Finite Group Databasebeta

← #74019

Subgroups — #74019

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×#930C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic, complemented
3C2×#895C211yesmaximal, characteristic, complemented
4C2×#892C211yesmaximal, characteristic, complemented
5C3×#380C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4972C211yesmaximal, characteristic, complemented
8#4960C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4672C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2414—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2212yes—
16C22×D12C2211yescharacteristic
17C22×D12C2212yes—
18C22×C4×D3C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2211yescharacteristic
23#930—24no—
24C4×#67—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36#895—24no—
37#892—24no—
38C3×#95C2212yes—
39C3⋊#95C2212yes—
40C2×#121—31no—
41C2×#121—31no—
42C2×C4×D4—31no—
43C2×#117—31no—
44#394—32no—
45#392—32no—
46#390—31no—
47#390—32no—
48#389—31no—
49#388—32no—
50#380D311yescharacteristic, complemented
51C24×C3C2311yescharacteristic
52C23×D3C2312yes—
53C23×D3C2311yescharacteristic
54C22×C3×C4—22no—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—22no—
66C2×#67—42no—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C2×D12—24no—
72C2×D12—22no—
73C2×D12—22no—
74C2×D12—22no—
75C2×D12—42no—
76C2×D12—42no—
77C2×C4×D3—22no—
78C2×C4×D3—22no—
79C2×C4×D3D414yescomplemented
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C22⋊C4—42no—
83#217—22no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3D414yescomplemented
91Dic3⋊C4—22no—
92Dic3⋊C4—42no—
93C4×Dic3—22no—
94C4×Dic3—24no—
95C22×D4—31no—
96C22×D4—32no—
97C22×D4—31no—
98C22×D4—32no—
99C23×C4—31no—
100C23×C4C2×D311yescharacteristic
101#121—64no—
102#121—64no—
103C4×D4—64no—
104#117—64no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4C2×D311yescharacteristic
109C2×C4⋊C4C2×D311yescharacteristic
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4C2×D312yes—
118C2×C42—32no—
119C2×C42—31no—
120#95—32no—
121#95C2×D312yes—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3#4112yes—
129C23×C3C2411yescharacteristic
130C22×D3—21no—
131C22×D3—21no—
132C22×D3—21no—
133C22×D3—21no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—22no—
142C22×D3—22no—
143C22×D3C2×D412yes—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—44no—
154C2×C3×C4C2×D412yes—
155C2×C3×C4C2×D412yes—
156C2×C3×C4#4112yes—
157#67—28no—
158#67—48no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—21no—
168C2×Dic3C2×D414yes—
169C2×Dic3C2×D412yes—
170C2×Dic3#4112yes—
171D12—48no—
172D12—44no—
173C4×D3—28no—
174C24—31no—
175C24—32no—
176C24C22×D311yescharacteristic
177C2×D4—64no—
178C2×D4—62no—
179C2×D4—64no—
180C2×D4—62no—
181C2×D4—62no—
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—31no—
193C22×C4—32no—
194C22×C4—31no—
195C22×C4—34no—
196C22×C4—31no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4C22×D312yes—
200C22×C4C22×D311yescharacteristic
201C22×C4C22×D312yes—
202C22×C4C22×D311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—34no—
206C4⋊C4—62no—
207C4⋊C4—122no—
208C22⋊C4—42no—
209C22⋊C4—64no—
210C22⋊C4—64no—
211C22⋊C4—64no—
212C22⋊C4—64no—
213C22⋊C4—62no—
214C22⋊C4—62no—
215C22⋊C4—122no—
216C42—64no—
217C42—62no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3C22×D411yescharacteristic
226C22×C3C22×D411yescharacteristic
227C22×C3C2×#4111yescharacteristic
228C22×C3C2×#4111yescharacteristic
229C22×C3C2×#4111yescharacteristic
230C22×C3H(5,𝔽2)11yescharacteristic
231C22×C3H(5,𝔽2)11yescharacteristic
232C2×D3—22no—
233C2×D3—22no—
234C2×D3—22no—
235C2×D3—41no—
236C2×D3—41no—
237C2×D3—41no—
238C2×D3—41no—
239C2×D3—42no—
240C2×D3—42no—
241C2×D3—42no—
242C2×D3—42no—
243C2×D3—42no—
244C2×D3—42no—
245C2×D3—42no—
246C3×C4—41no—
247C3×C4—42no—
248C3×C4—42no—
249C3×C4#12114yes—
250Dic3—22no—
251Dic3—22no—
252Dic3—24no—
253Dic3—41no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—32no—
261C23—61no—
262C23—61no—
263C23—61no—
264C23—61no—
265C23—61no—
266C23—61no—
267C23—62no—
268C23—62no—
269C23—62no—
270C23—62no—
271C23—62no—
272C23—62no—
273C23—62no—
274C23#23912yes—
275C23C23×D311yescharacteristic
276D4—68no—
277D4—128no—
278D4—128no—
279D4—124no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—34no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—44no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—68no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4#23512yes—
304C2×C4D3×D412yes—
305C2×C4D3×D412yes—
306C2×C3—42no—
307C2×C3C2×#12111yescharacteristic
308C2×C3#52511yescharacteristic
309C2×C3#53511yescharacteristic
310C2×C3#54611yescharacteristic
311C2×C3#55011yescharacteristic
312C2×C3#55311yescharacteristic
313C2×C3#55511yescharacteristic
314D3—42no—
315D3—82no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—62no—
324C22—62no—
325C22—62no—
326C22—121no—
327C22—121no—
328C22—121no—
329C22—121no—
330C22—122no—
331C22—122no—
332C22—122no—
333C22—122no—
334C22—122no—
335C22—122no—
336C22—122no—
337C22C2×#23511yescharacteristic
338C22C2×D3×D411yescharacteristic
339C22C2×D3×D411yescharacteristic
340C22#100411yescharacteristic
341C22C2×#23911yescharacteristic
342C22D3×#4111yescharacteristic
343C22#100911yescharacteristic
344C4—41no—
345C4—42no—
346C4—42no—
347C4—62no—
348C4—62no—
349C4—64no—
350C4—121no—
351C4#88414yes—
352C3#241411yescharacteristic, complemented, Sylow
353C2—42no—
354C2—122no—
355C2—242no—
356C2C2×#88411yescharacteristic
357C2#551411yescharacteristic
358C2#553211yescharacteristic
359C2#559911yescharacteristic
360C2#560711yescharacteristic
361C2#562711yescharacteristic
362C2#563011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
hypercenter#380D3
hypocenterC3#2414
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×#892C2semidirect
5. C3×#380C2semidirect
6. #4988C2semidirect
7. #4972C2semidirect
8. #4960C2semidirect
9. #4957C2semidirect
10. #4672C2semidirect
11. #4669C2semidirect
50. #380D3semidirect
79. C2×C4×D3D4semidirect
90. C4⋊Dic3D4semidirect
352. C3#2414semidirect

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×#8922
C3×#3802
#49882
#49722
#49602
#49572
#46722
#46692
#24143