Finite Group Databasebeta

← C2×#22560505 #30856954

Subgroups — C2×#22560505 #30856954

Order 17152.

2286 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11761611C211yesmaximal, characteristic, complemented
2C22×C4×D268C211yesmaximal, characteristic
3C22×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
4C22×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
5C22×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
6#22560505C218yesmaximal, direct product factor
7C2×#16863151C211yesmaximal, characteristic, complemented
8C2×#336×C67C211yesmaximal, characteristic, complemented
9C23×D268C411yescharacteristic
10C23×C4×D67C2211yescharacteristic
11C23×C4×D67C2211yescharacteristic
12C22×#11761611—24no—
13C22×#11761611—24no—
14C22×#11761611—24no—
15C22×#11761611C2214yescomplemented
16C23×C67⋊C8C2211yescharacteristic
17C23×C67⋊C8C2211yescharacteristic
18C2×C4×D268—24no—
19C2×C4×D268—24no—
20C2×C4×D268C414yes—
21C2×C4×D268C2214yes—
22C22×C4⋊Dic67C411yescharacteristic
23C22×(C2×D67)⋊C4C411yescharacteristic
24C22×(C2×D67)⋊C4C411yescharacteristic
25C2×(C2×D67)⋊C8—24no—
26C2×(C2×D67)⋊C8—24no—
27C2×(C2×D67)⋊C8—24no—
28C2×(C2×D67)⋊C8C2214yescomplemented
29C2×(C2×D67)⋊C8C2214yescomplemented
30C2×(C2×D67)⋊C8C2214yescomplemented
31#16863151C2214yescomplemented
32C23×C8×C67C2211yescharacteristic
33C23×C8×C67C2211yescharacteristic
34C22×C42×C67C2211yescharacteristic
35#336×C67C2214yescomplemented
36C24×D67C2×C411yescharacteristic
37C24×D67C2×C411yescharacteristic
38C22×D268—24no—
39C22×D268—24no—
40C22×D268—24no—
41C22×D268C2×C414yes—
42C23×Dic67C2×C411yescharacteristic
43C23×Dic67C2×C411yescharacteristic
44C22×C4×D67—22no—
45C22×C4×D67—22no—
46C22×C4×D67—22no—
47C22×C4×D67—22no—
48C22×C4×D67—22no—
49C22×C4×D67—22no—
50C22×C4×D67—22no—
51C22×C4×D67—22no—
52C22×C4×D67—22no—
53C22×C4×D67—22no—
54C22×C4×D67C2×C412yes—
55C22×C4×D67C2×C412yes—
56C22×C4×D67D414yes—
57C22×C4×D67D414yes—
58C22×C4×D67C2312yes—
59C22×C4×D67C2312yes—
60C2×#11761611—232no—
61C2×#11761611—28no—
62C2×#11761611—28no—
63C2×#11761611—28no—
64C22×C67⋊C8—22no—
65C22×C67⋊C8—22no—
66C22×C67⋊C8—22no—
67C22×C67⋊C8—22no—
68C22×C67⋊C8—22no—
69C22×C67⋊C8D414yescomplemented
70C22×C67⋊C8C2312yes—
71C22×C67⋊C8C2312yes—
72C4×D268—216no—
73C4×D268—28no—
74C4×D268—28no—
75C2×C4⋊Dic67—24no—
76C2×C4⋊Dic67C2×C414yes—
77C2×(C2×D67)⋊C4—24no—
78C2×(C2×D67)⋊C4—24no—
79C2×(C2×D67)⋊C4C2×C414yes—
80C2×(C2×D67)⋊C4C2×C414yes—
81(C2×D67)⋊C8—28no—
82(C2×D67)⋊C8—28no—
83(C2×D67)⋊C8—28no—
84C23×C4×C67C2×C411yescharacteristic
85C23×C4×C67C2×C411yescharacteristic
