Finite Group Databasebeta

← C4×#16863317 #30777643

Subgroups — C4×#16863317 #30777643

Order 17152.

1942 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic134C211yesmaximal, characteristic
2C2×#16863317C211yesmaximal, characteristic
3C2×C4×Dic67⋊C4C211yesmaximal, characteristic
4C2×C4×C4⋊Dic67C212yesmaximal
5#22558479C211yesmaximal, characteristic
6#22558474C211yesmaximal, characteristic
7#22558473C212yesmaximal
8#22558463C211yesmaximal, characteristic
9#22558461C212yesmaximal
10C4×C67⋊#95C212yesmaximal
11C4×#95×C67C211yesmaximal, characteristic
12C23×Dic134C2211yescharacteristic
13C2×C4×Dic134—28no—
14C2×C4×Dic134—24no—
15C22×C4⋊Dic67C2212yes—
16C22×C4⋊Dic67C2211yescharacteristic
17C22×C4⋊Dic67C2212yes—
18C22×Dic67⋊C4C2211yescharacteristic
19C22×Dic67⋊C4C2212yes—
20C22×Dic67⋊C4C2211yescharacteristic
21C22×C4×Dic67C2211yescharacteristic
22C22×C4×Dic67C2212yes—
23C22×C4×Dic67C2211yescharacteristic
24#16863317C418yesdirect product factor
25C4×Dic67⋊C4—24no—
26C4×C4⋊Dic67—28no—
27C2×C67⋊#95C2212yes—
28C2×C67⋊#95C2212yes—
29C2×C67⋊#95C2211yescharacteristic
30C2×C67⋊#95C2212yes—
31C2×C67⋊#95C2212yes—
32C2×C67⋊#95C2212yes—
33C2×C67⋊#95C2211yescharacteristic
34C2×C67⋊#95C2212yes—
35C22×C42×C67C2211yescharacteristic
36C22×C42×C67C2212yes—
37C2×#95×C67C2212yes—
38C2×#95×C67C2211yescharacteristic
39C2×#95×C67C2211yescharacteristic
40C22×Dic134—24no—
41C22×Dic134—24no—
42C22×Dic134—22no—
43C22×Dic134—22no—
44C22×Dic134C2×C414yes—
45C23×Dic67C2311yescharacteristic
46C23×Dic67C2312yes—
47C23×Dic67C2311yescharacteristic
48C23×Dic67C2311yescharacteristic
49C23×Dic67C2312yes—
50C23×Dic67C2311yescharacteristic
51C4×Dic134—232no—
52C2×C4⋊Dic67—24no—
53C2×C4⋊Dic67—24no—
54C2×C4⋊Dic67—24no—
55C2×C4⋊Dic67—24no—
56C2×C4⋊Dic67—24no—
57C2×C4⋊Dic67—22no—
58C2×C4⋊Dic67—24no—
59C2×C4⋊Dic67C2×C418yes—
60C2×Dic67⋊C4—24no—
61C2×Dic67⋊C4—24no—
62C2×Dic67⋊C4—22no—
63C2×Dic67⋊C4—22no—
64C2×Dic67⋊C4—22no—
65C2×Dic67⋊C4—22no—
66C2×Dic67⋊C4—24no—
67C2×Dic67⋊C4—22no—
68C2×Dic67⋊C4C2×C414yes—
69C2×C4×Dic67—22no—
70C2×C4×Dic67—24no—
71C2×C4×Dic67—24no—
72C2×C4×Dic67—24no—
73C2×C4×Dic67—22no—
74C2×C4×Dic67—22no—
75C2×C4×Dic67D418yes—
76C2×C4×Dic67Q814yescomplemented
77C67⋊#95—24no—
78C67⋊#95—24no—
79C67⋊#95—24no—
80C67⋊#95—24no—
81C67⋊#95—22no—
82C67⋊#95—22no—
83C67⋊#95—24no—
84C67⋊#95C2×C418yes—
85C23×C4×C67C2311yescharacteristic
86C23×C4×C67C2312yes—
87C23×C4×C67C2311yescharacteristic
88C23×C4×C67C2312yes—
89C23×C4×C67C2311yescharacteristic
90C2×C42×C67—24no—
91C2×C42×C67—24no—
92C2×C42×C67—24no—
93C2×C42×C67D414yes—
94C2×C42×C67Q818yescomplemented
95#95×C67—22no—
96#95×C67—24no—
97#95×C67C2×C414yes—
98C2×Dic134—28no—
99C2×Dic134—216no—
100C2×Dic134—28no—
