Finite Group Databasebeta

← #82584

Subgroups — #82584

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
1C2×C3×#120C211yesmaximal, characteristic
2C22×#217C211yesmaximal, characteristic, complemented
3C22×#217C211yesmaximal, characteristic, complemented
4C2×#935C211yesmaximal, characteristic, complemented
5#5222C212yesmaximal, complemented
6#5220C211yesmaximal, characteristic, complemented
7#5220C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4946C211yesmaximal, characteristic, complemented
10#4943C211yesmaximal, characteristic, complemented
11#4942C212yesmaximal, complemented
12#2412—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2211yescharacteristic
16C22×C3×D4C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2212yes—
19C23×Dic3C2211yescharacteristic
20C3×#120—24no—
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×#217—24no—
25C2×#217—24no—
26C2×#217C2212yes—
27C2×#217C2211yescharacteristic
28C2×#217C2212yes—
29C2×#217C2212yes—
30C2×#217C2211yescharacteristic
31C2×#217C2212yescomplemented
32C2×#217C2212yescomplemented
33#935—24no—
34C2×Dic3⋊C4C2212yescomplemented
35C2×Dic3⋊C4C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120D311yescharacteristic
42C22×C22⋊C4—31no—
43C22×C22⋊C4—31no—
44#397—32no—
45#394—32no—
46#386—31no—
47#386—32no—
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
59C24×C3C2311yescharacteristic
60C2×C3×D4—22no—
61C2×C3×D4—22no—
62C2×C3×D4—22no—
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4—24no—
66C2×C3×D4—42no—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×Dic3—22no—
71C22×Dic3—22no—
72C22×Dic3—22no—
73C22×Dic3—24no—
74C22×Dic3—22no—
75C22×Dic3—24no—
76C22×Dic3—42no—
77C22×Dic3—42no—
78C22×Dic3C2312yes—
79C22×Dic3C2312yes—
80C22×Dic3C2312yes—
81C22×Dic3C2312yes—
82C3×C22⋊C4—22no—
83C3×C22⋊C4—22no—
84C3×C22⋊C4—22no—
85#217—22no—
86#217—24no—
87#217—24no—
88#217—24no—
89#217—24no—
90#217—42no—
91#217D414yescomplemented
92Dic3⋊C4—24no—
93Dic3⋊C4—42no—
94C25C2×D311yescharacteristic
95C22×D4C2×D311yescharacteristic
96C22×D4C2×D311yescharacteristic
97C22×D4C2×D311yescharacteristic
98C23×C4—31no—
99C23×C4—32no—
100C23×C4—31no—
101#123—64no—
102#120—24no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—32no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—64no—
113C2×C22⋊C4—64no—
114C2×C22⋊C4C2×D311yescharacteristic
115C2×C22⋊C4C2×D311yescharacteristic
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—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—44no—
138C23×C3—48no—
139C23×C3—48no—
140C23×C3C2×D412yes—
141C23×C3C2×D412yes—
142C23×C3C2×D412yes—
143C23×C3C2×D412yes—
144C23×C3#4112yes—
145C23×C3#4114yes—
146C23×C3C2411yescharacteristic
147C3×D4—44no—
148C3×D4—44no—
149C3×D4—44no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—48no—
169C2×Dic3—44no—
170C2×Dic3—44no—
171C2×Dic3C2×D414yes—
172C24—22no—
173C24—24no—
174C24—22no—
175C24—42no—
176C24#6714yes—
177C24C22×D311yescharacteristic
178C24C22×D311yescharacteristic
179C24C22×D311yescharacteristic
180C24C22×D311yescharacteristic
181C2×D4—22no—
182C2×D4—24no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—42no—
188C22×C4—32no—
189C22×C4—32no—
190C22×C4—32no—
191C22×C4—32no—
192C22×C4—64no—
193C22×C4—62no—
194C22×C4—64no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—122no—
199C22×C4—122no—
200C22×C4C22×D311yescharacteristic
201C22×C4C22×D311yescharacteristic
202C22×C4C22×D311yescharacteristic
203C4⋊C4—64no—
204C4⋊C4—122no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—34no—
209C22⋊C4—64no—
210C22⋊C4—64no—
211C22⋊C4—62no—
212C22⋊C4—64no—
213C22⋊C4—64no—
214C22⋊C4—122no—
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—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3#12114yes—
235C22×C3#12314yescomplemented
236C22×C3C22×D411yescharacteristic
237C22×C3C22×D411yescharacteristic
238C22×C3C2×#4111yescharacteristic
239C22×C3C2×#4112yes—
240C22×C3H(5,𝔽2)12yes—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—41no—
244Dic3—24no—
245Dic3—42no—
246Dic3—42no—
247Dic3—42no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—28no—
254C23—28no—
255C23—21no—
256C23—22no—
257C23—21no—
258C23—22no—
259C23—21no—
260C23—22no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—44no—
265C23—48no—
266C23—48no—
267C23D3×D412yes—
268C23#23714yes—
269C23#23712yes—
270C23C2×#6712yes—
271C23C2×#6712yes—
272C23C2×#6712yes—
273C23C23×D311yescharacteristic
274D4—44no—
275D4—44no—
276D4—44no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—34no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—128no—
297C2×C4—124no—
298C2×C4—124no—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—42no—
302C2×C3—81no—
303C2×C3C2×#12111yescharacteristic
304C2×C3C2×#12311yescharacteristic
305C2×C3#53411yescharacteristic
306C2×C3#54612yes—
307C2×C3#55112yes—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—416no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22#93514yescomplemented
328C22#93914yes—
329C22C2×D3×D411yescharacteristic
330C22C2×#23712yes—
331C22C2×#23711yescharacteristic
332C22#100412yes—
333C22C22×#6711yescharacteristic
334C4—41no—
335C4—41no—
336C4—41no—
337C4—64no—
338C4—122no—
339C4—122no—
340C4—122no—
341C3#241211yescharacteristic, complemented, Sylow
342C2—24no—
343C2—24no—
344C2—42no—
345C2—81no—
346C2#558712yes—
347C2#559112yes—
348C2C2×#93511yescharacteristic
349C2C2×#93911yescharacteristic
350C2#580411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2412
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C22×#217C2semidirect
3. C22×#217C2semidirect
4. C2×#935C2semidirect
5. #5222C2semidirect
6. #5220C2semidirect
7. #5220C2semidirect
8. #4952C2semidirect
9. #4946C2semidirect
10. #4943C2semidirect
11. #4942C2semidirect
31. C2×#217C22semidirect
32. C2×#217C22semidirect
34. C2×Dic3⋊C4C22semidirect
35. C2×Dic3⋊C4C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
91. #217D4semidirect
235. C22×C3#123semidirect
327. C22#935semidirect
341. C3#2412semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C22×#2172
C2×#9352
#52222
#52202
#49522
#49462
#49432
#49422
#24123