Finite Group Databasebeta

← C2×#83694 #11675684

Subgroups — C2×#83694 #11675684

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×Q8×D12C211yesmaximal, characteristic
3#83694C2116yesmaximal, direct product factor
4C2×#5087C211yesmaximal, characteristic
5C2×#5078C211yesmaximal, characteristic, complemented
6C2×#5066C212yesmaximal
7C2×#5064C212yesmaximal, complemented
8C2×#5026C212yesmaximal
9C2×#5024C211yesmaximal, characteristic, complemented
10C2×#5021C212yesmaximal, complemented
11C2×#5020C211yesmaximal, characteristic, complemented
12C2×C4×#215C211yesmaximal, characteristic
13C2×#3386—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C3×#557—24no—
16C3×#557C2218yescomplemented
17C2×C3×C4⋊Q8—21no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—22no—
20C2×C3×#125C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C22×#215C2211yescharacteristic
26Q8×D12—24no—
27Q8×D12C2218yes—
28C2×#897—21no—
29C2×#897—22no—
30C2×C4×D12C2212yes—
31C2×C4×D12C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×#869C2212yes—
34C2×#835C2212yes—
35C2×#835C2211yescharacteristic
36#5087C2218yes—
37#5078C2218yescomplemented
38#5066C22116yes—
39#5064C22116yescomplemented
40#5026C22116yes—
41C4×#215C2218yes—
42#5024C2218yescomplemented
43#5021C22116yescomplemented
44#5020C2218yescomplemented
45C2×#809C2212yes—
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×#807C2212yes—
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2212yes—
54C2×#803C2212yes—
55C2×C4×C3⋊C8C2211yescharacteristic
56C2×#557D311yescharacteristic, complemented
57C2×D4×Q8—31no—
58#3386—316no—
59C2×#501—32no—
60C2×#500—31no—
61C2×#477—32no—
62C2×#475—31no—
63C2×#478—32no—
64C2×#474—31no—
65C2×#446—31no—
66C2×#440—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×Q8C2311yescharacteristic
71C2×D3×Q8—28no—
72C2×D3×Q8—48no—
73C22×D12C2311yescharacteristic
74C22×C4×D3—22no—
75C22×C4×D3—21no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C3×C4⋊Q8—28no—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8C2314yes—
81C3×#125—216no—
82C3×#125—24no—
83C3×#125C2318yes—
84C3×C4×Q8—28no—
85C3×C4×Q8—24no—
86C3×C4×Q8—22no—
87C3×C4×Q8D414yes—
88C3×C4×Q8C2314yes—
89C3×C4×Q8C2318yes—
90C3×C4×Q8C2314yes—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D412yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C42—22no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2312yes—
104C2×#215Q818yescomplemented
105C2×#215C2314yes—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2311yescharacteristic
115#897—28no—
116#897—216no—
117C4×D12—24no—
118C4×D12—22no—
119C4×D12C2314yes—
120C4×D12C2318yes—
121#869—24no—
122#869—22no—
123#869C2314yes—
124#869C2318yes—
125#835C2318yes—
126#835C2314yes—
127#809C2314yes—
128#809C2318yes—
129#808C2318yes—
130#808C2314yes—
131#807C2318yes—
132#807C2318yes—
133#807C2318yes—
134C4⋊(C3⋊C8)C2314yes—
135C4⋊(C3⋊C8)C2318yes—
136#803C2318yes—
137C4×C3⋊C8C2314yes—
138C23×Q8—61no—
139C22×SD16—31no—
140#557—24no—
141#557C2×D318yescomplemented
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—21no—
145C2×C4⋊Q8—31no—
146C2×C4⋊Q8—32no—
147C2×C4⋊Q8C2×D311yescharacteristic
148C2×#125—22no—
149C2×#125C2×D312yes—
150C2×C22⋊Q8—61no—
151C2×C22⋊Q8—62no—
152C2×C4×Q8—21no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×Q8C2×D312yes—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—32no—
157C2×C4×D4—31no—
158#501—316no—
159#500—38no—
160#478—316no—
161#477—316no—
162#475—38no—
163#474—38no—
164#446—38no—
