Finite Group Databasebeta

← C2×#11727690 #13381253

Subgroups — C2×#11727690 #13381253

Order 1792.

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×#6487C211yesmaximal, characteristic
2C22×C7×C22⋊C8C211yesmaximal, characteristic
3C2×#89984C211yesmaximal, characteristic
4C22×#6390C211yesmaximal, characteristic, complemented
5C22×#6390C211yesmaximal, characteristic, complemented
6C22×#6391C211yesmaximal, characteristic, complemented
7C2×#89346C211yesmaximal, characteristic, complemented
8#11727690C218yesmaximal, direct product factor
9C24×C4×C7C2211yescharacteristic
10C23×Dic14C2211yescharacteristic
11C23×C7×C8C2211yescharacteristic
12C23×C7×C8C2211yescharacteristic
13C22×C22⋊Dic7—21no—
14C22×C22⋊Dic7—21no—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C2×#6487—22no—
19C2×#6487—24no—
20C2×#6487—24no—
21C2×#6487C2214yes—
22C22×Dic7⋊C4—21no—
23C2×C7×C22⋊C8—24no—
24C2×C7×C22⋊C8C2214yes—
25#89984—22no—
26#89984C2214yes—
27C2×C7⋊#95—21no—
28C2×#6391—24no—
29C2×#6391C2214yescomplemented
30C2×#6390—24no—
31C2×#6390—24no—
32C2×#6390C2214yescomplemented
33C2×#6390C2214yescomplemented
34#89346C2214yescomplemented
35C2×#2035—71nomaximal, Sylow
36C25×C7D411yescharacteristic
37C23×C4×C7—22no—
38C23×C4×C7—22no—
39C23×C4×C7—22no—
40C23×C4×C7—22no—
41C23×C4×C7—22no—
42C23×C4×C7—22no—
43C23×C4×C7D411yescharacteristic
44C23×C4×C7D412yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—21no—
48C23×Dic7—21no—
49C23×Dic7—21no—
50C23×Dic7—21no—
51C22×Dic14—22no—
52C22×Dic14—22no—
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—21no—
56C22×Dic14—22no—
57C22×Dic14D414yes—
58C22×Dic14C2312yes—
59C22×C7×C8—22no—
60C22×C7×C8—22no—
61C22×C7×C8—22no—
62C22×C7×C8—22no—
63C22×C7×C8—22no—
64C22×C7×C8D414yes—
65C22×C7×C8C2312yes—
66C22×C7×C8C2312yes—
67C2×C22⋊Dic7—28no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7—44no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7D414yes—
80C2×C4⋊Dic7C2312yes—
81C2×C4⋊Dic7C2312yes—
82C2×C4⋊Dic7C2312yes—
83#6487—216no—
84#6487—28no—
85#6487—28no—
86C2×Dic7⋊C4—28no—
87C2×Dic7⋊C4—24no—
88C7×C22⋊C8—28no—
89C7⋊#95—24no—
90#6391—28no—
91#6390—28no—
92#6390—28no—
93C22×C22⋊Q8—71no—
94C22×#106—71no—
95C22×#103—71no—
96C22×#103—71no—
97C22×C22⋊C8D711yescharacteristic
98C2×#380—71no—
99#2035—78no—
100C2×#340—71no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7SD1614yes—
108C24×C7C2×D412yes—
109C24×C7C2×D411yescharacteristic
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—24no—
114C22×C4×C7—24no—
115C22×C4×C7—24no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—44no—
123C22×C4×C7—44no—
124C22×C4×C7—44no—
125C22×C4×C7—44no—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7#4112yes—
132C22×C4×C7#4112yes—
133C22×C4×C7#4112yes—
134C22×C4×C7C2411yescharacteristic
135C22×Dic7—24no—
136C22×Dic7—24no—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—42no—
142C22×Dic7—42no—
143C22×Dic7—42no—
144C22×Dic7—42no—
145C22×Dic7—42no—
146C22×Dic7—42no—
147C22×Dic7—42no—
148C22×Dic7—42no—
149C22×Dic7—42no—
150C22×Dic7—42no—
151C2×Dic14—28no—
152C2×Dic14—28no—
153C2×Dic14—22no—
154C2×Dic14—22no—
155C2×Dic14—22no—
156C2×Dic14—22no—
157C2×Dic14—24no—
158C2×Dic14—22no—
159C2×Dic14—44no—
160C2×Dic14—42no—
161C2×Dic14—42no—
162C2×Dic14—44no—
163C2×Dic14C2×D414yes—
164C2×C7×C8—28no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—44no—
171C2×C7×C8C2×D414yes—
172C22⋊Dic7—216no—
173C22⋊Dic7—48no—
174C4⋊Dic7—24no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—42no—
181C4⋊Dic7—42no—
182C4⋊Dic7C2×D414yes—
183Dic7⋊C4—216no—
184C23×Q8—71no—
185C24×C4C2×D711yescharacteristic
