no—
104#117—64no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4C2×D311yescharacteristic
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4C2×D311yescharacteristic
117C2×C22⋊C4C2×D311yescharacteristic
118C2×C42—32no—
119C2×C42C2×D311yescharacteristic
120#95—32no—
121#95—32no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
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×C3C2×D412yes—
135C23×C3#4112yes—
136C23×C3C2411yescharacteristic
137C22×D3—22no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—22no—
142C22×D3—22no—
143C22×D3—22no—
144C3×D4—28no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—28no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—44no—
157C2×C3×C4C2×D412yes—
158C2×C3×C4C2×D412yes—
159C2×C3×C4C2×D412yes—
160C2×C3×C4#4112yes—
161C2×C3×C4#4112yes—
162#67—48no—
163#67—48no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3C2×D414yes—
173D12—44no—
174C24—32no—
175C24C22×D311yescharacteristic
176C24C22×D311yescharacteristic
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—64no—
181C2×D4—64no—
182C2×D4—64no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—122no—
187C2×D4—122no—
188C2×D4—122no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—32no—
195C22×C4—32no—
196C22×C4—32no—
197C22×C4#6714yescomplemented
198C22×C4C22×D311yescharacteristic
199C22×C4C22×D311yescharacteristic
200C22×C4C22×D311yescharacteristic
201C22×C4C22×D311yescharacteristic
202C22×C4C22×D311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—34no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—122no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—64no—
211C22⋊C4—64no—
212C22⋊C4—64no—
213C22⋊C4—64no—
214C22⋊C4—122no—
215C22⋊C4—122no—
216C42—22no—
217C42—64no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3C22×D411yescharacteristic
233C22×C3C22×D411yescharacteristic
234C22×C3C2×#4111yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3H(5,𝔽2)11yescharacteristic
238C22×C3H(5,𝔽2)11yescharacteristic
239C2×D3—42no—
240C2×D3—42no—
241C2×D3—42no—
242C2×D3—42no—
243C2×D3—42no—
244C2×D3—42no—
245C2×D3—42no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4#12114yes—
251Dic3—24no—
252Dic3—42no—
253Dic3—42no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—62no—
267C23—62no—
268C23—62no—
269C23—62no—
270C23—62no—
271C23—62no—
272C23—62no—
273C23#23512yes—
274C23C2×#6712yes—
275C23C23×D311yescharacteristic
276D4—28no—
277D4—128no—
278D4—128no—
279D4—124no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—28no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—34no—
292C2×C4—44no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4#23512yes—
302C2×C4D3×D412yes—
303C2×C4#23712yes—
304C2×C4C2×#6712yes—
305C2×C4C2×#6712yes—
306C2×C3—42no—
307C2×C3—42no—
308C2×C3C2×#12111yescharacteristic
309C2×C3#52511yescharacteristic
310C2×C3#53511yescharacteristic
311C2×C3#54611yescharacteristic
312C2×C3#55011yescharacteristic
313C2×C3#55311yescharacteristic
314C2×C3#55511yescharacteristic
315D3—82no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—122no—
331C22—122no—
332C22—122no—
333C22—122no—
334C22—122no—
335C22—122no—
336C22—122no—
337C22C2×#23511yescharacteristic
338C22C2×#23511yescharacteristic
339C22C2×D3×D411yescharacteristic
340C22C2×#23711yescharacteristic
341C22#100911yescharacteristic
342C22#100911yescharacteristic
343C22C22×#6711yescharacteristic
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—64no—
349C4—122no—
350C4—122no—
351C4#93914yes—
352C3#241411yescharacteristic, complemented, Sylow
353C2—42no—
354C2—42no—
355C2—242no—
356C2#555011yescharacteristic
357C2#555311yescharacteristic
358C2#556311yescharacteristic
359C2#556511yescharacteristic
360C2#579111yescharacteristic
361C2C2×#93911yescharacteristic
362C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#2414
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×#930C2semidirect
3. C2×#930C2semidirect
4. C2×#926C2semidirect
5. #5223C2semidirect
6. #5212C2semidirect
7. #4990C2semidirect
8. #4959C2semidirect
9. #4957C2semidirect
10. #4939C2semidirect
11. #4928C2semidirect
42. C2×C4×D4D3semidirect
62. C22×C3×C4D4semidirect
92. C4⋊Dic3D4semidirect
197. C22×C4#67semidirect
352. C3#2414semidirect

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×#9302
C2×#9262
#52232
#52122
#49902
#49592
#49572
#49392
#49282
#24143