Finite Group Databasebeta

← C2×#81739 #11664460

Subgroups — C2×#81739 #11664460

Order 768.

2118 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5688C211yesmaximal, characteristic
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C2×#5082C211yesmaximal, characteristic, complemented
5C2×#5069C211yesmaximal, characteristic, complemented
6#81739C2116yesmaximal, direct product factor
7C2×#4802C214yesmaximal
8C2×#4792C214yesmaximal, complemented
9C2×#4703C211yesmaximal, characteristic, complemented
10C2×#4707C211yesmaximal, characteristic
11C2×#3139—31nomaximal, Sylow
12C22×D3×Q8C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×#944C2212yes—
16C22×ΓL(1,𝔽25)C2212yes—
17D3×C4⋊Q8—24no—
18D3×C4⋊Q8C2218yes—
19#5688—24no—
20#5688C2218yes—
21C2×#896—22no—
22C2×#892C2212yes—
23C2×D3×C4⋊C4—22no—
24C2×#888—22no—
25C2×#888C2212yes—
26C2×#887—22no—
27C2×C42×D3C2211yescharacteristic
28C2×#870C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C3×#491C2218yescomplemented
31C2×C3×#103C2214yes—
32C2×C3×#97C2211yescharacteristic
33C2×#835C2214yes—
34C2×#816C2214yes—
35C2×C8×Dic3C2212yes—
36#5082C2218yescomplemented
37#5069C2218yescomplemented
38C2×#810C2214yes—
39C2×#803C2211yescharacteristic
40#4802C22132yes—
41#4792C22132yescomplemented
42#4707C2218yes—
43#4703C2218yescomplemented
44C2×#533—31no—
45C22×C4⋊Q8—31no—
46#3139—316no—
47C2×#491—31no—
48C2×#491—31no—
49C2×#491—31no—
50C2×#491D311yescharacteristic, complemented
51C2×#487—34no—
52C2×#419—34no—
53C2×C4×M16—31no—
54C22×C3×Q8C2312yes—
55C2×D3×Q8—24no—
56C2×D3×Q8—216no—
57C2×D3×Q8C2318yes—
58C22×C4×D3—22no—
59C22×C4×D3D412yes—
60C22×C4×D3C2311yescharacteristic
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8C2314yes—
67C3×#125—22no—
68C3×#125C2314yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42C2311yescharacteristic
73#944—24no—
74#944C2318yes—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3C2312yes—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×C4×Dic3—22no—
82C2×C4×Dic3D412yes—
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85C2×ΓL(1,𝔽25)—28no—
86C2×ΓL(1,𝔽25)C2318yes—
87#896—216no—
88#892—24no—
89#892C2318yes—
90D3×C4⋊C4—216no—
91#888—24no—
92#888—216no—
93#888C2318yes—
94#887—216no—
95C42×D3D414yes—
96C42×D3C2314yes—
97#870—22no—
98#870C2314yes—
99#869—22no—
100#869C2314yes—
101C3×#103C23116yes—
102C3×#97C2314yes—
103#835C23116yes—
104#816C23116yes—
105C8×Dic3C2318yes—
106#810C23116yes—
107#803C2314yes—
108C23×Q8—32no—
109C22×M16—32no—
110#533—38no—
111#533—64no—
112C2×C4⋊Q8—32no—
113C2×C4⋊Q8—32no—
114C2×C4⋊Q8—38no—
115C2×C4⋊Q8—31no—
116C2×C4⋊Q8—62no—
117C2×C4⋊Q8—64no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×#125—31no—
120C2×#125C2×D311yescharacteristic
121C2×C22⋊Q8—62no—
122C2×C4×Q8—62no—
123C2×#117—32no—
124C22×C4⋊C4—62no—
