Finite Group Databasebeta

← C2×#83693 #11675622

Subgroups — C2×#83693 #11675622

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5579C211yesmaximal, characteristic
3#83693C2116yesmaximal, direct product factor
4C2×#5087C211yesmaximal, characteristic
5C2×#5080C211yesmaximal, characteristic, complemented
6C2×#5066C212yesmaximal
7C2×#5065C212yesmaximal, complemented
8C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×#5022C212yesmaximal, complemented
10C2×#5021C211yesmaximal, characteristic, complemented
11C2×C4×#215C211yesmaximal, characteristic
12C2×C4×#215C212yesmaximal
13C2×#3381—31nomaximal, Sylow
14C3×#557—24no—
15C3×#557C2218yescomplemented
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—22no—
19C2×C3×#125C2212yes—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×#946—21no—
25C22×#215C2211yescharacteristic
26#5579—24no—
27#5579C2218yes—
28C2×#896—22no—
29C2×D3×C4⋊C4—21no—
30C2×C4×D12C2211yescharacteristic
31C2×C4×D12C2212yes—
32C2×#870C2212yes—
33C2×#869C2211yescharacteristic
34C2×#835C2212yes—
35C2×#835C2211yescharacteristic
36#5087C2218yes—
37#5080C2218yescomplemented
38#5066C22116yes—
39#5065C22116yescomplemented
40C4×#215C22116yes—
41C4×#215C2218yes—
42Q8×C3⋊C8C2218yescomplemented
43#5022C22116yescomplemented
44#5021C2218yescomplemented
45C2×#809C2212yes—
46C2×#809C2211yescharacteristic
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×#808C2211yescharacteristic
50C2×#807C2212yes—
51C2×#807C2212yes—
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2212yes—
54C2×C4×C3⋊C8C2211yescharacteristic
55C2×C4×C3⋊C8C2212yes—
56C2×#557D311yescharacteristic, complemented
57C2×#554—31no—
58#3381—316no—
59C2×#499—32no—
60C2×#498—31no—
61C2×#479—32no—
62C2×#477—31no—
63C2×#478—32no—
64C2×#474—31no—
65C2×C8×Q8—31no—
66C2×C4×SD16—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×Q8C2311yescharacteristic
71C22×D12C2311yescharacteristic
72C22×C4×D3—21no—
73C22×C4×D3—22no—
74C22×Dic6—21no—
75C3×C4⋊Q8—28no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8C2314yes—
78C3×#125—216no—
79C3×#125—24no—
80C3×#125C2318yes—
81C3×C4×Q8—28no—
82C3×C4×Q8—24no—
83C3×C4×Q8—22no—
84C3×C4×Q8D414yes—
85C3×C4×Q8C2314yes—
86C3×C4×Q8C2318yes—
87C3×C4×Q8C2314yes—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C42—22no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2312yes—
101#946—28no—
102C2×#215Q818yescomplemented
103C2×#215C2314yes—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4—22no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#896—216no—
115D3×C4⋊C4—28no—
116C4×D12—24no—
117C4×D12—22no—
118C4×D12C2318yes—
119C4×D12C2314yes—
120#870—24no—
121#870C2318yes—
122#869—22no—
123#869C2314yes—
124#835C2318yes—
125#835C2314yes—
126#809C2314yes—
127#809C2318yes—
128#808C2318yes—
129#808C2314yes—
130#808C2318yes—
131#807C2318yes—
132#807C2318yes—
133C4⋊(C3⋊C8)C2314yes—
134C4⋊(C3⋊C8)C2318yes—
135C4×C3⋊C8C2318yes—
136C4×C3⋊C8C2314yes—
137C22×SD16—31no—
138#557—24no—
139#557C2×D318yescomplemented
140#554—38no—
141#554—64no—
142C2×C4⋊Q8—21no—
143C2×C4⋊Q8—31no—
144C2×C4⋊Q8C2×D311yescharacteristic
145C2×#125—22no—
146C2×#125—32no—
147C2×#125C2×D312yes—
148C2×C22⋊Q8—61no—
149C2×C22⋊Q8—62no—
150C2×C4×Q8—21no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×Q8C2×D312yes—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—32no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#499—316no—
