Finite Group Databasebeta

← C2×#71046 #10770849

Subgroups — C2×#71046 #10770849

Order 768.

1022 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5543C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic
3C2×#5094C211yesmaximal, characteristic, complemented
4C2×#5093C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#71046C218yesmaximal, direct product factor
7C2×#4481C211yesmaximal, characteristic, complemented
8C2×#4484C211yesmaximal, characteristic, complemented
9C2×#1643—31nomaximal, Sylow
10C22×C3×SD16—21no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×D4×Dic3—21no—
14C22×C4⋊Dic3—21no—
15C2×#924—21no—
16#5543—22no—
17#5543—22no—
18#5543—22no—
19#5543C2214yes—
20C2×#890—21no—
21C2×#878—22no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#461—22no—
25C3×#461C2214yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103—21no—
28#5094—22no—
29#5094C2214yescomplemented
30#5093—22no—
31#5093C2214yescomplemented
32C2×#810—21no—
33C2×#810—22no—
34C2×#808—21no—
35C2×C4⋊(C3⋊C8)C2211yescharacteristic
36C2×C4×C3⋊C8C2211yescharacteristic
37D4⋊(C3⋊C8)C2214yes—
38#4484C2214yescomplemented
39#4481C2214yescomplemented
40C2×#554—31no—
41C2×#487—31no—
42C2×#475—31no—
43C2×#461D311yescharacteristic
44#1643—38no—
45C2×#333—31no—
46C2×Q8⋊C8—31no—
47C2×D4⋊C8—31no—
48C22×C3×Q8—21no—
49C22×C3×D4D411yescharacteristic
50C23×C3×C4—21no—
51C23×Dic3—22no—
52C22×Dic6D411yescharacteristic
53C22×Dic6D412yes—
54C2×C3×SD16—24no—
55C2×C3×SD16—44no—
56C22×C3×C8—21no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8C2312yes—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×C4×D4D412yes—
64C3×C4×D4C2312yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71D4×Dic3—24no—
72D4×Dic3—24no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3D411yescharacteristic
79#924—24no—
80#924—24no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—21no—
83C2×C4×Dic3—21no—
84C22×C3⋊C8—21no—
85C22×C3⋊C8—21no—
86#890—24no—
87#890—24no—
88#878—28no—
89#878—28no—
90#869—21no—
91#869—21no—
92#869D412yes—
93#869C2312yes—
94C4×Dic6—21no—
95C4×Dic6—21no—
96C4×Dic6D412yes—
97C4×Dic6C2312yes—
98C3×C4⋊C8—21no—
99C3×C4⋊C8C2312yes—
100C3×#103—24no—
101#810—28no—
102#810—24no—
103#808—24no—
104C4⋊(C3⋊C8)—21no—
105C4⋊(C3⋊C8)C2312yes—
106C4×C3⋊C8—21no—
107C4×C3⋊C8C2312yes—
108C22×SD16—21no—
109#554—34no—
110#554—62no—
111#554—62no—
112#554—62no—
113C2×C4⋊Q8—31no—
114C2×C4⋊Q8C2×D311yescharacteristic
115C2×#125—61no—
116C2×C22⋊Q8—62no—
117C2×C22⋊Q8—61no—
118C2×C4×Q8—31no—
119C2×C4×D4—61no—
120C2×C4×D4C2×D311yescharacteristic
121C22×C4⋊C4—61no—
122#487—34no—
123#487—62no—
124#475—34no—
125#475—62no—
126#461—22no—
127#461C2×D314yes—
128C2×#106—61no—
129C2×C4⋊C8—31no—
130C2×C4⋊C8C2×D311yescharacteristic
131C2×#103—21no—
132C2×#103—61no—
133C2×#103—62no—
134C2×C4×C8—31no—
135#333—34no—
136Q8⋊C8—34no—
137D4⋊C8—34no—
138C24×C3—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8—41no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4SD1614yes—
146C2×C3×D4C2×D412yes—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—41no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C22×Dic3—28no—
160C22×Dic3—84no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—21no—
164C2×Dic6—22no—
165C2×Dic6—44no—
166C2×Dic6C2×D412yes—
167C2×Dic6C2×D414yes—
168C3×SD16—48no—
169C2×C3×C8—22no—
170C2×C3×C8—42no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—41no—
176C3×C4⋊C4C2×D412yes—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C42C2×D411yescharacteristic
180C3×C42C2×D411yescharacteristic
181C3×C42C2×D411yescharacteristic
182C3×C42C2411yescharacteristic
183#217—28no—
184C4⋊Dic3—28no—
185C4⋊Dic3—21no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3C2×D412yes—
191Dic3⋊C4—28no—
192Dic3⋊C4—22no—
193Dic3⋊C4—22no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—42no—
199C2×C3⋊C8—42no—
200C22×Q8—21no—
201C22×Q8—31no—
202C22×Q8—32no—
203C22×D4#6711yescharacteristic
204C23×C4—21no—
205C23×C4—62no—
206C2×SD16—24no—
207C2×SD16—44no—
208C22×C8—21no—
209C22×C8—61no—
210C22×C8—61no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—32no—
215C4⋊Q8—32no—
216C4⋊Q8—61no—
217C4⋊Q8—61no—
218C4⋊Q8C22×D312yes—
219#125—64no—
220#125—64no—
221C22⋊Q8—68no—
222C22⋊Q8—68no—
223C22⋊Q8—64no—
224C22⋊Q8—64no—
225C4×Q8—32no—
226C4×Q8—32no—
227C4×Q8—61no—
228C4×Q8—61no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—64no—
232C4×D4—64no—
233C4×D4#6712yes—
