Finite Group Databasebeta

← C2×#78152 #10964475

Subgroups — C2×#78152 #10964475

Order 768.

2918 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#896C211yesmaximal, characteristic
2C22×D3×C4⋊C4C211yesmaximal, characteristic
3C2×C3×#400C211yesmaximal, characteristic, complemented
4C2×#4985C211yesmaximal, characteristic, complemented
5C2×#4986C211yesmaximal, characteristic, complemented
6#78152C2116yesmaximal, direct product factor
7C2×#4664C212yesmaximal
8C2×#4664C212yesmaximal
9C2×#4659C211yesmaximal, characteristic, complemented
10C2×#4646C212yesmaximal, complemented
11C2×#4650C212yesmaximal, complemented
12C2×#4666C211yesmaximal, characteristic
13C2×#2503—31nomaximal, Sylow
14C23×C4×D3C2212yes—
15C23×C4×D3C2211yescharacteristic
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2211yescharacteristic
26C22×C4×Dic3C2212yes—
27C2×#896—28no—
28C2×#896C2218yes—
29C2×D3×C4⋊C4—28no—
30C2×D3×C4⋊C4C2218yes—
31C3×#400C2218yescomplemented
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4986C2218yescomplemented
43#4985C2218yescomplemented
44#4666C2218yes—
45#4664C22116yes—
46#4664C22116yes—
47#4659C2218yescomplemented
48#4650C22116yescomplemented
49#4646C22116yescomplemented
50C22×C22⋊Q8—31no—
51C23×C4⋊C4—31no—
52#2503—316no—
53C2×#400—31no—
54C2×#400—31no—
55C2×#400—31no—
56C2×#400D311yescharacteristic, complemented
57C2×#391—32no—
58C2×#382—32no—
59C2×#381—32no—
60C2×#381—32no—
61C2×C4×C22⋊C4—31no—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C22×C4×D3—28no—
74C22×C4×D3—28no—
75C22×C4×D3—216no—
76C22×C4×D3D418yes—
77C22×C4×D3C2318yes—
78C22×C4×D3C2314yes—
79C22×Dic6—24no—
80C22×Dic6—28no—
81C22×Dic6C2314yes—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4C2318yes—
90C2×(C2×D3)⋊C4C2314yes—
91C2×(C2×D3)⋊C4C2314yes—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3C2314yes—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—28no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4C2318yes—
98C2×Dic3⋊C4C2314yes—
99C2×Dic3⋊C4C2314yes—
100C2×C4×Dic3—28no—
101C2×C4×Dic3C2318yes—
102#896—232no—
103D3×C4⋊C4—232no—
104C3×#95C2318yes—
105C3×#95C2318yes—
106C3×#95C2314yes—
107C3⋊#95C2318yes—
108C3⋊#95C2318yes—
109C3⋊#95C2318yes—
110C3⋊#95C2318yes—
111C3⋊#95C2318yes—
112C3⋊#95C2318yes—
113C3⋊#95C2314yes—
114C23×Q8—31no—
115C24×C4—31no—
116C24×C4—32no—
117C2×C22⋊Q8—38no—
118C2×C22⋊Q8—68no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—38no—
124C22×C4⋊C4—68no—
125C22×C4⋊C4C2×D311yescharacteristic
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C22⋊C4—31no—
128C22×C22⋊C4—31no—
129C22×C22⋊C4—32no—
130C22×C42—32no—
131#400—38no—
132#400—38no—
133#400—38no—
134#400C2×D318yescomplemented
135#391—316no—
136#382—316no—
137#381—316no—
138#381—316no—
139C4×C22⋊C4—38no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95C2×D311yescharacteristic
148C2×#95C2×D312yes—
149C2×#95C2×D312yes—
150C24×C3C2411yescharacteristic
151C23×D3—24no—
152C23×D3—22no—
153C23×D3—22no—
154C23×D3C2×D414yes—
155C23×D3#4118yes—
156C23×D3C2412yes—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—28no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2414yes—
170C22×C3×C4C2412yes—
