Finite Group Databasebeta

← C2×#22571359 #31727879

Subgroups — C2×#22571359 #31727879

Order 17152.

2006 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863950C211yesmaximal, characteristic
2C2×C4⋊Q8×D67C211yesmaximal, characteristic
3#22571359C2116yesmaximal, direct product factor
4C2×#16863800C212yesmaximal, complemented
5C2×#16863745C212yesmaximal, complemented
6C2×#16863733C212yesmaximal
7C2×#16863730C212yesmaximal
8C2×#16863692C211yesmaximal, characteristic, complemented
9C2×#16863691C211yesmaximal, characteristic, complemented
10C2×#16863688C211yesmaximal, characteristic, complemented
11C2×#16863448C211yesmaximal, characteristic
12C2×#477×C67C211yesmaximal, characteristic, complemented
13C22×Q8×D67—21no—
14#16863950—24no—
15#16863950C2218yes—
16C2×#14524043C2211yescharacteristic
17C2×#14524042—21no—
18C2×#14524042C2212yes—
19C2×#14524042C2211yescharacteristic
20C2×#14524041—21no—
21C2×#14524041C2211yescharacteristic
22C4⋊Q8×D67—24no—
23C4⋊Q8×D67C2218yes—
24C2×#14524033—21no—
25C2×#14524023C2211yescharacteristic
26C2×#14524021—21no—
27C2×#14524019C2211yescharacteristic
28C2×C4×Dic134—21no—
29C2×C4×Dic134C2211yescharacteristic
30C2×#14524000—21no—
31C2×C4⋊C4×D67—21no—
32C2×C4⋊C4×D67C2212yes—
33C2×C42×D67C2211yescharacteristic
34#16863800C22116yescomplemented
35#16863745C22116yescomplemented
36#16863733C22116yes—
37#16863730C22116yes—
38#16863692C2218yescomplemented
39#16863691C2218yescomplemented
40#16863688C2218yescomplemented
41#16863448C2218yes—
42C2×#14523977C2212yes—
43C2×#14523976C2212yes—
44C2×#14523976C2212yes—
45C2×C4⋊(C67⋊C8)C2211yescharacteristic
46C2×#14523968C2212yes—
47C2×#14523967C2212yes—
48C2×#14523966C2212yes—
49C2×(C2×D67)⋊C8C2212yes—
50C2×C8×Dic67C2212yes—
51C2×C4⋊Q8×C67C2211yescharacteristic
52C2×C4×Q8×C67C2211yescharacteristic
53#477×C67C2218yescomplemented
54C2×#107×C67C2212yes—
55C2×C4⋊C8×C67C2211yescharacteristic
56C2×#103×C67C2212yes—
57C22×Dic134—21no—
58C22×Dic134—22no—
59C22×Dic134—21no—
60C22×Dic134C2311yescharacteristic
61C2×Q8×D67—28no—
62C2×Q8×D67—48no—
63C22×C4×D67—22no—
64C22×C4×D67D411yescharacteristic
65C22×C4×D67D411yescharacteristic
66C22×C4×D67C2311yescharacteristic
67C22×C67⋊C8C2312yes—
68#14524043—22no—
69#14524043C2314yes—
70#14524042—24no—
71#14524042—28no—
72#14524042—22no—
73#14524042C2318yes—
74#14524042C2314yes—
75#14524041—28no—
76#14524041—22no—
77#14524041C2314yes—
78#14524033—28no—
79#14524023—22no—
80#14524023C2314yes—
81#14524021—28no—
82#14524019—22no—
83#14524019C2314yes—
84C4×Dic134—28no—
85C4×Dic134—22no—
86C4×Dic134C2314yes—
87#14524000—28no—
88C2×C4⋊Dic67—21no—
89C2×C4⋊Dic67—21no—
