Finite Group Databasebeta

← C2×#77088 #11001709

Subgroups — C2×#77088 #11001709

Order 768.

2566 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic
2C22×#889C211yesmaximal, characteristic, complemented
3C2×C3×#394C211yesmaximal, characteristic
4C2×#5222C211yesmaximal, characteristic
5C2×#4979C211yesmaximal, characteristic, complemented
6C2×#4949C212yesmaximal, complemented
7C2×#4952C212yesmaximal, complemented
8C2×#4948C212yesmaximal, complemented
9C2×#4948C212yesmaximal, complemented
10#77088C2116yesmaximal, direct product factor
11C2×#4656C211yesmaximal, characteristic, complemented
12C2×#4650C211yesmaximal, characteristic, complemented
13C2×#2820—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×#217C2212yes—
20C22×#217C2212yes—
21C22×#217C2212yes—
22C22×#217C2212yes—
23C2×#936—28no—
24C2×#936C2218yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4⋊Dic3C2212yes—
27C22×Dic3⋊C4C2211yescharacteristic
28C22×Dic3⋊C4C2211yescharacteristic
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×#889—28no—
35C2×#889C2218yescomplemented
36C3×#394C2218yes—
37C2×C3×#95C2211yescharacteristic
38#5222C2218yes—
39C2×C3⋊#95C2211yescharacteristic
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4979C2218yescomplemented
44#4952C22116yescomplemented
45#4949C22116yescomplemented
46#4948C22116yescomplemented
47#4948C22116yescomplemented
48#4656C2218yescomplemented
49#4650C2218yescomplemented
50C22×#125—31no—
51C22×#124—31no—
52#2820—316no—
53C2×#399—32no—
54C2×#397—32no—
55C2×#396—32no—
56C2×#396—32no—
57C2×#395—31no—
58C2×#395—31no—
59C2×#394—31no—
60C2×#394D311yescharacteristic
61C2×#382—31no—
62C25×C3C2312yes—
63C22×C3×D4—24no—
64C22×C3×D4—44no—
65C22×C3×D4C2314yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C22×Dic6—24no—
77C22×Dic6—44no—
78C22×Dic6C2314yes—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4C2314yes—
81C2×C3×C22⋊C4—44no—
82C2×C3×C22⋊C4—44no—
83C2×C3×C22⋊C4C2318yes—
84C2×C3×C22⋊C4C2318yes—
85C2×#217—28no—
86C2×#217—28no—
87C2×#217—44no—
88C2×#217—44no—
89C2×#217C2318yes—
90C2×#217C2318yes—
91C2×#217C2318yes—
92C2×#217C2318yes—
93#936—232no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—44no—
96C2×C4⋊Dic3C2314yes—
97C2×C4⋊Dic3C2318yes—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—44no—
103C2×Dic3⋊C4—44no—
104C2×Dic3⋊C4C2314yes—
105C2×Dic3⋊C4C2314yes—
106C2×Dic3⋊C4C2314yes—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C2×C4×Dic3D418yes—
111C2×C4×Dic3C2314yes—
112#889—232no—
113C3×#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2314yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C23×Q8—31no—
119C23×D4C2×D311yescharacteristic
120C2×#125—38no—
121C2×#125—68no—
122C2×#124—38no—
123C2×#124—68no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4C2×D312yes—
138C22×C22⋊C4C2×D312yes—
139C22×C42—31no—
140#399—316no—
141#397—316no—
142#396—316no—
143#396—316no—
144#395—38no—
145#395—38no—
146#394—38no—
147#394C2×D318yes—
148#382—38no—
149C2×#95—31no—
150C2×#95—32no—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95C2×D311yescharacteristic
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—24no—
157C24×C3—24no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3C2414yes—
162C24×C3C2411yescharacteristic
163C2×C3×D4—28no—
164C2×C3×D4—48no—
165C2×C3×D4—416no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2414yes—
177C22×C3×C4C2412yes—
178C22×C3×C4C2412yes—
179C22×C3×C4C2412yes—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×D414yes—
196C22×Dic3C2414yes—
197C22×Dic3C2412yes—
198C22×Dic3C2414yes—
199C22×Dic3C2412yes—
200C22×Dic3C2412yes—
201C22×Dic3C2412yes—
202C2×Dic6—28no—
203C2×Dic6—48no—
204C2×Dic6—416no—
205C3×C4⋊C4—28no—
206C3×C22⋊C4—48no—
207C3×C22⋊C4—48no—
208#217—216no—
209#217—216no—
210#217—48no—
211#217—48no—
212C4⋊Dic3—28no—
213C4⋊Dic3—48no—
214Dic3⋊C4—28no—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217Dic3⋊C4—28no—
218Dic3⋊C4—48no—
219Dic3⋊C4—48no—
220C4×Dic3C2×D4116yes—
221C25C22×D312yes—
222C22×Q8—34no—
223C22×Q8—64no—
224C22×Q8—124no—
225C22×D4—24no—
226C22×D4—44no—
227C22×D4C22×D314yes—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—32no—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238#125—632no—
239#124—632no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—34no—
242C2×C4⋊C4—34no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—34no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—38no—
247C2×C4⋊C4—38no—
248C2×C4⋊C4—38no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—124no—
255C2×C4⋊C4—124no—
256C2×C4⋊C4—124no—
257C2×C4⋊C4C22×D314yes—
258C2×C22⋊C4—38no—
259C2×C22⋊C4—38no—
260C2×C22⋊C4—38no—