171C22×C3×C4C2414yes—
172C22×C3×C4C2412yes—
173C22×C3×C4C2412yes—
174C22×Dic3—22no—
175C22×Dic3—24no—
176C22×Dic3—22no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—28no—
181C22×Dic3—22no—
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3C2×D414yes—
185C22×Dic3#4118yes—
186C22×Dic3C2414yes—
187C22×Dic3C2412yes—
188C22×Dic3C2414yes—
189C22×Dic3C2412yes—
190C22×Dic3C2412yes—
191C2×C4×D3—232no—
192C2×C4×D3—216no—
193C2×C4×D3—216no—
194C2×C4×D3—232no—
195C2×C4×D3—432no—
196C2×C4×D3C2×D4116yes—
197C2×Dic6—28no—
198C2×Dic6—216no—
199C2×Dic6—416no—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—28no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—216no—
205C4⋊Dic3—28no—
206Dic3⋊C4—28no—
207Dic3⋊C4—216no—
208Dic3⋊C4—28no—
209C4×Dic3—216no—
210C25—31no—
211C22×Q8—34no—
212C22×Q8—64no—
213C22×Q8—68no—
214C23×C4—31no—
215C23×C4—38no—
216C23×C4—34no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—32no—
220C23×C4—32no—
221C23×C4—38no—
222C23×C4—68no—
223C23×C4—68no—
224C23×C4—616no—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D312yes—
228C23×C4C22×D312yes—
229C23×C4C22×D311yescharacteristic
230C22⋊Q8—632no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—34no—
236C2×C4⋊C4—34no—
237C2×C4⋊C4—632no—
238C2×C4⋊C4—68no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4C22×D314yes—
243C2×C4⋊C4C22×D314yes—
244C2×C22⋊C4—34no—
245C2×C22⋊C4—34no—
246C2×C22⋊C4—38no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—68no—
250C2×C42—38no—
251C2×C42—68no—
252#95—34no—
253#95—38no—
254#95—38no—
255#95—38no—
256#95—38no—
257#95—38no—
258#95—38no—
259#95C22×D314yes—
260#95C22×D318yes—
261#95C22×D318yes—
262C23×C3C22×D412yes—
263C23×C3C2×#4114yes—
264C23×C3C2×#4114yes—
265C23×C32−1+412yes—
266C23×C32−1+412yes—
267C23×C3C2511yescharacteristic
268C22×D3—28no—
269C22×D3—28no—
270C22×D3—216no—
271C22×D3—216no—
272C22×D3—24no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3#124116yes—
276C22×D3C22×D414yes—
277C22×D3C2×#4118yes—
278C2×C3×C4—28no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—28no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—44no—
291C2×C3×C4—48no—
292C2×C3×C4—44no—
293C2×C3×C4—48no—
294C2×C3×C4C22×D414yes—
295C2×Dic3—22no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—28no—
302C2×Dic3—22no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—44no—
306C2×Dic3—48no—
307C2×Dic3—44no—
308C2×Dic3—48no—
309C2×Dic3#123116yes—
310C2×Dic3C22×D414yes—
311C2×Dic3C2×#4118yes—
312C4×D3—232no—
313C4×D3—432no—
314Dic6—416no—
315C24—34no—
316C24—38no—
317C24—32no—
318C24—62no—
319C24—64no—
320C24—62no—
321C24C23×D311yescharacteristic
322C2×Q8—68no—
323C2×Q8—616no—
324C2×Q8—1216no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—34no—
337C22×C4—38no—
338C22×C4—32no—
339C22×C4—316no—
340C22×C4—34no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—34no—
344C22×C4—632no—
345C22×C4—616no—
346C22×C4—616no—
347C22×C4—632no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—64no—
351C22×C4—64no—
352C22×C4—62no—
353C22×C4—64no—
354C22×C4—64no—
355C22×C4—68no—
356C22×C4—64no—
357C22×C4—62no—
358C22×C4—1232no—
359C22×C4D3×D414yes—
360C22×C4C23×D314yes—
361C22×C4C23×D312yes—