90C2×C4⋊Dic67C2312yes—
91C2×C4⋊Dic67C2311yescharacteristic
92C2×Dic67⋊C4—22no—
93C2×Dic67⋊C4—22no—
94C2×Dic67⋊C4—21no—
95C2×Dic67⋊C4—21no—
96C2×Dic67⋊C4—21no—
97C2×Dic67⋊C4—21no—
98C2×Dic67⋊C4—21no—
99C4⋊C4×D67—28no—
100C4⋊C4×D67—24no—
101C4⋊C4×D67C2318yes—
102C2×(C2×D67)⋊C4—21no—
103C2×(C2×D67)⋊C4—21no—
104C2×C4×Dic67—21no—
105C2×C4×Dic67D411yescharacteristic
106C2×C4×Dic67D411yescharacteristic
107C2×C4×Dic67C2311yescharacteristic
108C42×D67D414yes—
109C42×D67C2314yes—
110#14523977C2318yes—
111#14523976C2318yes—
112#14523976C2318yes—
113C4⋊(C67⋊C8)C2314yes—
114#14523968C2318yes—
115#14523967C2318yes—
116#14523966C2318yes—
117(C2×D67)⋊C8C2318yes—
118C8×Dic67C2318yes—
119C22×Q8×C67—21no—
120C22×Q8×C67C2311yescharacteristic
121C22×C8×C67C2312yes—
122C4⋊Q8×C67—22no—
123C4⋊Q8×C67C2314yes—
124C4×Q8×C67—22no—
125C4×Q8×C67C2314yes—
126C2×C4⋊C4×C67—21no—
127C2×C4⋊C4×C67—21no—
128C2×C4⋊C4×C67C2311yescharacteristic
129C2×C4⋊C4×C67C2312yes—
130C2×C42×C67—21no—
131C2×C42×C67C2311yescharacteristic
132#107×C67C2318yes—
133C4⋊C8×C67C2314yes—
134#103×C67C2318yes—
135C23×D67C2×D411yescharacteristic
136C2×Dic134—22no—
137C2×Dic134—22no—
138C2×Dic134—21no—
139C2×Dic134—24no—
140C2×Dic134—24no—
141C2×Dic134—28no—
142C2×Dic134—44no—
143C2×Dic134—48no—
144C2×Dic134—44no—
145C2×Dic134C2412yes—
146Q8×D67—432no—
147C22×Dic67—21no—
148C22×Dic67—21no—
149C22×Dic67—22no—
150C22×Dic67C2×D411yescharacteristic
151C22×Dic67C2×D411yescharacteristic
152C22×Dic67C2×D411yescharacteristic
153C2×C4×D67—22no—
154C2×C4×D67—22no—
155C2×C4×D67—24no—
156C2×C4×D67—28no—
157C2×C4×D67—48no—
158C2×C4×D67Q1618yes—
159C2×C4×D67C2×D412yes—
160C2×C4×D67C2×D414yes—
161C2×C4×D67C2×D414yes—
162C2×C4×D67C2412yes—
163C2×C67⋊C8—24no—
164C2×C67⋊C8C2414yes—
165C4⋊Dic67—24no—
166C4⋊Dic67—22no—
167C4⋊Dic67—21no—
168C4⋊Dic67—24no—
169C4⋊Dic67C2414yes—
170C4⋊Dic67C2412yes—
171Dic67⋊C4—24no—
172Dic67⋊C4—28no—
173Dic67⋊C4—28no—
174Dic67⋊C4—24no—
175Dic67⋊C4—24no—
176Dic67⋊C4—24no—
177Dic67⋊C4—24no—
178(C2×D67)⋊C4—24no—
179(C2×D67)⋊C4—24no—
180C4×Dic67—24no—
181C4×Dic67C2×D412yes—
182C4×Dic67C2×D414yes—
183C4×Dic67C2×D414yes—
184C4×Dic67C2412yes—
185C2×Q8×C67—21no—
186C2×Q8×C67—24no—
187C2×Q8×C67—22no—
188C2×Q8×C67—22no—
189C2×Q8×C67—44no—
190C2×Q8×C67C2412yes—
191C22×C4×C67—21no—
192C22×C4×C67—21no—
193C22×C4×C67—22no—
194C22×C4×C67C2×D411yescharacteristic
195C22×C4×C67C2×D411yescharacteristic
196C22×C4×C67C2411yescharacteristic
197C2×C8×C67—24no—
198C2×C8×C67C2414yes—
199C4⋊C4×C67—24no—
200C4⋊C4×C67—24no—
201C4⋊C4×C67—21no—
202C4⋊C4×C67—22no—
203C4⋊C4×C67C2414yes—
204C4⋊C4×C67C2412yes—
205C42×C67—24no—
206C42×C67C2×D412yes—