101C2×Dic134—28no—
102C2×Dic134—216no—
103C22×Dic67—21no—
104C22×Dic67—22no—
105C22×Dic67—22no—
106C22×Dic67—21no—
107C22×Dic67—21no—
108C22×Dic67—21no—
109C22×Dic67—22no—
110C22×Dic67—22no—
111C22×Dic67—21no—
112C22×Dic67—21no—
113C22×Dic67—21no—
114C22×Dic67—22no—
115C22×Dic67—22no—
116C22×Dic67—22no—
117C22×Dic67—22no—
118C22×Dic67—22no—
119C22×Dic67—22no—
120C22×Dic67—22no—
121C22×Dic67—22no—
122C22×Dic67—21no—
123C22×Dic67—21no—
124C22×Dic67—21no—
125C22×Dic67—22no—
126C22×Dic67—22no—
127C22×Dic67—22no—
128C22×Dic67—22no—
129C22×Dic67C22×C412yes—
130C22×Dic67C22×C414yes—
131C22×Dic67C22×C412yes—
132C22×Dic67C2×D414yes—
133C22×Dic67C2×D414yes—
134C22×Dic67C2×Q812yes—
135C22×Dic67C2×Q812yes—
136C22×Dic67#4112yes—
137C22×Dic67#4114yes—
138C22×Dic67#4112yes—
139C4⋊Dic67—216no—
140C4⋊Dic67—28no—
141C4⋊Dic67—216no—
142Dic67⋊C4—28no—
143Dic67⋊C4—216no—
144Dic67⋊C4—28no—
145C4×Dic67—216no—
146C4×Dic67—28no—
147C4×Dic67—44no—
148C24×C67C2411yescharacteristic
149C22×C4×C67—22no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—22no—
157C22×C4×C67—22no—
158C22×C4×C67—22no—
159C22×C4×C67—22no—
160C22×C4×C67—21no—
161C22×C4×C67—21no—
162C22×C4×C67—22no—
163C22×C4×C67—22no—
164C22×C4×C67—22no—
165C22×C4×C67C22×C412yes—
166C22×C4×C67C22×C414yes—
167C22×C4×C67C2×D414yes—
168C22×C4×C67C2×D412yes—
169C22×C4×C67C2×D412yes—
170C22×C4×C67C2×D412yes—
171C22×C4×C67C2×Q812yes—
172C22×C4×C67C2×Q814yes—
173C22×C4×C67C2×Q814yes—
174C22×C4×C67C2×Q814yes—
175C22×C4×C67#4112yes—
176C22×C4×C67#4112yes—
177C22×C4×C67#4114yes—
178C42×C67—28no—
179C42×C67—216no—
180C23×C67C23×C411yescharacteristic
181C23×C67C22×D412yes—
182C23×C67C22×D411yescharacteristic
183C23×C67C22×Q811yescharacteristic
184C23×C67C22×Q812yes—
185C23×C67C2×#4112yes—
186C23×C67C2×#4111yescharacteristic
187C23×C67C2×#4111yescharacteristic
188C23×C67C2×#4111yescharacteristic
189C23×C67C2×#4111yescharacteristic
190C23×C67C2×#4112yes—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—22no—
195C2×C4×C67—24no—
196C2×C4×C67—22no—
197C2×C4×C67—24no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—24no—
201C2×C4×C67—22no—
202C2×C4×C67—24no—
203C2×C4×C67—22no—
204C2×C4×C67—22no—
205C2×C4×C67—24no—
206C2×C4×C67—24no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—42no—
213C2×C4×C67C4×D414yes—
214C2×C4×C67C4×Q818yescomplemented
215C2×C4×C67#12014yes—
216C2×C4×C67C22⋊Q818yes—
217C2×C4×C67C22⋊Q814yes—
218C2×C4×C67C4⋊Q818yes—
219C2×C4×C67C4⋊Q814yes—
220C2×Dic67—22no—
221C2×Dic67—24no—
222C2×Dic67—24no—
223C2×Dic67—22no—
224C2×Dic67—24no—
225C2×Dic67—22no—
226C2×Dic67—22no—
227C2×Dic67—24no—
228C2×Dic67—24no—
229C2×Dic67—24no—
230C2×Dic67—22no—
231C2×Dic67—22no—
232C2×Dic67—22no—
233C2×Dic67—24no—