86C23×C4×C67C2311yescharacteristic
87C22×C8×C67—22no—
88C22×C8×C67—22no—
89C22×C8×C67—22no—
90C22×C8×C67—22no—
91C22×C8×C67—22no—
92C22×C8×C67D414yescomplemented
93C22×C8×C67C2312yes—
94C22×C8×C67C2312yes—
95C2×C42×C67—22no—
96C2×C42×C67—22no—
97C2×C42×C67C2×C412yes—
98C2×C42×C67C2312yes—
99C23×D67—22no—
100C23×D67—22no—
101C23×D67—22no—
102C23×D67—22no—
103C23×D67—22no—
104C23×D67—22no—
105C23×D67—22no—
106C23×D67—22no—
107C23×D67—22no—
108C23×D67—22no—
109C23×D67—22no—
110C23×D67—22no—
111C23×D67C22⋊C414yes—
112C23×D67M1614yescomplemented
113C23×D67C22×C412yes—
114C23×D67C22×C412yes—
115C2×D268—232no—
116C2×D268—28no—
117C2×D268—28no—
118C2×D268—28no—
119C22×Dic67—22no—
120C22×Dic67—22no—
121C22×Dic67—22no—
122C22×Dic67—22no—
123C22×Dic67—22no—
124C22×Dic67C22⋊C414yes—
125C22×Dic67C22×C412yes—
126C22×Dic67C22×C412yes—
127C2×C4×D67—24no—
128C2×C4×D67—24no—
129C2×C4×D67—28no—
130C2×C4×D67—28no—
131C2×C4×D67—28no—
132C2×C4×D67—28no—
133C2×C4×D67—22no—
134C2×C4×D67—22no—
135C2×C4×D67—22no—
136C2×C4×D67—22no—
137C2×C4×D67—24no—
138C2×C4×D67—24no—
139C2×C4×D67—24no—
140C2×C4×D67—22no—
141C2×C4×D67—22no—
142C2×C4×D67—22no—
143C2×C4×D67—22no—
144C2×C4×D67—22no—
145C2×C4×D67—22no—
146C2×C4×D67—44no—
147C2×C4×D67—44no—
148C2×C4×D67—44no—
149C2×C4×D67—44no—
150C2×C4×D67C22⋊C418yes—
151C2×C4×D67C2×D414yes—
152C2×C4×D67C2×D414yes—
153#11761611—232no—
154C2×C67⋊C8—28no—
155C2×C67⋊C8—22no—
156C2×C67⋊C8—22no—
157C2×C67⋊C8—22no—
158C2×C67⋊C8—22no—
159C2×C67⋊C8—22no—
160C2×C67⋊C8—44no—
161C2×C67⋊C8C2×D414yescomplemented
162C4⋊Dic67—28no—
163(C2×D67)⋊C4—28no—
164(C2×D67)⋊C4—28no—
165C24×C67C22×C411yescharacteristic
166C22×C4×C67—22no—
167C22×C4×C67—22no—
168C22×C4×C67—22no—
169C22×C4×C67—22no—
170C22×C4×C67—22no—
171C22×C4×C67M1614yescomplemented
172C22×C4×C67C22×C412yes—
173C22×C4×C67C22×C412yes—
174C22×C4×C67C22×C411yescharacteristic
175C22×C4×C67C2×D412yes—
176C22×C4×C67C2×D412yes—
177C22×C4×C67C2×D412yes—
178C22×C4×C67C2×D412yes—
179C22×C4×C67#4112yes—
180C22×C4×C67#4112yes—
181C22×C4×C67C2411yescharacteristic
182C2×C8×C67—28no—
183C2×C8×C67—22no—
184C2×C8×C67—22no—
185C2×C8×C67—22no—
186C2×C8×C67—22no—
187C2×C8×C67—22no—
188C2×C8×C67—44no—
189C2×C8×C67C2×D414yescomplemented
190C42×C67—24no—
191C42×C67—22no—
192C42×C67—22no—
193C23×C67C2×C22⋊C412yes—
194C23×C67C4×D412yes—
195C23×C67C4×D412yes—
196C23×C67C2×M1612yes—
197C23×C67C2×M1612yes—
198C23×C67#13112yes—
199C23×C67#13112yes—
200C23×C67C23×C411yescharacteristic
201C22×D67—22no—
202C22×D67—22no—
203C22×D67—22no—
204C22×D67—22no—
