Finite Group Databasebeta

← C2×#22561412 #30883303

Subgroups — C2×#22561412 #30883303

Order 17152.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524022C211yesmaximal, characteristic
2C22×#14523976C211yesmaximal, characteristic, complemented
3C22×#14523966C211yesmaximal, characteristic, complemented
4C22×#14523966C211yesmaximal, characteristic, complemented
5C22×C22⋊(C67⋊C8)C211yesmaximal, characteristic
6#22561412C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C2×#380×C67C211yesmaximal, characteristic
9C23×Dic134C2211yescharacteristic
10C23×C67⋊C8C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524022—24no—
13C2×#14524022—24no—
14C2×#14524022—22no—
15C2×#14524022C2214yes—
16C22×C4⋊Dic67C2211yescharacteristic
17C22×Dic67⋊C4—21no—
18C22×C22⋊Dic67—21no—
19C2×#14523976—24no—
20C2×#14523976C2214yescomplemented
21C2×#14523966—24no—
22C2×#14523966—24no—
23C2×#14523966C2214yescomplemented
24C2×#14523966C2214yescomplemented
25C2×C22⋊(C67⋊C8)—24no—
26C2×C22⋊(C67⋊C8)C2214yes—
27#16863158C2214yescomplemented
28C24×C4×C67C2211yescharacteristic
29C22×C4⋊C4×C67C2211yescharacteristic
30C22×C4⋊C4×C67C2211yescharacteristic
31C22×C22⋊C4×C67—21no—
32#380×C67—22no—
33#380×C67C2214yes—
34C2×#95×C67—21no—
35C22×Dic134—22no—
36C22×Dic134—21no—
37C22×Dic134—22no—
38C22×Dic134—22no—
39C22×Dic134—22no—
40C22×Dic134—22no—
41C22×Dic134D414yes—
42C22×Dic134C2312yes—
43C23×Dic67—21no—
44C23×Dic67—21no—
45C22×C67⋊C8—22no—
46C22×C67⋊C8—22no—
47C22×C67⋊C8—22no—
48C22×C67⋊C8—22no—
49C22×C67⋊C8—22no—
50C22×C67⋊C8D414yes—
51C22×C67⋊C8C2312yes—
52C22×C67⋊C8C2312yes—
53#14524022—216no—
54#14524022—28no—
55#14524022—28no—
56C2×C4⋊Dic67—22no—
57C2×C4⋊Dic67—21no—
58C2×C4⋊Dic67D414yes—
59C2×C4⋊Dic67C2312yes—
60C2×Dic67⋊C4—28no—
61C2×Dic67⋊C4—24no—
62C2×C22⋊Dic67—28no—
63C2×C22⋊Dic67—24no—
64#14523976—28no—
65#14523966—28no—
66#14523966—28no—
67C22⋊(C67⋊C8)—28no—
68C25×C67D411yescharacteristic
69C23×C4×C67—22no—
70C23×C4×C67—22no—
71C23×C4×C67—22no—
72C23×C4×C67—22no—
73C23×C4×C67—22no—
74C23×C4×C67—22no—
75C23×C4×C67—21no—
76C23×C4×C67—21no—
77C23×C4×C67D411yescharacteristic
78C23×C4×C67D412yes—
79C23×C4×C67C2312yes—
80C23×C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67—22no—
82C2×C4⋊C4×C67—22no—
83C2×C4⋊C4×C67—22no—
84C2×C4⋊C4×C67—22no—
85C2×C4⋊C4×C67—21no—
86C2×C4⋊C4×C67—21no—
87C2×C4⋊C4×C67C2312yes—
88C2×C4⋊C4×C67C2312yes—
89C2×C22⋊C4×C67—24no—
90C2×C22⋊C4×C67—44no—
91#95×C67—24no—
92C2×Dic134—28no—
93C2×Dic134—28no—
94C2×Dic134—22no—
95C2×Dic134—22no—
96C2×Dic134—22no—
97C2×Dic134—22no—
98C2×Dic134—24no—
99C2×Dic134—22no—
100C2×Dic134—42no—
101C2×Dic134—44no—
102C2×Dic134—42no—
103C2×Dic134—44no—
104C2×Dic134C2×D414yes—
105C22×Dic67—24no—
106C22×Dic67—24no—
107C22×Dic67—22no—
108C22×Dic67—22no—
109C22×Dic67—42no—
110C22×Dic67—42no—
111C22×Dic67—42no—
112C22×Dic67—42no—
113C2×C67⋊C8—28no—
114C2×C67⋊C8—22no—
115C2×C67⋊C8—22no—
116C2×C67⋊C8—22no—
117C2×C67⋊C8—22no—
118C2×C67⋊C8—22no—
119C2×C67⋊C8—44no—
120C2×C67⋊C8C2×D414yes—
121C4⋊Dic67—24no—
122C4⋊Dic67—22no—
123C4⋊Dic67C2×D414yes—
