Finite Group Databasebeta

← C4×#16863693 #30863431

Subgroups — C4×#16863693 #30863431

Order 17152.

948 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863693C211yesmaximal, characteristic
2C42×D268C211yesmaximal, characteristic
3#22561167C211yesmaximal, characteristic, complemented
4#22560975C211yesmaximal, characteristic, complemented
5#22560798C211yesmaximal, characteristic, complemented
6#22560784C211yesmaximal, characteristic, complemented
7#22560469C211yesmaximal, characteristic
8#22560455C211yesmaximal, characteristic
9#22560216C211yesmaximal, characteristic
10C4×#14523976C211yesmaximal, characteristic, complemented
11C4×#14523974C211yesmaximal, characteristic, complemented
12C4×C4⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
13C4×#14523963C211yesmaximal, characteristic
14C4×#14523963C211yesmaximal, characteristic
15C4×#125×C67C211yesmaximal, characteristic, complemented
16C2×C4×D268C2211yescharacteristic
17C2×C4×D268C2211yescharacteristic
18C2×C4×D268C2211yescharacteristic
19C2×C42×D67—21no—
20#16863693C418yesdirect product factor
21C2×#14523976C2211yescharacteristic
22C2×#14523976C2211yescharacteristic
23C2×#14523974C2211yescharacteristic
24C2×#14523974C2211yescharacteristic
25C2×C4⋊(C67⋊C8)C2211yescharacteristic
26C2×C4⋊(C67⋊C8)C2211yescharacteristic
27C2×#14523963C2211yescharacteristic
28C2×#14523963C2211yescharacteristic
29C2×#14523963C2211yescharacteristic
30C2×#14523963C2211yescharacteristic
31C2×C4×C67⋊C8C2211yescharacteristic
32C2×C4×C67⋊C8C2211yescharacteristic
33C4×C4⋊Dic67C2211yescharacteristic
34C4×C4⋊Dic67C2211yescharacteristic
35C4×C4⋊Dic67C2211yescharacteristic
36C4×(C2×D67)⋊C4—21no—
37C4×(C2×D67)⋊C4—21no—
38C4×(C2×D67)⋊C4—21no—
39#16863158C2211yescharacteristic
40#16863158C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863151C2211yescharacteristic
48#16863151C2211yescharacteristic
49C2×#125×C67C2211yescharacteristic
50#389×C67C2211yescharacteristic
51#389×C67C2211yescharacteristic
52#387×C67—21no—
53#387×C67—21no—
54#382×C67C2211yescharacteristic
55C4×C4⋊C4×C67—21no—
56C4×C4⋊C4×C67—21no—
57C4×C4⋊C4×C67C2211yescharacteristic
58C4×C4⋊C4×C67C2211yescharacteristic
59C43×C67C2211yescharacteristic
60C22×D268C2311yescharacteristic
61C22×C4×D67—21no—
62C22×C4×D67—21no—
63C22×C4×D67—21no—
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C22×C67⋊C8C2311yescharacteristic
67C22×C67⋊C8C2311yescharacteristic
68C4×D268—22no—
69C4×D268—22no—
70C4×D268—22no—
71C4×D268—22no—
72C4×D268C2×C414yes—
73C4×D268Q814yes—
74C2×C4⋊Dic67C2311yescharacteristic
75C2×C4⋊Dic67C2311yescharacteristic
76C2×C4⋊Dic67C2311yescharacteristic
77C2×(C2×D67)⋊C4—21no—
78C2×(C2×D67)⋊C4—21no—
79C2×(C2×D67)⋊C4—21no—
80C2×C4×Dic67—21no—
81C2×C4×Dic67—21no—
82C2×C4×Dic67—21no—
83C42×D67—44no—
