Finite Group Databasebeta

← C2×#22561402 #30883284

Subgroups — C2×#22561402 #30883284

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×#14523974C211yesmaximal, characteristic, complemented
2C22×#14523967C211yesmaximal, characteristic, complemented
3C22×#14523967C211yesmaximal, characteristic, complemented
4C22×C22⋊(C67⋊C8)C211yesmaximal, characteristic
5C2×#16863256C211yesmaximal, characteristic
6#22561402C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C67C211yesmaximal, characteristic
9C23×C67⋊C8C2211yescharacteristic
10C23×C67⋊C8C2211yescharacteristic
11C22×C4⋊Dic67C2211yescharacteristic
12C22×C4⋊Dic67C2211yescharacteristic
13C22×C22⋊Dic67—21no—
14C2×#14523974—24no—
15C2×#14523974C2214yescomplemented
16C2×#14523967—24no—
17C2×#14523967—24no—
18C2×#14523967C2214yescomplemented
19C2×#14523967C2214yescomplemented
20C2×C22⋊(C67⋊C8)—24no—
21C2×C22⋊(C67⋊C8)C2214yes—
22#16863256—22no—
23#16863256C2214yes—
24C2×C67⋊#95—21no—
25#16863158C2214yescomplemented
26C23×Q8×C67C2211yescharacteristic
27C24×C4×C67C2211yescharacteristic
28C2×C22⋊Q8×C67—24no—
29C2×C22⋊Q8×C67—22no—
30C2×C22⋊Q8×C67—24no—
31C2×C22⋊Q8×C67C2214yes—
32C22×C4⋊C4×C67—21no—
33C22×C4⋊C4×C67C2211yescharacteristic
34C22×C22⋊C4×C67—21no—
35C23×Dic67—21no—
36C23×Dic67—21no—
37C22×C67⋊C8—22no—
38C22×C67⋊C8—22no—
39C22×C67⋊C8—22no—
40C22×C67⋊C8—22no—
41C22×C67⋊C8—22no—
42C22×C67⋊C8D414yes—
43C22×C67⋊C8C2312yes—
44C22×C67⋊C8C2312yes—
45C2×C4⋊Dic67—22no—
46C2×C4⋊Dic67—22no—
47C2×C4⋊Dic67—22no—
48C2×C4⋊Dic67—21no—
49C2×C4⋊Dic67—21no—
50C2×C4⋊Dic67—22no—
51C2×C4⋊Dic67C2312yes—
52C2×C4⋊Dic67C2312yes—
53C2×C22⋊Dic67—24no—
54C2×C22⋊Dic67—44no—
55#14523974—28no—
56#14523967—28no—
57#14523967—28no—
58C22⋊(C67⋊C8)—28no—
59C67⋊#95—24no—
60C25×C67D411yescharacteristic
61C22×Q8×C67—22no—
62C22×Q8×C67—21no—
63C22×Q8×C67—22no—
64C22×Q8×C67—22no—
65C22×Q8×C67—22no—
66C22×Q8×C67—22no—
67C22×Q8×C67D414yes—
68C22×Q8×C67C2312yes—
69C23×C4×C67—21no—
70C23×C4×C67—21no—
71C23×C4×C67—22no—
72C23×C4×C67—22no—
73C23×C4×C67—22no—
74C23×C4×C67—22no—
75C23×C4×C67—22no—
76C23×C4×C67—22no—
77C23×C4×C67D411yescharacteristic
78C23×C4×C67D412yes—
79C23×C4×C67C2312yes—
80C23×C4×C67C2311yescharacteristic
81C22⋊Q8×C67—216no—
82C22⋊Q8×C67—28no—
83C22⋊Q8×C67—28no—
84C2×C4⋊C4×C67—28no—
85C2×C4⋊C4×C67—24no—
86C2×C4⋊C4×C67—21no—
87C2×C4⋊C4×C67—22no—
88C2×C4⋊C4×C67D414yes—
89C2×C4⋊C4×C67C2312yes—
90C2×C22⋊C4×C67—28no—
91C2×C22⋊C4×C67—24no—
92C22×Dic67—22no—
93C22×Dic67—22no—
94C22×Dic67—42no—
95C22×Dic67—42no—
96C22×Dic67—42no—
97C22×Dic67—42no—
98C22×Dic67—42no—
99C22×Dic67—42no—
100C2×C67⋊C8—28no—
101C2×C67⋊C8—22no—
102C2×C67⋊C8—22no—
103C2×C67⋊C8—22no—
104C2×C67⋊C8—22no—
105C2×C67⋊C8—22no—
106C2×C67⋊C8—44no—
107C2×C67⋊C8C2×D414yes—
108C4⋊Dic67—22no—
109C4⋊Dic67—22no—