186C23×C8C2×D711yescharacteristic
187C23×C8C2×D711yescharacteristic
188C2×C22⋊Q8—74no—
189C2×C22⋊Q8—144no—
190C2×C22⋊Q8—144no—
191C2×C22⋊Q8—142no—
192C22×C4⋊C4—71no—
193C22×C4⋊C4—71no—
194C22×C4⋊C4—71no—
195C22×C4⋊C4—141no—
196C22×C22⋊C4—141no—
197C22×C22⋊C4—141no—
198C2×#106—74no—
199C2×#106—144no—
200C2×#103—74no—
201C2×#103—74no—
202C2×#103—144no—
203C2×#103—144no—
204C2×C22⋊C8—24no—
205C2×C22⋊C8C2×D714yes—
206#380—74no—
207#380—142no—
208C2×#95—141no—
209#340—74no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—28no—
217C23×C7—28no—
218C23×C7—28no—
219C23×C7—44no—
220C23×C7—44no—
221C23×C7—44no—
222C23×C7—44no—
223C23×C7#12012yes—
224C23×C7#12112yes—
225C23×C7#12312yes—
226C23×C7C2×SD1614yes—
227C23×C7C2×SD1612yes—
228C23×C7#13512yes—
229C23×C7#13712yes—
230C23×C7#13712yes—
231C23×C7C22×D411yescharacteristic
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—44no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—48no—
241C2×C4×C7#12012yes—
242C2×C4×C7#12112yes—
243C2×C4×C7#12114yes—
244C2×C4×C7#12114yes—
245C2×C4×C7#12314yes—
246C2×C4×C7#12312yes—
247C2×C4×C7#12414yes—
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—48no—
261C2×Dic7—48no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—84no—
269C2×Dic7—84no—
270Dic14—24no—
271Dic14—24no—
272Dic14—42no—
273Dic14—42no—
274Dic14—48no—
275C7×C8—24no—
276C7×C8—42no—
277C25D2811yescharacteristic
278C22×Q8—74no—
279C22×Q8—72no—
280C22×Q8—141no—
281C22×Q8—142no—
282C22×Q8—142no—
283C22×Q8—142no—
284C22×Q8—142no—
285C22×Q8—142no—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4—22no—
290C23×C4—22no—
291C23×C4—22no—
292C23×C4—141no—
293C23×C4—141no—
294C23×C4—141no—
295C23×C4—141no—
296C23×C4D2812yes—
297C23×C4D2811yescharacteristic
298C23×C4C22×D712yes—
299C23×C4C22×D711yescharacteristic
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8C22⋊D714yes—
306C22×C8C22×D712yes—
307C22×C8C22×D712yes—
308C22⋊Q8—148no—
309C22⋊Q8—148no—
310C22⋊Q8—1416no—
311C2×C4⋊C4—74no—
312C2×C4⋊C4—72no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—148no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—144no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4—141no—
325C2×C22⋊C4—148no—
326C2×C22⋊C4—144no—
327C2×C22⋊C4—144no—
328C2×C22⋊C4—284no—
329#106—148no—
330#103—148no—
331#103—148no—
332C22⋊C8—28no—
333#95—144no—
334C22×C7—24no—
335C22×C7—24no—
336C22×C7—24no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—416no—
342C22×C7#39414yes—
343C22×C7#44814yes—
344C22×C7#45214yes—
345C22×C7#46514yes—
346C22×C7#47014yes—
347C22×C7#48314yes—
348C22×C7#48514yes—
349C22×C7C2×#12011yescharacteristic
350C22×C7C2×#12111yescharacteristic
351C22×C7C2×#12311yescharacteristic
352C22×C7C22×SD1611yescharacteristic
353C22×C7C2×#13511yescharacteristic
354C22×C7C2×#13711yescharacteristic
355C22×C7C2×#13711yescharacteristic
356C4×C7—42no—
357C4×C7#39414yes—
358C4×C7C2×#12111yescharacteristic
359C4×C7C2×#12111yescharacteristic
360C4×C7C2×#12311yescharacteristic
361C4×C7C2×#12411yescharacteristic
362Dic7—44no—
363Dic7—44no—
364Dic7—82no—
365Dic7—82no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24#113214yes—
373C24C2×D2812yes—
374C24C2×D2811yescharacteristic
375C2×Q8—74no—
376C2×Q8—144no—
377C2×Q8—142no—
378C2×Q8—148no—
379C2×Q8—148no—
380C2×Q8—142no—
381C2×Q8—142no—
382C2×Q8—142no—
383C2×Q8—142no—
384C2×Q8—284no—
385C2×Q8—282no—
386C2×Q8—282no—
387C2×Q8—284no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—144no—
405C22×C4—144no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—142no—
410C22×C4—282no—
411C22×C4—282no—
412C22×C4—282no—
413C22×C4—282no—
414C22×C4—282no—
