Finite Group Databasebeta

← C2×#81940 #11664833

Subgroups — C2×#81940 #11664833

Order 768.

2118 conjugacy classes of subgroups.

The subgroup lattice has 470 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×#500C211yesmaximal, characteristic, complemented
4C2×#5078C211yesmaximal, characteristic, complemented
5C2×#5080C211yesmaximal, characteristic, complemented
6#81940C2116yesmaximal, direct product factor
7C2×#4881C214yesmaximal
8C2×#4876C214yesmaximal, complemented
9C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
10C2×#4680C211yesmaximal, characteristic, complemented
11C2×#3165—31nomaximal, Sylow
12C22×D3×Q8—21no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946—21no—
16C2×Q8×Dic3—21no—
17C2×#944—21no—
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⋊C4C2212yes—
27C2×#890C2212yes—
28C2×#888—21no—
29C2×#888C2212yes—
30C2×#887—21no—
31C2×C42×D3C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×#869C2211yescharacteristic
34C3×#500C2218yescomplemented
35C2×C3×#107C2214yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#817C2214yes—
38C2×#815C2212yes—
39#5080C2218yescomplemented
40#5078C2218yescomplemented
41C2×#808C2214yes—
42C2×#807C2214yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4881C22132yes—
45#4876C22132yescomplemented
46C4×ΓL(1,𝔽25)C2218yes—
47#4680C2218yescomplemented
48C2×#533—31no—
49C22×C4⋊Q8—31no—
50#3165—316no—
51C2×#501—34no—
52C2×#500—31no—
53C2×#500D311yescharacteristic, complemented
54C2×#498—31no—
55C2×#498—31no—
56C2×#428—34no—
57C2×C4×M16—31no—
58C22×C3×Q8—21no—
59C22×C3×Q8—21no—
60C2×D3×Q8—28no—
61C2×D3×Q8—48no—
62C22×C4×D3—22no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6—21no—
67C22×Dic6—21no—
68C22×Dic6—22no—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8C2314yes—
73C3×C4⋊Q8C2314yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C42C2311yescharacteristic
79#946—28no—
80Q8×Dic3—28no—
81#944—28no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2312yes—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96C2×ΓL(1,𝔽25)Q8116yescomplemented
97C2×ΓL(1,𝔽25)C2318yes—
98#897—28no—
99#892—24no—
100#892C2318yes—
101D3×C4⋊C4—24no—
102D3×C4⋊C4—28no—
103D3×C4⋊C4C2318yes—
104#890—24no—
105#890C2318yes—
106#888—24no—
107#888—28no—
108#888C2318yes—
109#887—28no—
110C42×D3D414yes—
111C42×D3C2314yes—
112#869—22no—
113#869—22no—
114#869C2314yes—
115#869C2314yes—
116C3×#107C23116yes—
117C3×C4×C8C2314yes—
118#817C23116yes—
119#815C2318yes—
120#808C23116yes—
121#807C23116yes—
122C4×C3⋊C8C2314yes—
123C23×Q8—61no—
124C22×M16—32no—
125#533—38no—
126#533—64no—
127C2×C4⋊Q8—38no—
128C2×C4⋊Q8—32no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8—31no—
131C2×C4⋊Q8—61no—
132C2×C4⋊Q8—64no—
133C2×C4⋊Q8—61no—
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—32no—
143C22×C4⋊C4—61no—
144C22×C42—31no—
145#501—332no—
146#500—38no—
147#500C2×D318yescomplemented
148#498—38no—
149#498—38no—
150#428—332no—
151C2×#107—34no—
152C2×#107C2×D314yes—
153C2×#106—34no—
154C2×#106—34no—
155C4×M16—38no—
156C2×#97—32no—
157C2×C4×C8—31no—
158C2×C4×C8C2×D311yescharacteristic
159C23×D3C2×D411yescharacteristic
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8—44no—
