Finite Group Databasebeta

← #82848

Subgroups — #82848

Order 384.

914 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#235C211yesmaximal, characteristic, complemented
2C2×C3×#123C211yesmaximal, characteristic, complemented
3C2×#939C212yesmaximal, complemented
4C2×C3⋊#95C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4989C212yesmaximal, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4958C212yesmaximal, complemented
10#4956C212yesmaximal, complemented
11#4947C211yesmaximal, characteristic, complemented
12#2430—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2212yes—
16C22×#67C2212yes—
17C23×Dic3C2212yes—
18C2×#235—24no—
19C2×#235—28no—
20C22×D12C2211yescharacteristic
21C22×C4×D3C2212yes—
22C22×Dic6C2211yescharacteristic
23C3×#123—24no—
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×#217C2212yes—
30#939—28no—
31C2×(C2×D3)⋊C4C2212yes—
32C2×(C2×D3)⋊C4C2212yes—
33C2×(C2×D3)⋊C4C2212yes—
34C2×Dic3⋊C4C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×#123D311yescharacteristic, complemented
42C2×#121—32no—
43#395—31no—
44#394—31no—
45#393—31no—
46#392—31no—
47#386—32no—
48#386—32no—
49#385—32no—
50C2×#95—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C23×D3C2312yes—
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C22×C3×C4—24no—
57C22×C3×C4D414yescomplemented
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—24no—
67C2×#67—42no—
68C2×#67D418yescomplemented
69C22×Dic3—24no—
70C22×Dic3—24no—
71C22×Dic3—42no—
72C22×Dic3C2312yes—
73C22×Dic3C2312yes—
74C22×Dic3C2312yes—
75#235—232no—
76C2×D12—24no—
77C2×D12—22no—
78C2×C4×D3—24no—
79C2×C4×D3—24no—
80C2×C4×D3—24no—
81C2×Dic6—22no—
82C2×Dic6—24no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—22no—
85C3×C22⋊C4—22no—
86C3×C22⋊C4—22no—
87C3×C22⋊C4—22no—
88#217—24no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—24no—
91(C2×D3)⋊C4—42no—
92Dic3⋊C4—24no—
93C2×#41—68no—
94C2×#41—64no—
95C22×Q8—31no—
96C22×D4—32no—
97C22×D4—32no—
98C22×D4—31no—
99C22×D4C2×D311yescharacteristic
100C23×C4—32no—
101C23×C4—32no—
102C23×C4C2×D311yescharacteristic
103#123—24no—
104#121—68no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4C2×D311yescharacteristic
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4C2×D311yescharacteristic
113C2×C22⋊C4C2×D311yescharacteristic
114C2×C22⋊C4C2×D311yescharacteristic
115#95—32no—
116#95—31no—
117#95—31no—
118#95—32no—
119#95—32no—
120C23×C3—22no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—22no—
124C23×C3—21no—
125C23×C3—22no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3C2×D414yes—
129C23×C3C2×D412yes—
130C23×C3C2411yescharacteristic
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—22no—
137C22×D3C2×D414yes—
138C3×D4—44no—
139C2×C3×C4—28no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4C2×D412yes—
150C2×C3×C4C2×D412yes—
151#67—216no—
152#67—48no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—48no—
161C2×Dic3C2×D414yes—
162C2×Dic3#4114yescomplemented
163D12—28no—
164C4×D3—216no—
165Dic6—28no—
166C24—32no—
167C24C22×D311yescharacteristic
168C24C22×D311yescharacteristic
169#41—632no—
170C2×Q8—64no—
171C2×Q8—62no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—38no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—64no—
180C2×D4—62no—
181C2×D4—122no—
182C22×C4—24no—
183C22×C4—32no—
184C22×C4—32no—
185C22×C4—32no—
186C22×C4—64no—
187C22×C4—64no—
188C22×C4—64no—
189C22×C4—64no—
190C22×C4—64no—
191C22×C4—122no—
192C22×C4#6714yescomplemented
193C22×C4C22×D311yescharacteristic
194C22×C4C22×D311yescharacteristic
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D311yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—64no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—64no—
205C22⋊C4—64no—
206C22⋊C4—64no—
207C22⋊C4—122no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3#12014yes—
218C22×C3C22×D412yes—
219C22×C3C22×D411yescharacteristic
220C22×C3C2×#4112yes—
221C22×C3H(5,𝔽2)11yescharacteristic
222C22×C32−1+411yescharacteristic
223C2×D3—24no—
224C2×D3—24no—
225C2×D3—24no—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C3×C4—22no—
231C3×C4—22no—
232C3×C4—41no—
233C3×C4—41no—
234C3×C4—41no—
235Dic3—24no—
236Dic3—24no—
237Dic3—42no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—34no—
247C23—62no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23D3×D414yes—
254C23C2×#6712yes—
255C23C23×D311yescharacteristic
256Q8—68no—
257D4—44no—
258D4—68no—
259D4—616no—
260D4—128no—
261C2×C4—28no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—34no—
272C2×C4—34no—
273C2×C4—616no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—128no—
282C2×C4C2×#6712yes—
283C2×C4C2×#6712yes—
284C2×C3—24no—
285C2×C3—81no—
286C2×C3C2×#12011yescharacteristic
287C2×C3#53211yescharacteristic
288C2×C3#53711yescharacteristic
289C2×C3#54612yes—
290C2×C3#54712yes—
291D3—44no—
292C22—24no—
293C22—24no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—64no—
302C22—64no—
303C22—64no—
304C22—122no—
305C22—122no—
306C22—122no—
307C22—122no—
308C22#93714yes—
309C22C2×D3×D412yes—
310C22D3×#4112yes—
311C22#100911yescharacteristic
312C22#101011yescharacteristic
313C22C22×#6711yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—64no—
320C4—64no—
321C4—122no—
322C3#243011yescharacteristic, complemented, Sylow
323C2—24no—
324C2—81no—
325C2—124no—
326C2#565412yes—
327C2#565512yes—
328C2C2×#93711yescharacteristic
329C2#583211yescharacteristic
330C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
hypercenterC2×#123D3
hypocenterC3#2430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×C3×#123C2semidirect
3. C2×#939C2semidirect
4. C2×C3⋊#95C2semidirect
5. #4991C2semidirect
6. #4989C2semidirect
7. #4979C2semidirect
8. #4960C2semidirect
9. #4958C2semidirect
10. #4956C2semidirect
11. #4947C2semidirect
41. C2×#123D3semidirect
57. C22×C3×C4D4semidirect
68. C2×#67D4semidirect
162. C2×Dic3#41semidirect
192. C22×C4#67semidirect
322. C3#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×C3×#1232
C2×#9392
C2×C3⋊#952
#49912
#49892
#49792
#49602
#49582
#49562
#49472
#24303