Finite Group Databasebeta

← C2×#19378 #6354065

Subgroups — C2×#19378 #6354065

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3252C211yesmaximal, characteristic
2C2×#3218C211yesmaximal, characteristic
3C2×#3165C211yesmaximal, characteristic, complemented
4C2×#2969C211yesmaximal, characteristic
5#19378C2116yesmaximal, direct product factor
6C2×#1694C211yesmaximal, characteristic, complemented
7C2×#1692C211yesmaximal, characteristic, complemented
8C2×#1693C211yesmaximal, characteristic, complemented
9C2×#1691C211yesmaximal, characteristic, complemented
10C2×#1674C211yesmaximal, characteristic, complemented
11C2×#1672C211yesmaximal, characteristic, complemented
12C2×#1673C211yesmaximal, characteristic, complemented
13C2×#1671C211yesmaximal, characteristic, complemented
14C2×#1569C211yesmaximal, characteristic, complemented
15C2×#1554C211yesmaximal, characteristic, complemented
16C2×#1546C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3252—24no—
21#3252C2218yes—
22#3218—24no—
23#3218C2218yes—
24#3165—24no—
25#3165C2218yescomplemented
26C2×#501—21no—
27C2×#501—21no—
28C2×#500C2211yescharacteristic
29C2×#500C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#498C2211yescharacteristic
32C2×#497—21no—
33C2×#491—21no—
34C2×#479—21no—
35C2×#477C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#470—21no—
38C2×#468—21no—
39C2×#464—21no—
40C2×#462C2211yescharacteristic
41C2×#462C2211yescharacteristic
42C2×#462C2211yescharacteristic
43C2×#461C2211yescharacteristic
44C2×#461C2211yescharacteristic
45C2×#461C2211yescharacteristic
46C2×#451—21no—
47C2×#448—21no—
48#2969—24no—
49#2969C2218yes—
50C2×#446C2211yescharacteristic
51C2×C8×Q8C2211yescharacteristic
52C2×#436—21no—
53C2×#436C2211yescharacteristic
54C2×#435—21no—
55C2×C8×D4C2211yescharacteristic
56C2×#433—21no—
57C2×#432C2211yescharacteristic
58C2×#431C2211yescharacteristic
59C2×#428—21no—
60C2×#428—21no—
61C2×#423C2211yescharacteristic
62C2×#405—21no—
63C2×C4×M16C2211yescharacteristic
64#1694C2218yescomplemented
65#1693C2218yescomplemented
66#1692C2218yescomplemented
67#1691C2218yescomplemented
68#1674C2218yescomplemented
69#1673C2218yescomplemented
70#1672C2218yescomplemented
71#1671C2218yescomplemented
72#1569C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1554C2218yescomplemented
78#1546C2218yescomplemented
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#332C2211yescharacteristic
86C2×#332C2211yescharacteristic
87C23×Q8—21no—
88C22×Q16—21no—
89C22×Q16—21no—
90C22×Q16—21no—
91C22×SD16—21no—
92C22×SD16—21no—
93C22×SD16—21no—
94C22×M16—21no—
95C22×M16—21no—
96C22×M16D411yescharacteristic
97C22×M16D411yescharacteristic
98C23×C8—21no—
99#533—22no—
100#533—22no—
101#533—22no—
102#533C2314yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2314yes—
114#529—22no—
115#529—22no—
116#529D414yes—
117#529C2314yes—
118C2×#126—21no—
119C2×#125—21no—
120C2×#125D411yescharacteristic
121C2×#124D411yescharacteristic
122C2×#123—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8D411yescharacteristic
126C2×#121D411yescharacteristic
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4C2311yescharacteristic
132C2×#117—21no—
133C2×#117—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C42C2311yescharacteristic
137#501—28no—
138#501—28no—
139#500—22no—
140#500—22no—
141#500C2314yes—
142#500C2314yes—
143#498—22no—
144#498—22no—
145#498C2314yes—
146#498C2314yes—
147#497—28no—
148#491—28no—
149#479—28no—
150#477—22no—
151#477C2314yes—
152#476—22no—
153#476C2314yes—
154#470—28no—
155#468—28no—
156#464—28no—
157#462—22no—
158#462—22no—
159#462—22no—
160#462C2314yes—
161#462C2314yes—
162#462C2314yes—
163#461—22no—
164#461—22no—
165#461—22no—
166#461C2314yes—
167#461C2314yes—