124Dic67⋊C4—216no—
125C22⋊Dic67—216no—
126C24×C67—22no—
127C24×C67—22no—
128C24×C67—22no—
129C24×C67—22no—
130C24×C67—22no—
131C24×C67—22no—
132C24×C67Q1614yes—
133C24×C67C2×D412yes—
134C24×C67C2×D411yescharacteristic
135C22×C4×C67—22no—
136C22×C4×C67—22no—
137C22×C4×C67—22no—
138C22×C4×C67—24no—
139C22×C4×C67—24no—
140C22×C4×C67—24no—
141C22×C4×C67—22no—
142C22×C4×C67—22no—
143C22×C4×C67—22no—
144C22×C4×C67—22no—
145C22×C4×C67—22no—
146C22×C4×C67—22no—
147C22×C4×C67—22no—
148C22×C4×C67—22no—
149C22×C4×C67—44no—
150C22×C4×C67—44no—
151C22×C4×C67—44no—
152C22×C4×C67—44no—
153C22×C4×C67—42no—
154C22×C4×C67—42no—
155C22×C4×C67—42no—
156C22×C4×C67—42no—
157C22×C4×C67—42no—
158C22×C4×C67—42no—
159C22×C4×C67C2×D412yes—
160C22×C4×C67C2×D412yes—
161C22×C4×C67C2×D412yes—
162C22×C4×C67C2×D412yes—
163C22×C4×C67C2×D411yescharacteristic
164C22×C4×C67#4112yes—
165C22×C4×C67#4112yes—
166C22×C4×C67#4112yes—
167C22×C4×C67C2411yescharacteristic
168C4⋊C4×C67—22no—
169C4⋊C4×C67—22no—
170C4⋊C4×C67—22no—
171C4⋊C4×C67—22no—
172C4⋊C4×C67—42no—
173C4⋊C4×C67—42no—
174C22⋊C4×C67—48no—
175C23×C67—22no—
176C23×C67—22no—
177C23×C67—22no—
178C23×C67—22no—
179C23×C67—22no—
180C23×C67—22no—
181C23×C67—28no—
182C23×C67—28no—
183C23×C67—28no—
184C23×C67—44no—
185C23×C67—44no—
186C23×C67—44no—
187C23×C67—44no—
188C23×C67#12012yes—
189C23×C67#12112yes—
190C23×C67#12312yes—
191C23×C67C2×Q1614yes—
192C23×C67C2×Q1612yes—
193C23×C67#13512yes—
194C23×C67AGL(1,ℤ/8)12yes—
195C23×C67#13712yes—
196C23×C67C22×D411yescharacteristic
197C2×C4×C67—22no—
198C2×C4×C67—22no—
199C2×C4×C67—22no—
200C2×C4×C67—44no—
201C2×C4×C67—42no—
202C2×C4×C67—42no—
203C2×C4×C67—42no—
204C2×C4×C67—42no—
205C2×C4×C67—48no—
206C2×C4×C67—42no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—84no—
213C2×C4×C67—84no—
214C2×C4×C67#12012yes—
215C2×C4×C67#12112yes—
216C2×C4×C67#12114yes—
217C2×C4×C67#12114yes—
218C2×C4×C67#12314yes—
219C2×C4×C67#12312yes—
220C2×C4×C67#12414yes—
221C2×C4×C67C22×D411yescharacteristic
222C2×C4×C67C22×D411yescharacteristic
223C2×C4×C67C22×D411yescharacteristic
224C2×C4×C67C2×#4111yescharacteristic
225C2×C4×C67C2×#4111yescharacteristic
226C2×C4×C67C2×#4111yescharacteristic
227C2×Dic67—24no—
228C2×Dic67—24no—
229C2×Dic67—42no—
230C2×Dic67—42no—
231C2×Dic67—42no—
232C2×Dic67—42no—
233C2×Dic67—48no—
234C2×Dic67—48no—
235Dic134—24no—
236Dic134—24no—
237Dic134—42no—
238Dic134—42no—
239Dic134—48no—
240C67⋊C8—24no—
241C67⋊C8—42no—
242C22×C67—24no—
243C22×C67—24no—
244C22×C67—24no—
245C22×C67—42no—
246C22×C67—42no—
247C22×C67—42no—
248C22×C67—42no—
249C22×C67—416no—
250C22×C67#39414yes—
251C22×C67#44914yes—
252C22×C67#45114yes—
253C22×C67#46714yes—
254C22×C67#46814yes—
255C22×C67#48414yes—
256C22×C67#48514yes—
257C22×C67C2×#12011yescharacteristic
258C22×C67C2×#12111yescharacteristic
259C22×C67C2×#12311yescharacteristic
260C22×C67C22×Q1611yescharacteristic
261C22×C67C2×#13511yescharacteristic
262C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C67C2×#13711yescharacteristic
264C4×C67—42no—
265C4×C67—82no—