125C22×C42—31no—
126#491—38no—
127#491—38no—
128#491—38no—
129#491C2×D318yescomplemented
130#487—332no—
131#419—332no—
132C2×#103—34no—
133C2×#103—34no—
134C2×#103—34no—
135C2×#103C2×D314yes—
136C4×M16—38no—
137C2×#97—31no—
138C2×#97C2×D311yescharacteristic
139C2×C4×C8—32no—
140C23×D3C2×D411yescharacteristic
141C2×C3×Q8—22no—
142C2×C3×Q8—28no—
143C2×C3×Q8C2414yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2411yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D412yes—
152D3×Q8—232no—
153D3×Q8—416no—
154C2×C4×D3—28no—
155C2×C4×D3—28no—
156C2×C4×D3—48no—
157C2×C4×D3C2×D418yes—
158C2×C4×D3C2×D412yes—
159C2×C4×D3#4118yes—
160C2×C4×D3C2412yes—
161C2×Dic6—28no—
162C2×Dic6—22no—
163C2×Dic6—28no—
164C2×Dic6—28no—
165C2×Dic6—48no—
166C2×Dic6—48no—
167C2×Dic6C2414yes—
168C2×C3×C8—24no—
169C2×C3×C8C2414yes—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4C2414yes—
174C3×C42C2×D412yes—
175C3×C42C2412yes—
176(C2×D3)⋊C4—28no—
177C4⋊Dic3—28no—
178C4⋊Dic3—22no—
179C4⋊Dic3C2414yes—
180Dic3⋊C4—28no—
181Dic3⋊C4—28no—
182Dic3⋊C4—216no—
183C4×Dic3—28no—
184C4×Dic3C2×D412yes—
185C4×Dic3C2×D418yes—
186C4×Dic3C2412yes—
187C2×C3⋊C8—24no—
188C2×C3⋊C8C2414yes—
189ΓL(1,𝔽25)—216no—
190C22×Q8—38no—
191C22×Q8—32no—
192C22×Q8—62no—
193C22×Q8—62no—
194C22×Q8—64no—
195C22×Q8—616no—
196C22×Q8C22×D312yes—
197C23×C4—31no—
198C23×C4—32no—
199C23×C4—62no—
200C2×M16—38no—
201C2×M16—68no—
202C22×C8—32no—
203C22×C8C22×D312yes—
204C4⋊Q8—22no—
205C4⋊Q8—38no—
206C4⋊Q8—38no—
207C4⋊Q8—34no—
208C4⋊Q8—616no—
209C4⋊Q8—64no—
210C4⋊Q8—64no—
211C4⋊Q8—62no—
212C4⋊Q8C22×D314yes—
213#125—22no—
214#125—34no—
215#125—62no—
216#125C22×D314yes—
217C22⋊Q8—616no—
218C4×Q8—616no—
219#117—38no—
220#117—64no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—616no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—62no—
231C2×C42—32no—
232C2×C42—31no—
233C2×C42—34no—
234C2×C42—34no—
235C2×C42—62no—
236C2×C42C22×D311yescharacteristic
237#103—316no—
238#103—316no—
239#103—316no—
240#103C22×D3116yes—
241#97—34no—
242#97C22×D314yes—
243C4×C8—38no—
244C23×C3C22×D411yescharacteristic
245C22×D3—24no—
246C22×D3#12414yes—
247C22×D3C22×D412yes—
248C3×Q8—28no—
249C3×Q8—44no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4C22×D414yes—
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4114yes—
258C2×C3×C4C2511yescharacteristic
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—44no—
262C2×Dic3—44no—
263C2×Dic3#12414yes—
264C2×Dic3#13718yes—
265C2×Dic3C22×D412yes—
266C2×Dic3C22×D414yes—
267C4×D3—216no—
268C4×D3—416no—
269C4×D3#12418yes—
270C4×D3C2×#4118yes—
271Dic6—28no—
272Dic6—44no—
273Dic6—416no—
274Dic6—416no—
275C3×C8—24no—
276C3⋊C8—24no—
277C24—31no—
278C2×Q8—22no—
279C2×Q8—28no—
280C2×Q8—34no—
281C2×Q8—68no—
282C2×Q8—68no—
283C2×Q8—632no—
284C2×Q8—62no—
285C2×Q8—68no—
286C2×Q8—128no—