110C4⋊Dic67—22no—
111C4⋊Dic67—22no—
112C4⋊Dic67—42no—
113C4⋊Dic67—42no—
114C22⋊Dic67—48no—
115C24×C67—22no—
116C24×C67—22no—
117C24×C67—22no—
118C24×C67—22no—
119C24×C67—22no—
120C24×C67—22no—
121C24×C67SD1614yes—
122C24×C67C2×D412yes—
123C24×C67C2×D411yescharacteristic
124C2×Q8×C67—24no—
125C2×Q8×C67—22no—
126C2×Q8×C67—28no—
127C2×Q8×C67—28no—
128C2×Q8×C67—22no—
129C2×Q8×C67—22no—
130C2×Q8×C67—22no—
131C2×Q8×C67—22no—
132C2×Q8×C67—44no—
133C2×Q8×C67—42no—
134C2×Q8×C67—42no—
135C2×Q8×C67—44no—
136C2×Q8×C67C2×D414yes—
137C22×C4×C67—24no—
138C22×C4×C67—24no—
139C22×C4×C67—22no—
140C22×C4×C67—22no—
141C22×C4×C67—22no—
142C22×C4×C67—22no—
143C22×C4×C67—22no—
144C22×C4×C67—24no—
145C22×C4×C67—24no—
146C22×C4×C67—24no—
147C22×C4×C67—22no—
148C22×C4×C67—22no—
149C22×C4×C67—22no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—22no—
153C22×C4×C67—42no—
154C22×C4×C67—42no—
155C22×C4×C67—42no—
156C22×C4×C67—42no—
157C22×C4×C67—44no—
158C22×C4×C67—44no—
159C22×C4×C67—44no—
160C22×C4×C67—44no—
161C22×C4×C67C2×D412yes—
162C22×C4×C67C2×D412yes—
163C22×C4×C67C2×D411yescharacteristic
164C22×C4×C67C2×D412yes—
165C22×C4×C67C2×D412yes—
166C22×C4×C67#4112yes—
167C22×C4×C67#4112yes—
168C22×C4×C67#4112yes—
169C22×C4×C67C2411yescharacteristic
170C4⋊C4×C67—216no—
171C4⋊C4×C67—24no—
172C4⋊C4×C67—22no—
173C4⋊C4×C67C2×D414yes—
174C22⋊C4×C67—216no—
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×SD1614yes—
192C23×C67C2×SD1612yes—
193C23×C67#13512yes—
194C23×C67#13712yes—
195C23×C67#13712yes—
196C23×C67C22×D411yescharacteristic
197Q8×C67—24no—
198Q8×C67—24no—
199Q8×C67—42no—
200Q8×C67—42no—
201Q8×C67—48no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—44no—
212C2×C4×C67—48no—
213C2×C4×C67—48no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—48no—
219C2×C4×C67#12012yes—
220C2×C4×C67#12114yes—
221C2×C4×C67#12114yes—
222C2×C4×C67#12112yes—
223C2×C4×C67#12314yes—
224C2×C4×C67#12312yes—
225C2×C4×C67#12414yes—
226C2×C4×C67C22×D411yescharacteristic
227C2×C4×C67C22×D411yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C2×#4111yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C2×C4×C67C2×#4111yescharacteristic
232C2×Dic67—42no—
233C2×Dic67—42no—
234C2×Dic67—42no—
235C2×Dic67—42no—
236C2×Dic67—42no—
237C2×Dic67—42no—
238C2×Dic67—84no—
239C2×Dic67—84no—
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#44814yes—
252C22×C67#45214yes—
253C22×C67#46514yes—
254C22×C67#47014yes—
255C22×C67#48314yes—
256C22×C67#48514yes—
257C22×C67C2×#12011yescharacteristic
258C22×C67C2×#12111yescharacteristic
259C22×C67C2×#12311yescharacteristic
260C22×C67C22×SD1611yescharacteristic
261C22×C67C2×#13511yescharacteristic
262C22×C67C2×#13711yescharacteristic
263C22×C67C2×#13711yescharacteristic
264C4×C67—42no—
265C4×C67—44no—