205C22×D67—22no—
206C22×D67—22no—
207C22×D67—22no—
208C22×D67—22no—
209C22×D67—22no—
210C22×D67—22no—
211C22×D67—22no—
212C22×D67—22no—
213C22×D67—28no—
214C22×D67—28no—
215C22×D67—28no—
216C22×D67—28no—
217C22×D67—28no—
218C22×D67—28no—
219C22×D67—44no—
220C22×D67—44no—
221C22×D67—44no—
222C22×D67—44no—
223C22×D67—44no—
224C22×D67—44no—
225C22×D67—44no—
226C22×D67—44no—
227C22×D67C2×C22⋊C414yes—
228C22×D67C2×M1614yescomplemented
229C2×C4×C67—22no—
230C2×C4×C67—22no—
231C2×C4×C67—28no—
232C2×C4×C67—22no—
233C2×C4×C67—22no—
234C2×C4×C67—22no—
235C2×C4×C67—44no—
236C2×C4×C67C2×C22⋊C412yes—
237C2×C4×C67C4×D412yes—
238C2×C4×C67C4×D412yes—
239C2×C4×C67#12014yes—
240C2×C4×C67#12114yes—
241C2×C4×C67#12114yes—
242C2×C4×C67#12314yes—
243C2×C4×C67C2×M1614yescomplemented
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67C2×#4111yescharacteristic
249C2×C4×C67C2×#4111yescharacteristic
250C2×Dic67—22no—
251C2×Dic67—22no—
252C2×Dic67—28no—
253C2×Dic67—22no—
254C2×Dic67—22no—
255C2×Dic67—22no—
256C2×Dic67—44no—
257C2×Dic67C2×C22⋊C414yes—
258D268—232no—
259C4×D67—24no—
260C4×D67—24no—
261C4×D67—24no—
262C4×D67—24no—
263C4×D67—216no—
264C4×D67—42no—
265C4×D67—42no—
266C4×D67—42no—
267C4×D67—42no—
268C4×D67—48no—
269C8×C67—24no—
270C8×C67—42no—
271C67⋊C8—24no—
272C67⋊C8—42no—
273C22×C67#38614yes—
274C22×C67#40714yes—
275C22×C67#40814yes—
276C22×C67#43514yes—
277C22×C67#43514yes—
278C22×C67#43614yes—
279C22×C67#43614yes—
280C22×C67C22×C22⋊C411yescharacteristic
281C22×C67C2×C4×D411yescharacteristic
282C22×C67C2×C4×D411yescharacteristic
283C22×C67C22×M1611yescharacteristic
284C22×C67C22×M1611yescharacteristic
285C22×C67C2×#13111yescharacteristic
286C22×C67C2×#13111yescharacteristic
287C2×D67—24no—
288C2×D67—24no—
289C2×D67—24no—
290C2×D67—24no—
291C2×D67—24no—
292C2×D67—24no—
293C2×D67—42no—
294C2×D67—416no—
295C2×D67—416no—
296C2×D67—42no—
297C2×D67—42no—
298C2×D67—42no—
299C2×D67—42no—
300C2×D67—42no—
301C2×D67—42no—
302C2×D67—42no—
303C4×C67—24no—
304C4×C67—42no—
305C4×C67#38614yes—
306C4×C67C2×#12011yescharacteristic
307C4×C67C2×#12111yescharacteristic
308C4×C67C2×#12111yescharacteristic
309C4×C67C2×#12311yescharacteristic
310Dic67—24no—
311Dic67—42no—
312C2×#1870—671nomaximal, Sylow
313C2×C67#187018yescomplemented
314C2×C67C2×#38611yescharacteristic
315C2×C67C2×#40711yescharacteristic
316C2×C67C2×#40811yescharacteristic
317C2×C67C2×#43511yescharacteristic
318C2×C67C2×#43511yescharacteristic
319C2×C67C2×#43611yescharacteristic
320C2×C67C2×#43611yescharacteristic
321D67—44no—
322D67—44no—
