Finite Group Databasebeta

← C2×#22570022 #31723360

Subgroups — C2×#22570022 #31723360

Order 17152.

2358 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
1C22×#14524042C211yesmaximal, characteristic, complemented
2C2×#16863864C211yesmaximal, characteristic
3#22570022C2116yesmaximal, direct product factor
4C2×#16863690C212yesmaximal, complemented
5C2×#16863680C212yesmaximal, complemented
6C2×#16863685C212yesmaximal, complemented
7C2×#16863675C212yesmaximal, complemented
8C2×#16863439C211yesmaximal, characteristic, complemented
9C2×#16863419C211yesmaximal, characteristic, complemented
10C2×#16863398C211yesmaximal, characteristic
11C2×#16863395C211yesmaximal, characteristic
12C2×#428×C67C211yesmaximal, characteristic, complemented
13C22×#11761628C2211yescharacteristic
14C23×Dic134—21no—
15C22×#11761612C2211yescharacteristic
16C2×#14524042—21no—
17C2×#14524042—21no—
18C2×#14524042—24no—
19C2×#14524042C2212yes—
20C2×#14524042C2212yes—
21C2×#14524042C2218yescomplemented
22#16863864—24no—
23#16863864C2218yes—
24C2×#14524008C2211yescharacteristic
25C2×#14524008C2211yescharacteristic
26C2×#14524005C2211yescharacteristic
27C2×#14524005C2211yescharacteristic
28C2×(C22⋊D67)⋊C4—21no—
29C2×(C22⋊D67)⋊C4—21no—
30C2×#14524000—21no—
31C22×C4⋊Dic67C2211yescharacteristic
32C22×Dic67⋊C4—21no—
33C22×C4×Dic67C2211yescharacteristic
34C2×C42×D67—21no—
35#16863690C22116yescomplemented
36#16863685C22116yescomplemented
37#16863680C22116yescomplemented
38#16863675C22116yescomplemented
39#16863439C2218yescomplemented
40C2×#14523976C2212yes—
41C2×#14523974C2212yes—
42#16863419C2218yescomplemented
43C2×Dic67⋊C8C2212yes—
44C2×Dic67⋊C8C2212yes—
45#16863398C2218yes—
46#16863395C2218yes—
47C2×#14523968C2212yes—
48C2×#14523966C2212yes—
49C2×#14523965C2212yes—
50C2×#14523963C2212yes—
51C22×M16×C67C2211yescharacteristic
52C2×#117×C67C2211yescharacteristic
53C22×C4⋊C4×C67C2211yescharacteristic
54#428×C67C2218yescomplemented
55C2×#107×C67C2212yes—
56C2×#106×C67C2212yes—
57C2×#11761628—22no—
58C2×#11761628—24no—
59C2×#11761628Q818yescomplemented
60C2×#11761628C2314yes—
61C22×Dic134—22no—
62C22×Dic134—22no—
63C22×Dic134—28no—
64C22×Dic134—48no—
65C22×Dic134C2311yescharacteristic
66C22×D268C2311yescharacteristic
67C22×C22⋊D67—21no—
68C23×Dic67—21no—
69C23×Dic67D411yescharacteristic
70C23×Dic67D411yescharacteristic
71C22×C4×D67—21no—
72C22×C4×D67—21no—
73C22×C4×D67—21no—
74C2×#11761612—24no—
75C2×#11761612C2314yes—
76C22×C67⋊C8C2312yes—
77#14524042—24no—
78#14524042—24no—
79#14524042—28no—
80#14524042—28no—
81#14524042C2318yes—
82#14524042C2318yes—
83#14524008—22no—
84#14524008—22no—
85#14524008C2314yes—
86#14524008C2314yes—
87#14524005—22no—
88#14524005—22no—
89#14524005C2314yes—
90#14524005C2314yes—
91(C22⋊D67)⋊C4—28no—
92(C22⋊D67)⋊C4—28no—
93#14524000—28no—
94C2×C4⋊Dic67—21no—
95C2×C4⋊Dic67—22no—
96C2×C4⋊Dic67C2312yes—
97C2×C4⋊Dic67C2314yes—
