Finite Group Databasebeta

← C2×#81600 #11664458

Subgroups — C2×#81600 #11664458

Order 768.

2118 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5728C211yesmaximal, characteristic
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C2×#5093C211yesmaximal, characteristic, complemented
5C2×#5079C211yesmaximal, characteristic, complemented
6#81600C2116yesmaximal, direct product factor
7C2×#4802C214yesmaximal
8C2×#4797C214yesmaximal, complemented
9C2×#4681C211yesmaximal, characteristic, complemented
10C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
11C2×#3139—31nomaximal, Sylow
12C22×D3×Q8C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946—21no—
16C2×Q8×Dic3—21no—
17C2×#944C2212yes—
18C22×ΓL(1,𝔽25)C2212yes—
19#5728—24no—
20#5728C2218yes—
21D3×C4⋊Q8—24no—
22D3×C4⋊Q8C2218yes—
23C2×#897—21no—
24C2×#892C2212yes—
25C2×D3×C4⋊C4—21no—
26C2×D3×C4⋊C4—21no—
27C2×#890C2212yes—
28C2×#888—21no—
29C2×#888—21no—
30C2×#887—21no—
31C2×C42×D3C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×#869C2211yescharacteristic
34C3×#487C2218yescomplemented
35C2×C3×#103C2214yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#835C2214yes—
38C2×#816C2214yes—
39C2×#815C2212yes—
40#5093C2218yescomplemented
41#5079C2218yescomplemented
42C2×#810C2214yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4802C22132yes—
45#4797C22132yescomplemented
46C4×ΓL(1,𝔽25)C2218yes—
47#4681C2218yescomplemented
48C2×#533—31no—
49C22×C4⋊Q8—31no—
50#3139—316no—
51C2×#491—34no—
52C2×#487—31no—
53C2×#487—31no—
54C2×#487—31no—
55C2×#487D311yescharacteristic, complemented
56C2×#419—34no—
57C2×C4×M16—31no—
58C22×C3×Q8—21no—
59C22×C3×Q8C2312yes—
60C2×D3×Q8—24no—
61C2×D3×Q8—216no—
62C2×D3×Q8C2318yes—
63C22×C4×D3—22no—
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×Dic6—21no—
68C22×Dic6—22no—
69C22×Dic6C2312yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8C2314yes—
74C3×C4⋊Q8C2314yes—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C42C2311yescharacteristic
80#946—28no—
81Q8×Dic3—28no—
82#944—24no—
83#944C2318yes—
84C2×(C2×D3)⋊C4—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C2×ΓL(1,𝔽25)—28no—
99C2×ΓL(1,𝔽25)C2318yes—
100#897—28no—
101#892—24no—
102#892C2318yes—
103D3×C4⋊C4—28no—
104D3×C4⋊C4—28no—
105#890—24no—
106#890C2318yes—
107#888—28no—
108#888—28no—
109#887—28no—
110C42×D3D414yes—
111C42×D3C2314yes—
112#869—22no—
113#869—22no—
114#869C2314yes—
115#869C2314yes—
116C3×#103C23116yes—
117C3×C4×C8C2314yes—
118#835C23116yes—
119#816C23116yes—
120#815C2318yes—
121#810C23116yes—
122C4×C3⋊C8C2314yes—
123C23×Q8—32no—
124C22×M16—32no—
125#533—38no—
126#533—64no—
127C2×C4⋊Q8—32no—
128C2×C4⋊Q8—38no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8—31no—
131C2×C4⋊Q8—61no—
132C2×C4⋊Q8—61no—
133C2×C4⋊Q8—64no—
134C2×C4⋊Q8C2×D311yescharacteristic
135C2×C4⋊Q8C2×D311yescharacteristic
136C2×#125—32no—
137C2×C22⋊Q8—61no—
138C2×C22⋊Q8—61no—
139C2×C4×Q8—61no—
140C2×C4×Q8—61no—
141C2×#117—32no—
142C22×C4⋊C4—61no—
143C22×C4⋊C4—61no—
144C22×C42—31no—
145#491—332no—
146#487—38no—
147#487—38no—
148#487—38no—
149#487C2×D318yescomplemented
150#419—332no—
151C2×#103—34no—
152C2×#103—34no—
153C2×#103—34no—
