Finite Group Databasebeta

← C2×#22572625 #31734318

Subgroups — C2×#22572625 #31734318

Order 17152.

1918 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864119C211yesmaximal, characteristic, complemented
2C2×#16864112C211yesmaximal, characteristic
3#22572625C2116yesmaximal, direct product factor
4C2×#16863792C211yesmaximal, characteristic, complemented
5C2×#16863736C212yesmaximal, complemented
6C2×#16863723C212yesmaximal, complemented
7C2×#16863677C211yesmaximal, characteristic, complemented
8C2×#16863628C212yesmaximal, complemented
9C2×C4×#11761615C211yesmaximal, characteristic, complemented
10C2×#16863461C211yesmaximal, characteristic, complemented
11C2×#16863433C212yesmaximal, complemented
12C2×#477×C67C211yesmaximal, characteristic, complemented
13C22×#11761615C2211yescharacteristic
14C22×C8×D67C2212yes—
15#16864119—24no—
16#16864119C2218yescomplemented
17#16864112—24no—
18#16864112C2218yes—
19C2×#14524043C2211yescharacteristic
20C2×#14524038—22no—
21C2×#14524032—22no—
22C2×#14524031—21no—
23C2×#14524019C2212yes—
24C2×#14524018—21no—
25C2×#14524018C2212yes—
26C2×C4×Dic134—21no—
27C2×C4×D268C2211yescharacteristic
28C2×C4×D268C2211yescharacteristic
29C2×#14524001—21no—
30C2×#14524001C2212yes—
31C2×#14524001C2212yes—
32#16863792C2218yescomplemented
33#16863736C22116yescomplemented
34#16863723C22116yescomplemented
35#16863677C2218yescomplemented
36#16863628C22116yescomplemented
37C4×#11761615C2218yescomplemented
38#16863461C2218yescomplemented
39#16863433C22116yescomplemented
40C2×#14523977C2212yes—
41C2×#14523974C2211yescharacteristic
42C2×#14523974C2212yes—
43C2×#14523974C2212yes—
44C2×#14523967C2211yescharacteristic
45C2×#14523965C2212yes—
46C2×#14523963C2211yescharacteristic
47C2×(C2×D67)⋊C8C2212yes—
48C2×C4×C67⋊C8C2211yescharacteristic
49C2×C4⋊Q8×C67C2211yescharacteristic
50C2×C4×Q8×C67C2211yescharacteristic
51#477×C67C2218yescomplemented
52C2×#107×C67C2212yes—
53C2×C4⋊C8×C67C2211yescharacteristic
54C2×#103×C67C2212yes—
55C22×Dic134—21no—
56C22×D268—22no—
57C22×D268D411yescharacteristic
58C22×D268C2311yescharacteristic
59C22×C4×D67—21no—
60C22×C4×D67C2312yes—
61C2×#11761615—24no—
62C2×#11761615C2314yes—
63C22×C67⋊C8C2311yescharacteristic
64C22×C67⋊C8C2311yescharacteristic
65C2×C8×D67—28no—
66C2×C8×D67C2318yes—
67#14524043—22no—
68#14524043C2314yes—
69#14524038—216no—
70#14524032—216no—
71#14524031—28no—
72#14524019—24no—
73#14524019C2318yes—
74#14524018—28no—
75#14524018—24no—
76#14524018C2318yes—
77C4×Dic134—28no—
78C4×D268—22no—
79C4×D268D414yes—
80C4×D268C2314yes—
81C4×D268C2314yes—
82#14524001—24no—
83#14524001—24no—
84#14524001—28no—
85#14524001C2318yes—
86#14524001C2318yes—
87C2×C4⋊Dic67—21no—
88C2×C4⋊Dic67D411yescharacteristic
89C2×C4⋊Dic67C2311yescharacteristic
90C2×C4⋊Dic67C2312yes—
91C2×Dic67⋊C4—22no—
92C2×(C2×D67)⋊C4—22no—
93C2×(C2×D67)⋊C4—22no—
94C2×(C2×D67)⋊C4—22no—
95C2×(C2×D67)⋊C4—22no—
96C2×(C2×D67)⋊C4—21no—
97C2×(C2×D67)⋊C4D412yes—
98C2×C4×Dic67—22no—
99C2×C4×Dic67—22no—
100#14523977C2318yes—
101#14523974C2318yes—
102#14523974C2318yes—
103#14523974C2314yes—
104#14523967C2314yes—
105#14523965C2318yes—
106#14523963C2314yes—
