Finite Group Databasebeta

← C2×#74116 #11001073

Subgroups — C2×#74116 #11001073

Order 768.

3354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#929C211yesmaximal, characteristic
2C23×C4×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#381C211yesmaximal, characteristic, complemented
4C2×#4958C212yesmaximal
5C2×#4950C212yesmaximal, complemented
6#74116C2116yesmaximal, direct product factor
7C2×#4670C214yesmaximal, complemented
8C2×#4653C214yesmaximal, complemented
9C2×#2853—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C23×#67C2211yescharacteristic
12C24×Dic3C2212yes—
13C22×C3×C22⋊C4C2212yes—
14C22×#217C2211yescharacteristic
15C2×#929—28no—
16C2×#929C2218yes—
17C22×(C2×D3)⋊C4C2214yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2214yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3—28no—
22C22×C4×Dic3C2214yes—
23C22×C4×Dic3C2218yescomplemented
24C3×#381C2218yescomplemented
25C2×C3×#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28#4958C22116yes—
29#4950C22116yescomplemented
30#4670C22132yescomplemented
31#4653C22132yescomplemented
32C22×#123—31no—
33C23×C42—31no—
34#2853—316no—
35C2×#388—34no—
36C2×#387—34no—
37C2×#386—32no—
38C2×#385—32no—
39C2×#381D311yescharacteristic, complemented
40C25×C3C2311yescharacteristic
41C24×D3C2311yescharacteristic
42C23×C3×C4—28no—
43C23×C3×C4D418yes—
44C23×C3×C4C2314yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2314yes—
47C22×#67—24no—
48C22×#67—28no—
49C22×#67C2314yes—
50C23×Dic3—28no—
51C23×Dic3—216no—
52C23×Dic3C2318yes—
53C23×Dic3C2311yescharacteristic
54C23×Dic3C2312yes—
55C23×Dic3C2314yes—
56C2×C3×C22⋊C4—28no—
57C2×C3×C22⋊C4C2318yes—
58C2×#217—24no—
59C2×#217C2314yes—
60#929—232no—
61C2×(C2×D3)⋊C4—28no—
62C2×(C2×D3)⋊C4—216no—
63C2×(C2×D3)⋊C4C23116yes—
64C2×(C2×D3)⋊C4C2318yes—
65C2×C4⋊Dic3—216no—
66C2×C4⋊Dic3C23116yes—
67C2×Dic3⋊C4—28no—
68C2×Dic3⋊C4C2318yes—
69C2×C4×Dic3—216no—
70C2×C4×Dic3—232no—
71C2×C4×Dic3C23116yes—
72C3×#95C23116yes—
73C3⋊#95C23116yes—
74C3⋊#95C23116yes—
75C23×D4—31no—
76C24×C4—32no—
77C24×C4C2×D311yescharacteristic
78C2×#123—38no—
79C2×#123—68no—
80C22×C4⋊C4—32no—
81C22×C4⋊C4—34no—
82C22×C22⋊C4—31no—
83C22×C22⋊C4—32no—
84C22×C22⋊C4—34no—
85C22×C22⋊C4C2×D312yes—
86C22×C42—38no—
87C22×C42—34no—
88C22×C42—68no—
89#388—332no—
90#387—332no—
91#386—316no—
92#385—316no—
93#381C2×D318yescomplemented
94C2×#95—34no—
95C2×#95—34no—
96C2×#95C2×D314yes—
97C24×C3—28no—
98C24×C3C2×D414yes—
99C24×C3#4118yes—
100C24×C3C2412yes—
101C24×C3C2411yescharacteristic
102C23×D3—22no—
103C23×D3—28no—
104C23×D3—24no—
105C23×D3C2412yes—
106C22×C3×C4—232no—
107C22×C3×C4—216no—
108C22×C3×C4—28no—
109C22×C3×C4—28no—
110C22×C3×C4—28no—
111C22×C3×C4—28no—
112C22×C3×C4C2×D418yes—
113C22×C3×C4C2×D4116yes—
114C22×C3×C4C2418yes—
115C22×C3×C4C2414yes—
116C2×#67—28no—
117C2×#67—216no—
118C2×#67—416no—
119C22×Dic3—264no—
120C22×Dic3—216no—
121C22×Dic3—28no—
122C22×Dic3—28no—
123C22×Dic3—232no—
124C22×Dic3—22no—
125C22×Dic3—24no—
126C22×Dic3—28no—
127C22×Dic3—24no—
128C22×Dic3#41116yes—
129C22×Dic3C2412yes—
130C22×Dic3C2414yes—
131C22×Dic3C2418yes—
132C3×C22⋊C4—216no—
133#217—28no—
134(C2×D3)⋊C4—216no—
135(C2×D3)⋊C4—232no—
136C4⋊Dic3—232no—
137Dic3⋊C4—216no—
138C4×Dic3—232no—
139C25—31no—
140C25C22×D311yescharacteristic
141C22×D4—34no—
142C22×D4—64no—
143C22×D4—68no—
144C23×C4—28no—
145C23×C4—34no—
146C23×C4—38no—
147C23×C4—32no—
148C23×C4—31no—
149C23×C4—616no—
150C23×C4—68no—
151C23×C4D1218yes—
152C23×C4C22×D314yes—
153C23×C4C22×D314yes—
154C23×C4C22×D312yes—
155#123—632no—
156C2×C4⋊C4—38no—
157C2×C4⋊C4—316no—
158C2×C4⋊C4—616no—
159C2×C4⋊C4—68no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—34no—
162C2×C22⋊C4—38no—
163C2×C22⋊C4—316no—
164C2×C22⋊C4—616no—
165C2×C22⋊C4—64no—
166C2×C22⋊C4—68no—
167C2×C22⋊C4C22×D318yes—
