Finite Group Databasebeta

← C2×#83558 #11675688

Subgroups — C2×#83558 #11675688

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×D4×Q8C211yesmaximal, characteristic
2C2×#5567C211yesmaximal, characteristic, complemented
3#83558C2116yesmaximal, direct product factor
4C2×#5091C212yesmaximal, complemented
5C2×#5094C211yesmaximal, characteristic, complemented
6C2×#5081C212yesmaximal, complemented
7C2×#5078C211yesmaximal, characteristic, complemented
8C2×#5024C211yesmaximal, characteristic, complemented
9C2×C4×#214C211yesmaximal, characteristic
10C2×#5009C212yesmaximal
11C2×#5007C211yesmaximal, characteristic
12C2×#5018C212yesmaximal
13C2×#3386—31nomaximal, Sylow
14C23×C3×Q8—21no—
15C3×D4×Q8—24no—
16C3×D4×Q8C2218yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×#944—21no—
25C22×#214C2211yescharacteristic
26#5567—24no—
27#5567C2218yescomplemented
28C2×#889—22no—
29C2×#887—21no—
30C2×#870C2212yes—
31C2×#869C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C2×#832C2212yes—
35C2×#832C2211yescharacteristic
36#5094C2218yescomplemented
37#5091C22116yescomplemented
38#5081C22116yescomplemented
39#5078C2218yescomplemented
40#5024C2218yescomplemented
41#5018C22116yes—
42C4×#214C2218yes—
43#5009C22116yes—
44#5007C2218yes—
45C2×#810C2212yes—
46C2×#810C2211yescharacteristic
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×#557—31no—
57C2×D4×Q8D311yescharacteristic
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—28no—
69C22×C3×Q8—22no—
70C22×C3×Q8—21no—
71C22×C3×Q8—48no—
72C22×C3×Q8D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—22no—
75C23×C3×C4—21no—
76C22×Dic6—21no—
77C22×Dic6C2311yescharacteristic
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—24no—
80C3×C4⋊Q8C2314yes—
81C3×C4⋊Q8C2318yes—
82C3×C22⋊Q8—28no—
83C3×C22⋊Q8—216no—
84C3×C4×Q8D414yes—
85C3×C4×Q8C2314yes—
86C3×C4×D4—22no—
87C3×C4×D4—24no—
88C3×C4×D4C2314yes—
89C3×C4×D4C2318yes—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2312yes—
100#944—28no—
101C2×#214Q818yescomplemented
102C2×#214C2314yes—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2312yes—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—22no—
108C2×Dic3⋊C4—22no—
109C2×Dic3⋊C4—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—21no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2311yescharacteristic
115#889—216no—
116#887—28no—
117#870—24no—
118#870C2318yes—
119#869—22no—
120#869C2314yes—
121C4×Dic6—24no—
122C4×Dic6—22no—
123C4×Dic6C2314yes—
124C4×Dic6C2318yes—
125#832C2318yes—
126#832C2314yes—
127#810C2314yes—
128#810C2318yes—
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—21no—
139C22×SD16—31no—
140#557—38no—
141#557—64no—
142D4×Q8—24no—
143D4×Q8C2×D318yes—
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8—61no—
146C2×C4⋊Q8C2×D311yescharacteristic
147C2×C4⋊Q8C2×D312yes—
148C2×#125—32no—
149C2×#125—62no—
150C2×C22⋊Q8—21no—
151C2×C22⋊Q8—22no—
152C2×C4×Q8—31no—
153C2×C4×Q8—32no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4C2×D312yes—
157C2×C4×D4C2×D311yescharacteristic
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—
180C24×C3—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—28no—
