Finite Group Databasebeta

← C2×#68327 #10767372

Subgroups — C2×#68327 #10767372

Order 768.

1200 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5623C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic
3C2×#4739C211yesmaximal, characteristic
4C2×#4656C211yesmaximal, characteristic
5C2×C3×#328C211yesmaximal, characteristic
6C2×#4469C211yesmaximal, characteristic
7C2×#4452C211yesmaximal, characteristic
8#68327C218yesmaximal, direct product factor
9C2×#1605—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#944—21no—
13C22×#216—21no—
14C22×C4⋊Dic3—21no—
15C2×#924C2211yescharacteristic
16C22×Dic3⋊C4C2211yescharacteristic
17C22×Dic12—21no—
18#5623—22no—
19#5623—22no—
20#5623—22no—
21#5623C2214yes—
22C2×#889—21no—
23C2×#889—21no—
24C2×#888—21no—
25C2×#878—21no—
26C2×#877—21no—
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95—21no—
30C2×#830C2211yescharacteristic
31C2×#816—21no—
32#5050—22no—
33#5050C2214yes—
34C2×#810—21no—
35#4739—22no—
36#4739C2214yes—
37#4656—22no—
38#4656C2214yes—
39C3×#328C2214yes—
40#4469C2214yes—
41#4452C2214yes—
42C2×#544—31no—
43C2×#451—31no—
44C2×#451—31no—
45C2×#395—31no—
46#1605—38no—
47C2×#328—31no—
48C2×#328D311yescharacteristic
49C2×#324—31no—
50C22×C3×Q8D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C23×Dic3—21no—
54C23×Dic3D411yescharacteristic
55C23×Dic3D411yescharacteristic
56C22×Dic6—21no—
57C22×Dic6—21no—
58C22×Dic6—21no—
59C22×Dic6—21no—
60C22×Dic6—21no—
61C22×Dic6—21no—
62C22×Dic6—44no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2312yes—
65C22×C3×C8—21no—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8D412yes—
69C3×C22⋊Q8C2312yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×#217—21no—
74#944—24no—
75#944—24no—
76C2×#216—24no—
77C2×#216—44no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—44no—
81C2×C4⋊Dic3D411yescharacteristic
82#924—21no—
83#924—21no—
84#924D412yes—
85#924C2312yes—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4D412yes—
96C2×Dic3⋊C4C2312yes—
97C2×C4×Dic3—21no—
98C2×C4×Dic3—21no—
99C22×C3⋊C8—21no—
100C2×Dic12—24no—
101C2×Dic12—44no—
102#889—24no—
103#889—24no—
104#889—24no—
105#889—24no—
106#888—24no—
107#888—24no—
108#878—24no—
109#878—24no—
110#877—24no—
111#877—24no—
112C3×C22⋊C8—21no—
113C3×C22⋊C8C2312yes—
114C3×#95—21no—
115C3×#95C2312yes—
116C3⋊#95—24no—
117#830—21no—
118#830C2312yes—
119#816—24no—
120#810—24no—
121C23×Q8—31no—
122C22×Q16—61no—
123C22×Q16—61no—
124#544—34no—
125#544—62no—
126#544—62no—
127#544—62no—
128C2×C4⋊Q8—61no—
129C2×C4⋊Q8—61no—
130C2×#125—61no—
131C2×#125—61no—
132C2×C22⋊Q8—31no—
133C2×C22⋊Q8—61no—
134C2×C22⋊Q8C2×D311yescharacteristic
135C2×#117—61no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—61no—
138#451—34no—
139#451—34no—
140#451—62no—
141#451—62no—
142C2×#103—61no—
143C2×#103—61no—
144C2×C22⋊C8—31no—
145C2×C22⋊C8C2×D311yescharacteristic
146#395—34no—
147#395—62no—
148C2×#95—61no—
149C2×#95C2×D311yescharacteristic
150#328—34no—
151#328C2×D314yes—
152#324—34no—
153C24×C3C2×D411yescharacteristic
154C2×C3×Q8—21no—
155C2×C3×Q8—42no—
156C2×C3×Q8C2×D412yes—
157C22×C3×C4—24no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—42no—
180C22×Dic3—41no—
181C22×Dic3—41no—
182C22×Dic3—42no—
183C22×Dic3Q1614yescomplemented
184C22×Dic3C2×D412yes—
185C22×Dic3C2×D412yes—
186C2×Dic6—24no—
187C2×Dic6—24no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—24no—
191C2×Dic6—21no—
192C2×Dic6—44no—
193C2×Dic6—44no—
194C2×Dic6—44no—
195C2×Dic6—44no—
196C2×Dic6—42no—
197C2×Dic6—42no—
198C2×Dic6—41no—
199C2×Dic6—41no—
200C2×Dic6—42no—
201C2×Dic6—82no—
202C2×Dic6C2×D412yes—
203C2×C3×C8—22no—
204C2×C3×C8—42no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211#217—22no—
212#217—22no—
213#216—48no—
214C4⋊Dic3—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3—48no—
217C4⋊Dic3C2×D412yes—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—24no—
221Dic3⋊C4—24no—
222Dic3⋊C4—22no—
223Dic3⋊C4—22no—
224Dic3⋊C4—22no—
225Dic3⋊C4—22no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228C4×Dic3—24no—
229C4×Dic3—24no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—42no—
232Dic12—48no—
233C22×Q8—31no—
234C22×Q8—32no—
235C22×Q8—61no—
236C22×Q8—61no—
237C22×Q8—61no—
238C22×Q8—61no—
239C22×Q8—61no—
240C22×Q8—61no—
241C22×Q8—124no—
242C22×Q8D1211yescharacteristic
243C23×C4—21no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—61no—
247C23×C4C22×D311yescharacteristic
248C2×Q16—64no—
249C2×Q16—64no—
250C2×Q16—124no—
251C2×Q16—124no—
252C22×C8—21no—
253C22×C8—61no—
254C4⋊Q8—64no—
255C4⋊Q8—64no—
256C4⋊Q8—64no—
257C4⋊Q8—64no—
258#125—64no—
259#125—64no—
260#125—64no—
261#125—64no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—32no—