323C23×M16—671no—
324C22×C4×D4—671no—
325C22×C22⋊C8—671no—
326C22×C22⋊C8—671no—
327C22×C22⋊C8—671no—
328#1870—678no—
329C2×#336—671no—
330C2×#336D6711yescharacteristic, complemented
331C67C2×#187011yescharacteristic, complemented, Sylow
332C23×D4—671no—
333C24×C4—671no—
334C24×C4—671no—
335C22×M16—674no—
336C22×M16—1344no—
337C22×M16—1344no—
338C22×M16—1344no—
339C23×C8—671no—
340C23×C8—671no—
341C23×C8C2×D6711yescharacteristic
342C23×C8C2×D6711yescharacteristic
343C2×C4×D4—674no—
344C2×C4×D4—674no—
345C2×C4×D4—1344no—
346C2×C4×D4—1344no—
347C22×C4⋊C4—671no—
348C22×C22⋊C4—671no—
349C22×C22⋊C4—671no—
350C22×C42C2×D6711yescharacteristic
351C2×C22⋊C8—674no—
352C2×C22⋊C8—674no—
353C2×C22⋊C8—674no—
354C2×C22⋊C8—1344no—
355C2×C22⋊C8—1344no—
356C2×C22⋊C8—1344no—
357#336—674no—
358#336C2×D6714yescomplemented
359C25—671no—
360C25—671no—
361C22×D4—674no—
362C22×D4—1344no—
363C22×D4—1344no—
364C22×D4—1344no—
365C23×C4—671no—
366C23×C4—672no—
367C23×C4—674no—
368C23×C4—674no—
369C23×C4—672no—
370C23×C4—671no—
371C23×C4—672no—
372C23×C4—672no—
373C23×C4—1342no—
374C23×C4—1342no—
375C23×C4—1342no—
376C23×C4—1342no—
377C23×C4—1342no—
378C23×C4—1342no—
379C23×C4—1342no—
380C23×C4—1342no—
381C23×C4—1342no—
382C23×C4—1342no—
383C23×C4C4×D6711yescharacteristic
384C23×C4C4×D6711yescharacteristic
385C23×C4C22×D6711yescharacteristic
386C2×M16—1348no—
387C2×M16—1348no—
388C2×M16—13432no—
389C2×M16—1348no—
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—674no—
396C22×C8—672no—
397C22×C8—672no—
398C22×C8—1342no—
399C22×C8—1342no—
400C22×C8—1342no—
401C22×C8—1342no—
402C22×C8—1342no—
403C22×C8D26814yescomplemented
404C22×C8C22×D6712yes—
405C22×C8C22×D6712yes—
406C4×D4—13416no—
407C4×D4—1348no—
408C4×D4—1348no—
409C2×C4⋊C4—674no—
410C2×C4⋊C4—1344no—
411C2×C22⋊C4—674no—
412C2×C22⋊C4—674no—
413C2×C22⋊C4—1344no—
414C2×C22⋊C4—1344no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42C4×D6712yes—
418C2×C42C22×D6712yes—
419C22⋊C8—1348no—
420C22⋊C8—1348no—
421C22⋊C8—1348no—
422C24—674no—
423C24—674no—
424C24—672no—
425C24—672no—
426C24—1342no—
427C24—1342no—
428C24—1342no—
429C24—1342no—
430C24—1342no—
431C24—1342no—
432C24—1342no—
433C24—1342no—
434C24—1342no—
435C24—1342no—
436C24—1342no—
437C24—1342no—
438C24C2×C4×D6711yescharacteristic
439C2×D4—1348no—
440C2×D4—1348no—
441C2×D4—13432no—
442C2×D4—1348no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—2684no—
449C22×C4—2684no—
450C22×C4—2684no—
451C22×C4—2684no—
452C22×C4—674no—
453C22×C4—672no—
454C22×C4—678no—
455C22×C4—674no—
456C22×C4—674no—
457C22×C4—672no—