98C2×Dic67⋊C4—22no—
99C2×Dic67⋊C4—22no—
100C2×Dic67⋊C4—28no—
101C2×Dic67⋊C4—21no—
102C2×Dic67⋊C4—21no—
103C2×(C2×D67)⋊C4—21no—
104C2×(C2×D67)⋊C4—21no—
105C2×C4×Dic67—21no—
106C2×C4×Dic67—21no—
107C2×C4×Dic67D411yescharacteristic
108C2×C4×Dic67D411yescharacteristic
109C2×C4×Dic67D414yes—
110C2×C4×Dic67C2311yescharacteristic
111C2×C4×Dic67C2311yescharacteristic
112C2×C4×Dic67C2314yes—
113C42×D67—28no—
114#14523976C2318yes—
115#14523974C2318yes—
116Dic67⋊C8C2318yes—
117Dic67⋊C8C2318yes—
118#14523968C2318yes—
119#14523966C2318yes—
120#14523965C2318yes—
121#14523963C2318yes—
122C23×C4×C67—21no—
123C23×C4×C67C2311yescharacteristic
124C2×M16×C67—24no—
125C2×M16×C67C2314yes—
126C22×C8×C67C2312yes—
127#117×C67—22no—
128#117×C67C2314yes—
129C2×C4⋊C4×C67—21no—
130C2×C4⋊C4×C67—22no—
131C2×C4⋊C4×C67C2312yes—
132C2×C4⋊C4×C67C2314yes—
133C2×C4⋊C4×C67C2311yescharacteristic
134C2×C4⋊C4×C67C2311yescharacteristic
135C2×C22⋊C4×C67—21no—
136C2×C42×C67—21no—
137#107×C67C2318yes—
138#106×C67C2318yes—
139C23×D67—21no—
140#11761628—216no—
141#11761628—28no—
142#11761628C2×Q8116yescomplemented
143C2×Dic134—21no—
144C2×Dic134—22no—
145C2×Dic134—28no—
146C2×Dic134—28no—
147C2×Dic134—48no—
148C2×Dic134—432no—
149C2×Dic134—48no—
150C2×Dic134C2×Q814yescomplemented
151C2×Dic134C2412yes—
152C2×D268—21no—
153C2×D268—22no—
154C2×D268C2×Q814yescomplemented
155C2×D268C2412yes—
156C2×C22⋊D67—28no—
157C2×C22⋊D67—24no—
158C22×Dic67—21no—
159C22×Dic67—24no—
160C22×Dic67—24no—
161C22×Dic67—24no—
162C22×Dic67—22no—
163C22×Dic67—44no—
164C22×Dic67C2×D411yescharacteristic
165C22×Dic67C2×D411yescharacteristic
166C22×Dic67C2×D414yes—
167C22×Dic67C2×D414yes—
168C22×Dic67C2×D411yescharacteristic
169C22×Dic67C2×D411yescharacteristic
170C2×C4×D67—28no—
171C2×C4×D67—24no—
172C2×C4×D67—24no—
173C2×C4×D67—24no—
174C2×C4×D67—44no—
175C2×C4×D67—44no—
176#11761612—28no—
177C2×C67⋊C8—24no—
178C2×C67⋊C8C2414yes—
179C4⋊Dic67—24no—
180C4⋊Dic67—22no—
181C4⋊Dic67C2414yes—
182Dic67⋊C4—28no—
183Dic67⋊C4—28no—
184Dic67⋊C4—24no—
185Dic67⋊C4—24no—
186(C2×D67)⋊C4—24no—
187(C2×D67)⋊C4—24no—
188C4×Dic67—24no—
189C4×Dic67—24no—
190C4×Dic67C2×D414yes—
191C4×Dic67C2×D414yes—
192C4×Dic67C2×D412yes—
193C4×Dic67C2×D412yes—
194C4×Dic67C2412yes—
195C4×Dic67C2412yes—
196C24×C67C2×D411yescharacteristic
197C22×C4×C67—21no—
198C22×C4×C67—21no—
199C22×C4×C67—24no—
200C22×C4×C67—22no—
201C22×C4×C67—44no—
202C22×C4×C67C2×D411yescharacteristic
203C22×C4×C67C2×D412yes—
204C22×C4×C67C2×Q814yes—
205C22×C4×C67#4114yes—
206C22×C4×C67C2411yescharacteristic
207C22×C4×C67C2412yes—
208M16×C67—28no—
209C2×C8×C67—24no—
210C2×C8×C67C2414yes—
211C4⋊C4×C67—24no—
212C4⋊C4×C67—22no—
213C4⋊C4×C67—21no—
214C4⋊C4×C67—21no—
215C4⋊C4×C67C2412yes—
216C4⋊C4×C67C2414yes—
217C4⋊C4×C67C2412yes—
218C22⋊C4×C67—24no—
219C42×C67—24no—
220C23×C67—22no—
221C23×C67—22no—
222C23×C67C22⋊Q814yes—
223C23×C67C22×D412yes—
224C23×C67C22×D411yescharacteristic