265C22⋊Q8—32no—
266C22⋊Q8—64no—
267C22⋊Q8—64no—
268C22⋊Q8—61no—
269C22⋊Q8—61no—
270C22⋊Q8D1212yes—
271C22⋊Q8C22×D312yes—
272#117—64no—
273#117—64no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—124no—
290C2×C4⋊C4D1211yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—61no—
293C2×C42—61no—
294C2×C42—61no—
295#103—64no—
296#103—64no—
297C22⋊C8—21no—
298C22⋊C8—32no—
299C22⋊C8—61no—
300C22⋊C8C22×D312yes—
301#95—21no—
302#95—64no—
303#95C22×D312yes—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3#12011yescharacteristic
307C23×C3#12011yescharacteristic
308C23×C3C2×Q1612yes—
309C23×C3#13712yes—
310C23×C3C22×D411yescharacteristic
311C3×Q8—42no—
312C3×Q8—81no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—48no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340C2×Dic3—44no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—44no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—81no—
348C2×Dic3C2×Q1614yescomplemented
349Dic6—24no—
350Dic6—24no—
351Dic6—42no—
352Dic6—42no—
353Dic6—44no—
354Dic6—44no—
355Dic6—42no—
356Dic6—81no—
357Dic6—84no—
358C3×C8—42no—
359C3⋊C8—42no—
360C24D3×D411yescharacteristic
361C2×Q8—21no—
362C2×Q8—32no—
363C2×Q8—42no—
364C2×Q8—64no—
365C2×Q8—64no—
366C2×Q8—64no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—122no—
371C2×Q8—122no—
372C2×Q8—122no—
373C2×Q8—124no—
374C2×Q8—124no—
375C2×Q8—124no—
376C2×Q8—124no—
377C2×Q8—121no—
378C2×Q8—121no—
379C2×Q8—242no—
380C2×Q8C2×D1212yes—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—34no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—62no—
394C22×C4—64no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—122no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4—122no—
404C22×C4C2×D1211yescharacteristic
405C22×C4C2×D1211yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4D3×D411yescharacteristic
408C22×C4D3×D411yescharacteristic
409C22×C4C23×D311yescharacteristic
410Q16—128no—
411Q16—128no—
412C2×C8—22no—
413C2×C8—42no—
414C2×C8—62no—
415C2×C8—122no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—32no—
420C4⋊C4—64no—
421C4⋊C4—64no—
422C4⋊C4—64no—
423C4⋊C4—64no—
424C4⋊C4—64no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—61no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—128no—
433C4⋊C4C2×D1212yes—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—62no—
437C22⋊C4—62no—
438C42—64no—
439C42—64no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3#45112yes—
447C22×C3#45112yes—
448C22×C3#45612yes—
449C22×C3B(4,𝔽2)12yes—
450C22×C3C2×#12011yescharacteristic
451C22×C3C22×Q1611yescharacteristic
452C22×C3C2×#13711yescharacteristic
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—42no—
460Dic3—24no—
461Dic3—42no—
462Dic3—44no—
463Dic3—42no—
464Dic3—42no—
465C23—22no—
466C23—22no—
467C23#88211yescharacteristic
468C23#88211yescharacteristic
469C23#91512yes—
470C23D3×Q1612yes—
471C23C2×D3×D411yescharacteristic
472Q8—42no—
473Q8—64no—
474Q8—64no—
475Q8—81no—
476Q8—122no—
477Q8—122no—
478Q8—122no—
479Q8—124no—
480Q8—124no—
481Q8—241no—
482Q8—244no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—34no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—122no—
507C2×C4—124no—
508C2×C4—122no—
509C2×C4—124no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—241no—
515C2×C4#88211yescharacteristic
516C2×C4#88211yescharacteristic
517C2×C4C22×D1211yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C8—42no—
520C8—122no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—22no—
524C2×C3#160514yescomplemented
525C2×C3C2×#45111yescharacteristic
526C2×C3C2×#45611yescharacteristic
527C2×C3C2×B(4,𝔽2)11yescharacteristic
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—24no—
533C22—24no—
534C22#474112yes—
535C22#480912yes—
536C22#480912yes—
537C22#482612yes—
538C22C2×#88211yescharacteristic
539C22C2×#91511yescharacteristic
540C22C2×D3×Q1611yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—44no—
546C4—42no—
547C4—42no—
548C4—64no—
549C4—124no—
550C4—122no—
551C4—122no—
552C4—122no—
553C3C2×#160511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—22no—
557C2#6832714yesdirect product factor
558C2C2×#474111yescharacteristic
559C2C2×#480911yescharacteristic
560C2C2×#482611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#328D3
hypocenterC3C2×#1605
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
8. #68327C2direct (N × H)
183. C22×Dic3Q16semidirect
348. C2×Dic3C2×Q16semidirect
524. C2×C3#1605semidirect
553. C3C2×#1605semidirect
557. C2#68327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56232
C2×#50502
C2×#47392
C2×#46562
C2×C3×#3282
C2×#44692
C2×#44522
#683272
C2×#16053