458C22×C4—1342no—
459C22×C4—1342no—
460C22×C4—1342no—
461C22×C4—1344no—
462C22×C4—1344no—
463C22×C4—1344no—
464C22×C4—1342no—
465C22×C4—1342no—
466C22×C4—1342no—
467C22×C4—1342no—
468C22×C4—1342no—
469C22×C4—1342no—
470C22×C4—1342no—
471C22×C4—1342no—
472C22×C4—1344no—
473C22×C4—1344no—
474C22×C4—1348no—
475C22×C4—1348no—
476C22×C4—1348no—
477C22×C4—1348no—
478C22×C4—1342no—
479C22×C4—1342no—
480C22×C4—1342no—
481C22×C4—1342no—
482C22×C4#1176161114yescomplemented
483C22×C4C2×C4×D6712yes—
484C22×C4C2×C4×D6712yes—
485C22×C4C2×C4×D6711yescharacteristic
486C22×C4D4×D6712yes—
487C22×C4D4×D6712yes—
488C22×C4D4×D6712yes—
489C22×C4C2×D26812yes—
490C22×C4#1176162812yes—
491C22×C4#1176162912yes—
492C22×C4C23×D6711yescharacteristic
493M16—13432no—
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8—2684no—
502C2×C8—674no—
503C2×C8—1342no—
504C2×C8—1342no—
505C2×C8—1342no—
506C2×C8—1348no—
507C2×C8—1342no—
508C2×C8—1342no—
509C2×C8C2×D26814yescomplemented
510C4⋊C4—1348no—
511C22⋊C4—1348no—
512C22⋊C4—1348no—
513C42—24no—
514C42—22no—
515C42—22no—
516C23—2684no—
517C23—2684no—
518C23—2684no—
519C23—2684no—
520C23—2684no—
521C23—2684no—
522C23—2684no—
523C23—2684no—
524C23—674no—
525C23—674no—
526C23—1348no—
527C23—1342no—
528C23—1342no—
529C23—1342no—
530C23—1342no—
531C23—1342no—
532C23—1348no—
533C23—1348no—
534C23—1348no—
535C23—1342no—
536C23—1342no—
537C23—1342no—
538C23—1342no—
539C23—1342no—
540C23—1342no—
541C23—1342no—
542C23—1348no—
543C23—1348no—
544C23C22⋊C4×D6712yes—
545C23#1452400512yes—
546C23C4×D26812yes—
547C23M16×D6712yes—
548C23C2×#1176161112yes—
549C23#1452404912yes—
550C23#1452405112yes—
551C23C22×C4×D6711yescharacteristic
552D4—13432no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—2684no—
561C2×C4—2688no—
562C2×C4—2682no—
563C2×C4—2682no—
564C2×C4—2682no—
565C2×C4—2682no—
566C2×C4—674no—
567C2×C4—1342no—
568C2×C4—1342no—
569C2×C4—1342no—
570C2×C4—1348no—
571C2×C4—1342no—
572C2×C4—1342no—
573C2×C4—13416no—
574C2×C4—1344no—
575C2×C4—1344no—
576C2×C4—1344no—
577C2×C4—1344no—
578C2×C4C22⋊C4×D6712yes—
579C2×C4#1452400512yes—
580C2×C4C4×D26812yes—
581C2×C4#1452401314yes—
582C2×C4#1452401614yes—
583C2×C4#1452401814yes—
584C2×C4#1452402814yes—
585C2×C4C2×#1176161114yescomplemented
586C2×C4C2×D4×D6711yescharacteristic
587C2×C4C2×D4×D6711yescharacteristic
588C2×C4C2×D4×D6711yescharacteristic
589C2×C4C22×D26811yescharacteristic
590C2×C4C2×#1176162811yescharacteristic
591C2×C4C2×#1176162911yescharacteristic
592C8—24no—
593C8—42no—
594C8—2682no—
595C8—1344no—
596C22—2682no—
