Finite Group Databasebeta

← #82591

Subgroups — #82591

Order 384.

900 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#120C211yesmaximal, characteristic, complemented
2C2×#953C211yesmaximal, characteristic, complemented
3C22×#217C211yesmaximal, characteristic, complemented
4C2×#939C211yesmaximal, characteristic, complemented
5C2×#935C211yesmaximal, characteristic, complemented
6C2×#886C211yesmaximal, characteristic, complemented
7#5223C212yesmaximal, complemented
8#5221C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4949C212yesmaximal, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#2539—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2212yes—
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C22×#67C2211yescharacteristic, complemented
19C23×Dic3C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C3×#120—24no—
22C2×C3×C22⋊C4C2212yes—
23C2×C3×C22⋊C4C2211yescharacteristic
24#953—24no—
25C2×#217—24no—
26C2×#217C2212yes—
27C2×#217C2212yes—
28C2×#217C2212yes—
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic, complemented
31C2×#217C2212yescomplemented
32#939—24no—
33#935—24no—
34C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
35C2×(C2×D3)⋊C4C2212yescomplemented
36C2×C4⋊Dic3C2212yescomplemented
37C2×Dic3⋊C4C2211yescharacteristic, complemented
38C2×Dic3⋊C4C2212yescomplemented
39C2×C4×Dic3C2212yescomplemented
40#886—24no—
41C3⋊#95C2212yescomplemented
42C2×#123—31no—
43C2×#123—31no—
44C2×#121—31no—
45C2×#120—31no—
46C2×#120D311yescharacteristic, complemented
47C22×C22⋊C4—31no—
48#399—32no—
49#392—32no—
50#390—32no—
51#390—32no—
52C4×C22⋊C4—31no—
53C24×C3—22no—
54C24×C3—24no—
55C24×C3—22no—
56C24×C3—22no—
57C24×C3—42no—
58C24×C3C2311yescharacteristic
59C24×C3C2312yes—
60C24×C3C2311yescharacteristic
61C23×D3C2311yescharacteristic
62C2×C3×D4—22no—
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4—24no—
66C2×C3×D4—24no—
67C2×C3×D4—42no—
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2312yes—
70C2×#67—22no—
71C2×#67—22no—
72C2×#67—22no—
73C2×#67—24no—
74C2×#67—22no—
75C2×#67—24no—
76C2×#67—42no—
77C2×#67—42no—
78C22×Dic3—22no—
79C22×Dic3—22no—
80C22×Dic3—22no—
81C22×Dic3—42no—
82C22×Dic3D414yescomplemented
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2312yes—
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2312yes—
87C22×Dic3C2311yescharacteristic
88C3×C22⋊C4—24no—
89C3×C22⋊C4D414yescomplemented
90#217—24no—
91#217—22no—
92#217—24no—
93#217—24no—
94#217—24no—
95#217D414yescomplemented
96(C2×D3)⋊C4—24no—
97(C2×D3)⋊C4—22no—
98C4⋊Dic3—24no—
99Dic3⋊C4—24no—
100Dic3⋊C4—22no—
101C4×Dic3—42no—
102C25C2×D311yescharacteristic
103C22×D4—31no—
104C22×D4—32no—
105C22×D4—31no—
106C22×D4C2×D311yescharacteristic
107C22×D4C2×D312yes—
108C23×C4—31no—
109C23×C4—31no—
110#123—64no—
111#123—64no—
112#121—64no—
113#120—24no—
114#120—64no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—32no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—32no—
124C2×C22⋊C4—31no—
125C2×C22⋊C4—32no—
126C2×C22⋊C4—64no—
127C2×C22⋊C4C2×D312yes—
128C2×C22⋊C4C2×D311yescharacteristic
129C2×C42—32no—
130#95—32no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—28no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3—48no—
146C23×C3—48no—
147C23×C3—44no—
148C23×C3—44no—
149C23×C3C2×D412yes—
150C23×C3C2×D412yes—
151C23×C3C2×D412yes—
152C23×C3#4114yes—
153C23×C3C2411yescharacteristic
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3—22no—
158C22×D3—22no—
159C3×D4—48no—
160C3×D4—44no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4C2×D414yes—
167#67—48no—
168#67—44no—
169#67—44no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3—22no—
173C2×Dic3—22no—
174C2×Dic3—22no—
175C2×Dic3—21no—
176C2×Dic3—21no—
177C2×Dic3—22no—
178C2×Dic3—22no—
179C2×Dic3—21no—