165#440—316no—
166C4×SD16—38no—
167C2×#107—32no—
168C2×#107—32no—
169C2×#107—32no—
170C2×#106—32no—
171C2×#106—31no—
172C2×C4⋊C8—32no—
173C2×C4⋊C8—31no—
174C2×#103—32no—
175C2×#103—31no—
176C2×#102—31no—
177C2×#102—32no—
178C2×#97—32no—
179C2×C4×C8—31no—
180C23×D3—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—21no—
183C2×C3×Q8—24no—
184C2×C3×Q8—22no—
185C2×C3×Q8—44no—
186C2×C3×Q8C2×D414yes—
187C2×C3×Q8C2×Q814yescomplemented
188C2×C3×Q8C2412yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—22no—
199D3×Q8—432no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12C2×Q814yes—
203C2×D12C2412yes—
204C2×C4×D3—28no—
205C2×C4×D3—24no—
206C2×C4×D3—48no—
207C2×C4×D3—44no—
208C2×Dic6—28no—
209C2×Dic6—24no—
210C2×Dic6—48no—
211C2×Dic6—44no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4C2×D414yes—
220C3×C4⋊C4C2×D418yes—
221C3×C4⋊C4C2414yes—
222C3×C4⋊C4C2412yes—
223C3×C42—28no—
224C3×C42—24no—
225C3×C42C2×D414yes—
226C3×C42C2×D412yes—
227C3×C42C2412yes—
228C3×C42C2414yes—
229#215C2×Q8116yescomplemented
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3—22no—
234C4⋊Dic3—28no—
235C4⋊Dic3—24no—
236C4⋊Dic3C2412yes—
237C4⋊Dic3C2414yes—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—24no—
240C2×C3⋊C8C2×Q814yescomplemented
241C2×C3⋊C8C2412yes—
242C2×C3⋊C8C2414yes—
243C2×C3⋊C8C2412yes—
244C22×Q8—21no—
245C22×Q8—68no—
246C22×Q8—62no—
247C22×Q8—61no—
248C22×Q8—128no—
249C22×Q8#6711yescharacteristic
250C22×Q8C22×D311yescharacteristic
251C22×D4—31no—
252C23×C4—62no—
253C23×C4—61no—
254C2×SD16—38no—
255C2×SD16—34no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—32no—
259C4⋊Q8—28no—
260C4⋊Q8—22no—
261C4⋊Q8—34no—
262C4⋊Q8—38no—
263C4⋊Q8—64no—
264C4⋊Q8—62no—
265C4⋊Q8C22×D314yes—
266#125—216no—
267#125—24no—
268#125C22×D318yes—
269C22⋊Q8—68no—
270C22⋊Q8—616no—
271C4×Q8—28no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8#6714yes—
275C4×Q8C22×D314yes—
276C4×Q8C22×D318yes—
277C4×Q8C22×D314yes—
278C4×D4—34no—
279C4×D4—38no—
280C4×D4—64no—
281C4×D4—62no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4#6711yescharacteristic
292C2×C4⋊C4#6712yes—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—61no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42C22×D312yes—
300C2×C42C22×D311yescharacteristic
301#107—38no—
302#107—38no—
303#107—38no—
304#106—34no—
305#106—38no—
306C4⋊C8—34no—
307C4⋊C8—38no—
308#103—34no—
309#103—38no—
310#102—34no—
311#102—38no—
312#97—38no—
313C4×C8—34no—
314C23×C3C22×D411yescharacteristic
315C22×D3—24no—
316C22×D3—22no—
317C22×D3—42no—
318C22×D3—42no—
319C3×Q8—24no—
320C3×Q8—28no—
321C3×Q8—22no—
322C3×Q8—48no—
323C3×Q8C22×Q814yescomplemented
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—44no—
332C2×C3×C4—42no—
333C2×C3×C4AGL(1,ℤ/8)14yes—
334C2×C3×C4C22×D414yes—
335C2×C3×C4C22×D412yes—
336C2×C3×C4C22×D414yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D412yes—
339C2×C3×C4C22×Q812yes—
340C2×C3×C42−1+412yes—
341C2×C3×C4C2511yescharacteristic
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—42no—
345C2×Dic3—44no—
346D12—24no—
347D12—22no—
348D12C22×Q814yes—
349C4×D3—416no—
350C4×D3—48no—
351Dic6—416no—
352Dic6—48no—
353C3⋊C8—22no—
354C3⋊C8—24no—
355C3⋊C8C22×Q814yescomplemented
356C24—61no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—24no—
361C2×Q8—44no—
362C2×Q8—68no—
363C2×Q8—64no—
364C2×Q8—128no—
365C2×Q8—1232no—
366C2×Q8—124no—
367C2×Q8D3×Q814yescomplemented
368C2×Q8C2×#6714yes—
369C2×Q8C23×D312yes—
370C2×D4—34no—