415C22×C4—282no—
416C22×C4—282no—
417C22×C4—282no—
418C22×C4—282no—
419C22×C4—282no—
420C22×C4C2×D2812yes—
421C22×C4C2×D2811yescharacteristic
422C22×C4#115712yes—
423C22×C4D4×D712yes—
424C22×C4D4×D712yes—
425C22×C4#115912yes—
426C22×C4#115912yes—
427C22×C4C2×C22⋊D712yes—
428C22×C4C23×D711yescharacteristic
429C2×C8—28no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—44no—
436C2×C8C2×C22⋊D714yes—
437C4⋊C4—74no—
438C4⋊C4—1416no—
439C4⋊C4—144no—
440C4⋊C4—142no—
441C4⋊C4—142no—
442C4⋊C4—142no—
443C4⋊C4—142no—
444C4⋊C4—142no—
445C4⋊C4—282no—
446C4⋊C4—282no—
447C22⋊C4—1416no—
448C22⋊C4—288no—
449C2×C7—44no—
450C2×C7#203518yescomplemented
451C2×C7C2×#39411yescharacteristic
452C2×C7C2×#44811yescharacteristic
453C2×C7C2×#45211yescharacteristic
454C2×C7C2×#46511yescharacteristic
455C2×C7C2×#47011yescharacteristic
456C2×C7C2×#48311yescharacteristic
457C2×C7C2×#48511yescharacteristic
458C23—28no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—28no—
466C23—28no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23#644512yes—
472C23#644912yes—
473C23C2×#113214yes—
474C23C2×#113212yes—
475C23#646712yes—
476C23#647212yes—
477C23#647212yes—
478C23#649312yes—
479C23C22×D2811yescharacteristic
480Q8—144no—
481Q8—144no—
482Q8—282no—
483Q8—282no—
484Q8—288no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—144no—
495C2×C4—144no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—288no—
501C2×C4—288no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—564no—
509C2×C4—564no—
510C2×C4#644512yes—
511C2×C4#644614yes—
512C2×C4#644814yes—
513C2×C4#644912yes—
514C2×C4#649312yes—
515C2×C4#650114yes—
516C2×C4#650214yes—
517C2×C4C2×#115711yescharacteristic
518C2×C4C2×D4×D711yescharacteristic
519C2×C4C2×D4×D711yescharacteristic
520C2×C4C2×#115911yescharacteristic
521C2×C4C2×#115911yescharacteristic
522C2×C4C22×C22⋊D711yescharacteristic
523C8—24no—
524C8—42no—
525C7C2×#203511yescharacteristic, complemented, Sylow
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—416no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22#8949314yes—
535C22#8949514yes—
536C22#8950514yes—
537C22#8951014yes—
538C22#8973314yes—
539C22#8988614yes—
540C22#8990914yes—
541C22C2×#644511yescharacteristic
542C22C2×#644911yescharacteristic
543C22C22×#113211yescharacteristic
544C22C2×#646711yescharacteristic
545C22C2×#647211yescharacteristic
546C22C2×#647211yescharacteristic
547C22C2×#649311yescharacteristic
548C4—42no—
549C4—284no—
550C4—284no—
551C4—562no—
552C4—562no—
553C4#8973314yes—
554C4C2×#644611yescharacteristic
555C4C2×#644811yescharacteristic
556C4C2×#650111yescharacteristic
557C4C2×#650211yescharacteristic
558C2—44no—
559C2#1172769018yesdirect product factor
560C2C2×#8949311yescharacteristic
561C2C2×#8949511yescharacteristic
562C2C2×#8950511yescharacteristic
563C2C2×#8951011yescharacteristic
564C2C2×#8973311yescharacteristic
565C2C2×#8988611yescharacteristic
566C2C2×#8990911yescharacteristic
subgroupG/H
centerC24C2×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×C22⋊C8C2
socleC24×C7C2×D4
prime radicalC25×C7D4
hypercenterC22×C22⋊C8D7
hypocenterC7C2×#2035
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C22×#6390C2semidirect
5. C22×#6390C2semidirect
6. C22×#6391C2semidirect
7. C2×#89346C2semidirect
8. #11727690C2direct (N × H)
29. C2×#6391C22semidirect
32. C2×#6390C22semidirect
33. C2×#6390C22semidirect
34. #89346C22semidirect
450. C2×C7#2035semidirect
525. C7C2×#2035semidirect
559. C2#11727690direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C22×C7×C22⋊C82
C2×#899842
C22×#63902
C22×#63912
C2×#893462
#117276902
C2×#20357