107(C2×D67)⋊C8C2318yes—
108C4×C67⋊C8C2314yes—
109C22×Q8×C67—21no—
110C22×Q8×C67C2311yescharacteristic
111C22×C8×C67C2312yes—
112C4⋊Q8×C67—22no—
113C4⋊Q8×C67C2314yes—
114C4×Q8×C67—22no—
115C4×Q8×C67C2314yes—
116C2×C4⋊C4×C67—21no—
117C2×C4⋊C4×C67—21no—
118C2×C4⋊C4×C67C2311yescharacteristic
119C2×C4⋊C4×C67C2312yes—
120C2×C42×C67—21no—
121C2×C42×C67C2311yescharacteristic
122#107×C67C2318yes—
123C4⋊C8×C67C2314yes—
124#103×C67C2318yes—
125C23×D67—21no—
126C23×D67C2×D412yes—
127C2×Dic134—24no—
128C2×Dic134—44no—
129C2×D268—24no—
130C2×D268—22no—
131C2×D268—22no—
132C2×D268—21no—
133C2×D268—28no—
134C2×D268—84no—
135C2×D268C2×D414yes—
136C2×D268C2412yes—
137C22×Dic67—21no—
138C22×Dic67—22no—
139C22×Dic67C2×D412yes—
140C2×C4×D67—24no—
141C2×C4×D67—24no—
142C2×C4×D67—24no—
143C2×C4×D67—44no—
144C2×C4×D67C2×D414yes—
145C2×C4×D67C2414yes—
146#11761615—28no—
147C2×C67⋊C8—22no—
148C2×C67⋊C8#4114yes—
149C2×C67⋊C8C2412yes—
150C2×C67⋊C8C2412yes—
151C8×D67—216no—
152C4⋊Dic67—24no—
153C4⋊Dic67—21no—
154C4⋊Dic67—22no—
155C4⋊Dic67C2×D414yes—
156C4⋊Dic67C2414yes—
157C4⋊Dic67C2412yes—
158Dic67⋊C4—28no—
159(C2×D67)⋊C4—28no—
160(C2×D67)⋊C4—28no—
161(C2×D67)⋊C4—28no—
162(C2×D67)⋊C4—24no—
163(C2×D67)⋊C4—28no—
164(C2×D67)⋊C4C2×D418yes—
165C4×Dic67—28no—
166C4×Dic67—28no—
167C2×Q8×C67—21no—
168C2×Q8×C67—24no—
169C2×Q8×C67—22no—
170C2×Q8×C67—22no—
171C2×Q8×C67—82no—
172C2×Q8×C67C2412yes—
173C22×C4×C67—21no—
174C22×C4×C67—21no—
175C22×C4×C67—22no—
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2411yescharacteristic
179C2×C8×C67—24no—
180C2×C8×C67C2414yes—
181C4⋊C4×C67—24no—
182C4⋊C4×C67—24no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—22no—
185C4⋊C4×C67C2414yes—
186C4⋊C4×C67C2412yes—
187C42×C67—24no—
188C42×C67C2×D412yes—
189C42×C67C2412yes—
190C23×C67C22×D411yescharacteristic
191C22×D67—24no—
192C22×D67—24no—
193C22×D67—24no—
194C22×D67—22no—
195C22×D67—42no—
196C22×D67—42no—
197C22×D67—42no—
198C22×D67C22×D414yes—
199Q8×C67—22no—
200Q8×C67—22no—
201Q8×C67—42no—
202Q8×C67—84no—
203C2×C4×C67—22no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—24no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—44no—
210C2×C4×C67#13514yes—
211C2×C4×C67C22×D412yes—
212C2×C4×C67C22×D411yescharacteristic
213C2×C4×C67C22×D412yes—
214C2×C4×C67C2×#4112yes—
215C2×C4×C672−1+412yes—
216C2×C4×C67C2511yescharacteristic
217C2×Dic67—24no—
218C2×Dic67—22no—
219C2×Dic67—24no—
220C2×Dic67—42no—
221C2×Dic67—44no—
222C2×Dic67C22×D414yes—
223D268—28no—
224D268—22no—
225D268—22no—
226D268—42no—
227D268—88no—
228C4×D67—28no—
229C4×D67—24no—
230C4×D67—24no—
231C4×D67—48no—
232Dic134—48no—
233C8×C67—24no—
234C67⋊C8—22no—
235C67⋊C8C2×#4114yes—
236C22×C67#54712yes—
237C22×C67C2×#13512yes—
238C22×C67#57712yes—
239C22×C67C23×D411yescharacteristic
240C2×D67—24no—
241C2×D67—24no—
242C2×D67—24no—
243C2×D67—48no—
244C2×D67—42no—
245C2×D67—42no—
246C2×D67—42no—
247C2×D67—84no—
248C4×C67—22no—
249C4×C67—42no—
250C4×C67—44no—
251C4×C67—42no—
252C4×C67#54712yes—
253C4×C67C2×#13514yes—
254C4×C67C22×#4111yescharacteristic
255C4×C67C2×2−1+411yescharacteristic
256Dic67—24no—
257Dic67—44no—
258Dic67—42no—