84#14523976—22no—
85#14523976C2×C414yescomplemented
86#14523974—22no—
87#14523974C2×C414yescomplemented
88C4⋊(C67⋊C8)—22no—
89C4⋊(C67⋊C8)C2×C414yescomplemented
90#14523963—22no—
91#14523963—22no—
92#14523963C2×C414yes—
93#14523963C2×C414yes—
94C4×C67⋊C8—22no—
95C4×C67⋊C8—22no—
96C67⋊#95—21no—
97C67⋊#95—21no—
98C67⋊#95—21no—
99#125×C67—22no—
100#125×C67C2×C414yescomplemented
101C2×C4⋊C4×C67—21no—
102C2×C4⋊C4×C67—21no—
103C2×C4⋊C4×C67—21no—
104C2×C4⋊C4×C67—21no—
105C2×C4⋊C4×C67C2311yescharacteristic
106C2×C4⋊C4×C67C2311yescharacteristic
107C2×C4⋊C4×C67C2311yescharacteristic
108C2×C4⋊C4×C67C2311yescharacteristic
109C2×C42×C67—21no—
110C2×C42×C67—21no—
111C2×C42×C67D411yescharacteristic
112C2×C42×C67D411yescharacteristic
113C2×C42×C67D411yescharacteristic
114C2×C42×C67D411yescharacteristic
115C2×C42×C67C2311yescharacteristic
116C2×C42×C67C2311yescharacteristic
117C2×C42×C67C2311yescharacteristic
118#95×C67—21no—
119#95×C67—21no—
120#95×C67—21no—
121#95×C67—21no—
122#95×C67—21no—
123#95×C67—21no—
124C23×D67—21no—
125C2×D268—21no—
126C2×D268—21no—
127C2×D268—21no—
128C2×D268C22×C412yes—
129C2×D268C2×Q812yes—
130C2×D268#4112yes—
131C22×Dic67—21no—
132C22×Dic67—21no—
133C22×Dic67—21no—
134C2×C4×D67—24no—
135C2×C4×D67—24no—
136C2×C4×D67—24no—
137C2×C4×D67—42no—
138C2×C4×D67—42no—
139C2×C4×D67—42no—
140C2×C4×D67—42no—
141C2×C4×D67—42no—
142C2×C4×D67—42no—
143C2×C67⋊C8—21no—
144C2×C67⋊C8—21no—
145C2×C67⋊C8—21no—
146C2×C67⋊C8—21no—
147C2×C67⋊C8—21no—
148C2×C67⋊C8—21no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8C22×C412yes—
154C2×C67⋊C8C22×C412yes—
155C4⋊Dic67—21no—
156C4⋊Dic67—21no—
157C4⋊Dic67—21no—
158C4⋊Dic67C22×C412yes—
159C4⋊Dic67C2×Q812yes—
160C4⋊Dic67#4112yes—
161(C2×D67)⋊C4—24no—
162(C2×D67)⋊C4—24no—
163(C2×D67)⋊C4—24no—
164C4×Dic67—42no—
165C4×Dic67—42no—
166C4×Dic67—42no—
167C22×C4×C67—21no—
168C22×C4×C67—21no—
169C22×C4×C67—21no—
170C22×C4×C67—21no—
171C22×C4×C67C2×D411yescharacteristic
172C22×C4×C67C2×D411yescharacteristic
173C22×C4×C67C2×D411yescharacteristic
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2411yescharacteristic
178C4⋊C4×C67—24no—
179C4⋊C4×C67—24no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—42no—
187C4⋊C4×C67—42no—
188C4⋊C4×C67C22×C412yes—
189C4⋊C4×C67C22×C412yes—
190C42×C67—22no—
191C42×C67—22no—
192C42×C67—22no—
193C42×C67—21no—
194C42×C67—21no—
195C42×C67—41no—
196C42×C67—42no—
197C42×C67—42no—
198C42×C67C22×C412yes—
199C42×C67C2×Q812yes—
200C42×C67#4112yes—
201C42×C67#4112yes—
202C23×C67C22×D411yescharacteristic
203C22×D67—22no—
204C22×D67—22no—
205C22×D67—22no—
206C22×D67—41no—
207C22×D67—41no—
208C22×D67—41no—
209C22×D67—41no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—22no—
219C2×C4×C67—22no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67C4×D412yes—