207C42×C67C2412yes—
208C23×C67C22×D411yescharacteristic
209C22×D67—22no—
210C22×D67—22no—
211C22×D67C2×Q1614yes—
212C22×D67C22×D412yes—
213Q8×C67—22no—
214Q8×C67—22no—
215Q8×C67—48no—
216Q8×C67—42no—
217C2×C4×C67—22no—
218C2×C4×C67—22no—
219C2×C4×C67—22no—
220C2×C4×C67—24no—
221C2×C4×C67—42no—
222C2×C4×C67—44no—
223C2×C4×C67—42no—
224C2×C4×C67C2×Q1614yes—
225C2×C4×C67C22×D412yes—
226C2×C4×C67C22×D411yescharacteristic
227C2×C4×C67C22×D412yes—
228C2×C4×C672−1+412yes—
229C2×C4×C672−1+412yes—
230C2×C4×C67C2511yescharacteristic
231C2×Dic67—22no—
232C2×Dic67—22no—
233C2×Dic67—22no—
234C2×Dic67—24no—
235C2×Dic67—42no—
236C2×Dic67—44no—
237C2×Dic67—42no—
238C2×Dic67C2×Q1614yes—
239C2×Dic67AGL(1,ℤ/8)14yes—
240C2×Dic67C22×D412yes—
241C2×Dic67C22×D412yes—
242C2×Dic67C22×D412yes—
243C4×D67—24no—
244C4×D67—22no—
245C4×D67—22no—
246C4×D67—28no—
247C4×D67—416no—
248C4×D67C2×Q16116yes—
249Dic134—22no—
250Dic134—22no—
251Dic134—48no—
252Dic134—42no—
253Dic134—416no—
254Dic134—48no—
255C8×C67—24no—
256C67⋊C8—24no—
257C22×C67#53612yes—
258C22×C67C22×Q1612yes—
259C22×C67C2×AGL(1,ℤ/8)12yes—
260C22×C67C23×D411yescharacteristic
261C2×D67—22no—
262C2×D67—28no—
263C2×D67—22no—
264C2×D67C22×Q1614yes—
265C4×C67—22no—
266C4×C67—42no—
267C4×C67—44no—
268C4×C67—42no—
269C4×C67#53612yes—
270C4×C67C22×Q1614yes—
271C4×C67C2×2−1+411yescharacteristic
272C4×C67C2×2−1+411yescharacteristic
273Dic67—22no—
274Dic67—42no—
275Dic67—44no—
276Dic67—42no—
277Dic67C22×Q1614yes—
278Dic67C2×AGL(1,ℤ/8)14yes—
279C2×#3230—671nomaximal, Sylow
280C2×C67#323014yescomplemented
281C2×C67C2×#53611yescharacteristic
282C2×C67C23×Q1611yescharacteristic
283C2×C67C22×AGL(1,ℤ/8)11yescharacteristic
284D67—24no—
285C2×#533—671no—
286C22×C4⋊Q8—671no—
287#3230—6716no—
288C2×#500—672no—
289C2×#487—672no—
290C2×#484—672no—
291C2×#484—672no—
292C2×#477—671no—
293C2×#477—671no—
294C2×#477—671no—
295C2×#477D6711yescharacteristic, complemented
296C2×#431—671no—
297C67C2×#323011yescharacteristic, complemented, Sylow
298C23×Q8—1341no—
299#533—678no—
300#533—1344no—
301C2×C4⋊Q8—678no—
302C2×C4⋊Q8—672no—
303C2×C4⋊Q8—671no—
304C2×C4⋊Q8—671no—
305C2×C4⋊Q8—671no—
306C2×C4⋊Q8—1341no—
307C2×C4⋊Q8—1344no—
308C2×C4⋊Q8—1341no—
309C2×C4⋊Q8C2×D6711yescharacteristic
310C2×#125—1341no—
311C2×C22⋊Q8—671no—
312C2×C22⋊Q8—671no—
313C2×C22⋊Q8—1341no—
314C2×C4×Q8—671no—
315C2×C4×Q8—1341no—
316C2×C4×Q8C2×D6711yescharacteristic
317C2×#117—1341no—
318C22×C4⋊C4—672no—
319C22×C4⋊C4—1341no—
320C22×C42—671no—
321#500—6716no—
322#487—6716no—
323#484—6716no—
324#484—6716no—
325#477—678no—
326#477—678no—
327#477—678no—
328#477C2×D6718yescomplemented
329#431—678no—