259C2×#3340—671nomaximal, Sylow
260C2×C67#334014yescomplemented
261C2×C67C2×#54711yescharacteristic
262C2×C67C22×#13511yescharacteristic
263C2×C67C2×#57711yescharacteristic
264D67—44no—
265D67—82no—
266C2×#556—671no—
267C2×#555—671no—
268#3340—6716no—
269C2×#498—671no—
270C2×#485—672no—
271C2×#482—672no—
272C2×#477D6711yescharacteristic, complemented
273C2×#474—671no—
274C2×#465—672no—
275C2×C4×SD16—671no—
276C2×C8×D4—671no—
277C2×#427—672no—
278C67C2×#334011yescharacteristic, complemented, Sylow
279C22×SD16—671no—
280C23×C8—672no—
281#556—678no—
282#556—1344no—
283#555—678no—
284#555—1344no—
285C2×C4⋊Q8—671no—
286C2×C4⋊Q8C2×D6711yescharacteristic
287C2×#126—1342no—
288C2×#124—1341no—
289C2×#124—1342no—
290C2×C22⋊Q8—672no—
291C2×#121—672no—
292C2×#121—1341no—
293C2×C4×Q8—1341no—
294C2×C4×Q8C2×D6711yescharacteristic
295C2×C4×D4—671no—
296C2×C4×D4—671no—
297C2×#117—672no—
298C2×#117—672no—
299C2×#117—1341no—
300#498—678no—
301#485—6716no—
302#482—6716no—
303#477C2×D6718yescomplemented
304#474—678no—
305#465—6716no—
306C4×SD16—678no—
307C8×D4—678no—
308#427—6716no—
309C2×#107—672no—
310C2×#107C2×D6712yes—
311C2×#106—672no—
312C2×#106—672no—
313C2×#106—671no—
314C2×C4⋊C8C2×D6711yescharacteristic
315C2×#103—671no—
316C2×#103C2×D6712yes—
317C2×#102—672no—
318C2×#102—671no—
319C2×C22⋊C8—672no—
320C2×C4×C8—671no—
321C22×Q8—21no—
322C22×Q8—1341no—
323C22×Q8C22×D6711yescharacteristic
324C22×D4—671no—
325C22×D4—671no—
326C22×D4—1342no—
327C23×C4—672no—
328C23×C4—1341no—
329C2×SD16—674no—
330C2×SD16—1344no—
331C22×C8—671no—
332C22×C8—671no—
333C22×C8—678no—
334C22×C8—1348no—
335C22×C8C22×D6712yes—
336C4⋊Q8—22no—
337C4⋊Q8—674no—
338C4⋊Q8—1342no—
339C4⋊Q8C22×D6714yes—
340#126—13416no—
341#124—1348no—
342#124—13416no—
343C22⋊Q8—678no—
344C22⋊Q8—1344no—
345#121—678no—
346#121—1344no—
347#121—1348no—
348C4×Q8—22no—
349C4×Q8—1348no—
350C4×Q8C22×D6714yes—
351C4×D4—674no—
352C4×D4—674no—
353C4×D4—674no—
354C4×D4—1342no—
355#117—678no—
356#117—678no—
357#117—1348no—
358#117—1344no—
359#117—1344no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—671no—
363C2×C4⋊C4—671no—
364C2×C4⋊C4—672no—
365C2×C4⋊C4—1342no—
366C2×C4⋊C4—1341no—
367C2×C4⋊C4C22×D6711yescharacteristic
368C2×C4⋊C4C22×D6712yes—
369C2×C22⋊C4—672no—
370C2×C22⋊C4—1342no—
371C2×C22⋊C4—1342no—
372C2×C22⋊C4—1342no—
373C2×C22⋊C4—1342no—
374C2×C22⋊C4—1341no—
375C2×C42—21no—
376C2×C42—1342no—
377C2×C42—1342no—
378C2×C42C22×D6711yescharacteristic
379#107—678no—
380#107C22×D6718yes—
381#106—674no—
382#106—678no—
383#106—678no—
384C4⋊C8C22×D6714yes—
385#103—674no—
386#103C22×D6718yes—
387#102—678no—
388#102—674no—
389C22⋊C8—678no—
390C4×C8—674no—
391C24—672no—
392C24—1341no—
393C2×Q8—21no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—82no—
398C2×Q8—2684no—
399C2×Q8—1344no—
400C2×Q8C23×D6712yes—
401C2×D4—5364no—
402C2×D4—674no—
403C2×D4—672no—
404C2×D4—1348no—
405C2×D4—1344no—
406C2×D4—1341no—
407C2×D4—1342no—
408C2×D4—1342no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—2684no—
413C22×C4—672no—
414C22×C4—674no—
415C22×C4—674no—
416C22×C4—1341no—
417C22×C4—1342no—
418C22×C4—1344no—
419C22×C4—1344no—