362C22×C4C23×D312yes—
363C22×C4C23×D314yes—
364C22×C4C23×D312yes—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—616no—
368C4⋊C4—68no—
369C4⋊C4—68no—
370C4⋊C4—68no—
371C22⋊C4—68no—
372C22⋊C4—68no—
373C22⋊C4—616no—
374C42—616no—
375C22×C3C2×#12314yes—
376C22×C3C2×#12414yes—
377C22×C3#53614yes—
378C22×C3#55218yes—
379C22×C3#55218yes—
380C22×C3C23×D411yescharacteristic
381C22×C3C22×#4112yes—
382C22×C3C22×#4112yes—
383C22×C3C2×2−1+411yescharacteristic
384C22×C3C2×2−1+411yescharacteristic
385C2×D3—24no—
386C2×D3—24no—
387C2×D3—28no—
388C2×D3—28no—
389C2×D3—232no—
390C2×D3C2×#12418yes—
391C3×C4—24no—
392C3×C4—42no—
393C3×C4—44no—
394C3×C4—42no—
395C3×C4—44no—
396Dic3—42no—
397Dic3—44no—
398Dic3—42no—
399Dic3—44no—
400Dic3C2×#12318yes—
401C23—316no—
402C23—34no—
403C23—38no—
404C23—68no—
405C23—616no—
406C23—62no—
407C23—64no—
408C23—62no—
409C23—616no—
410C23—68no—
411C23C2×D3×D412yes—
412C23#100712yes—
413C23D3×#4114yes—
414C23D3×#4114yes—
415C23#101012yes—
416C23C24×D311yescharacteristic
417Q8—1216no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—316no—
431C2×C4—34no—
432C2×C4—38no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—632no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—64no—
441C2×C4—64no—
442C2×C4—62no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—68no—
446C2×C4—64no—
447C2×C4—62no—
448C2×C4—1232no—
449C2×C4—124no—
450C2×C4—128no—
451C2×C4—128no—
452C2×C4—124no—
453C2×C4C2×D3×D414yes—
454C2×C3#250318yescomplemented
455C2×C3C22×#12311yescharacteristic
456C2×C3C22×#12411yescharacteristic
457C2×C3C2×#53611yescharacteristic
458C2×C3C2×#55212yes—
459C2×C3C2×#55212yes—
460D3—28no—
461C22—38no—
462C22—632no—
463C22—68no—
464C22—68no—
465C22—64no—
466C22—64no—
467C22#562814yes—
468C22D3×#12314yes—
469C22D3×#12414yes—
470C22#569318yes—
471C22#570818yes—
472C22C22×D3×D411yescharacteristic
473C22C2×#100711yescharacteristic
474C22C2×D3×#4112yes—
475C22C2×D3×#4112yes—
476C22C2×#101011yescharacteristic
477C4—24no—
478C4—38no—
479C4—44no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—124no—
484C4—124no—
485C4—122no—
486C4—122no—
487C3C2×#250311yescharacteristic, complemented, Sylow
488C2—68no—
489C2#7815218yesdirect product factor
490C2C2×#562811yescharacteristic
491C2C2×D3×#12311yescharacteristic
492C2C2×D3×#12411yescharacteristic
493C2C2×#569312yes—
494C2C2×#570812yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#400C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#400D3
hypocenterC3C2×#2503
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
3. C2×C3×#400C2semidirect
4. C2×#4985C2semidirect
5. C2×#4986C2semidirect
6. #78152C2direct (N × H)
9. C2×#4659C2semidirect
10. C2×#4646C2semidirect
11. C2×#4650C2semidirect
31. C3×#400C22semidirect
42. #4986C22semidirect
43. #4985C22semidirect
47. #4659C22semidirect
48. #4650C22semidirect
49. #4646C22semidirect
56. C2×#400D3semidirect
134. #400C2×D3semidirect
454. C2×C3#2503semidirect
487. C3C2×#2503semidirect
489. C2#78152direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8962
C22×D3×C4⋊C42
C2×C3×#4002
C2×#49852
C2×#49862
#781522
C2×#46642
C2×#46592
C2×#46462
C2×#46502
C2×#46662
C2×#25033