330C2×#107—672no—
331C2×#107—672no—
332C2×#107—672no—
333C2×#107C2×D6712yes—
334C2×C4⋊C8—671no—
335C2×C4⋊C8C2×D6711yescharacteristic
336C2×#103—672no—
337C2×#103—672no—
338C2×#103—672no—
339C2×#103C2×D6712yes—
340C2×C22⋊C8—672no—
341C2×C4×C8—672no—
342C22×Q8—21no—
343C22×Q8—2688no—
344C22×Q8—671no—
345C22×Q8—1341no—
346C22×Q8—1342no—
347C22×Q8—1348no—
348C22×Q8—1341no—
349C22×Q8C22×D6711yescharacteristic
350C23×C4—671no—
351C23×C4—671no—
352C23×C4—671no—
353C23×C4—1342no—
354C22×C8—672no—
355C22×C8C22×D6712yes—
356C4⋊Q8—22no—
357C4⋊Q8—678no—
358C4⋊Q8—674no—
359C4⋊Q8—674no—
360C4⋊Q8—674no—
361C4⋊Q8—1348no—
362C4⋊Q8—1344no—
363C4⋊Q8—1342no—
364C4⋊Q8—1348no—
365C4⋊Q8—1342no—
366C4⋊Q8—1342no—
367C4⋊Q8C22×D6714yes—
368#125—1348no—
369C22⋊Q8—674no—
370C22⋊Q8—674no—
371C22⋊Q8—1342no—
372C22⋊Q8—1348no—
373C22⋊Q8—1342no—
374C4×Q8—22no—
375C4×Q8—674no—
376C4×Q8—1348no—
377C4×Q8—1342no—
378C4×Q8C22×D6714yes—
379#117—1348no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—672no—
383C2×C4⋊C4—671no—
384C2×C4⋊C4—678no—
385C2×C4⋊C4—1344no—
386C2×C4⋊C4—1348no—
387C2×C4⋊C4—1342no—
388C2×C4⋊C4—1342no—
389C2×C4⋊C4—1341no—
390C2×C4⋊C4—1341no—
391C2×C4⋊C4—1341no—
392C2×C4⋊C4—1341no—
393C2×C4⋊C4—1341no—
394C2×C4⋊C4—1341no—
395C2×C4⋊C4—1341no—
396C2×C4⋊C4C22×D6712yes—
397C2×C4⋊C4C22×D6711yescharacteristic
398C2×C22⋊C4—1341no—
399C2×C22⋊C4—1341no—
400C2×C42—21no—
401C2×C42—671no—
402C2×C42—671no—
403C2×C42—671no—
404C2×C42—674no—
405C2×C42—674no—
406C2×C42—1341no—
407C2×C42C22×D6711yescharacteristic
408#107—678no—
409#107—678no—
410#107—678no—
411#107C22×D6718yes—
412C4⋊C8—674no—
413C4⋊C8C22×D6714yes—
414#103—678no—
415#103—678no—
416#103—678no—
417#103C22×D6718yes—
418C22⋊C8—678no—
419C4×C8—678no—
420C24—671no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—2684no—
427C2×Q8—2688no—
428C2×Q8—2684no—
429C2×Q8—26832no—
430C2×Q8—672no—
431C2×Q8—1344no—
432C2×Q8—1348no—
433C2×Q8—1342no—
434C2×Q8—1342no—
435C2×Q8—1341no—
436C2×Q8—1344no—
437C2×Q8C23×D6712yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—2688no—
442C22×C4—671no—
443C22×C4—671no—
444C22×C4—678no—
445C22×C4—674no—
446C22×C4—671no—
447C22×C4—672no—
448C22×C4—674no—
449C22×C4—672no—
450C22×C4—1348no—
451C22×C4—1344no—
452C22×C4—1341no—
453C22×C4—1341no—
454C22×C4—1342no—
455C22×C4—1342no—
456C22×C4—1342no—
457C22×C4D4×D6711yescharacteristic
458C22×C4D4×D6711yescharacteristic
459C22×C4C23×D6711yescharacteristic
460C2×C8—24no—
461C2×C8—674no—
462C2×C8—1344no—
463C2×C8C23×D6714yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—674no—
469C4⋊C4—672no—
470C4⋊C4—1348no—
471C4⋊C4—1348no—
472C4⋊C4—1344no—