154C2×#103C2×D314yes—
155C4×M16—38no—
156C2×#97—32no—
157C2×C4×C8—31no—
158C2×C4×C8C2×D311yescharacteristic
159C23×D3C2×D411yescharacteristic
160C2×C3×Q8—22no—
161C2×C3×Q8—24no—
162C2×C3×Q8—28no—
163C2×C3×Q8—44no—
164C2×C3×Q8C2414yes—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175D3×Q8—232no—
176D3×Q8—416no—
177C2×C4×D3—24no—
178C2×C4×D3—24no—
179C2×C4×D3—28no—
180C2×C4×D3—48no—
181C2×C4×D3C2×D414yes—
182C2×C4×D3C2×D414yes—
183C2×C4×D3C2×D412yes—
184C2×C4×D3#4118yes—
185C2×C4×D3C2412yes—
186C2×Dic6—28no—
187C2×Dic6—22no—
188C2×Dic6—24no—
189C2×Dic6—28no—
190C2×Dic6—44no—
191C2×Dic6—48no—
192C2×Dic6C2414yes—
193C2×C3×C8—24no—
194C2×C3×C8C2414yes—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4C2414yes—
200C3×C42C2×D412yes—
201C3×C42C2412yes—
202(C2×D3)⋊C4—28no—
203C4⋊Dic3—22no—
204C4⋊Dic3—24no—
205C4⋊Dic3—24no—
206C4⋊Dic3—24no—
207C4⋊Dic3—28no—
208C4⋊Dic3C2414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211Dic3⋊C4—28no—
212C4×Dic3—28no—
213C4×Dic3C2×D412yes—
214C4×Dic3C2×D414yes—
215C4×Dic3C2×D414yes—
216C4×Dic3C2412yes—
217C2×C3⋊C8—24no—
218C2×C3⋊C8C2414yes—
219ΓL(1,𝔽25)—216no—
220C22×Q8—21no—
221C22×Q8—38no—
222C22×Q8—32no—
223C22×Q8—62no—
224C22×Q8—64no—
225C22×Q8—616no—
226C22×Q8—61no—
227C22×Q8C22×D312yes—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—62no—
232C2×M16—38no—
233C2×M16—68no—
234C22×C8—32no—
235C22×C8C22×D312yes—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—38no—
239C4⋊Q8—34no—
240C4⋊Q8—34no—
241C4⋊Q8—68no—
242C4⋊Q8—68no—
243C4⋊Q8—64no—
244C4⋊Q8—62no—
245C4⋊Q8—62no—
246C4⋊Q8C22×D314yes—
247C4⋊Q8C22×D314yes—
248#125—38no—
249#125—64no—
250C22⋊Q8—68no—
251C22⋊Q8—68no—
252C4×Q8—68no—
253C4×Q8—68no—
254#117—38no—
255#117—64no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—68no—
261C2×C4⋊C4—68no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D312yes—
270C2×C22⋊C4—62no—
271C2×C42—31no—
272C2×C42—31no—
273C2×C42—31no—
274C2×C42—34no—
275C2×C42—34no—
276C2×C42—62no—
277C2×C42C22×D311yescharacteristic
278#103—316no—
279#103—316no—
280#103—316no—
281#103C22×D3116yes—
282#97—38no—
283C4×C8—34no—
284C4×C8C22×D314yes—
285C23×C3C22×D411yescharacteristic
286C22×D3—22no—
287C22×D3—22no—
288C22×D3#12414yes—
289C22×D3C22×D412yes—
290C3×Q8—28no—
291C3×Q8—44no—
292C3×Q8—48no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4#13714yes—
298C2×C3×C4#13714yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D412yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4C2×#4114yes—
303C2×C3×C4C2511yescharacteristic
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—44no—
309C2×Dic3—44no—
310C2×Dic3#12414yes—
311C2×Dic3C22×D412yes—
312C2×Dic3C22×D412yes—
313C2×Dic3C22×D412yes—
314C4×D3—28no—
315C4×D3—28no—
316C4×D3—416no—
317C4×D3#12418yes—
318C4×D3C2×#4118yes—
319Dic6—28no—
320Dic6—44no—
321Dic6—48no—
322Dic6—416no—
323C3×C8—24no—
324C3⋊C8—24no—
325C24—31no—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—28no—
329C2×Q8—34no—
330C2×Q8—44no—
331C2×Q8—68no—
332C2×Q8—632no—
333C2×Q8—64no—
334C2×Q8—62no—
335C2×Q8—68no—
336C2×Q8—128no—
337C2×Q8—1216no—
338C2×Q8—124no—
339C2×Q8C23×D314yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—31no—