225C22×D67—24no—
226C22×D67—22no—
227C22×D67—42no—
228C22×D67—42no—
229C2×C4×C67—22no—
230C2×C4×C67—22no—
231C2×C4×C67—24no—
232C2×C4×C67—48no—
233C2×C4×C67—44no—
234C2×C4×C67—42no—
235C2×C4×C67—42no—
236C2×C4×C67C22⋊Q814yes—
237C2×C4×C67C22⋊Q818yes—
238C2×C4×C67C22×D412yes—
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C22×Q812yes—
241C2×C4×C67C22×Q814yes—
242C2×C4×C67C2×#4112yes—
243C2×C4×C67C2×#4114yes—
244C2×C4×C67C2511yescharacteristic
245C22⋊D67—216no—
246C2×Dic67—24no—
247C2×Dic67—22no—
248C2×Dic67—22no—
249C2×Dic67—22no—
250C2×Dic67—28no—
251C2×Dic67—28no—
252C2×Dic67—24no—
253C2×Dic67—48no—
254C2×Dic67—44no—
255C2×Dic67AGL(1,ℤ/8)14yes—
256C2×Dic67#13714yes—
257C2×Dic67C22×D412yes—
258C2×Dic67C22×D412yes—
259C2×Dic67C22×D412yes—
260C2×Dic67C22×D412yes—
261D268—24no—
262D268—22no—
263D268C22×Q814yescomplemented
264C4×D67—216no—
265C4×D67—48no—
266C4×D67—48no—
267Dic134—24no—
268Dic134—22no—
269Dic134—416no—
270Dic134—416no—
271Dic134C22×Q814yescomplemented
272C8×C67—24no—
273C67⋊C8—24no—
274C22×C67—28no—
275C22×C67—22no—
276C22×C67—22no—
277C22×C67C2×C22⋊Q814yes—
278C22×C67C2×C22⋊Q812yes—
279C22×C67C2×AGL(1,ℤ/8)12yes—
280C22×C67C2×#13712yes—
281C22×C67C23×D411yescharacteristic
282C2×D67—24no—
283C2×D67—48no—
284C2×D67—42no—
285C2×D67—42no—
286C4×C67—44no—
287C4×C67—42no—
288C4×C67—42no—
289C4×C67C2×C22⋊Q812yes—
290C4×C67C2×C22⋊Q814yes—
291C4×C67C23×Q811yescharacteristic
292C4×C67C22×#4111yescharacteristic
293Dic67—24no—
294Dic67—22no—
295Dic67—22no—
296Dic67—44no—
297Dic67C2×AGL(1,ℤ/8)14yes—
298Dic67C2×#13714yes—
299C2×#3079—671nomaximal, Sylow
300C2×C67—24no—
301C2×C67#307914yescomplemented
302C2×C67C22×C22⋊Q811yescharacteristic
303C2×C67C22×AGL(1,ℤ/8)11yescharacteristic
304C2×C67C22×#13711yescharacteristic
305D67—44no—
306C22×C4⋊Q8—671no—
307C2×C4×#41—671no—
308#3079—6716no—
309C2×#477—672no—
310C2×#475—672no—
311C2×#476—672no—
312C2×#474—672no—
313C2×#428—671no—
314C2×#428D6711yescharacteristic, complemented
315C2×#423—671no—
316C2×#417—671no—
317C2×#417—671no—
318C67C2×#307911yescharacteristic, complemented, Sylow
319C22×#41—671no—
320C23×Q8—1341no—
321C22×M16—671no—
322C22×M16C2×D6711yescharacteristic
323C2×C4⋊Q8—672no—
324C2×C4⋊Q8—672no—
325C2×C4⋊Q8—678no—
326C2×C4⋊Q8—1341no—
327C2×C4⋊Q8—1344no—
328C2×C4⋊Q8—1341no—
329C4×#41—678no—
330C4×#41—1344no—
331C2×C4×Q8—671no—
332C2×C4×Q8—671no—
333C2×C4×D4—671no—
334C2×C4×D4—671no—
335C2×C4×D4—1341no—
336C2×C4×D4—1341no—
337C2×#117—1341no—
338C2×#117C2×D6711yescharacteristic
339C22×C4⋊C4—671no—
340C22×C4⋊C4—1341no—
341C22×C4⋊C4C2×D6711yescharacteristic
342C22×C42—671no—
343C22×C42—1341no—
344#477—6716no—
345#476—6716no—
346#475—6716no—
347#474—6716no—
348#428—678no—
349#428C2×D6718yescomplemented
350C2×#107—672no—
351C2×#107C2×D6712yes—
352C2×#106—672no—
353C2×#106C2×D6712yes—
354#423—678no—
355C2×C4⋊C8—672no—
356C2×C4⋊C8—672no—
357#417—678no—