163C2×C3×Q8—44no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3C2×D411yescharacteristic
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174D3×Q8—432no—
175C2×C4×D3—24no—
176C2×C4×D3—24no—
177C2×C4×D3—28no—
178C2×C4×D3—48no—
179C2×C4×D3C2×D414yes—
180C2×C4×D3C2×D414yes—
181C2×C4×D3C2×D412yes—
182C2×C4×D3C2×Q818yes—
183C2×C4×D3C2412yes—
184C2×Dic6—24no—
185C2×Dic6—24no—
186C2×Dic6—28no—
187C2×Dic6—48no—
188C2×Dic6—44no—
189C2×Dic6—44no—
190C2×C3×C8C2×Q818yescomplemented
191C2×C3×C8C2414yes—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C2414yes—
197C3×C4⋊C4C2414yes—
198C3×C42C2×D412yes—
199C3×C42C2412yes—
200(C2×D3)⋊C4—28no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—28no—
206C4⋊Dic3C2414yes—
207C4⋊Dic3C2414yes—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211C4×Dic3—28no—
212C4×Dic3C2×D412yes—
213C4×Dic3C2×D414yes—
214C4×Dic3C2×D414yes—
215C4×Dic3C2412yes—
216C2×C3⋊C8C2×Q818yescomplemented
217C2×C3⋊C8C2414yes—
218ΓL(1,𝔽25)C2×Q8132yescomplemented
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—62no—
222C22×Q8—68no—
223C22×Q8—61no—
224C22×Q8—61no—
225C22×Q8—128no—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—62no—
230C2×M16—316no—
231C2×M16—38no—
232C22×C8—32no—
233C22×C8C22×D312yes—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—38no—
237C4⋊Q8—34no—
238C4⋊Q8—34no—
239C4⋊Q8—68no—
240C4⋊Q8—64no—
241C4⋊Q8—68no—
242C4⋊Q8—62no—
243C4⋊Q8—62no—
244C4⋊Q8C22×D314yes—
245C4⋊Q8C22×D314yes—
246#125—38no—
247#125—64no—
248C22⋊Q8—68no—
249C22⋊Q8—68no—
250C4×Q8—68no—
251C4×Q8—68no—
252#117—38no—
253#117—64no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—38no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—68no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4C22×D312yes—
268C2×C4⋊C4C22×D312yes—
269C2×C22⋊C4—62no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—31no—
273C2×C42—34no—
274C2×C42—34no—
275C2×C42—62no—
276C2×C42C22×D311yescharacteristic
277#107—316no—
278#107C22×D3116yes—
279#106—316no—
280#106—316no—
281#97—38no—
282C4×C8—34no—
283C4×C8C22×D314yes—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—22no—
287C22×D3C4⋊Q814yes—
288C22×D3C22×D412yes—
289C3×Q8—48no—
290C3×Q8—48no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4AGL(1,ℤ/8)14yes—
296C2×C3×C4#13714yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D412yes—
299C2×C3×C4C22×D412yes—
300C2×C3×C4C22×Q814yes—
301C2×C3×C4C2511yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—44no—
307C2×Dic3—44no—
308C2×Dic3C4⋊Q814yes—
309C2×Dic3C22×D412yes—
310C2×Dic3C22×D412yes—
311C2×Dic3C22×D412yes—
312C4×D3—28no—
313C4×D3—28no—
314C4×D3—416no—
315C4×D3C4⋊Q818yes—
316C4×D3C22×Q818yes—
317Dic6—416no—
318Dic6—48no—
319Dic6—48no—
320C3×C8C22×Q818yescomplemented
321C3⋊C8C22×Q818yescomplemented
322C24—31no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—68no—
328C2×Q8—64no—
329C2×Q8—64no—
330C2×Q8—128no—
331C2×Q8—1232no—
332C2×Q8—124no—
333C2×Q8—124no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—31no—
337C22×C4—32no—
338C22×C4—38no—
339C22×C4—32no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—34no—
343C22×C4—34no—
344C22×C4—68no—
345C22×C4—64no—
346C22×C4—64no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—128no—
350C22×C4D3×D411yescharacteristic