266C4×C67—44no—
267C4×C67#39414yes—
268C4×C67C2×#12111yescharacteristic
269C4×C67C2×#12111yescharacteristic
270C4×C67C2×#12311yescharacteristic
271C4×C67C2×#12411yescharacteristic
272Dic67—82no—
273Dic67—82no—
274C2×#2035—671nomaximal, Sylow
275C2×C67—44no—
276C2×C67#203518yescomplemented
277C2×C67C2×#39411yescharacteristic
278C2×C67C2×#44811yescharacteristic
279C2×C67C2×#45211yescharacteristic
280C2×C67C2×#46511yescharacteristic
281C2×C67C2×#47011yescharacteristic
282C2×C67C2×#48311yescharacteristic
283C2×C67C2×#48511yescharacteristic
284C22×C22⋊Q8D6711yescharacteristic
285C22×#106—671no—
286C22×#103—671no—
287C22×#103—671no—
288C22×C22⋊C8—671no—
289C2×#380—671no—
290#2035—678no—
291C2×#340—671no—
292C67C2×#203511yescharacteristic, complemented, Sylow
293C23×Q8C2×D6711yescharacteristic
294C24×C4C2×D6711yescharacteristic
295C23×C8—671no—
296C23×C8—671no—
297C2×C22⋊Q8—24no—
298C2×C22⋊Q8—22no—
299C2×C22⋊Q8—24no—
300C2×C22⋊Q8C2×D6714yes—
301C22×C4⋊C4—21no—
302C22×C4⋊C4—671no—
303C22×C4⋊C4—671no—
304C22×C4⋊C4C2×D6711yescharacteristic
305C22×C22⋊C4—21no—
306C22×C22⋊C4—1341no—
307C2×#106—674no—
308C2×#106—1344no—
309C2×#103—674no—
310C2×#103—674no—
311C2×#103—1344no—
312C2×#103—1344no—
313C2×C22⋊C8—674no—
314C2×C22⋊C8—1344no—
315#380—674no—
316#380—1342no—
317C2×#95—1341no—
318#340—674no—
319C25C22⋊D6711yescharacteristic
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8C22⋊D6714yes—
327C22×Q8C22×D6712yes—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
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—216no—
351C22⋊Q8—28no—
352C22⋊Q8—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—672no—
358C2×C4⋊C4—672no—
359C2×C4⋊C4—1341no—
360C2×C4⋊C4—1341no—
361C2×C4⋊C4—1342no—
362C2×C4⋊C4—1342no—
363C2×C4⋊C4—1342no—
364C2×C4⋊C4—1342no—
365C2×C4⋊C4C22⋊D6714yes—
366C2×C4⋊C4C22×D6712yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—2684no—
370C2×C22⋊C4—1344no—
371#106—1348no—
372#103—1348no—
373#103—1348no—
374C22⋊C8—1348no—
375#95—1344no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#1176161514yes—
383C24C2×C22⋊D6712yes—
384C24C2×C22⋊D6711yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D6714yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—2682no—
423C22×C4—2682no—
424C22×C4—2682no—
425C22×C4—2682no—
426C22×C4—2682no—
427C22×C4—2682no—
428C22×C4—1342no—
429C22×C4—1342no—
430C22×C4D4×D6712yes—
431C22×C4C2×C22⋊D6712yes—
432C22×C4C2×C22⋊D6711yescharacteristic
433C22×C4C2×C22⋊D6712yes—
434C22×C4C2×C22⋊D6712yes—
435C22×C4#1176162712yes—
436C22×C4#1176162712yes—
437C22×C4#1176162712yes—
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—216no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—2682no—
451C4⋊C4—2682no—
452C4⋊C4—1342no—
453C4⋊C4—1342no—
454C4⋊C4—1342no—