234C2×Dic67—24no—
235C2×Dic67—22no—
236C2×Dic67—22no—
237C2×Dic67—24no—
238C2×Dic67—24no—
239C2×Dic67—24no—
240C2×Dic67—24no—
241C2×Dic67—41no—
242C2×Dic67—41no—
243C2×Dic67—42no—
244C2×Dic67—42no—
245C2×Dic67—42no—
246C2×Dic67—41no—
247C2×Dic67—41no—
248C2×Dic67—42no—
249C2×Dic67—42no—
250C2×Dic67—42no—
251C2×Dic67—41no—
252C2×Dic67—42no—
253C2×Dic67—41no—
254C2×Dic67C4×D418yes—
255C2×Dic67C4×Q814yescomplemented
256Dic134—232no—
257C22×C67C2×C4×D412yes—
258C22×C67C2×C4×D411yescharacteristic
259C22×C67C2×C4×Q811yescharacteristic
260C22×C67C2×C4×Q812yes—
261C22×C67C4×#4111yescharacteristic
262C22×C67C2×#12011yescharacteristic
263C22×C67C2×C22⋊Q812yes—
264C22×C67C2×C22⋊Q811yescharacteristic
265C22×C67#52512yes—
266C22×C67#52512yes—
267C22×C67#52511yescharacteristic
268C22×C67#52511yescharacteristic
269C22×C67#52912yes—
270C22×C67#52911yescharacteristic
271C22×C67C2×C4⋊Q812yes—
272C22×C67C2×C4⋊Q811yescharacteristic
273C22×C67#53212yes—
274C22×C67#53211yescharacteristic
275C22×C67#53312yes—
276C22×C67#53312yes—
277C22×C67#53312yes—
278C22×C67#53311yescharacteristic
279C22×C67#53312yes—
280C4×C67—24no—
281C4×C67—28no—
282C4×C67—42no—
283C4×C67—44no—
284C4×C67#39518yescomplemented
285Dic67—28no—
286Dic67—24no—
287Dic67—44no—
288Dic67—42no—
289Dic67—42no—
290Dic67—42no—
291C4×#395—671nomaximal, Sylow
292C2×C67C4×#12011yescharacteristic
293C2×C67C4×C22⋊Q812yes—
294C2×C67C4×C22⋊Q811yescharacteristic
295C2×C67C4×C4⋊Q812yes—
296C2×C67C4×C4⋊Q811yescharacteristic
297C2×C67C2×#39511yescharacteristic
298C2×C67#238811yescharacteristic
299C2×C67#239611yescharacteristic
300C2×C67#239612yes—
301C2×C67#239911yescharacteristic
302C2×C67#239912yes—
303C22×C4×Q8—671no—
304C2×#395—671no—
305C2×C4×C4⋊C4—672no—
306C2×C4×C4⋊C4—671no—
307#1446—671no—
308#1444—672no—
309#1444—671no—
310#1441—672no—
311#1441—671no—
312C4×#95—672no—
313C4×#95D6711yescharacteristic
314C67C4×#39511yescharacteristic, complemented, Sylow
315C23×Q8—671no—
316C2×C4×Q8—1348no—
317C2×C4×Q8—1344no—
318C22×C4⋊C4—671no—
319C22×C4⋊C4—672no—
320C22×C4⋊C4—672no—
321C22×C4⋊C4—671no—
322C22×C4⋊C4—671no—
323C22×C4⋊C4—672no—
324C22×C42—672no—
325C22×C42—671no—
326C22×C42—671no—
327C22×C42C2×D6712yes—
328C22×C42C2×D6711yescharacteristic
329#395—678no—
330C4×C4⋊C4—1348no—
331C4×C4⋊C4—1344no—
332C2×#95—672no—
333C2×#95—672no—
334C2×#95—672no—
335C2×#95—672no—
336C2×#95—672no—
337C2×#95—671no—
338C2×#95—672no—
339C2×#95—671no—
340C2×#95C2×D6712yes—
341C2×#95C2×D6711yescharacteristic
342C2×#95C2×D6711yescharacteristic
343C22×Q8—674no—
344C22×Q8—1344no—
345C22×Q8—1344no—
346C22×Q8—1342no—
347C22×Q8—1342no—
348C23×C4—672no—
349C23×C4—671no—
350C23×C4—671no—
351C23×C4—672no—
352C23×C4—671no—