351C22×C4D3×D411yescharacteristic
352C22×C4C23×D311yescharacteristic
353M16—332no—
354C2×C8—38no—
355C2×C8—34no—
356C2×C8D3×Q818yescomplemented
357C2×C8C23×D314yes—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—34no—
363C4⋊C4—34no—
364C4⋊C4—68no—
365C4⋊C4—68no—
366C4⋊C4—68no—
367C4⋊C4—68no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—64no—
371C4⋊C4—64no—
372C4⋊C4C23×D314yes—
373C4⋊C4C23×D314yes—
374C22⋊C4—68no—
375C42—34no—
376C42—34no—
377C42—32no—
378C42—32no—
379C42—68no—
380C42D3×D412yes—
381C42C23×D312yes—
382C22×C3C2×C4⋊Q812yes—
383C22×C3C2×AGL(1,ℤ/8)12yes—
384C22×C3C2×#13712yes—
385C22×C3C23×D411yescharacteristic
386C2×D3—28no—
387C2×D3—22no—
388C2×D3—22no—
389C2×D3C2×C4⋊Q814yes—
390C3×C4—44no—
391C3×C4—44no—
392C3×C4C2×C4⋊Q812yes—
393C3×C4C2×AGL(1,ℤ/8)14yes—
394C3×C4C2×#13714yes—
395C3×C4C23×Q812yes—
396Dic3—22no—
397Dic3—22no—
398Dic3—44no—
399Dic3—44no—
400Dic3C2×C4⋊Q814yes—
401C23—32no—
402C23—34no—
403C23—62no—
404C23—62no—
405C23C2×D3×D411yescharacteristic
406Q8—48no—
407Q8—48no—
408Q8—1216no—
409Q8—128no—
410Q8—128no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—32no—
414C2×C4—34no—
415C2×C4—38no—
416C2×C4—38no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—68no—
424C2×C4—68no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—1216no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4#91114yes—
431C2×C4#91814yes—
432C2×C4C2×D3×D411yescharacteristic
433C2×C4C2×D3×D412yes—
434C2×C4C2×D3×D412yes—
435C2×C4C2×D3×Q814yes—
436C2×C4C24×D311yescharacteristic
437C8—38no—
438C8C2×D3×Q818yescomplemented
439C2×C3#316514yescomplemented
440C2×C3C22×C4⋊Q811yescharacteristic
441C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
442C2×C3C22×#13711yescharacteristic
443D3—24no—
444C22—34no—
445C22—68no—
446C22—62no—
447C22—62no—
448C22D3×C4⋊Q812yes—
449C22C2×#91112yes—
450C22C2×#91812yes—
451C22C22×D3×D411yescharacteristic
452C4—34no—
453C4—44no—
454C4—44no—
455C4—62no—
456C4—62no—
457C4—124no—
458C4—124no—
459C4D3×C4⋊Q812yes—
460C4C2×#91114yes—
461C4C2×#91814yes—
462C4C22×D3×Q812yes—
463C3C2×#316511yescharacteristic, complemented, Sylow
464C2—64no—
465C2#8194014yesdirect product factor
466C2C2×D3×C4⋊Q811yescharacteristic
467C2C22×#91111yescharacteristic
468C2C22×#91811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#500C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#500D3
hypocenterC3C2×#3165
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
3. C2×C3×#500C2semidirect
4. C2×#5078C2semidirect
5. C2×#5080C2semidirect
6. #81940C2direct (N × H)
8. C2×#4876C2semidirect
10. C2×#4680C2semidirect
34. C3×#500C22semidirect
39. #5080C22semidirect
40. #5078C22semidirect
45. #4876C22semidirect
47. #4680C22semidirect
53. C2×#500D3semidirect
96. C2×ΓL(1,𝔽25)Q8semidirect
147. #500C2×D3semidirect
190. C2×C3×C8C2×Q8semidirect
216. C2×C3⋊C8C2×Q8semidirect
218. ΓL(1,𝔽25)C2×Q8semidirect
320. C3×C8C22×Q8semidirect
321. C3⋊C8C22×Q8semidirect
356. C2×C8D3×Q8semidirect
438. C8C2×D3×Q8semidirect
439. C2×C3#3165semidirect
463. C3C2×#3165semidirect
465. C2#81940direct (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×#5002
C2×#50782
C2×#50802
#819402
C2×#48812
C2×#48762
C2×C4×ΓL(1,𝔽25)2
C2×#46802
C2×#31653