287C2×Q8—128no—
288C2×Q8—1216no—
289C2×Q8C23×D314yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—31no—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4—38no—
296C22×C4—32no—
297C22×C4—38no—
298C22×C4—68no—
299C22×C4—68no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—128no—
303C22×C4D3×D412yes—
304C22×C4C23×D311yescharacteristic
305M16—616no—
306C2×C8—24no—
307C2×C8—34no—
308C2×C8—64no—
309C2×C8C23×D314yes—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—22no—
313C4⋊C4—34no—
314C4⋊C4—616no—
315C4⋊C4—68no—
316C4⋊C4—62no—
317C4⋊C4—68no—
318C4⋊C4—68no—
319C4⋊C4C23×D314yes—
320C22⋊C4—68no—
321C42—38no—
322C42—32no—
323C42—32no—
324C42—68no—
325C42D3×D412yes—
326C42C23×D312yes—
327C22×C3C2×#12412yes—
328C22×C3C2×#13714yes—
329C22×C3C23×D411yescharacteristic
330C2×D3—28no—
331C2×D3—24no—
332C2×D3C2×#12414yes—
333C3×C4—24no—
334C3×C4—44no—
335C3×C4—44no—
336C3×C4C2×#12412yes—
337C3×C4C22×#4112yes—
338Dic3—44no—
339Dic3—44no—
340Dic3C2×#12414yes—
341Dic3C2×#13718yes—
342C23—34no—
343C23—32no—
344C23—64no—
345C23C2×D3×D411yescharacteristic
346Q8—28no—
347Q8—44no—
348Q8—68no—
349Q8—1216no—
350Q8—1216no—
351Q8—124no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—34no—
356C2×C4—32no—
357C2×C4—34no—
358C2×C4—38no—
359C2×C4—38no—
360C2×C4—38no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—616no—
364C2×C4—64no—
365C2×C4—64no—
366C2×C4—1216no—
367C2×C4—124no—
368C2×C4—124no—
369C2×C4C2×D3×D414yes—
370C2×C4C2×D3×D411yescharacteristic
371C2×C4D3×#4114yes—
372C2×C4C24×D311yescharacteristic
373C8—24no—
374C8—64no—
375C2×C3#313914yescomplemented
376C2×C3C22×#12411yescharacteristic
377C2×C3C22×#13712yes—
378D3—24no—
379C22—34no—
380C22—68no—
381C22—64no—
382C22D3×#12412yes—
383C22D3×#13714yes—
384C22C22×D3×D411yescharacteristic
385C4—24no—
386C4—34no—
387C4—38no—
388C4—44no—
389C4—44no—
390C4—124no—
391C4—124no—
392C4D3×#12412yes—
393C4C2×D3×#4112yes—
394C3C2×#313911yescharacteristic, complemented, Sylow
395C2—64no—
396C2#8173914yesdirect product factor
397C2C2×D3×#12411yescharacteristic
398C2C2×D3×#13712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#491C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#491D3
hypocenterC3C2×#3139
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
3. C2×C3×#491C2semidirect
4. C2×#5082C2semidirect
5. C2×#5069C2semidirect
6. #81739C2direct (N × H)
8. C2×#4792C2semidirect
9. C2×#4703C2semidirect
30. C3×#491C22semidirect
36. #5082C22semidirect
37. #5069C22semidirect
41. #4792C22semidirect
43. #4703C22semidirect
50. C2×#491D3semidirect
129. #491C2×D3semidirect
375. C2×C3#3139semidirect
394. C3C2×#3139semidirect
396. C2#81739direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#56882
C2×C3×#4912
C2×#50822
C2×#50692
#817392
C2×#48022
C2×#47922
C2×#47032
C2×#47072
C2×#31393