168#461C2314yes—
169#451—28no—
170#448—28no—
171#446—22no—
172#446C2314yes—
173C8×Q8—22no—
174C8×Q8C2314yes—
175#436—28no—
176#436—22no—
177#436C2314yes—
178#435—28no—
179C8×D4—22no—
180C8×D4C2314yes—
181#433—28no—
182#432—22no—
183#432C2314yes—
184#431—22no—
185#431C2314yes—
186#428—28no—
187#428—28no—
188C2×#107—21no—
189C2×#107—21no—
190C2×#107—21no—
191C2×#107—21no—
192C2×#107—21no—
193C2×#106—21no—
194C2×#106—21no—
195C2×#106—21no—
196C2×#106—21no—
197C2×#106—21no—
198#423—22no—
199#423C2314yes—
200C2×C4⋊C8—21no—
201C2×C4⋊C8—21no—
202C2×C4⋊C8—21no—
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2311yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2311yescharacteristic
209C2×#103—21no—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#102—21no—
217C2×C22⋊C8—21no—
218C2×C22⋊C8—21no—
219C2×C22⋊C8—21no—
220C2×C22⋊C8—21no—
221#405—28no—
222C4×M16D414yes—
223C4×M16C2314yes—
224C2×#97—21no—
225C2×#97—21no—
226C2×#97D411yescharacteristic
227C2×#97D411yescharacteristic
228C2×C4×C8—21no—
229C2×C4×C8—21no—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232#335C2314yes—
233#335C2314yes—
234#334C2314yes—
235#334C2314yes—
236#333C2314yes—
237#333C2314yes—
238#332C2314yes—
239#332C2314yes—
240#332C2314yes—
241#332C2314yes—
242#332C2314yes—
243#332C2314yes—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—21no—
249C22×Q8—42no—
250C22×Q8—41no—
251C22×Q8—82no—
252C22×Q8—82no—
253C22×Q8C2×D411yescharacteristic
254C22×D4—21no—
255C22×D4C2×D411yescharacteristic
256C23×C4—21no—
257C23×C4—41no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4#4111yescharacteristic
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—24no—
264C2×Q16—82no—
265C2×Q16—82no—
266C2×Q16—82no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—24no—
270C2×SD16—82no—
271C2×SD16—82no—
272C2×SD16—82no—
273C2×M16—24no—
274C2×M16—24no—
275C2×M16—24no—
276C2×M16—24no—
277C2×M16—44no—
278C2×M16—82no—
279C2×M16C2×D414yes—
280C2×M16C2×D414yes—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—24no—
288C22×C8—44no—
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—24no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—24no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—24no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—42no—
309C4⋊Q8—42no—
310C4⋊Q8—42no—
311C4⋊Q8C2412yes—
312C4⋊Q8C2412yes—
313C4⋊Q8C2412yes—
314C4⋊Q8C2412yes—
315#126—24no—
316#126—24no—
317#125—22no—
318#125—24no—
319#125—42no—
320#125C2×D414yes—
321#124—22no—
322#124C2×D414yes—
323#123—24no—
324#123—24no—
325C22⋊Q8—22no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328C22⋊Q8—42no—
329C22⋊Q8—42no—
330C22⋊Q8C2×D414yes—
331#121—22no—
332#121C2×D414yes—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—24no—
336C4×Q8—24no—
337C4×Q8—42no—
338C4×Q8—42no—
339C4×Q8C2×D412yes—
340C4×Q8C2412yes—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4C2×D412yes—
346C4×D4C2412yes—
347#117—24no—
348#117—24no—
349#117—24no—
350#117—42no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2×D411yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—41no—
379C2×C42—21no—
380C2×C42—41no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42#4111yescharacteristic
386C2×C42#4112yes—
387C2×C42C2411yescharacteristic
388C2×C42C2412yes—
389#107—24no—
390#107—24no—
391#107—24no—
392#107—24no—
393#107—24no—
394#106—24no—
395#106—24no—
396#106—24no—
397#106—24no—
398#106—24no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8C2412yes—
409C4⋊C8C2412yes—
410C4⋊C8C2412yes—
411C4⋊C8C2412yes—
412C4⋊C8C2412yes—
413C4⋊C8C2412yes—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421#102—24no—
422C22⋊C8—24no—
423C22⋊C8—24no—
424C22⋊C8—24no—
425C22⋊C8—24no—
426#97—24no—
427#97—24no—
428#97C2×D414yes—
