Finite Group Databasebeta

← #82108

Subgroups — #82108

Order 384.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#953C211yesmaximal, characteristic
3C22×#217C211yesmaximal, characteristic, complemented
4C2×#886C211yesmaximal, characteristic, complemented
5#5213C212yesmaximal, complemented
6#5212C212yesmaximal, complemented
7#4961C212yesmaximal, complemented
8#4958C211yesmaximal, characteristic, complemented
9#4953C212yesmaximal, complemented
10#4950C211yesmaximal, characteristic, complemented
11#4943C211yesmaximal, characteristic, complemented
12#2412—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C23×C3×C4C2212yes—
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C2×C3×C22⋊C4—24no—
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×C3×C22⋊C4C2211yescharacteristic
24#953—24no—
25C2×#217—24no—
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2212yes—
29C2×#217C2211yescharacteristic
30C2×#217C2212yes—
31C2×(C2×D3)⋊C4C2212yescomplemented
32C2×(C2×D3)⋊C4C2212yescomplemented
33C2×C4⋊Dic3C2212yescomplemented
34C2×C4⋊Dic3C2212yescomplemented
35#886—24no—
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120—31no—
42C22×C22⋊C4—31no—
43C22×C22⋊C4D311yescharacteristic, complemented
44#397—32no—
45#394—32no—
46#386—32no—
47#386—31no—
48#385—31no—
49#381—31no—
50#380—32no—
51C24×C3—22no—
52C24×C3—22no—
53C24×C3—24no—
54C24×C3—42no—
55C24×C3D414yes—
56C24×C3C2311yescharacteristic
57C24×C3C2311yescharacteristic
58C24×C3C2311yescharacteristic
59C23×D3C2311yescharacteristic
60C22×C3×C4—24no—
61C22×C3×C4—24no—
62C22×C3×C4—42no—
63C22×C3×C4C2312yes—
64C22×C3×C4C2312yes—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—22no—
70C2×#67—24no—
71C2×#67—42no—
72C22×Dic3—22no—
73C22×Dic3—22no—
74C22×Dic3—22no—
75C22×Dic3—22no—
76C22×Dic3—42no—
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2312yes—
81C22×Dic3C2312yes—
82C3×C22⋊C4—22no—
83C3×C22⋊C4—24no—
84C3×C22⋊C4D414yescomplemented
85#217—24no—
86#217—24no—
87#217—22no—
88#217—22no—
89#217—22no—
90(C2×D3)⋊C4—24no—
91(C2×D3)⋊C4—42no—
92C4⋊Dic3—24no—
93C4⋊Dic3—42no—
94C25C2×D311yescharacteristic
95C22×D4—31no—
96C22×D4—31no—
97C22×D4—31no—
98C23×C4—31no—
99C23×C4—31no—
100C23×C4C2×D312yes—
101#123—64no—
102#120—64no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—32no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—64no—
114C2×C22⋊C4C2×D311yescharacteristic
115C2×C22⋊C4C2×D312yes—
116C2×C22⋊C4C2×D311yescharacteristic
117#95—32no—
118#95—32no—
119#95—32no—
120#95—32no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—28no—
127C23×C3—28no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—44no—
133C23×C3—48no—
134C23×C3—48no—
135C23×C3C2×D412yes—
136C23×C3C2×D412yes—
137C23×C3C2×D412yes—
138C23×C3C2×D412yes—
139C23×C3#4112yes—
140C23×C3#4114yes—
141C23×C3C2411yescharacteristic
142C22×D3—21no—
143C22×D3—21no—
144C22×D3—21no—
145C22×D3—22no—
146C22×D3—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—48no—
153C2×C3×C4C2×D414yes—
154#67—44no—
155#67—44no—
156#67—44no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—22no—
165C2×Dic3—21no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—44no—
171C2×Dic3—44no—
172C24—22no—
173C24—24no—
174C24—22no—
175C24—31no—
176C24—42no—
177C24D1214yes—
178C24C22×D311yescharacteristic
179C24C22×D311yescharacteristic
180C24C22×D311yescharacteristic
181C2×D4—62no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—64no—
186C2×D4—62no—