183C2×C3×Q8—24no—
184C2×C3×Q8—48no—
185C2×C3×Q8—432no—
186C2×C3×Q8—44no—
187C2×C3×Q8C2×D414yes—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4C2×Q814yes—
191C2×C3×D4C2412yes—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—28no—
195C22×C3×C4—24no—
196C22×C3×C4—48no—
197C22×C3×C4—44no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C2×Dic6—21no—
207C2×Dic6—22no—
208C2×Dic6—24no—
209C2×Dic6—44no—
210C2×Dic6C2×Q814yescomplemented
211C2×Dic6C2412yes—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4C2×D414yes—
217C3×C4⋊C4C2×D418yes—
218C3×C4⋊C4C2412yes—
219C3×C4⋊C4C2414yes—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—24no—
222C3×C42C2×D412yes—
223C3×C42C2×D414yes—
224C3×C42C2412yes—
225C3×C42C2414yes—
226#214C2×Q8116yescomplemented
227C4⋊Dic3—22no—
228C4⋊Dic3—21no—
229C4⋊Dic3—24no—
230C4⋊Dic3C2414yes—
231C4⋊Dic3C2412yes—
232Dic3⋊C4—24no—
233Dic3⋊C4—28no—
234Dic3⋊C4—28no—
235Dic3⋊C4—28no—
236C4×Dic3—28no—
237C4×Dic3—24no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—24no—
240C2×C3⋊C8C2×Q814yescomplemented
241C2×C3⋊C8C2412yes—
242C2×C3⋊C8C2414yes—
243C2×C3⋊C8C2412yes—
244C22×Q8—28no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—31no—
248C22×Q8—48no—
249C22×Q8—61no—
250C22×Q8#6711yescharacteristic
251C22×D4C22×D311yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C2×SD16—38no—
255C2×SD16—34no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—32no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—34no—
262C4⋊Q8—68no—
263C4⋊Q8—62no—
264C4⋊Q8C22×D314yes—
265C4⋊Q8C22×D318yes—
266#125—38no—
267#125—616no—
268#125—64no—
269C22⋊Q8—28no—
270C22⋊Q8—216no—
271C4×Q8—34no—
272C4×Q8—38no—
273C4×Q8—68no—
274C4×Q8—64no—
275C4×Q8—62no—
276C4×Q8#6714yes—
277C4×Q8C22×D314yes—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4C22×D314yes—
281C4×D4C22×D318yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4#6711yescharacteristic
292C2×C4⋊C4#6712yes—
293C2×C4⋊C4C22×D312yes—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C42—62no—
298C2×C42—61no—
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×C3—24no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3C22×D411yescharacteristic
319C3×Q8—28no—
320C3×Q8—416no—
321C3×Q8—48no—
322C3×D4—24no—
323C3×D4—22no—
324C3×D4C22×Q814yes—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—416no—
331C2×C3×C4—44no—
332C2×C3×C4—42no—
333C2×C3×C4—48no—
334C2×C3×C4AGL(1,ℤ/8)14yes—
335C2×C3×C4C22×D414yes—
336C2×C3×C4C22×D414yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D412yes—
340C2×C3×C4C22×Q812yes—
341C2×C3×C42−1+412yes—
342C2×C3×C4C2511yescharacteristic
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—24no—
347C2×Dic3—44no—
348C2×Dic3—42no—
349Dic6—24no—
350Dic6—22no—
351Dic6—48no—
352Dic6C22×Q814yescomplemented
353C3⋊C8—22no—
354C3⋊C8—24no—
355C3⋊C8C22×Q814yescomplemented
356C24—21no—
357C2×Q8—24no—
358C2×Q8—28no—
359C2×Q8—24no—
360C2×Q8—34no—
361C2×Q8—32no—
362C2×Q8—48no—
363C2×Q8—432no—
364C2×Q8—44no—
365C2×Q8—62no—
366C2×Q8—61no—
367C2×Q8—64no—
368C2×Q8—124no—
369C2×Q8C2×#6714yes—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4D3×Q814yes—