234C2×C4×C67C4×D412yes—
235C2×C4×C67C22⋊Q812yes—
236C2×C4×C67C22⋊Q812yes—
237C2×C4×C67#13512yes—
238C2×C4×C67#13512yes—
239C2×C4×C67AGL(1,ℤ/8)12yes—
240C2×C4×C67C23×C411yescharacteristic
241C2×C4×C67C22×Q811yescharacteristic
242C2×C4×C67C2×#4111yescharacteristic
243C2×C4×C67C2×#4111yescharacteristic
244C2×C4×C67C2×#4111yescharacteristic
245C2×C4×C67C2×#4111yescharacteristic
246C2×Dic67—22no—
247C2×Dic67—22no—
248C2×Dic67—22no—
249C2×Dic67—41no—
250C2×Dic67—41no—
251C2×Dic67—41no—
252C2×Dic67—41no—
253C2×Dic67—41no—
254C2×Dic67—41no—
255D268—22no—
256D268—22no—
257D268—22no—
258D268C4×Q814yes—
259C4×D67—48no—
260C4×D67—48no—
261C4×D67—48no—
262C67⋊C8—22no—
263C67⋊C8—22no—
264C67⋊C8—41no—
265C67⋊C8—41no—
266C22×C67C2×C4×D411yescharacteristic
267C22×C67C2×C22⋊Q811yescharacteristic
268C22×C67#52511yescharacteristic
269C22×C67C2×#13511yescharacteristic
270C22×C67C2×#13511yescharacteristic
271C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C67#57511yescharacteristic
273C2×D67—24no—
274C2×D67—42no—
275C2×D67—42no—
276C2×D67—42no—
277C2×D67—42no—
278C2×D67—42no—
279C2×D67—42no—
280C4×C67—22no—
281C4×C67—22no—
282C4×C67—22no—
283C4×C67—22no—
284C4×C67—41no—
285C4×C67—42no—
286C4×C67—42no—
287C4×C67—81no—
288C4×C67—81no—
289C4×C67#47814yescomplemented
290C4×C67C2×C4×Q811yescharacteristic
291C4×C67C4×#4111yescharacteristic
292C4×C67#52911yescharacteristic
293C4×C67#53311yescharacteristic
294Dic67—42no—
295Dic67—42no—
296Dic67—42no—
297C4×#478—671nomaximal, Sylow
298C2×C67C4×C22⋊Q811yescharacteristic
299C2×C67C4×#13511yescharacteristic
300C2×C67C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C67C2×#47811yescharacteristic
302C2×C67#307611yescharacteristic
303C2×C67#310211yescharacteristic
304C2×C67#310511yescharacteristic
305D67—44no—
306C2×#478—671no—
307C4×#125D6711yescharacteristic, complemented
308C42×D4—671no—
309#1989—671no—
310#1949—671no—
311#1921—671no—
312#1919—671no—
313#1864—671no—
314#1862—671no—
315#1804—671no—
316C4×#107—671no—
317C4×#106—671no—
318C4×C4⋊C8—671no—
319C4×#102—671no—
320C4×#102—671no—
321C67C4×#47811yescharacteristic, complemented, Sylow
322C2×#125C2×D6711yescharacteristic
323C2×C4×D4—671no—
324C2×C4×D4—671no—
325C2×C4×D4—671no—
326C22×C42—1341no—
327#478—678no—
328C2×#107—671no—
329C2×#107—671no—
330C2×#106—671no—
331C2×#106—671no—
332C2×C4⋊C8—671no—
333C2×C4⋊C8—671no—
334C2×#102—671no—
335C2×#102—671no—
336C2×#102—671no—
337C2×#102—671no—
338C2×C4×C8—671no—
339C2×C4×C8—671no—
340#389C2×D6711yescharacteristic
341#389C2×D6711yescharacteristic
342#387—21no—
343#387—21no—
344#382C2×D6711yescharacteristic
345C4×C4⋊C4—21no—
346C4×C4⋊C4—21no—
347C4×C4⋊C4—671no—
348C4×C4⋊C4—671no—
349C4×C4⋊C4—671no—
350C4×C4⋊C4C2×D6711yescharacteristic