187C2×D4—122no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—31no—
191C22×C4—32no—
192C22×C4—32no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—42no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—122no—
201C22×C4C22×D312yes—
202C22×C4C22×D312yes—
203C4⋊C4—64no—
204C4⋊C4—122no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—64no—
208C22⋊C4—64no—
209C22⋊C4—62no—
210C22⋊C4—62no—
211C22⋊C4—62no—
212C22⋊C4—64no—
213C22⋊C4—122no—
214C22⋊C4#6714yescomplemented
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—24no—
220C22×C3—24no—
221C22×C3—24no—
222C22×C3—24no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—416no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3#12114yes—
230C22×C3#12314yescomplemented
231C22×C3C22×D411yescharacteristic
232C22×C3C22×D411yescharacteristic
233C22×C3C2×#4111yescharacteristic
234C22×C3C2×#4112yes—
235C22×C3H(5,𝔽2)12yes—
236C2×D3—41no—
237C2×D3—41no—
238C2×D3—41no—
239C2×D3—42no—
240C2×D3—42no—
241C3×C4—24no—
242C3×C4—42no—
243Dic3—42no—
244Dic3—41no—
245Dic3—41no—
246Dic3—42no—
247Dic3—41no—
248C23—21no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—28no—
254C23—28no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—44no—
260C23—48no—
261C23—48no—
262C23—61no—
263C23—61no—
264C23—62no—
265C23—62no—
266C23—61no—
267C23C2×D1212yes—
268C23C2×D1212yes—
269C23C2×D1212yes—
270C23#23512yes—
271C23#23714yes—
272C23C2×#6712yes—
273C23C23×D311yescharacteristic
274D4—124no—
275D4—124no—
276D4—124no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—48no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—62no—
289C2×C4—61no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—61no—
296C2×C4—124no—
297C2×C4—124no—
298C2×C4C2×#6714yes—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—42no—
302C2×C3C2×#12111yescharacteristic
303C2×C3C2×#12311yescharacteristic
304C2×C3#53411yescharacteristic
305C2×C3#54612yes—
306C2×C3#55112yes—
307D3—81no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—22no—
314C22—24no—
315C22—24no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—416no—
320C22—42no—
321C22—42no—
322C22—121no—
323C22—121no—
324C22—122no—
325C22—121no—
326C22—122no—
327C22#88614yescomplemented
328C22#93014yes—
329C22C22×D1211yescharacteristic
330C22C2×#23511yescharacteristic
331C22C2×#23712yes—
332C22#100412yes—
333C22C22×#6711yescharacteristic
334C4—24no—
335C4—42no—
336C4—121no—
337C4—122no—
338C4—121no—
339C4—122no—
340C4—121no—
341C3#241211yescharacteristic, complemented, Sylow
342C2—24no—
343C2—24no—
344C2—42no—
345C2—241no—
346C2C2×#88611yescharacteristic
347C2#549311yescharacteristic
348C2#549512yes—
349C2C2×#93011yescharacteristic
350C2#580612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC2×#953C2
hypercenterC22×C22⋊C4D3
hypocenterC3#2412
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊C4C2semidirect
3. C22×#217C2semidirect
4. C2×#886C2semidirect
5. #5213C2semidirect
6. #5212C2semidirect
7. #4961C2semidirect
8. #4958C2semidirect
9. #4953C2semidirect
10. #4950C2semidirect
11. #4943C2semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×C4⋊Dic3C22semidirect
34. C2×C4⋊Dic3C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
43. C22×C22⋊C4D3semidirect
84. C3×C22⋊C4D4semidirect
214. C22⋊C4#67semidirect
230. C22×C3#123semidirect
327. C22#886semidirect
341. C3#2412semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
C2×#9532
C22×#2172
C2×#8862
#52132
#52122
#49612
#49582
#49532
#49502
#49432
#24123