234C4×D4C22×D312yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4#6711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—62no—
248C2×C42—61no—
249C2×C42C22×D311yescharacteristic
250#106—64no—
251C4⋊C8—21no—
252C4⋊C8—32no—
253C4⋊C8—61no—
254C4⋊C8C22×D312yes—
255#103—24no—
256#103—64no—
257#103—68no—
258C4×C8—32no—
259C4×C8—61no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3#12011yescharacteristic
265C3×Q8—42no—
266C3×Q8—42no—
267C3×Q8—82no—
268C3×D4—22no—
269C3×D4—24no—
270C3×D4C2×SD1614yes—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—82no—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4C2×SD1612yes—
283C2×C3×C4#13512yes—
284C2×C3×C4#13712yes—
285C2×C3×C4#13712yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—24no—
292C2×Dic3—42no—
293C2×Dic3—44no—
294C2×Dic3—42no—
295C2×Dic3—88no—
296Dic6—22no—
297Dic6—22no—
298Dic6—42no—
299Dic6—44no—
300Dic6—82no—
301C3×C8—42no—
302C3⋊C8—42no—
303C3⋊C8—42no—
304C24—21no—
305C2×Q8—22no—
306C2×Q8—32no—
307C2×Q8—34no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—61no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—62no—
315C2×Q8—124no—
316C2×D4—21no—
317C2×D4—22no—
318C2×D4#21414yes—
319C2×D4C2×#6712yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4—68no—
330C22×C4—244no—
331C22×C4D3×D411yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4C2×#6711yescharacteristic
334SD16—48no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—62no—
338C2×C8—62no—
339C2×C8—122no—
340C2×C8—122no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—32no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—68no—
348C4⋊C4—68no—
349C4⋊C4—61no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—62no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4C2×#6712yes—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—68no—
360C42—62no—
361C42—62no—
362C42D3×D411yescharacteristic
363C42D3×D411yescharacteristic
364C42C2×#6711yescharacteristic
365C42C23×D311yescharacteristic
366C22×C3—24no—
367C22×C3—48no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3#44812yes—
371C22×C3#45212yes—
372C22×C3#45512yes—
373C22×C3C2×#12011yescharacteristic
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—42no—
377C3×C4—82no—
378C3×C4#44812yes—
379C3×C4#45212yes—
380C3×C4C22×SD1611yescharacteristic
381C3×C4C2×#13511yescharacteristic
382C3×C4C2×#13711yescharacteristic
383C3×C4C2×#13711yescharacteristic
384Dic3—42no—
385Dic3—44no—
386Dic3—42no—
387C23—22no—
388C23—24no—
389C23—42no—
390C23—42no—
391C23#93711yescharacteristic
392Q8—42no—
393Q8—42no—
394Q8—62no—
395Q8—62no—
396Q8—82no—
397Q8—124no—
398Q8—122no—
399Q8—242no—
400D4—24no—
401D4—22no—
402D4C2×#21414yes—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—64no—
413C2×C4—82no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—124no—
417C2×C4—248no—
418C2×C4#91512yes—
419C2×C4#91612yes—
420C2×C4C2×#21412yes—
421C2×C4#93711yescharacteristic
422C2×C4#93711yescharacteristic
423C2×C4#93711yescharacteristic
424C2×C4#95112yes—
425C2×C4C2×D3×D411yescharacteristic
426C2×C4C2×D3×D411yescharacteristic
427C2×C4C22×#6711yescharacteristic
428C8—42no—
429C8—122no—
430C8—122no—
431C2×C3—44no—
432C2×C3#164314yescomplemented
433C2×C3C2×#44811yescharacteristic
434C2×C3C2×#45211yescharacteristic
435C2×C3C2×#45511yescharacteristic
436C22—24no—
437C22—42no—
438C22—48no—
439C22—42no—
440C22#504012yes—
441C22#517312yes—
442C22#520212yes—
443C22C2×#93711yescharacteristic
444C4—21no—
445C4—21no—
446C4—42no—
447C4—82no—
448C4—122no—
449C4—124no—
450C4—122no—
451C4#504012yes—
452C4#517312yes—
453C4C2×#91511yescharacteristic
454C4C2×#91611yescharacteristic
455C4C22×#21411yescharacteristic
456C4C2×#95111yescharacteristic
457C3C2×#164311yescharacteristic, complemented, Sylow
458C2—44no—
459C2#7104614yesdirect product factor
460C2C2×#504011yescharacteristic
461C2C2×#517311yescharacteristic
462C2C2×#520211yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC2×C3×#461C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenterC2×#461D3
hypocenterC3C2×#1643
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. C2×#5094C2semidirect
4. C2×#5093C2semidirect
6. #71046C2direct (N × H)
7. C2×#4481C2semidirect
8. C2×#4484C2semidirect
29. #5094C22semidirect
31. #5093C22semidirect
38. #4484C22semidirect
39. #4481C22semidirect
432. C2×C3#1643semidirect
457. C3C2×#1643semidirect
459. C2#71046direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55432
C2×C3×#4612
C2×#50942
C2×#50932
C2×D4⋊(C3⋊C8)2
#710462
C2×#44812
C2×#44842
C2×#16433