351C4×C4⋊C4C2×D6711yescharacteristic
352C4×C22⋊C4—1341no—
353C4×C22⋊C4—1341no—
354C4×C22⋊C4—1341no—
355C43C2×D6711yescharacteristic
356#340—671no—
357#340—671no—
358#340—671no—
359#340—671no—
360#340—671no—
361#340—671no—
362#340—671no—
363#340—671no—
364#336—671no—
365#336—671no—
366C22×D4—671no—
367C23×C4—1341no—
368C23×C4—1341no—
369C23×C4—1341no—
370C22×C8—671no—
371C22×C8—671no—
372C22×C8—671no—
373C22×C8—671no—
374#125—22no—
375#125C4×D6714yescomplemented
376C4×D4—674no—
377C4×D4—674no—
378C4×D4—1342no—
379C4×D4—1342no—
380C4×D4—1342no—
381C4×D4—1342no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—671no—
387C2×C4⋊C4—671no—
388C2×C4⋊C4—671no—
389C2×C4⋊C4C22×D6711yescharacteristic
390C2×C4⋊C4C22×D6711yescharacteristic
391C2×C4⋊C4C22×D6711yescharacteristic
392C2×C4⋊C4C22×D6711yescharacteristic
393C2×C22⋊C4—1341no—
394C2×C22⋊C4—1341no—
395C2×C22⋊C4—1341no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—2684no—
399C2×C42—1341no—
400C2×C42—1341no—
401C2×C42—1341no—
402C2×C42C22⋊D6711yescharacteristic
403C2×C42C22⋊D6711yescharacteristic
404C2×C42C22⋊D6711yescharacteristic
405C2×C42C22⋊D6711yescharacteristic
406C2×C42C22×D6711yescharacteristic
407C2×C42C22×D6711yescharacteristic
408C2×C42C22×D6711yescharacteristic
409#107—674no—
410#107—1342no—
411#106—674no—
412#106—1342no—
413C4⋊C8—674no—
414C4⋊C8—1342no—
415#102—674no—
416#102—674no—
417#102—1342no—
418#102—1342no—
419C4×C8—1342no—
420C4×C8—1342no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1341no—
428#95—1341no—
429#95—1341no—
430C24—1341no—
431C2×D4—672no—
432C2×D4—672no—
433C2×D4—672no—
434C2×D4—1341no—
435C2×D4—1341no—
436C2×D4—1341no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2682no—
442C22×C4—2682no—
443C22×C4—2682no—
444C22×C4—2682no—
445C22×C4—2682no—
446C22×C4—2682no—
447C22×C4—1344no—
448C22×C4—1341no—
449C22×C4—1341no—
450C22×C4—1341no—
451C22×C4—1344no—
452C22×C4—1344no—
453C22×C4C2×C22⋊D6711yescharacteristic
454C22×C4C2×C22⋊D6711yescharacteristic
455C22×C4C2×C22⋊D6711yescharacteristic
456C22×C4C2×C22⋊D6711yescharacteristic
457C22×C4C2×C22⋊D6711yescharacteristic
458C22×C4C2×C22⋊D6711yescharacteristic
459C22×C4C23×D6711yescharacteristic
460C2×C8—672no—
461C2×C8—672no—
462C2×C8—1341no—
463C2×C8—1341no—
464C2×C8—1341no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8—1341no—
470C2×C8—1341no—
471C2×C8—1341no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—672no—
483C4⋊C4—672no—
484C4⋊C4—672no—
485C4⋊C4—1341no—
486C4⋊C4—1341no—
487C4⋊C4—1341no—
488C4⋊C4C2×C4×D6712yes—
489C4⋊C4C2×C4×D6712yes—
490C22⋊C4—1344no—
491C22⋊C4—1344no—
492C22⋊C4—1344no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—2682no—
502C42—2682no—
503C42—2682no—
504C42C2×C4×D6712yes—
505C42Q8×D6712yes—
506C42#1176162812yes—