373C2×D4C23×D312yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—62no—
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—22no—
398C4⋊C4—21no—
399C4⋊C4—32no—
400C4⋊C4—34no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—64no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—68no—
407C4⋊C4—64no—
408C4⋊C4C2×#6714yes—
409C4⋊C4C2×#6718yes—
410C4⋊C4C23×D312yes—
411C4⋊C4C23×D314yes—
412C22⋊C4—28no—
413C22⋊C4—24no—
414C42—68no—
415C42—64no—
416C42C2×#6714yes—
417C42C2×#6712yes—
418C42C23×D312yes—
419C42C23×D314yes—
420C22×C3—24no—
421C22×C3—48no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3D4×Q812yes—
425C22×C3C2×AGL(1,ℤ/8)12yes—
426C22×C3#57812yes—
427C22×C3C23×D411yescharacteristic
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—22no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4D4×Q812yes—
434C3×C4C2×AGL(1,ℤ/8)14yes—
435C3×C4C23×Q811yescharacteristic
436C3×C4C2×2−1+411yescharacteristic
437Dic3—24no—
438Dic3—44no—
439Dic3—42no—
440C23—22no—
441C23—24no—
442C23—42no—
443C23—42no—
444C23C22×#6711yescharacteristic
445Q8—28no—
446Q8—34no—
447Q8—416no—
448Q8—48no—
449Q8—62no—
450Q8—64no—
451Q8—128no—
452D4—22no—
453D4—24no—
454D4C2×D3×Q814yes—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—48no—
464C2×C4—62no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—64no—
468C2×C4—124no—
469C2×C4—122no—
470C2×C4#93214yes—
471C2×C4C2×D3×Q812yes—
472C2×C4#100712yes—
473C2×C4C22×#6714yes—
474C2×C4C22×#6712yes—
475C2×C4C22×#6714yes—
476C2×C4C22×#6712yes—
477C2×C4C22×#6711yescharacteristic
478C2×C4C24×D311yescharacteristic
479C8—34no—
480C8—62no—
481C8—64no—
482C2×C3—44no—
483C2×C3#338614yescomplemented
484C2×C3C2×D4×Q811yescharacteristic
485C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
486C2×C3C2×#57811yescharacteristic
487C22—24no—
488C22—48no—
489C22—42no—
490C22—42no—
491C22C2×#93212yes—
492C22Q8×#6712yes—
493C22#583812yes—
494C22C23×#6711yescharacteristic
495C4—24no—
496C4—24no—
497C4—22no—
498C4—44no—
499C4—42no—
500C4—64no—
501C4—124no—
502C4—122no—
503C4C2×#93214yes—
504C4Q8×#6712yes—
505C4C22×D3×Q811yescharacteristic
506C4C2×#100711yescharacteristic
507C3C2×#338611yescharacteristic, complemented, Sylow
508C2—44no—
509C2#8355814yesdirect product factor
510C2C22×#93211yescharacteristic
511C2C2×Q8×#6711yescharacteristic
512C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3386
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#5567C2semidirect
3. #83558C2direct (N × H)
4. C2×#5091C2semidirect
5. C2×#5094C2semidirect
6. C2×#5081C2semidirect
7. C2×#5078C2semidirect
8. C2×#5024C2semidirect
27. #5567C22semidirect
36. #5094C22semidirect
37. #5091C22semidirect
38. #5081C22semidirect
39. #5078C22semidirect
40. #5024C22semidirect
101. C2×#214Q8semidirect
210. C2×Dic6C2×Q8semidirect
226. #214C2×Q8semidirect
240. C2×C3⋊C8C2×Q8semidirect
352. Dic6C22×Q8semidirect
355. C3⋊C8C22×Q8semidirect
483. C2×C3#3386semidirect
507. C3C2×#3386semidirect
509. C2#83558direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×#55672
#835582
C2×#50912
C2×#50942
C2×#50812
C2×#50782
C2×#50242
C2×C4×#2142
C2×#50092
C2×#50072
C2×#50182
C2×#33863