266C4×C67—82no—
267C4×C67#39414yes—
268C4×C67C2×#12111yescharacteristic
269C4×C67C2×#12111yescharacteristic
270C4×C67C2×#12311yescharacteristic
271C4×C67C2×#12411yescharacteristic
272Dic67—44no—
273Dic67—44no—
274C2×#2036—671nomaximal, Sylow
275C2×C67—44no—
276C2×C67#203618yescomplemented
277C2×C67C2×#39411yescharacteristic
278C2×C67C2×#45111yescharacteristic
279C2×C67C2×#44911yescharacteristic
280C2×C67C2×#46711yescharacteristic
281C2×C67C2×#46811yescharacteristic
282C2×C67C2×#48511yescharacteristic
283C2×C67C2×#48411yescharacteristic
284C22×C22⋊Q8—671no—
285C22×#107—671no—
286C22×#103—671no—
287C22×#103—671no—
288C22×C22⋊C8—671no—
289C2×#380D6711yescharacteristic
290#2036—678no—
291C2×#340—671no—
292C67C2×#203611yescharacteristic, complemented, Sylow
293C23×Q8—671no—
294C24×C4C2×D6711yescharacteristic
295C23×C8—671no—
296C23×C8—671no—
297C2×C22⋊Q8—674no—
298C2×C22⋊Q8—1344no—
299C2×C22⋊Q8—1344no—
300C2×C22⋊Q8—1342no—
301C22×C4⋊C4—671no—
302C22×C4⋊C4—1341no—
303C22×C4⋊C4C2×D6711yescharacteristic
304C22×C4⋊C4C2×D6711yescharacteristic
305C22×C22⋊C4—21no—
306C22×C22⋊C4—1341no—
307C2×#107—674no—
308C2×#107—1344no—
309C2×#103—674no—
310C2×#103—674no—
311C2×#103—1344no—
312C2×#103—1344no—
313C2×C22⋊C8—674no—
314C2×C22⋊C8—1344no—
315#380—22no—
316#380C2×D6714yes—
317C2×#95—21no—
318#340—674no—
319C25C22⋊D6711yescharacteristic
320C22×Q8—674no—
321C22×Q8—672no—
322C22×Q8—1342no—
323C22×Q8—1342no—
324C22×Q8—1342no—
325C22×Q8—1342no—
326C22×Q8—1342no—
327C22×Q8—1341no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—1341no—
337C23×C4—1341no—
338C23×C4C22⋊D6712yes—
339C23×C4C22⋊D6711yescharacteristic
340C23×C4C22×D6712yes—
341C23×C4C22×D6711yescharacteristic
342C22×C8—674no—
343C22×C8—672no—
344C22×C8—672no—
345C22×C8—1342no—
346C22×C8—1342no—
347C22×C8—1342no—
348C22×C8—1342no—
349C22×C8—1342no—
350C22⋊Q8—1348no—
351C22⋊Q8—1348no—
352C22⋊Q8—13416no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—674no—
360C2×C4⋊C4—672no—
361C2×C4⋊C4—1348no—
362C2×C4⋊C4—1342no—
363C2×C4⋊C4—1341no—
364C2×C4⋊C4—1344no—
365C2×C4⋊C4C22×D6712yes—
366C2×C4⋊C4C22×D6712yes—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—1348no—
370C2×C22⋊C4—1344no—
371#107—1348no—
372#103—1348no—
373#103—1348no—
374C22⋊C8—1348no—
375#95—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#1176161814yes—
383C24C2×C22⋊D6712yes—
384C24C2×C22⋊D6711yescharacteristic
385C2×Q8—2682no—
386C2×Q8—2684no—
387C2×Q8—2682no—
388C2×Q8—2684no—
389C2×Q8—674no—
390C2×Q8—1348no—
391C2×Q8—1348no—
392C2×Q8—1342no—
393C2×Q8—1342no—
394C2×Q8—1342no—
395C2×Q8—1342no—
396C2×Q8—1344no—
397C2×Q8—1342no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—2682no—
423C22×C4—2682no—
424C22×C4—2682no—
425C22×C4—2682no—
426C22×C4—1344no—
427C22×C4—1344no—
428C22×C4—1342no—
429C22×C4—1342no—
430C22×C4D4×D6712yes—
431C22×C4D4×D6712yes—
432C22×C4D4×D6712yes—
433C22×C4C2×C22⋊D6712yes—
434C22×C4C2×C22⋊D6711yescharacteristic
435C22×C4#1176162712yes—
436C22×C4#1176162812yes—
437C22×C4#1176162812yes—
438C22×C4C23×D6711yescharacteristic