358#417—678no—
359C2×#103—672no—
360C2×#103—672no—
361C2×#102—672no—
362C2×#102—672no—
363C2×#41—678no—
364C2×#41—674no—
365C2×#41—1342no—
366C2×#41—1344no—
367C22×Q8—2688no—
368C22×Q8—671no—
369C22×Q8—1348no—
370C22×Q8—1342no—
371C22×Q8—1342no—
372C22×D4—671no—
373C22×D4—1341no—
374C23×C4—21no—
375C23×C4—671no—
376C23×C4—671no—
377C23×C4—1341no—
378C23×C4—1341no—
379C23×C4—1341no—
380C23×C4—1341no—
381C23×C4C22×D6711yescharacteristic
382C2×M16—24no—
383C2×M16—674no—
384C2×M16—1344no—
385C2×M16C22×D6714yes—
386C22×C8—672no—
387C22×C8C22×D6712yes—
388C4⋊Q8—678no—
389C4⋊Q8—678no—
390C4⋊Q8—1348no—
391C4⋊Q8—1348no—
392C4⋊Q8—1344no—
393C4⋊Q8—1344no—
394C4×Q8—674no—
395C4×Q8—674no—
396C4×Q8—1342no—
397C4×Q8—1342no—
398C4×D4—674no—
399C4×D4—674no—
400C4×D4—1342no—
401C4×D4—1342no—
402C4×D4—1348no—
403C4×D4—1348no—
404#117—22no—
405#117—1348no—
406#117C22×D6714yes—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—672no—
410C2×C4⋊C4—674no—
411C2×C4⋊C4—1348no—
412C2×C4⋊C4—1342no—
413C2×C4⋊C4—1342no—
414C2×C4⋊C4—1341no—
415C2×C4⋊C4—1341no—
416C2×C4⋊C4—1342no—
417C2×C4⋊C4—1341no—
418C2×C4⋊C4C22×D6712yes—
419C2×C4⋊C4C22×D6714yes—
420C2×C4⋊C4C22×D6711yescharacteristic
421C2×C4⋊C4C22×D6711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1341no—
424C2×C22⋊C4—1341no—
425C2×C42—21no—
426C2×C42—671no—
427C2×C42—674no—
428C2×C42—671no—
429C2×C42—671no—
430C2×C42—671no—
431C2×C42—674no—
432C2×C42—1348no—
433C2×C42—1341no—
434C2×C42—1341no—
435#107—678no—
436#107C22×D6718yes—
437#106—678no—
438#106C22×D6718yes—
439C4⋊C8—678no—
440C4⋊C8—678no—
441#103—678no—
442#103—678no—
443#102—678no—
444#102—678no—
445C24—1341no—
446C24D4×D6711yescharacteristic
447#41—6716no—
448#41—13416no—
449#41—1348no—
450C2×Q8—2688no—
451C2×Q8—26832no—
452C2×Q8—2688no—
453C2×Q8—674no—
454C2×Q8—672no—
455C2×Q8—1348no—
456C2×Q8—1348no—
457C2×Q8—1341no—
458C2×Q8—1342no—
459C2×D4—674no—
460C2×D4—672no—
461C2×D4—1348no—
462C2×D4—1344no—
463C2×D4—1341no—
464C2×D4—1342no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—2684no—
471C22×C4—2684no—
472C22×C4—2684no—
473C22×C4—671no—
474C22×C4—671no—
475C22×C4—674no—
476C22×C4—674no—
477C22×C4—671no—
478C22×C4—671no—
479C22×C4—1344no—
480C22×C4—1344no—
481C22×C4—1344no—
482C22×C4—1342no—
483C22×C4—1344no—
484C22×C4—1344no—
485C22×C4—1341no—
486C22×C4—1348no—
487C22×C4—1344no—
488C22×C4D4×D6712yes—
489C22×C4D4×D6711yescharacteristic
490C22×C4Q8×D6714yes—
491C22×C4#1176162814yes—
492C22×C4C23×D6712yes—
493C22×C4C23×D6711yescharacteristic
494M16—28no—
495M16—1348no—
496C2×C8—24no—
497C2×C8—674no—
498C2×C8—1344no—
499C2×C8C23×D6714yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—674no—
505C4⋊C4—1348no—
506C4⋊C4—1348no—
507C4⋊C4—1344no—
508C4⋊C4—1344no—
509C4⋊C4—1344no—
510C4⋊C4—1342no—
511C4⋊C4C23×D6714yes—
512C4⋊C4C23×D6712yes—
513C4⋊C4C23×D6712yes—
514C22⋊C4—24no—
515C22⋊C4—1344no—
516C22⋊C4—1344no—
517C42—24no—
