Finite Group Databasebeta

← C2×#30799 #7501876

Subgroups — C2×#30799 #7501876

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#481C211yesmaximal, characteristic
3C2×#2490C211yesmaximal, characteristic
4C2×#2440C211yesmaximal, characteristic, complemented
5#30799C2116yesmaximal, direct product factor
6C2×#2093C211yesmaximal, characteristic, complemented
7C2×#2076C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#2034C211yesmaximal, characteristic
11C2×#2023C211yesmaximal, characteristic, complemented
12C2×#1948C211yesmaximal, characteristic, complemented
13C2×#1936C211yesmaximal, characteristic, complemented
14C2×#1924C211yesmaximal, characteristic, complemented
15C2×#1919C211yesmaximal, characteristic, complemented
16C2×#1852C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×#121C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#501—28no—
23C2×#501C2218yescomplemented
24C2×#481—28no—
25C2×#481C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2490—24no—
41#2490C2218yes—
42#2440—24no—
43#2440C2218yescomplemented
44C2×#398—21no—
45C2×#397—21no—
46C2×#395—21no—
47C2×#394—21no—
48C2×#390—21no—
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2093C2218yescomplemented
61#2076C2218yescomplemented
62#2067C2218yescomplemented
63#2059C2218yescomplemented
64#2034C2218yes—
65#2023C2218yescomplemented
66#1948C2218yescomplemented
67#1936C2218yescomplemented
68#1924C2218yescomplemented
69#1919C2218yescomplemented
70#1852C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—42no—
91C2×C4⋊Q8C2314yes—
92C2×#125—24no—
93C2×#125—22no—
94C2×#125—42no—
95C2×#125C2314yes—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—48no—
98C2×#121—22no—
99C2×#121—24no—
100C2×#121—42no—
101C2×#121C2314yes—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—42no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#501—232no—
134#481—232no—
135C2×#107—24no—
136C2×#107—24no—
137C2×#107—42no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106—24no—
142C2×#106—24no—
143C2×#106D418yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2314yes—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102—24no—
152C2×#102—42no—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8—24no—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97—24no—
162C2×#97C2314yes—
163#398—28no—
164#397—28no—
165#395—28no—
166#394—28no—
167#390—28no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#388—28no—
177#385—28no—
178#384—22no—
179#384C2314yes—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4D414yes—
187C4×C22⋊C4C2314yes—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C25C2×D411yescharacteristic
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—82no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—82no—
220C22×D4C2412yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8C2×D414yes—
257C22×C8#4114yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265#125—28no—
266#125—44no—
267#125—48no—
268C22⋊Q8—432no—
269#121—28no—
270#121—44no—
271#121—48no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4C2×D414yes—
336C2×C22⋊C4C2×D414yes—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—82no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347#107—28no—
348#107—28no—
349#107—44no—
350#106—28no—
351#106—28no—
352#106C2×D4116yes—
353C4⋊C8—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—44no—
358C22⋊C8—28no—
359C22⋊C8—28no—
360#97—28no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24#13714yes—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—84no—
388C2×Q8—816no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—84no—
405C2×D4—816no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—84no—
464C22×C4—816no—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+412yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C22×C4C2511yescharacteristic
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8C22×D414yes—
494C2×C8C2×#4114yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—84no—
522C4⋊C4C22×D414yes—
523C4⋊C4C22×D414yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—84no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—28no—
545C23—28no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23#53612yes—
568C23D4212yes—
569C23#54712yes—
570C23C2×AGL(1,ℤ/8)12yes—
571C23C2×#13714yes—
572C23C2×#13712yes—
573C23#57612yes—
574C23#57712yes—
575C23C23×D411yescharacteristic
576Q8—816no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—816no—
582D4—84no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—816no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4#53612yes—
637C2×C4#53614yes—
638C2×C4#54214yes—
639C2×C4D4212yes—
640C2×C4#54712yes—
641C2×C4#54814yes—
642C2×C4#56314yes—
643C2×C4C2×AGL(1,ℤ/8)14yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C8—24no—
651C8—24no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—416no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—168no—
670C22#267414yes—
671C22#323914yes—
672C22#324114yes—
673C22#326614yes—
674C22#327414yes—
675C22#334514yes—
676C22#334614yes—
677C22C2×#53611yescharacteristic
678C22C2×D4211yescharacteristic
679C22C2×#54711yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C22×#13711yescharacteristic
682C22C2×#57611yescharacteristic
683C22C2×#57711yescharacteristic
684C4—24no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#267414yes—
697C4C2×#53611yescharacteristic
698C4C2×#54211yescharacteristic
699C4C2×#54811yescharacteristic
700C4C2×#56311yescharacteristic
701C2—44no—
702C2—162no—
703C2#3079918yesdirect product factor
704C2C2×#267411yescharacteristic
705C2C2×#323911yescharacteristic
706C2C2×#324111yescharacteristic
707C2C2×#326611yescharacteristic
708C2C2×#327411yescharacteristic
709C2C2×#334511yescharacteristic
710C2C2×#334611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
4. C2×#2440C2semidirect
5. #30799C2direct (N × H)
6. C2×#2093C2semidirect
7. C2×#2076C2semidirect
8. C2×#2067C2semidirect
9. C2×#2059C2semidirect
11. C2×#2023C2semidirect
12. C2×#1948C2semidirect
13. C2×#1936C2semidirect
14. C2×#1924C2semidirect
15. C2×#1919C2semidirect
16. C2×#1852C2semidirect
23. C2×#501C22semidirect
43. #2440C22semidirect
60. #2093C22semidirect
61. #2076C22semidirect
62. #2067C22semidirect
63. #2059C22semidirect
65. #2023C22semidirect
66. #1948C22semidirect
67. #1936C22semidirect
68. #1924C22semidirect
69. #1919C22semidirect
70. #1852C22semidirect
703. C2#30799direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4812
C2×#24902
C2×#24402
#307992
C2×#20932
C2×#20762
C2×#20672
C2×#20592
C2×#20342
C2×#20232
C2×#19482
C2×#19362
C2×#19242
C2×#19192
C2×#18522