597C22—2682no—
598C22—2682no—
599C22—2682no—
600C22—2682no—
601C22—2682no—
602C22—2682no—
603C22—2682no—
604C22—26816no—
605C22—26816no—
606C22—1344no—
607C22—1344no—
608C22—1344no—
609C22—1344no—
610C22—1344no—
611C22—1344no—
612C22#1686327414yes—
613C22#1686335814yes—
614C22#1686336114yes—
615C22#1686346614yes—
616C22#1686346914yes—
617C22#1686347314yes—
618C22#1686347514yes—
619C22C2×C22⋊C4×D6711yescharacteristic
620C22C2×#1452400511yescharacteristic
621C22C2×C4×D26811yescharacteristic
622C22C2×M16×D6711yescharacteristic
623C22C22×#1176161111yescharacteristic
624C22C2×#1452404911yescharacteristic
625C22C2×#1452405111yescharacteristic
626C4—24no—
627C4—42no—
628C4—2682no—
629C4—1344no—
630C4#1686327414yes—
631C4C2×#1452401311yescharacteristic
632C4C2×#1452401611yescharacteristic
633C4C2×#1452401811yescharacteristic
634C4C2×#1452402811yescharacteristic
635C2—2684no—
636C2—2684no—
637C2#2256050518yesdirect product factor
638C2C2×#1686327411yescharacteristic
639C2C2×#1686335811yescharacteristic
640C2C2×#1686336111yescharacteristic
641C2C2×#1686346611yescharacteristic
642C2C2×#1686346911yescharacteristic
643C2C2×#1686347311yescharacteristic
644C2C2×#1686347511yescharacteristic
subgroupG/H
centerC23×C4C22×D67
commutatorC23×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#336×C67C2
socleC24×C67C22×C4
prime radicalC23×D268C4
cosocleC22×C4×C67C24
hypercenterC2×#336D67
hypocenterC67C2×#1870
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23×C4 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
1. C23×#11761611C2semidirect
3. C22×(C2×D67)⋊C8C2semidirect
4. C22×(C2×D67)⋊C8C2semidirect
5. C22×(C2×D67)⋊C8C2semidirect
6. #22560505C2direct (N × H)
7. C2×#16863151C2semidirect
8. C2×#336×C67C2semidirect
15. C22×#11761611C22semidirect
28. C2×(C2×D67)⋊C8C22semidirect
29. C2×(C2×D67)⋊C8C22semidirect
30. C2×(C2×D67)⋊C8C22semidirect
31. #16863151C22semidirect
35. #336×C67C22semidirect
69. C22×C67⋊C8D4semidirect
92. C22×C8×C67D4semidirect
112. C23×D67M16semidirect
161. C2×C67⋊C8C2×D4semidirect
171. C22×C4×C67M16semidirect
189. C2×C8×C67C2×D4semidirect
228. C22×D67C2×M16semidirect
243. C2×C4×C67C2×M16semidirect
313. C2×C67#1870semidirect
330. C2×#336D67semidirect
331. C67C2×#1870semidirect
358. #336C2×D67semidirect
403. C22×C8D268semidirect
482. C22×C4#11761611semidirect
509. C2×C8C2×D268semidirect
585. C2×C4C2×#11761611semidirect
637. C2#22560505direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616112
C22×C4×D2682
C22×(C2×D67)⋊C82
#225605052
C2×#168631512
C2×#336×C672
C2×#187067