158#498—38no—
159#479—316no—
160#478—316no—
161#477—38no—
162#474—38no—
163C8×Q8—38no—
164C4×SD16—38no—
165C4×SD16—316no—
166C2×#107—32no—
167C2×#107—32no—
168C2×#106—32no—
169C2×#106—32no—
170C2×#106—31no—
171C2×C4⋊C8—32no—
172C2×C4⋊C8—31no—
173C2×#103—32no—
174C2×#103—31no—
175C2×#102—31no—
176C2×#102—32no—
177C2×C4×C8—31no—
178C2×C4×C8—32no—
179C23×D3—21no—
180C2×C3×Q8—24no—
181C2×C3×Q8—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—22no—
184C2×C3×Q8—44no—
185C2×C3×Q8C2×D414yes—
186C2×C3×Q8C2×Q814yes—
187C2×C3×Q8C2412yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—22no—
198C2×D12—21no—
199C2×D12—22no—
200C2×D12C2×Q814yes—
201C2×D12C2412yes—
202C2×C4×D3—24no—
203C2×C4×D3—28no—
204C2×C4×D3—44no—
205C2×C4×D3—84no—
206C2×Dic6—24no—
207C2×Dic6—82no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4C2×D414yes—
216C3×C4⋊C4C2×D418yes—
217C3×C4⋊C4C2412yes—
218C3×C4⋊C4C2414yes—
219C3×C42—28no—
220C3×C42—24no—
221C3×C42C2×D412yes—
222C3×C42C2×D414yes—
223C3×C42C2412yes—
224C3×C42C2414yes—
225#215C2×Q8116yescomplemented
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—28no—
228C4⋊Dic3—22no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3C2412yes—
232C4⋊Dic3C2414yes—
233Dic3⋊C4—28no—
234Dic3⋊C4—28no—
235C2×C3⋊C8—24no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8C2×Q814yes—
238C2×C3⋊C8C2414yes—
239C2×C3⋊C8C2412yes—
240C2×C3⋊C8C2412yes—
241C22×Q8—21no—
242C22×Q8—61no—
243C22×Q8#6711yescharacteristic
244C22×Q8C22×D311yescharacteristic
245C22×D4—31no—
246C23×C4—61no—
247C23×C4—62no—
248C2×SD16—38no—
249C2×SD16—34no—
250C22×C8—31no—
251C22×C8—31no—
252C22×C8—32no—
253C4⋊Q8—28no—
254C4⋊Q8—22no—
255C4⋊Q8—34no—
256C4⋊Q8—62no—
257C4⋊Q8C22×D314yes—
258#125—216no—
259#125—24no—
260#125—38no—
261#125—64no—
262#125C22×D318yes—
263C22⋊Q8—68no—
264C22⋊Q8—616no—
265C4×Q8—28no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8#6714yes—
269C4×Q8C22×D314yes—
270C4×Q8C22×D318yes—
271C4×Q8C22×D314yes—
272C4×D4—34no—
273C4×D4—38no—
274C4×D4—64no—
275C4×D4—62no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—68no—
287C2×C4⋊C4#6711yescharacteristic
288C2×C4⋊C4#6712yes—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D312yes—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—62no—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42C22×D312yes—
296C2×C42C22×D311yescharacteristic
297#107—38no—
298#107—38no—
299#106—38no—
300#106—34no—
301#106—38no—
302C4⋊C8—34no—
303C4⋊C8—38no—
304#103—38no—
305#103—34no—
306#102—38no—
307#102—34no—
308C4×C8—34no—
309C4×C8—38no—
310C23×C3C22×D411yescharacteristic
311C22×D3—24no—
312C22×D3—22no—
313C22×D3—42no—
314C22×D3—42no—
315C3×Q8—24no—
316C3×Q8—28no—
317C3×Q8—22no—
318C3×Q8—48no—
319C3×Q8C22×Q814yes—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—44no—
328C2×C3×C4—42no—
329C2×C3×C4#13514yes—
330C2×C3×C4C22×D414yes—
331C2×C3×C4C22×D414yes—
332C2×C3×C4C22×D412yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×Q812yes—
336C2×C3×C42−1+412yes—
337C2×C3×C4C2511yescharacteristic
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—42no—
341C2×Dic3—44no—
342D12—24no—
343D12—22no—
344D12C22×Q814yes—
345C4×D3—48no—
346C4×D3—88no—
347Dic6—84no—
348C3⋊C8—24no—
349C3⋊C8—22no—
350C3⋊C8C22×Q814yes—
351C24—61no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—24no—