353C23×C4—671no—
354C23×C4C22×D6711yescharacteristic
355C23×C4C22×D6712yes—
356C23×C4C22×D6711yescharacteristic
357C23×C4C22×D6712yes—
358C23×C4C22×D6711yescharacteristic
359C4×Q8—13432no—
360C2×C4⋊C4—674no—
361C2×C4⋊C4—678no—
362C2×C4⋊C4—1344no—
363C2×C4⋊C4—1344no—
364C2×C4⋊C4—1344no—
365C2×C4⋊C4—1342no—
366C2×C4⋊C4—1342no—
367C2×C4⋊C4—1344no—
368C2×C4⋊C4—1344no—
369C2×C4⋊C4—1344no—
370C2×C4⋊C4—1344no—
371C2×C4⋊C4—1344no—
372C2×C4⋊C4—1342no—
373C2×C4⋊C4—1342no—
374C2×C4⋊C4—1342no—
375C2×C4⋊C4—1342no—
376C2×C4⋊C4—1344no—
377C2×C42—24no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—678no—
381C2×C42—674no—
382C2×C42—1344no—
383C2×C42—1342no—
384C2×C42—1344no—
385C2×C42—1344no—
386C2×C42—1342no—
387C2×C42—1342no—
388C2×C42Dic13418yescomplemented
389C2×C42D26814yes—
390#95—22no—
391#95—24no—
392#95—678no—
393#95—1344no—
394#95—1344no—
395#95—1344no—
396#95—1344no—
397#95—1342no—
398#95—1342no—
399#95—1344no—
400#95C4×D6714yes—
401C24C23×D6711yescharacteristic
402C2×Q8—13416no—
403C2×Q8—1348no—
404C2×Q8—13416no—
405C2×Q8—1348no—
406C2×Q8—1348no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—672no—
424C22×C4—674no—
425C22×C4—674no—
426C22×C4—672no—
427C22×C4—672no—
428C22×C4—674no—
429C22×C4—672no—
430C22×C4—674no—
431C22×C4—672no—
432C22×C4—672no—
433C22×C4—1342no—
434C22×C4—1342no—
435C22×C4—1342no—
436C22×C4—1341no—
437C22×C4—1341no—
438C22×C4—1341no—
439C22×C4—1342no—
440C22×C4—1342no—
441C22×C4—1342no—
442C22×C4—1342no—
443C22×C4—1342no—
444C22×C4—1341no—
445C22×C4—1341no—
446C22×C4—1342no—
447C22×C4—1342no—
448C22×C4—1341no—
449C22×C4—1342no—
450C22×C4—1342no—
451C22×C4—1342no—
452C22×C4—1341no—
453C22×C4—1342no—
454C22×C4—1341no—
455C22×C4—1342no—
456C22×C4—1341no—
457C22×C4—1341no—
458C22×C4—1342no—
459C22×C4C2×C4×D6712yes—
460C22×C4C2×C4×D6714yes—
461C22×C4D4×D6714yes—
462C22×C4C2×D26812yes—
463C22×C4C2×D26812yes—
464C22×C4C2×D26812yes—
465C22×C4Q8×D6712yes—
466C22×C4C2×Dic13414yes—
467C22×C4C2×Dic13414yes—
468C22×C4C2×Dic13414yes—
469C22×C4#1176162712yes—
470C22×C4#1176162812yes—
471C22×C4#1176162814yes—
472C4⋊C4—13416no—
473C4⋊C4—13416no—
474C4⋊C4—1348no—
475C4⋊C4—1348no—
476C4⋊C4—13416no—
477C4⋊C4—1348no—
478C42—28no—
479C42—216no—
480C42—2684no—
481C42—13416no—
482C42—1348no—
483C23C22×C4×D6711yescharacteristic
484C23C2×D4×D6712yes—
485C23C22×D26811yescharacteristic
486C23C2×Q8×D6711yescharacteristic
487C23C22×Dic13412yes—
488C23#41×D6712yes—
489C23#41×D6711yescharacteristic
490C23#41×D6711yescharacteristic
491C23C2×#1176162711yescharacteristic
492C23C2×#1176162811yescharacteristic
493C23C2×#1176162812yes—
494Q8—13432no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—2681no—
518C2×C4—2681no—
519C2×C4—2682no—