439C2×C8—2684no—
440C2×C8—674no—
441C2×C8—1342no—
442C2×C8—1342no—
443C2×C8—1342no—
444C2×C8—1348no—
445C2×C8—1342no—
446C2×C8—1342no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—674no—
454C4⋊C4—13416no—
455C4⋊C4—1344no—
456C4⋊C4—1342no—
457C22⋊C4—48no—
458C22⋊C4—13416no—
459C23—28no—
460C23—22no—
461C23—22no—
462C23—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1452401112yes—
473C23#1452401512yes—
474C23#1452402612yes—
475C23C2×#1176161814yes—
476C23C2×#1176161812yes—
477C23#1452406312yes—
478C23#1452406912yes—
479C23#1452407312yes—
480C23C22×C22⋊D6711yescharacteristic
481Q8—2688no—
482Q8—2682no—
483Q8—2682no—
484Q8—1344no—
485Q8—1344no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—2688no—
504C2×C4—2688no—
505C2×C4—2682no—
506C2×C4—2682no—
507C2×C4—2682no—
508C2×C4—2682no—
509C2×C4—1344no—
510C2×C4—1344no—
511C2×C4#1452401112yes—
512C2×C4#1452401512yes—
513C2×C4#1452401614yes—
514C2×C4#1452401614yes—
515C2×C4#1452402514yes—
516C2×C4#1452402612yes—
517C2×C4#1452402914yes—
518C2×C4C2×D4×D6711yescharacteristic
519C2×C4C2×D4×D6711yescharacteristic
520C2×C4C2×D4×D6711yescharacteristic
521C2×C4C2×#1176162711yescharacteristic
522C2×C4C2×#1176162811yescharacteristic
523C2×C4C2×#1176162811yescharacteristic
524C8—2682no—
525C8—1344no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—416no—
533C22—42no—
534C22#1686331114yes—
535C22#1686353914yes—
536C22#1686355114yes—
537C22#1686363514yes—
538C22#1686364014yes—
539C22#1686373114yes—
540C22#1686373714yes—
541C22C2×#1452401111yescharacteristic
542C22C2×#1452401511yescharacteristic
543C22C2×#1452402611yescharacteristic
544C22C22×#1176161811yescharacteristic
545C22C2×#1452406311yescharacteristic
546C22C2×#1452406911yescharacteristic
547C22C2×#1452407311yescharacteristic
548C4—42no—
549C4—82no—
550C4—82no—
551C4—2684no—
552C4—2684no—
553C4#1686331114yes—
554C4C2×#1452401611yescharacteristic
555C4C2×#1452401611yescharacteristic
556C4C2×#1452402511yescharacteristic
557C4C2×#1452402911yescharacteristic
558C2—44no—
559C2#2256141218yesdirect product factor
560C2C2×#1686331111yescharacteristic
561C2C2×#1686355111yescharacteristic
562C2C2×#1686353911yescharacteristic
563C2C2×#1686363511yescharacteristic
564C2C2×#1686364011yescharacteristic
565C2C2×#1686373711yescharacteristic
566C2C2×#1686373111yescharacteristic
subgroupG/H
centerC24C2×C22⋊D67
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC2×#380×C67C2
socleC24×C67C2×D4
prime radicalC25×C67D4
hypercenterC2×#380D67
hypocenterC67C2×#2036
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#14523976C2semidirect
3. C22×#14523966C2semidirect
4. C22×#14523966C2semidirect
6. #22561412C2direct (N × H)
7. C2×#16863158C2semidirect
20. C2×#14523976C22semidirect
23. C2×#14523966C22semidirect
24. C2×#14523966C22semidirect
27. #16863158C22semidirect
276. C2×C67#2036semidirect
292. C67C2×#2036semidirect
559. C2#22561412direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240222
C22×#145239762
C22×#145239662
C22×C22⋊(C67⋊C8)2
#225614122
C2×#168631582
C2×#380×C672
C2×#203667