343C22×C4—32no—
344C22×C4—38no—
345C22×C4—32no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—34no—
349C22×C4—34no—
350C22×C4—68no—
351C22×C4—64no—
352C22×C4—64no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—128no—
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4C23×D311yescharacteristic
359M16—616no—
360C2×C8—24no—
361C2×C8—34no—
362C2×C8—64no—
363C2×C8C23×D314yes—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—34no—
369C4⋊C4—68no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—68no—
373C4⋊C4—62no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4C23×D314yes—
378C22⋊C4—68no—
379C42—34no—
380C42—34no—
381C42—32no—
382C42—32no—
383C42—68no—
384C42D3×D412yes—
385C42C23×D312yes—
386C22×C3C2×#12412yes—
387C22×C3C2×#13712yes—
388C22×C3C2×#13712yes—
389C22×C3C23×D411yescharacteristic
390C2×D3—28no—
391C2×D3—22no—
392C2×D3—22no—
393C2×D3C2×#12414yes—
394C3×C4—44no—
395C3×C4—44no—
396C3×C4C2×#12412yes—
397C3×C4C2×#13714yes—
398C3×C4C2×#13714yes—
399C3×C4C22×#4112yes—
400Dic3—22no—
401Dic3—22no—
402Dic3—44no—
403Dic3—44no—
404Dic3C2×#12414yes—
405C23—34no—
406C23—32no—
407C23—62no—
408C23—62no—
409C23C2×D3×D411yescharacteristic
410Q8—28no—
411Q8—44no—
412Q8—48no—
413Q8—68no—
414Q8—1216no—
415Q8—124no—
416Q8—128no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—34no—
420C2×C4—32no—
421C2×C4—38no—
422C2×C4—38no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—68no—
430C2×C4—68no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—1216no—
434C2×C4—124no—
435C2×C4—124no—
436C2×C4#91514yes—
437C2×C4#91814yes—
438C2×C4C2×D3×D411yescharacteristic
439C2×C4C2×D3×D412yes—
440C2×C4C2×D3×D412yes—
441C2×C4D3×#4114yes—
442C2×C4C24×D311yescharacteristic
443C8—24no—
444C8—64no—
445C2×C3#313914yescomplemented
446C2×C3C22×#12411yescharacteristic
447C2×C3C22×#13711yescharacteristic
448C2×C3C22×#13711yescharacteristic
449D3—24no—
450C22—34no—
451C22—62no—
452C22—62no—
453C22—68no—
454C22D3×#12412yes—
455C22C2×#91512yes—
456C22C2×#91812yes—
457C22C22×D3×D411yescharacteristic
458C4—34no—
459C4—44no—
460C4—44no—
461C4—62no—
462C4—62no—
463C4—124no—
464C4—124no—
465C4D3×#12412yes—
466C4C2×#91514yes—
467C4C2×#91814yes—
468C4C2×D3×#4112yes—
469C3C2×#313911yescharacteristic, complemented, Sylow
470C2—64no—
471C2#8160014yesdirect product factor
472C2C2×D3×#12411yescharacteristic
473C2C22×#91511yescharacteristic
474C2C22×#91811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#487C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#487D3
hypocenterC3C2×#3139
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. C2×C3×#487C2semidirect
4. C2×#5093C2semidirect
5. C2×#5079C2semidirect
6. #81600C2direct (N × H)
8. C2×#4797C2semidirect
9. C2×#4681C2semidirect
34. C3×#487C22semidirect
40. #5093C22semidirect
41. #5079C22semidirect
45. #4797C22semidirect
47. #4681C22semidirect
55. C2×#487D3semidirect
149. #487C2×D3semidirect
445. C2×C3#3139semidirect
469. C3C2×#3139semidirect
471. C2#81600direct (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×#57282
C2×C3×#4872
C2×#50932
C2×#50792
#816002
C2×#48022
C2×#47972
C2×#46812
C2×C4×ΓL(1,𝔽25)2
C2×#31393