180C2×Dic3—44no—
181C2×Dic3—44no—
182C2×Dic3C2×D414yes—
183C2×Dic3C2×D412yes—
184C2×Dic3C2×D412yes—
185C24—22no—
186C24—24no—
187C24—22no—
188C24—22no—
189C24—31no—
190C24—42no—
191C24C22×D311yescharacteristic
192C24C22×D312yes—
193C24C22×D311yescharacteristic
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—42no—
200C2×D4—64no—
201C2×D4—62no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—64no—
205C2×D4—62no—
206C2×D4—122no—
207C2×D4—122no—
208C22×C4—31no—
209C22×C4—32no—
210C22×C4—31no—
211C22×C4—34no—
212C22×C4—32no—
213C22×C4—31no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4—62no—
217C22×C4—122no—
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D312yes—
220C4⋊C4—62no—
221C4⋊C4—64no—
222C4⋊C4—64no—
223C22⋊C4—24no—
224C22⋊C4—34no—
225C22⋊C4—62no—
226C22⋊C4—64no—
227C22⋊C4—64no—
228C22⋊C4—64no—
229C22⋊C4—62no—
230C22⋊C4—64no—
231C22⋊C4—64no—
232C22⋊C4#6714yescomplemented
233C42—122no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—24no—
241C22×C3—416no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3#12714yes—
252C22×C3C22×D411yescharacteristic
253C22×C3C22×D411yescharacteristic
254C22×C3C22×D411yescharacteristic
255C22×C3C2×#4112yes—
256C22×C3H(5,𝔽2)12yes—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—42no—
261C2×D3—42no—
262C3×C4—24no—
263C3×C4—41no—
264Dic3—22no—
265Dic3—22no—
266Dic3—24no—
267Dic3—42no—
268Dic3—41no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—22no—
282C23—21no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—44no—
287C23—61no—
288C23—61no—
289C23—62no—
290C23—62no—
291C23—61no—
292C23D3×D412yes—
293C23D3×D412yes—
294C23#23714yes—
295C23C2×#6712yes—
296C23C23×D311yescharacteristic
297D4—48no—
298D4—44no—
299D4—128no—
300D4—124no—
301D4—124no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—34no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4—61no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—124no—
321C2×C4—124no—
322C2×C4C2×#6714yes—
323C2×C3—24no—
324C2×C3—44no—
325C2×C3—81no—
326C2×C3#52511yescharacteristic
327C2×C3C2×#12711yescharacteristic
328C2×C3#53411yescharacteristic
329C2×C3#54612yes—
330C2×C3#54612yes—
331D3—81no—
332C22—24no—
333C22—22no—
334C22—22no—
335C22—24no—
336C22—24no—
337C22—24no—
338C22—24no—
339C22—416no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—121no—
350C22—122no—
351C22—121no—
352C22—121no—
353C22—122no—
354C22#94014yes—
355C22C2×D3×D411yescharacteristic
356C22C2×D3×D411yescharacteristic
357C22C2×#23712yes—
358C22#100412yes—
359C22C22×#6711yescharacteristic
360C4—24no—
361C4—41no—
362C4—62no—
363C4—62no—
364C4—64no—
365C4—122no—
366C4—121no—
367C3#253911yescharacteristic, complemented, Sylow
368C2—24no—
369C2—44no—
370C2—81no—
371C2—241no—
372C2#558611yescharacteristic
373C2#558911yescharacteristic
374C2#559112yes—
375C2C2×#94011yescharacteristic
376C2#580612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2539
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×C3×#120C2semidirect
2. C2×#953C2semidirect
3. C22×#217C2semidirect
4. C2×#939C2semidirect
5. C2×#935C2semidirect
6. C2×#886C2semidirect
7. #5223C2semidirect
8. #5221C2semidirect
9. #4957C2semidirect
10. #4949C2semidirect
11. Dic3×C22⋊C4C2semidirect
18. C22×#67C22semidirect
30. C2×#217C22semidirect
31. C2×#217C22semidirect
34. C2×(C2×D3)⋊C4C22semidirect
35. C2×(C2×D3)⋊C4C22semidirect
36. C2×C4⋊Dic3C22semidirect
37. C2×Dic3⋊C4C22semidirect
38. C2×Dic3⋊C4C22semidirect
39. C2×C4×Dic3C22semidirect
41. C3⋊#95C22semidirect
46. C2×#120D3semidirect
82. C22×Dic3D4semidirect
89. C3×C22⋊C4D4semidirect
95. #217D4semidirect
232. C22⋊C4#67semidirect
367. C3#2539semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C2×#9532
C22×#2172
C2×#9392
C2×#9352
C2×#8862
#52232
#52212
#49572
#49492
Dic3×C22⋊C42
#25393