507C42#1176162912yes—
508C23—2681no—
509C23—2681no—
510C23—2681no—
511C23—2681no—
512C23—1342no—
513C23—1342no—
514C23—1342no—
515C23C22×C22⋊D6711yescharacteristic
516D4—674no—
517D4—1342no—
518D4—1342no—
519D4—1342no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—2688no—
544C2×C4—2688no—
545C2×C4—2681no—
546C2×C4—2681no—
547C2×C4—2681no—
548C2×C4—2681no—
549C2×C4—2681no—
550C2×C4—2681no—
551C2×C4—2688no—
552C2×C4—1342no—
553C2×C4—1342no—
554C2×C4—1342no—
555C2×C4C4×C22⋊D6712yes—
556C2×C4C4×C22⋊D6712yes—
557C2×C4#1452401912yes—
558C2×C4#1452401912yes—
559C2×C4#1452406312yes—
560C2×C4#1452406312yes—
561C2×C4#1452406812yes—
562C2×C4C22×C4×D6711yescharacteristic
563C2×C4C2×Q8×D6711yescharacteristic
564C2×C4#41×D6711yescharacteristic
565C2×C4#41×D6711yescharacteristic
566C2×C4C2×#1176162811yescharacteristic
567C2×C4C2×#1176162911yescharacteristic
568C8—2681no—
569C8—2681no—
570C8—1342no—
571C8—1342no—
572C22—2682no—
573C22—2682no—
574C22—2682no—
575C22—2682no—
576C22—2682no—
577C22—2682no—
578C22—1344no—
579C22C2×C4×C22⋊D6711yescharacteristic
580C22C2×#1452401911yescharacteristic
581C22#1686390311yescharacteristic
582C22C2×#1452406311yescharacteristic
583C22C2×#1452406311yescharacteristic
584C22C2×#1452406811yescharacteristic
585C22#1686419411yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—2682no—
596C4—2682no—
597C4—2682no—
598C4#1686369314yesdirect product factor
599C4C4×Q8×D6711yescharacteristic
600C4C4×#1176162911yescharacteristic
601C4#1686393311yescharacteristic
602C4#1686395011yescharacteristic
603C2—2684no—
604C2C4×#1452401911yescharacteristic
605C2C4×#1452406311yescharacteristic
606C2C4×#1452406811yescharacteristic
607C2C2×#1686369311yescharacteristic
608C2#2256999111yescharacteristic
609C2#2257022111yescharacteristic
610C2#2257025211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC4×#125×C67C2
socleC23×C67C22×D4
prime radicalC22×D268C23
cosocleC22×C4×C67C24
hypercenterC4×#125D67
hypocenterC67C4×#478
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #22561167C2semidirect
4. #22560975C2semidirect
5. #22560798C2semidirect
6. #22560784C2semidirect
10. C4×#14523976C2semidirect
11. C4×#14523974C2semidirect
12. C4×C4⋊(C67⋊C8)C2semidirect
15. C4×#125×C67C2semidirect
20. #16863693C4direct (N × H)
85. #14523976C2×C4semidirect
87. #14523974C2×C4semidirect
89. C4⋊(C67⋊C8)C2×C4semidirect
100. #125×C67C2×C4semidirect
289. C4×C67#478semidirect
307. C4×#125D67semidirect
321. C67C4×#478semidirect
375. #125C4×D67semidirect
598. C4#16863693direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168636932
C42×D2682
#225611672
#225609752
#225607982
#225607842
#225604692
#225604552
#225602162
C4×#145239762
C4×#145239742
C4×C4⋊(C67⋊C8)2
C4×#145239632
C4×#125×C672
C4×#47867