356C2×Q8—44no—
357C2×Q8—64no—
358C2×Q8—242no—
359C2×Q8D3×Q814yes—
360C2×Q8C2×#6714yes—
361C2×Q8C23×D312yes—
362C2×D4—34no—
363C2×D4—32no—
364C2×D4—62no—
365C2×D4—61no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—64no—
372C22×C4—68no—
373C22×C4—124no—
374C22×C4—244no—
375C22×C4C2×#6712yes—
376C22×C4C2×#6712yes—
377C22×C4C2×#6711yescharacteristic
378C22×C4C2×#6711yescharacteristic
379C22×C4C23×D311yescharacteristic
380SD16—316no—
381C2×C8—34no—
382C2×C8—32no—
383C2×C8—32no—
384C2×C8—34no—
385C2×C8—62no—
386C2×C8—64no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—34no—
395C4⋊C4—32no—
396C4⋊C4—61no—
397C4⋊C4—62no—
398C4⋊C4—68no—
399C4⋊C4—64no—
400C4⋊C4—68no—
401C4⋊C4C2×#6714yes—
402C4⋊C4C2×#6718yes—
403C4⋊C4C23×D314yes—
404C4⋊C4C23×D312yes—
405C22⋊C4—64no—
406C22⋊C4—68no—
407C42—28no—
408C42—24no—
409C42C2×#6714yes—
410C42C2×#6712yes—
411C42C23×D314yes—
412C42C23×D312yes—
413C22×C3D4×Q812yes—
414C22×C3C2×#13512yes—
415C22×C3#57712yes—
416C22×C3C23×D411yescharacteristic
417C2×D3—24no—
418C2×D3—48no—
419C2×D3—42no—
420C2×D3—42no—
421C3×C4—24no—
422C3×C4—24no—
423C3×C4—22no—
424C3×C4—24no—
425C3×C4—44no—
426C3×C4—42no—
427C3×C4D4×Q812yes—
428C3×C4C2×#13514yes—
429C3×C4C23×Q811yescharacteristic
430C3×C4C2×2−1+411yescharacteristic
431Dic3—42no—
432Dic3—44no—
433C23—62no—
434C23—64no—
435C23—122no—
436C23—122no—
437C23C22×#6711yescharacteristic
438Q8—22no—
439Q8—24no—
440Q8—28no—
441Q8—48no—
442Q8—244no—
443Q8C2×D3×Q814yes—
444D4—34no—
445D4—62no—
446D4—64no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—122no—
459C2×C4—124no—
460C2×C4—128no—
461C2×C4—248no—
462C2×C4#95014yes—
463C2×C4C2×D3×Q812yes—
464C2×C4#100712yes—
465C2×C4C22×#6714yes—
466C2×C4C22×#6714yes—
467C2×C4C22×#6712yes—
468C2×C4C22×#6711yescharacteristic
469C2×C4C22×#6712yes—
470C2×C4C24×D311yescharacteristic
471C8—34no—
472C8—62no—
473C8—64no—
474C2×C3#338114yescomplemented
475C2×C3C2×D4×Q811yescharacteristic
476C2×C3C22×#13511yescharacteristic
477C2×C3C2×#57711yescharacteristic
478D3—44no—
479C22—64no—
480C22—122no—
481C22—128no—
482C22—122no—
483C22Q8×#6712yes—
484C22C2×#95012yes—
485C22#583712yes—
486C22C23×#6711yescharacteristic
487C4—24no—
488C4—24no—
489C4—24no—
490C4—22no—
491C4—44no—
492C4—42no—
493C4—122no—
494C4—124no—
495C4Q8×#6712yes—
496C4C2×#95014yes—
497C4C22×D3×Q811yescharacteristic
498C4C2×#100711yescharacteristic
499C3C2×#338111yescharacteristic, complemented, Sylow
500C2—124no—
501C2#8369314yesdirect product factor
502C2C2×Q8×#6711yescharacteristic
503C2C22×#95011yescharacteristic
504C2C2×#583711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#557C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3C2×#3381
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
3. #83693C2direct (N × H)
5. C2×#5080C2semidirect
7. C2×#5065C2semidirect
8. C2×Q8×C3⋊C8C2semidirect
9. C2×#5022C2semidirect
10. C2×#5021C2semidirect
15. C3×#557C22semidirect
37. #5080C22semidirect
39. #5065C22semidirect
42. Q8×C3⋊C8C22semidirect
43. #5022C22semidirect
44. #5021C22semidirect
56. C2×#557D3semidirect
102. C2×#215Q8semidirect
139. #557C2×D3semidirect
225. #215C2×Q8semidirect
474. C2×C3#3381semidirect
499. C3C2×#3381semidirect
501. C2#83693direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#55792
#836932
C2×#50872
C2×#50802
C2×#50662
C2×#50652
C2×Q8×C3⋊C82
C2×#50222
C2×#50212
C2×C4×#2152
C2×#33813