371C2×D4—32no—
372C2×D4—62no—
373C2×D4—61no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—68no—
380C22×C4—64no—
381C22×C4—128no—
382C22×C4—124no—
383C22×C4C2×#6712yes—
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6712yes—
386C22×C4C2×#6711yescharacteristic
387C22×C4C23×D311yescharacteristic
388SD16—316no—
389C2×C8—34no—
390C2×C8—32no—
391C2×C8—32no—
392C2×C8—34no—
393C2×C8—62no—
394C2×C8—64no—
395C4⋊C4—28no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—34no—
404C4⋊C4—62no—
405C4⋊C4—61no—
406C4⋊C4—68no—
407C4⋊C4—64no—
408C4⋊C4C2×#6714yes—
409C4⋊C4C2×#6718yes—
410C4⋊C4C23×D312yes—
411C4⋊C4C23×D314yes—
412C22⋊C4—68no—
413C22⋊C4—64no—
414C42—28no—
415C42—24no—
416C42C2×#6714yes—
417C42C2×#6712yes—
418C42C23×D312yes—
419C42C23×D314yes—
420C22×C3D4×Q812yes—
421C22×C3C2×AGL(1,ℤ/8)12yes—
422C22×C3#57812yes—
423C22×C3C23×D411yescharacteristic
424C2×D3—24no—
425C2×D3—48no—
426C2×D3—42no—
427C2×D3—42no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—22no—
432C3×C4—44no—
433C3×C4—42no—
434C3×C4D4×Q812yes—
435C3×C4C2×AGL(1,ℤ/8)14yes—
436C3×C4C23×Q811yescharacteristic
437C3×C4C2×2−1+411yescharacteristic
438Dic3—42no—
439Dic3—44no—
440C23—62no—
441C23—64no—
442C23—122no—
443C23—122no—
444C23C22×#6711yescharacteristic
445Q8—22no—
446Q8—24no—
447Q8—28no—
448Q8—48no—
449Q8—1216no—
450Q8—128no—
451Q8C2×D3×Q814yescomplemented
452D4—34no—
453D4—62no—
454D4—64no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—64no—
465C2×C4—62no—
466C2×C4—1216no—
467C2×C4—124no—
468C2×C4—122no—
469C2×C4—128no—
470C2×C4#94914yes—
471C2×C4C2×D3×Q812yes—
472C2×C4#100712yes—
473C2×C4C22×#6714yes—
474C2×C4C22×#6712yes—
475C2×C4C22×#6714yes—
476C2×C4C22×#6711yescharacteristic
477C2×C4C22×#6712yes—
478C2×C4C24×D311yescharacteristic
479C8—34no—
480C8—62no—
481C8—64no—
482C2×C3#338614yescomplemented
483C2×C3C2×D4×Q811yescharacteristic
484C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
485C2×C3C2×#57811yescharacteristic
486D3—44no—
487C22—64no—
488C22—122no—
489C22—128no—
490C22—122no—
491C22Q8×#6712yes—
492C22C2×#94912yes—
493C22#583812yes—
494C22C23×#6711yescharacteristic
495C4—24no—
496C4—24no—
497C4—24no—
498C4—22no—
499C4—44no—
500C4—42no—
501C4—124no—
502C4—122no—
503C4Q8×#6712yes—
504C4C2×#94914yes—
505C4C22×D3×Q811yescharacteristic
506C4C2×#100711yescharacteristic
507C3C2×#338611yescharacteristic, complemented, Sylow
508C2—124no—
509C2#8369414yesdirect product factor
510C2C2×Q8×#6711yescharacteristic
511C2C22×#94911yescharacteristic
512C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#557C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3C2×#3386
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. #83694C2direct (N × H)
5. C2×#5078C2semidirect
7. C2×#5064C2semidirect
9. C2×#5024C2semidirect
10. C2×#5021C2semidirect
11. C2×#5020C2semidirect
16. C3×#557C22semidirect
37. #5078C22semidirect
39. #5064C22semidirect
42. #5024C22semidirect
43. #5021C22semidirect
44. #5020C22semidirect
56. C2×#557D3semidirect
104. C2×#215Q8semidirect
141. #557C2×D3semidirect
187. C2×C3×Q8C2×Q8semidirect
229. #215C2×Q8semidirect
240. C2×C3⋊C8C2×Q8semidirect
323. C3×Q8C22×Q8semidirect
355. C3⋊C8C22×Q8semidirect
367. C2×Q8D3×Q8semidirect
451. Q8C2×D3×Q8semidirect
482. C2×C3#3386semidirect
507. C3C2×#3386semidirect
509. C2#83694direct (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×Q8×D122
#836942
C2×#50872
C2×#50782
C2×#50662
C2×#50642
C2×#50262
C2×#50242
C2×#50212
C2×#50202
C2×C4×#2152
C2×#33863