168C2×C42—316no—
169C2×C42—616no—
170C2×C42—632no—
171#95—316no—
172#95—316no—
173#95C22×D3116yes—
174C23×C3—216no—
175C23×C3—232no—
176C23×C3—28no—
177C23×C3#123116yes—
178C23×C3C22×D414yes—
179C23×C3C22×D412yes—
180C23×C3C2×#4118yes—
181C23×C3C2×#4114yes—
182C23×C3C2×#4118yes—
183C23×C3C2511yescharacteristic
184C22×D3—22no—
185C22×D3—28no—
186C22×D3—24no—
187C22×D3—44no—
188C22×D3—48no—
189C22×D3—416no—
190C2×C3×C4—232no—
191C2×C3×C4—216no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—28no—
195C2×C3×C4—28no—
196C2×C3×C4—416no—
197C2×C3×C4C22×D418yes—
198#67—416no—
199C2×Dic3—264no—
200C2×Dic3—232no—
201C2×Dic3—28no—
202C2×Dic3—28no—
203C2×Dic3—22no—
204C2×Dic3—24no—
205C2×Dic3—28no—
206C2×Dic3—24no—
207C2×Dic3—44no—
208C2×Dic3—48no—
209C2×Dic3C2×#41116yes—
210C24—28no—
211C24—32no—
212C24—62no—
213C24—64no—
214C24—68no—
215C24C2×D1214yes—
216C24#23718yes—
217C24C23×D312yes—
218C24C23×D311yescharacteristic
219C2×D4—68no—
220C2×D4—616no—
221C2×D4—1216no—
222C22×C4—232no—
223C22×C4—216no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—316no—
229C22×C4—38no—
230C22×C4—32no—
231C22×C4—34no—
232C22×C4—632no—
233C22×C4—664no—
234C22×C4—616no—
235C22×C4—68no—
236C22×C4—68no—
237C22×C4—64no—
238C22×C4—62no—
239C22×C4—64no—
240C22×C4—68no—
241C22×C4C2×D1218yes—
242C22×C4C2×D12116yes—
243C22×C4C23×D318yes—
244C22×C4C23×D314yes—
245C4⋊C4—632no—
246C4⋊C4—616no—
247C22⋊C4—216no—
248C22⋊C4—632no—
249C22⋊C4—68no—
250C22⋊C4—616no—
251C42—632no—
252C22×C3—232no—
253C22×C3—28no—
254C22×C3—216no—
255C22×C3C2×#12318yes—
256C22×C3C2×#12314yes—
257C22×C3#52518yes—
258C22×C3#529116yes—
259C22×C3C23×D411yescharacteristic
260C22×C3C22×#4112yes—
261C22×C3C22×#4114yes—
262C2×D3—42no—
263C2×D3—44no—
264C2×D3—48no—
265C2×D3—88no—
266C3×C4—28no—
267C3×C4—48no—
268Dic3—216no—
269Dic3—42no—
270Dic3—44no—
271C23—232no—
272C23—28no—
273C23—216no—
274C23—62no—
275C23—64no—
276C23—68no—
277C23—1216no—
278C23—124no—
279C23—128no—
280C23#886116yes—
281C23C22×D1214yes—
282C23C22×D1212yes—
283C23C2×#23718yes—
284C23C2×#23714yes—
285C23D3×#4118yes—
286C23C24×D311yescharacteristic
287D4—1216no—
288C2×C4—28no—
289C2×C4—232no—
290C2×C4—216no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—316no—
295C2×C4—416no—
296C2×C4—664no—
297C2×C4—68no—
298C2×C4—68no—
299C2×C4—632no—
300C2×C4—64no—
301C2×C4—62no—
302C2×C4—64no—
303C2×C4—68no—
304C2×C4—124no—
305C2×C4—128no—
306C2×C4C22×D1218yes—
307C2×C3—28no—
308C2×C3#285318yescomplemented
309C2×C3C22×#12311yescharacteristic
310C2×C3C2×#52512yes—
311C2×C3C2×#52914yes—
312D3—82no—
313C22—232no—
314C22—28no—
315C22—216no—
316C22—122no—
317C22—124no—
318C22—128no—
319C22—248no—
320C22C2×#88618yes—
321C22C2×#88614yes—
322C22#552418yes—
323C22#5649116yes—
324C22C23×D1211yescharacteristic
325C22C22×#23712yes—
326C22C2×D3×#4114yes—
327C4—28no—
328C4—48no—
329C4—616no—
330C4—122no—
331C4—124no—
332C3C2×#285311yescharacteristic, complemented, Sylow
333C2—28no—
334C2—242no—
335C2#7411618yesdirect product factor
336C2C22×#88611yescharacteristic
337C2C2×#552412yes—
338C2C2×#564914yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#381C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#381D3
hypocenterC3C2×#2853
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic3C2semidirect
3. C2×C3×#381C2semidirect
5. C2×#4950C2semidirect
6. #74116C2direct (N × H)
7. C2×#4670C2semidirect
8. C2×#4653C2semidirect
23. C22×C4×Dic3C22semidirect
24. C3×#381C22semidirect
29. #4950C22semidirect
30. #4670C22semidirect
31. #4653C22semidirect
39. C2×#381D3semidirect
93. #381C2×D3semidirect
308. C2×C3#2853semidirect
332. C3C2×#2853semidirect
335. C2#74116direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9292
C23×C4×Dic32
C2×C3×#3812
C2×#49582
C2×#49502
#741162
C2×#46702
C2×#46532
C2×#28533