261C2×C22⋊C4—38no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—68no—
265C2×C22⋊C4—68no—
266C2×C22⋊C4—124no—
267C2×C22⋊C4—124no—
268C2×C22⋊C4C22×D318yes—
269C2×C22⋊C4C22×D318yes—
270C2×C42—38no—
271C2×C42—34no—
272#95—34no—
273#95—38no—
274#95—38no—
275#95—38no—
276#95C22×D314yes—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—48no—
285C23×C3—48no—
286C23×C3—48no—
287C23×C3—48no—
288C23×C3—48no—
289C23×C3—48no—
290C23×C3—48no—
291C23×C3C22×D412yes—
292C23×C3H(5,𝔽2)14yes—
293C23×C3H(5,𝔽2)12yes—
294C23×C32−1+414yes—
295C23×C32−1+412yes—
296C23×C3C2511yescharacteristic
297C3×D4—416no—
298C2×C3×C4—28no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—48no—
311C2×C3×C4C22×D414yes—
312C2×Dic3—28no—
313C2×Dic3—28no—
314C2×Dic3—22no—
315C2×Dic3—24no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—48no—
329C2×Dic3—44no—
330C2×Dic3—44no—
331C2×Dic3—48no—
332C2×Dic3C22×D414yes—
333C2×Dic3C22×D414yes—
334Dic6—416no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24C23×D314yes—
343C24C23×D311yescharacteristic
344C2×Q8—68no—
345C2×Q8—128no—
346C2×Q8—1216no—
347C2×D4—28no—
348C2×D4—48no—
349C2×D4—416no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—34no—
364C22×C4—32no—
365C22×C4—34no—
366C22×C4—32no—
367C22×C4—62no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—64no—
375C22×C4—64no—
376C22×C4—64no—
377C22×C4—64no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—64no—
381C22×C4D3×D414yes—
382C22×C4C23×D314yes—
383C22×C4C23×D312yes—
384C22×C4C23×D312yes—
385C22×C4C23×D312yes—
386C4⋊C4—28no—
387C4⋊C4—68no—
388C4⋊C4—68no—
389C4⋊C4—68no—
390C4⋊C4—68no—
391C4⋊C4—68no—
392C4⋊C4—128no—
393C4⋊C4—128no—
394C4⋊C4—128no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—616no—
398C22⋊C4—616no—
399C22⋊C4—128no—
400C22⋊C4—128no—
401C42—316no—
402C22×C3—44no—
403C22×C3—44no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—816no—
410C22×C3#53514yes—
411C22×C3#53614yes—
412C22×C3#53714yes—
413C22×C3#56218yes—
414C22×C3#56318yes—
415C22×C3C23×D411yescharacteristic
416C22×C3C2×H(5,𝔽2)12yes—
417C22×C3C2×H(5,𝔽2)11yescharacteristic
418C22×C3C2×2−1+412yes—
419C22×C3C2×2−1+411yescharacteristic
420C3×C4—24no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—44no—
424Dic3—24no—
425Dic3—24no—
426Dic3—44no—
427Dic3—42no—
428Dic3—42no—
429Dic3—44no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—48no—
438C23—48no—
439C23—48no—
440C23—48no—
441C23—48no—
442C23—48no—
443C23—48no—
444C23C2×D3×D412yes—
445C23#100414yes—
446C23#100412yes—
447C23#101012yes—
448C23#101014yes—
449C23C24×D311yescharacteristic
450Q8—1216no—
451D4—416no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—34no—
463C2×C4—34no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—68no—
468C2×C4—62no—
469C2×C4—64no—
470C2×C4—62no—
471C2×C4—68no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—62no—
482C2×C4—64no—
483C2×C4—128no—
484C2×C4—124no—
485C2×C4—124no—
486C2×C4—128no—
487C2×C4C2×D3×D414yes—
488C2×C3—84no—
489C2×C3#282018yescomplemented
490C2×C3C2×#53511yescharacteristic
491C2×C3C2×#53611yescharacteristic
492C2×C3C2×#53711yescharacteristic
493C2×C3C2×#56212yes—
494C2×C3C2×#56312yes—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—816no—
503C22#559214yes—
504C22#560814yes—
505C22#562118yes—
506C22#565118yes—
507C22#567514yes—
508C22C22×D3×D411yescharacteristic
509C22C2×#100412yes—
510C22C2×#100411yescharacteristic
511C22C2×#101012yes—
512C22C2×#101011yescharacteristic
513C4—24no—
514C4—42no—
515C4—42no—
516C4—44no—
517C4—64no—
518C4—64no—
519C4—124no—
520C4—122no—
521C4—122no—
522C4—124no—
523C3C2×#282011yescharacteristic, complemented, Sylow
524C2—84no—
525C2#7708818yesdirect product factor
526C2C2×#559211yescharacteristic
527C2C2×#560811yescharacteristic
528C2C2×#562112yes—
529C2C2×#565112yes—
530C2C2×#567511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#394C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#394D3
hypocenterC3C2×#2820
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#889C2semidirect
5. C2×#4979C2semidirect
6. C2×#4949C2semidirect
7. C2×#4952C2semidirect
8. C2×#4948C2semidirect
9. C2×#4948C2semidirect
10. #77088C2direct (N × H)
11. C2×#4656C2semidirect
12. C2×#4650C2semidirect
35. C2×#889C22semidirect
43. #4979C22semidirect
44. #4952C22semidirect
45. #4949C22semidirect
46. #4948C22semidirect
47. #4948C22semidirect
48. #4656C22semidirect
49. #4650C22semidirect
489. C2×C3#2820semidirect
523. C3C2×#2820semidirect
525. C2#77088direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9362
C22×#8892
C2×C3×#3942
C2×#52222
C2×#49792
C2×#49492
C2×#49522
C2×#49482
#770882
C2×#46562
C2×#46502
C2×#28203