Finite Group Databasebeta

← C2×#12621 #492988

Subgroups — C2×#12621 #492988

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C22×#352C212yesmaximal
3C2×C4×#100C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#12621C218yesmaximal, direct product factor
6C2×#1390C212yesmaximal
7C22×#123C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic, complemented
10C22×#100C2212yes—
11C22×#99C2212yes—
12C2×C4×M16—21no—
13#2322—22no—
14#2322—22no—
15#2322C414yescomplemented
16#2322C2214yes—
17C2×#389—21no—
18C2×#388—21no—
19C2×#387—21no—
20C2×#386C2211yescharacteristic
21C2×#385—21no—
22C2×#385C411yescharacteristic, complemented
23C2×#381C411yescharacteristic, complemented
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C411yescharacteristic, complemented
26C2×#352C4116yescomplemented
27C2×#352C2218yes—
28C4×#100—22no—
29C4×#100C2214yes—
30C4×#99—22no—
31C4×#99C2214yes—
32C2×#343—21no—
33C2×#342—21no—
34#1390C2218yes—
35C23×D4C2311yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C22×M16—22no—
41C2×#123—21no—
42C2×#123—21no—
43C2×#123—24no—
44C2×#123C2×C418yes—
45C2×#123C2×C412yes—
46C2×#123C2312yes—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4—42no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4—41no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4C2×C414yescomplemented
63C22×C4⋊C4D411yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42—41no—
73C22×C42D411yescharacteristic
74C2×#100—22no—
75C2×#100—24no—
76C2×#100C2×C418yes—
77C2×#100C2314yes—
78C2×#99—22no—
79C2×#99—24no—
80C2×#99C2×C418yes—
81C2×#99C2314yes—
82C4×M16—24no—
83C4×M16—42no—
84C2×#97—41no—
85C2×C4×C8—41no—
86#389—24no—
87#389—24no—
88#388—24no—
89#388—24no—
90#387—24no—
91#387—24no—
92#386—21no—
93#386—21no—
94#386C2×C412yes—
95#386C2312yes—
96#385—24no—
97#385—24no—
98#385—22no—
99#385C2×C414yescomplemented
100#381—22no—
101#381C2×C414yescomplemented
102C4×C22⋊C4—22no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—42no—
105C4×C22⋊C4C2×C414yescomplemented
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—41no—
112#352C2×C4132yescomplemented
113#343—24no—
114#342—24no—
115C25C22×C411yescharacteristic
116C25C2×D411yescharacteristic
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4C22×C411yescharacteristic
121C22×D4C22×C414yes—
122C22×D4C2×D411yescharacteristic
123C22×D4C2×D411yescharacteristic
124C22×D4#4114yes—
125C22×D4C2411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C4214yescomplemented
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C2×M16—28no—
161C2×M16—24no—
162C2×M16—44no—
163C2×M16—42no—
164C22×C8—42no—
165#123—24no—
166#123—28no—
167#123—28no—
168#123—24no—
169#123C4218yescomplemented
170#123C22×C418yes—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4C4214yescomplemented
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22×C412yes—
221C2×C22⋊C4C22×C412yes—
222C2×C22⋊C4C22×C412yes—
223C2×C22⋊C4C2×D412yes—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—44no—
229C2×C42—82no—
230C2×C42—84no—
231C2×C42C22⋊C412yes—
232C2×C42C2×D412yes—
233#100—28no—
234#100—24no—
235#100C22×C418yes—
236#99—28no—
237#99—24no—
238#99C22×C418yes—
239#97—44no—
240C4×C8—44no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—21no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C2×C4212yes—
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262C24#11712yes—
263C24C4×D414yes—
264C24C23×C411yescharacteristic
265C24C22×D411yescharacteristic
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—41no—
278C2×D4C2×C4212yes—
279C2×D4#11712yes—
280C2×D4C4×D414yes—
281C2×D4C4×D414yes—
282C2×D4C23×C412yes—
283C2×D4C2×#4112yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4#9911yescharacteristic
354C22×C4#9911yescharacteristic
355C22×C4#9912yes—
356C22×C4C2×C4214yescomplemented
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C4×D414yes—
363C22×C4C22×D411yescharacteristic
364M16—28no—
365M16—44no—
366M16—48no—
367C2×C8—44no—
368C2×C8—48no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—48no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—84no—
377C22⋊C4—22no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—24no—
381C22⋊C4—28no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—48no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4C2×C4214yescomplemented
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—84no—
398C42—88no—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—21no—
413C23—21no—
414C23—24no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—41no—
419C23—44no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—48no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—48no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—81no—
434C23—81no—
435C23—84no—
436C23C4×C22⋊C412yes—
437C23C4×C22⋊C412yes—
438C23C4×C22⋊C412yes—
439C23C2×#9911yescharacteristic
440C23C2×#9911yescharacteristic
441C23#41012yes—
442C23C22×C4211yescharacteristic
443C23C22×C22⋊C411yescharacteristic
444C23C2×#11711yescharacteristic
445C23C2×C4×D412yes—
446D4—42no—
447D4—42no—
448D4—44no—
449D4—44no—
450D4—42no—
451D4—42no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—816no—
506C2×C4—84no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4C4×C22⋊C412yes—
510C2×C4C2×#9912yes—
511C2×C4C2×#9912yes—
512C2×C4C2×C4×D412yes—
513C8—48no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—24no—
519C22—24no—
520C22—21no—
521C22—22no—
522C22—22no—
523C22—21no—
524C22—22no—
525C22—41no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—48no—
543C22—44no—
544C22—44no—
545C22—81no—
546C22—81no—
547C22—84no—
548C22C4×#9912yes—
549C22C4×#9912yes—
550C22#212112yes—
551C22#212312yes—
552C22C2×C4×C22⋊C411yescharacteristic
553C22C22×#9911yescharacteristic
554C22C2×#41011yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—44no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—82no—
565C4—88no—
566C4—82no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—42no—
571C2—41no—
572C2—42no—
573C2—41no—
574C2—42no—
575C2#1262114yesdirect product factor
576C2C2×C4×#9911yescharacteristic
577C2C2×#212111yescharacteristic
578C2C2×#212311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12621C2direct (N × H)
9. C23×C4⋊C4C4semidirect
15. #2322C4semidirect
22. C2×#385C4semidirect
23. C2×#381C4semidirect
25. C2×C4×C22⋊C4C4semidirect
26. C2×#352C4semidirect
62. C22×C4⋊C4C2×C4semidirect
99. #385C2×C4semidirect
101. #381C2×C4semidirect
105. C4×C22⋊C4C2×C4semidirect
112. #352C2×C4semidirect
148. C23×C4C42semidirect
169. #123C42semidirect
217. C2×C22⋊C4C42semidirect
356. C22×C4C2×C42semidirect
392. C22⋊C4C2×C42semidirect
575. C2#12621direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23222
C22×#3522
C2×C4×#1002
C2×C4×#992
#126212
C2×#13902