Finite Group Databasebeta

← C2×#12794 #493693

Subgroups — C2×#12794 #493693

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C22×#352C212yesmaximal
3C2×#1800C211yesmaximal, characteristic
4C2×#1799C211yesmaximal, characteristic
5#12794C218yesmaximal, direct product factor
6C2×#1390C212yesmaximal
7C22×#123C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C42C411yescharacteristic, complemented
10C2×#423—21no—
11C22×#100C2212yes—
12C22×#99C2212yes—
13C4×#123—22no—
14C4×#123—22no—
15C4×#123C414yescomplemented
16C4×#123C2214yes—
17C2×#388—21no—
18C2×#387—21no—
19C2×#386C2211yescharacteristic
20C2×#385C411yescharacteristic, complemented
21C2×#381C411yescharacteristic, complemented
22C2×#380—21no—
23C2×C4×C4⋊C4—21no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C411yescharacteristic, complemented
26C2×#352—28no—
27C2×#352C2218yes—
28#1800—22no—
29#1800C2214yes—
30#1799—22no—
31#1799C2214yes—
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—22no—
44C2×#123—22no—
45C2×#123C2×C412yes—
46C2×#123D414yes—
47C2×#123Q814yes—
48C2×#123C2312yes—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—42no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×C4×D4C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—41no—
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—42no—
72C22×C42—41no—
73C22×C42—41no—
74C22×C42C2×C414yescomplemented
75C22×C42D411yescharacteristic
76#423—24no—
77#423—42no—
78C2×C4⋊C8—41no—
79C2×C4⋊C8—41no—
80C2×#100—24no—
81C2×#100—24no—
82C2×#100—22no—
83C2×#100C2314yes—
84C2×#99—24no—
85C2×#99—24no—
86C2×#99—22no—
87C2×#99C2314yes—
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—22no—
97#385C2×C414yescomplemented
98#381—22no—
99#381C2×C414yescomplemented
100#380—24no—
101#380—42no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4C2×C414yescomplemented
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—41no—
114#352—216no—
115#343—24no—
116#342—24no—
117C25C22×C411yescharacteristic
118C25C2×D411yescharacteristic
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—42no—
122C22×D4C22×C411yescharacteristic
123C22×D4C2×D411yescharacteristic
124C22×D4C2×D412yes—
125C22×D4C2×D411yescharacteristic
126C22×D4C2×Q812yes—
127C22×D4#4112yes—
128C22×D4#4112yes—
129C22×D4C2411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—42no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C4⋊C414yescomplemented
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—44no—
167C2×M16—44no—
168C22×C8—42no—
169#123—24no—
170#123—24no—
171#123—24no—
172#123—24no—
173#123—24no—
174#123—22no—
175#123—22no—
176#123C4⋊C418yescomplemented
177#123C2×D414yes—
178#123C2×Q814yes—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—84no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
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—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C4⋊C414yescomplemented
221C2×C22⋊C4C22×C412yes—
222C2×C22⋊C4C22×C412yes—
223C2×C22⋊C4C22×C412yes—
224C2×C22⋊C4C2×D412yes—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42C22⋊C412yes—
242C2×C42C2×D412yes—
243C4⋊C8—44no—
244C4⋊C8—44no—
245#100—24no—
246#100—24no—
247#100—44no—
248#99—24no—
249#99—24no—
250#99—44no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—44no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24C2×C4⋊C412yes—
272C24#11712yes—
273C24#12112yes—
274C24C22⋊Q812yes—
275C24C23×C411yescharacteristic
276C24C22×D411yescharacteristic
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4C2×C4⋊C412yes—
294C2×D4#11712yes—
295C2×D4#12112yes—
296C2×D4#12112yes—
297C2×D4C22⋊Q812yes—
298C2×D4C22⋊Q812yes—
299C2×D4C22×D411yescharacteristic
300C2×D4C22×Q811yescharacteristic
301C2×D4C2×#4111yescharacteristic
302C2×D4C2×#4111yescharacteristic
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—44no—
331C22×C4—48no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—48no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—84no—
369C22×C4—82no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4#9912yes—
374C22×C4#9911yescharacteristic
375C22×C4#9911yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C4⋊C414yescomplemented
382C22×C4#12112yes—
383C22×C4C22⋊Q812yes—
384C22×C4C22×D411yescharacteristic
385M16—44no—
386M16—44no—
387M16—84no—
388C2×C8—44no—
389C2×C8—84no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—28no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—88no—
397C22⋊C4—22no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—28no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—84no—
413C22⋊C4C2×C4⋊C414yescomplemented
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—48no—
419C42—48no—
420C42—44no—
421C42—44no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—84no—
462C23#38012yes—
463C23#38012yes—
464C23#38012yes—
465C23C2×#9911yescharacteristic
466C23C2×#9911yescharacteristic
467C23#41012yes—
468C23C22×C22⋊C411yescharacteristic
469C23C22×C4⋊C411yescharacteristic
470C23C2×#11711yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×C22⋊Q811yescharacteristic
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—168no—
542C2×C4#35214yescomplemented
543C2×C4#38012yes—
544C2×C4C2×#9912yes—
545C2×C4C2×#9912yes—
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C8—84no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—24no—
554C22—24no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22#179912yes—
588C22#179912yes—
589C22C2×#35212yes—
590C22#212012yes—
591C22C2×#38011yescharacteristic
592C22C22×#9911yescharacteristic
593C22C2×#41011yescharacteristic
594C4—22no—
595C4—21no—
596C4—21no—
597C4—22no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—42no—
603C4—44no—
604C4—82no—
605C4—164no—
606C4C2×#35214yescomplemented
607C2—21no—
608C2—21no—
609C2—22no—
610C2—41no—
611C2—41no—
612C2—41no—
613C2—42no—
614C2—41no—
615C2—42no—
616C2#1279414yesdirect product factor
617C2C2×#179911yescharacteristic
618C2C22×#35211yescharacteristic
619C2C2×#212011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ 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. #12794C2direct (N × H)
9. C23×C42C4semidirect
15. C4×#123C4semidirect
20. C2×#385C4semidirect
21. C2×#381C4semidirect
25. C2×C4×C22⋊C4C4semidirect
74. C22×C42C2×C4semidirect
97. #385C2×C4semidirect
99. #381C2×C4semidirect
107. C4×C22⋊C4C2×C4semidirect
158. C23×C4C4⋊C4semidirect
176. #123C4⋊C4semidirect
220. C2×C22⋊C4C4⋊C4semidirect
381. C22×C4C2×C4⋊C4semidirect
413. C22⋊C4C2×C4⋊C4semidirect
542. C2×C4#352semidirect
606. C4C2×#352semidirect
616. C2#12794direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C22×#3522
C2×#18002
C2×#17992
#127942
C2×#13902