455C4⋊C4—1342no—
456C4⋊C4C2×C22⋊D6714yes—
457C22⋊C4—216no—
458C22⋊C4—2688no—
459C23—28no—
460C23—22no—
461C23—28no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1452401212yes—
473C23#1452401512yes—
474C23#1452402412yes—
475C23C2×#1176161514yes—
476C23C2×#1176161512yes—
477C23#1452406312yes—
478C23#1452407312yes—
479C23#1452407412yes—
480C23C22×C22⋊D6711yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—2682no—
504C2×C4—2682no—
505C2×C4—2682no—
506C2×C4—2682no—
507C2×C4—2682no—
508C2×C4—2682no—
509C2×C4—5364no—
510C2×C4—5364no—
511C2×C4#1452401212yes—
512C2×C4#1452401414yes—
513C2×C4#1452401512yes—
514C2×C4#1452401514yes—
515C2×C4#1452402412yes—
516C2×C4#1452402414yes—
517C2×C4#1452403014yes—
518C2×C4C2×D4×D6711yescharacteristic
519C2×C4C22×C22⋊D6711yescharacteristic
520C2×C4C22×C22⋊D6711yescharacteristic
521C2×C4C2×#1176162711yescharacteristic
522C2×C4C2×#1176162711yescharacteristic
523C2×C4C2×#1176162711yescharacteristic
524C8—2682no—
525C8—1344no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—416no—
533C22—42no—
534C22#1686331014yes—
535C22#1686353514yes—
536C22#1686355814yes—
537C22#1686362714yes—
538C22#1686365214yes—
539C22#1686372414yes—
540C22#1686373414yes—
541C22C2×#1452401211yescharacteristic
542C22C2×#1452401511yescharacteristic
543C22C2×#1452402411yescharacteristic
544C22C22×#1176161511yescharacteristic
545C22C2×#1452406311yescharacteristic
546C22C2×#1452407311yescharacteristic
547C22C2×#1452407411yescharacteristic
548C4—42no—
549C4—44no—
550C4—44no—
551C4—5362no—
552C4—5362no—
553C4#1686331014yes—
554C4C2×#1452401411yescharacteristic
555C4C2×#1452401511yescharacteristic
556C4C2×#1452402411yescharacteristic
557C4C2×#1452403011yescharacteristic
558C2—44no—
559C2#2256140218yesdirect product factor
560C2C2×#1686331011yescharacteristic
561C2C2×#1686353511yescharacteristic
562C2C2×#1686355811yescharacteristic
563C2C2×#1686362711yescharacteristic
564C2C2×#1686365211yescharacteristic
565C2C2×#1686372411yescharacteristic
566C2C2×#1686373411yescharacteristic
subgroupG/H
centerC24C2×C22⋊D67
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC22×C22⋊Q8×C67C2
socleC24×C67C2×D4
prime radicalC25×C67D4
hypercenterC22×C22⋊Q8D67
hypocenterC67C2×#2035
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523974C2semidirect
2. C22×#14523967C2semidirect
3. C22×#14523967C2semidirect
6. #22561402C2direct (N × H)
7. C2×#16863158C2semidirect
15. C2×#14523974C22semidirect
18. C2×#14523967C22semidirect
19. C2×#14523967C22semidirect
25. #16863158C22semidirect
276. C2×C67#2035semidirect
292. C67C2×#2035semidirect
559. C2#22561402direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145239742
C22×#145239672
C22×C22⋊(C67⋊C8)2
C2×#168632562
#225614022
C2×#168631582
C22×C22⋊Q8×C672
C2×#203567