420C22×C4—1344no—
421C22×C4D4×D6711yescharacteristic
422C22×C4D4×D6711yescharacteristic
423C22×C4C23×D6711yescharacteristic
424SD16—1348no—
425C2×C8—24no—
426C2×C8—674no—
427C2×C8—672no—
428C2×C8—672no—
429C2×C8—13416no—
430C2×C8—1342no—
431C2×C8C23×D6714yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—674no—
437C4⋊C4—674no—
438C4⋊C4—672no—
439C4⋊C4—1348no—
440C4⋊C4—1341no—
441C4⋊C4—1342no—
442C4⋊C4—1344no—
443C4⋊C4C23×D6712yes—
444C4⋊C4C23×D6714yes—
445C22⋊C4—678no—
446C22⋊C4—1348no—
447C22⋊C4—1348no—
448C22⋊C4—1348no—
449C22⋊C4—1348no—
450C22⋊C4—1344no—
451C42—24no—
452C42—1348no—
453C42—1348no—
454C42D4×D6712yes—
455C42C23×D6712yes—
456C23—2682no—
457C23—2682no—
458C23—2682no—
459C23—674no—
460C23—1342no—
461C23—1344no—
462C23—1344no—
463C23—1344no—
464C23C2×D4×D6711yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—84no—
469Q8—2688no—
470D4—2682no—
471D4—5368no—
472D4—1348no—
473D4—1342no—
474D4—1342no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—2682no—
483C2×C4—2684no—
484C2×C4—2688no—
485C2×C4—674no—
486C2×C4—1342no—
487C2×C4—1344no—
488C2×C4—1344no—
489C2×C4—1348no—
490C2×C4—1344no—
491C2×C4—1344no—
492C2×C4#1452406614yes—
493C2×C4C2×D4×D6712yes—
494C2×C4C2×D4×D6711yescharacteristic
495C2×C4C2×D4×D6712yes—
496C2×C4#41×D6712yes—
497C2×C4#1452409112yes—
498C2×C4C24×D6711yescharacteristic
499C8—24no—
500C8—674no—
501C8—1342no—
502C22—2688no—
503C22—2682no—
504C22—2682no—
505C22—2682no—
506C22—5364no—
507C22—1344no—
508C22—1344no—
509C22—1344no—
510C22#1686404812yes—
511C22C2×#1452406612yes—
512C22#1686420612yes—
513C22C22×D4×D6711yescharacteristic
514C4—22no—
515C4—42no—
516C4—44no—
517C4—42no—
518C4—2682no—
519C4—2684no—
520C4—1344no—
521C4#1686404812yes—
522C4C2×#1452406614yes—
523C4C2×#41×D6711yescharacteristic
524C4C2×#1452409111yescharacteristic
525C2—2684no—
526C2—5362no—
527C2#2257262514yesdirect product factor
528C2C2×#1686404811yescharacteristic
529C2C22×#1452406611yescharacteristic
530C2C2×#1686420611yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#477×C67C2
socleC23×C67C22×D4
prime radicalC2×#16864112C2
hypercenterC2×#477D67
hypocenterC67C2×#3340
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
1. C2×#16864119C2semidirect
3. #22572625C2direct (N × H)
4. C2×#16863792C2semidirect
5. C2×#16863736C2semidirect
6. C2×#16863723C2semidirect
7. C2×#16863677C2semidirect
8. C2×#16863628C2semidirect
9. C2×C4×#11761615C2semidirect
10. C2×#16863461C2semidirect
11. C2×#16863433C2semidirect
12. C2×#477×C67C2semidirect
16. #16864119C22semidirect
32. #16863792C22semidirect
33. #16863736C22semidirect
34. #16863723C22semidirect
35. #16863677C22semidirect
36. #16863628C22semidirect
37. C4×#11761615C22semidirect
38. #16863461C22semidirect
39. #16863433C22semidirect
51. #477×C67C22semidirect
260. C2×C67#3340semidirect
272. C2×#477D67semidirect
278. C67C2×#3340semidirect
303. #477C2×D67semidirect
527. C2#22572625direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168641192
C2×#168641122
#225726252
C2×#168637922
C2×#168637362
C2×#168637232
C2×#168636772
C2×#168636282
C2×C4×#117616152
C2×#168634612
C2×#168634332
C2×#477×C672
C2×#334067