473C4⋊C4—1344no—
474C4⋊C4—1344no—
475C4⋊C4—1344no—
476C4⋊C4—1344no—
477C4⋊C4—1342no—
478C4⋊C4—1341no—
479C4⋊C4—1344no—
480C4⋊C4—1344no—
481C4⋊C4C23×D6714yes—
482C4⋊C4C23×D6712yes—
483C22⋊C4—1344no—
484C22⋊C4—1344no—
485C42—24no—
486C42—674no—
487C42—674no—
488C42—672no—
489C42—672no—
490C42—1344no—
491C42D4×D6712yes—
492C42C23×D6712yes—
493C23—674no—
494C23—672no—
495C23—1342no—
496C23—1342no—
497C23C2×D4×D6711yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—48no—
501Q8—42no—
502Q8—2688no—
503Q8—26816no—
504Q8—2688no—
505Q8—2682no—
506Q8—1342no—
507Q8—1342no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—26816no—
516C2×C4—2684no—
517C2×C4—2682no—
518C2×C4—2682no—
519C2×C4—674no—
520C2×C4—674no—
521C2×C4—672no—
522C2×C4—672no—
523C2×C4—6716no—
524C2×C4—672no—
525C2×C4—1348no—
526C2×C4—1342no—
527C2×C4—1342no—
528C2×C4—1344no—
529C2×C4—1342no—
530C2×C4—1344no—
531C2×C4—1342no—
532C2×C4—1342no—
533C2×C4Q16×D6714yes—
534C2×C4C2×D4×D6712yes—
535C2×C4C2×D4×D6712yes—
536C2×C4C2×D4×D6711yescharacteristic
537C2×C4#1452409012yes—
538C2×C4#1452409112yes—
539C2×C4C24×D6711yescharacteristic
540C8—24no—
541C8—1344no—
542C22—674no—
543C22—1342no—
544C22—1348no—
545C22—1342no—
546C22#1686396912yes—
547C22C2×Q16×D6712yes—
548C22AGL(1,ℤ/8)×D6712yes—
549C22C22×D4×D6711yescharacteristic
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—2684no—
555C4—2682no—
556C4—2682no—
557C4—674no—
558C4—674no—
559C4—1342no—
560C4#1686396912yes—
561C4C2×Q16×D6714yes—
562C4C2×#1452409011yescharacteristic
563C4C2×#1452409111yescharacteristic
564C2—1344no—
565C2#2257135914yesdirect product factor
566C2C2×#1686396911yescharacteristic
567C2C22×Q16×D6711yescharacteristic
568C2C2×AGL(1,ℤ/8)×D6711yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#477×C67C2
socleC23×C67C22×D4
prime radicalC23×D67C2×D4
hypercenterC2×#477D67
hypocenterC67C2×#3230
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #22571359C2direct (N × H)
4. C2×#16863800C2semidirect
5. C2×#16863745C2semidirect
8. C2×#16863692C2semidirect
9. C2×#16863691C2semidirect
10. C2×#16863688C2semidirect
12. C2×#477×C67C2semidirect
34. #16863800C22semidirect
35. #16863745C22semidirect
38. #16863692C22semidirect
39. #16863691C22semidirect
40. #16863688C22semidirect
53. #477×C67C22semidirect
280. C2×C67#3230semidirect
295. C2×#477D67semidirect
297. C67C2×#3230semidirect
328. #477C2×D67semidirect
565. C2#22571359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168639502
C2×C4⋊Q8×D672
#225713592
C2×#168638002
C2×#168637452
C2×#168637332
C2×#168637302
C2×#168636922
C2×#168636912
C2×#168636882
C2×#168634482
C2×#477×C672
C2×#323067