429#97C2×D414yes—
430C4×C8—24no—
431C4×C8—24no—
432C4×C8C2×D412yes—
433C4×C8C2×D412yes—
434C4×C8C2412yes—
435C4×C8C2412yes—
436C24—21no—
437C24#12111yescharacteristic
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—44no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—84no—
457C2×Q8—84no—
458C2×Q8—84no—
459C2×Q8—88no—
460C2×Q8—82no—
461C2×Q8C22×D412yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—82no—
467C2×D4C22×D412yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—84no—
486C22×C4#12112yes—
487C22×C4#12112yes—
488C22×C4#12111yescharacteristic
489C22×C4#12111yescharacteristic
490C22×C4#12111yescharacteristic
491C22×C4#12112yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4111yescharacteristic
497C22×C4C2×#4112yes—
498Q16—84no—
499Q16—84no—
500Q16—84no—
501SD16—84no—
502SD16—84no—
503SD16—84no—
504M16—28no—
505M16—28no—
506M16—48no—
507M16—84no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—42no—
517C2×C8—42no—
518C2×C8—42no—
519C2×C8—42no—
520C2×C8—42no—
521C2×C8—42no—
522C2×C8—48no—
523C2×C8#13714yes—
524C2×C8#13714yes—
525C2×C8C22×D412yes—
526C2×C8C22×D412yes—
527C2×C8C22×D412yes—
528C2×C8C22×D412yes—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4C22×D412yes—
557C4⋊C4C22×D412yes—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—44no—
565C42—22no—
566C42—22no—
567C42—44no—
568C42#12112yes—
569C42#12112yes—
570C42#12112yes—
571C42C22×D412yes—
572C42C22×D411yescharacteristic
573C42C22×D412yes—
574C42C22×D411yescharacteristic
575C42C2×#4111yescharacteristic
576C42C2×#4112yes—
577C42C2511yescharacteristic
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C2×#12112yes—
586C23C2×#12111yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—82no—
597Q8—82no—
598Q8—88no—
599Q8—82no—
600Q8—82no—
601Q8—82no—
602D4—22no—
603D4—22no—
604D4—42no—
605D4—84no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—88no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12112yes—
640C2×C4C2×#12111yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#13712yes—
643C2×C4C2×#13712yes—
644C2×C4#57612yes—
645C2×C4#57712yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C8—22no—
650C8—22no—
651C8—42no—
652C8—42no—
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C8C2×#13714yes—
658C8C2×#13714yes—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—84no—
667C22#303912yes—
668C22#306212yes—
669C22#306912yes—
670C22C22×#12111yescharacteristic
671C4—22no—
672C4—22no—
673C4—21no—
674C4—21no—
675C4—22no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#303912yes—
684C4#306212yes—
685C4C22×#13711yescharacteristic
686C4C22×#13711yescharacteristic
687C4C2×#57611yescharacteristic
688C4C2×#57711yescharacteristic
689C2—42no—
690C2—82no—
691C2#1937814yesdirect product factor
692C2C2×#303911yescharacteristic
693C2C2×#306211yescharacteristic
694C2C2×#306911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#3165C2semidirect
5. #19378C2direct (N × H)
6. C2×#1694C2semidirect
7. C2×#1692C2semidirect
8. C2×#1693C2semidirect
9. C2×#1691C2semidirect
10. C2×#1674C2semidirect
11. C2×#1672C2semidirect
12. C2×#1673C2semidirect
13. C2×#1671C2semidirect
14. C2×#1569C2semidirect
15. C2×#1554C2semidirect
16. C2×#1546C2semidirect
25. #3165C22semidirect
64. #1694C22semidirect
65. #1693C22semidirect
66. #1692C22semidirect
67. #1691C22semidirect
68. #1674C22semidirect
69. #1673C22semidirect
70. #1672C22semidirect
71. #1671C22semidirect
72. #1569C22semidirect
77. #1554C22semidirect
78. #1546C22semidirect
691. C2#19378direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32522
C2×#32182
C2×#31652
C2×#29692
#193782
C2×#16942
C2×#16922
C2×#16932
C2×#16912
C2×#16742
C2×#16722
C2×#16732
C2×#16712
C2×#15692
C2×#15542
C2×#15462