520C2×C4—2682no—
521C2×C4—2682no—
522C2×C4—2681no—
523C2×C4—2681no—
524C2×C4—2682no—
525C2×C4—2682no—
526C2×C4—2682no—
527C2×C4—2681no—
528C2×C4—2682no—
529C2×C4—2681no—
530C2×C4—678no—
531C2×C4—674no—
532C2×C4—1344no—
533C2×C4—1342no—
534C2×C4—1344no—
535C2×C4—1342no—
536C2×C4—1344no—
537C2×C4—1344no—
538C2×C4—1342no—
539C2×C4—1344no—
540C2×C4—1344no—
541C2×C4—1344no—
542C2×C4—1342no—
543C2×C4—1344no—
544C2×C4—1342no—
545C2×C4—1344no—
546C2×C4—1342no—
547C2×C4—1344no—
548C2×C4—1344no—
549C2×C4—1342no—
550C2×C4—1344no—
551C2×C4—1342no—
552C2×C4—1342no—
553C2×C4C4×D26814yes—
554C2×C4C4×Dic13418yescomplemented
555C2×C4#1452401314yes—
556C2×C4#1452402018yes—
557C2×C4#1452402314yes—
558C2×C4#1452404218yes—
559C2×C4#1452404314yes—
560C22C4×D4×D6712yes—
561C22C2×C4×D26811yescharacteristic
562C22C4×Q8×D6711yescharacteristic
563C22C2×C4×Dic13412yes—
564C22C4×#1176162711yescharacteristic
565C22C2×#1452401311yescharacteristic
566C22C2×#1452402012yes—
567C22C2×#1452402311yescharacteristic
568C22#1686390512yes—
569C22#1686390512yes—
570C22#1686390811yescharacteristic
571C22#1686390811yescharacteristic
572C22#1686392612yes—
573C22#1686392811yescharacteristic
574C22C2×#1452404212yes—
575C22C2×#1452404311yescharacteristic
576C22#1686394312yes—
577C22#1686394611yescharacteristic
578C22#1686394912yes—
579C22#1686394912yes—
580C22#1686395012yes—
581C22#1686395011yescharacteristic
582C22#1686395312yes—
583C4—24no—
584C4—28no—
585C4—42no—
586C4—44no—
587C4—2682no—
588C4—2684no—
589C4—2682no—
590C4—2682no—
591C4—1348no—
592C4—1344no—
593C4#1686331718yesdirect product factor
594C2C4×#1452401311yescharacteristic
595C2C4×#1452402012yes—
596C2C4×#1452402311yescharacteristic
597C2C4×#1452404212yes—
598C2C4×#1452404311yescharacteristic
599C2C2×#1686331711yescharacteristic
600C2#2256342511yescharacteristic
601C2#2256348111yescharacteristic
602C2#2256348312yes—
603C2#2256350211yescharacteristic
604C2#2256350312yes—
subgroupG/H
centerC23×C4C22×D67
commutatorC23×C67C23×C4
FrattiniC24C23×D67
FittingC4×#95×C67C2
socle, prime radical, cosocleC24×C67C24
hypercenterC4×#95D67
hypocenterC67C4×#395
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23×C4 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
24. #16863317C4direct (N × H)
76. C2×C4×Dic67Q8semidirect
94. C2×C42×C67Q8semidirect
214. C2×C4×C67C4×Q8semidirect
255. C2×Dic67C4×Q8semidirect
284. C4×C67#395semidirect
314. C67C4×#395semidirect
388. C2×C42Dic134semidirect
554. C2×C4C4×Dic134semidirect
593. C4#16863317direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Dic1342
C2×#168633172
C2×C4×Dic67⋊C42
C2×C4×C4⋊Dic672
#225584792
#225584742
#225584732
#225584632
#225584612
C4×C67⋊#952
C4×#95×C672
C4×#39567