518C42—674no—
519C42—674no—
520C42—672no—
521C42—672no—
522C42—672no—
523C42—672no—
524C42—1344no—
525C42—1344no—
526C23—22no—
527C23—22no—
528C23—2682no—
529C23—2682no—
530C23—1344no—
531C23—1342no—
532C23#1452402114yes—
533C23C2×D4×D6712yes—
534C23C2×D4×D6711yescharacteristic
535Q8—26816no—
536Q8—26816no—
537Q8—674no—
538Q8—1344no—
539Q8—1342no—
540D4—674no—
541D4—13416no—
542D4—1344no—
543D4—1342no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—2688no—
552C2×C4—2688no—
553C2×C4—2688no—
554C2×C4—2684no—
555C2×C4—674no—
556C2×C4—674no—
557C2×C4—672no—
558C2×C4—672no—
559C2×C4—672no—
560C2×C4—672no—
561C2×C4—1344no—
562C2×C4—1348no—
563C2×C4—1348no—
564C2×C4—1344no—
565C2×C4—1342no—
566C2×C4—1342no—
567C2×C4—1342no—
568C2×C4—13416no—
569C2×C4#1452402114yes—
570C2×C4#1452402118yes—
571C2×C4C2×D4×D6712yes—
572C2×C4C2×D4×D6711yescharacteristic
573C2×C4C2×Q8×D6712yes—
574C2×C4C2×Q8×D6714yes—
575C2×C4C2×#1176162812yes—
576C2×C4C2×#1176162814yes—
577C2×C4C24×D6711yescharacteristic
578C8—24no—
579C8—1344no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—2688no—
584C22—2682no—
585C22—2682no—
586C22—1344no—
587C22C2×#1452402114yes—
588C22C2×#1452402112yes—
589C22AGL(1,ℤ/8)×D6712yes—
590C22#137×D6712yes—
591C22C22×D4×D6711yescharacteristic
592C4—44no—
593C4—42no—
594C4—42no—
595C4—2684no—
596C4—674no—
597C4—674no—
598C4—1342no—
599C4—1342no—
600C4—1344no—
601C4C2×#1452402114yes—
602C4C2×#1452402112yes—
603C4C22×Q8×D6711yescharacteristic
604C4C22×#1176162811yescharacteristic
605C2—24no—
606C2—2684no—
607C2#2257002214yesdirect product factor
608C2C22×#1452402111yescharacteristic
609C2C2×AGL(1,ℤ/8)×D6711yescharacteristic
610C2C2×#137×D6711yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#428×C67C2
socleC23×C67C22×D4
prime radicalC22×#11761628C22
hypercenterC2×#428D67
hypocenterC67C2×#3079
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C22×#14524042C2semidirect
3. #22570022C2direct (N × H)
4. C2×#16863690C2semidirect
5. C2×#16863680C2semidirect
6. C2×#16863685C2semidirect
7. C2×#16863675C2semidirect
8. C2×#16863439C2semidirect
9. C2×#16863419C2semidirect
12. C2×#428×C67C2semidirect
21. C2×#14524042C22semidirect
35. #16863690C22semidirect
36. #16863685C22semidirect
37. #16863680C22semidirect
38. #16863675C22semidirect
39. #16863439C22semidirect
42. #16863419C22semidirect
54. #428×C67C22semidirect
59. C2×#11761628Q8semidirect
142. #11761628C2×Q8semidirect
150. C2×Dic134C2×Q8semidirect
154. C2×D268C2×Q8semidirect
263. D268C22×Q8semidirect
271. Dic134C22×Q8semidirect
301. C2×C67#3079semidirect
314. C2×#428D67semidirect
318. C67C2×#3079semidirect
349. #428C2×D67semidirect
607. C2#22570022direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240422
C2×#168638642
#225700222
C2×#168636902
C2×#168636802
C2×#168636852
C2×#168636752
C2×#168634392
C2×#168634192
C2×#168633982
C2×#168633952
C2×#428×C672
C2×#307967