Finite Group Databasebeta

← C2×#22589533 #32024905

Subgroups — C2×#22589533 #32024905

Order 18176.

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×#14556059C211yesmaximal, characteristic, complemented
2C22×#14556052C211yesmaximal, characteristic, complemented
3C22×#14556052C211yesmaximal, characteristic, complemented
4C22×C22⋊(C71⋊C8)C211yesmaximal, characteristic
5C2×#17985859C211yesmaximal, characteristic
6#22589533C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C71C211yesmaximal, characteristic
9C23×C71⋊C8C2211yescharacteristic
10C23×C71⋊C8C2211yescharacteristic
11C22×C4⋊Dic71C2211yescharacteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C22⋊Dic71—21no—
14C2×#14556059—24no—
15C2×#14556059C2214yescomplemented
16C2×#14556052—24no—
17C2×#14556052—24no—
18C2×#14556052C2214yescomplemented
19C2×#14556052C2214yescomplemented
20C2×C22⋊(C71⋊C8)—24no—
21C2×C22⋊(C71⋊C8)C2214yes—
22#17985859—22no—
23#17985859C2214yes—
24C2×C71⋊#95—21no—
25#17985761C2214yescomplemented
26C23×Q8×C71C2211yescharacteristic
27C24×C4×C71C2211yescharacteristic
28C2×C22⋊Q8×C71—24no—
29C2×C22⋊Q8×C71—22no—
30C2×C22⋊Q8×C71—24no—
31C2×C22⋊Q8×C71C2214yes—
32C22×C4⋊C4×C71—21no—
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C22⋊C4×C71—21no—
35C23×Dic71—21no—
36C23×Dic71—21no—
37C22×C71⋊C8—22no—
38C22×C71⋊C8—22no—
39C22×C71⋊C8—22no—
40C22×C71⋊C8—22no—
41C22×C71⋊C8—22no—
42C22×C71⋊C8D414yes—
43C22×C71⋊C8C2312yes—
44C22×C71⋊C8C2312yes—
45C2×C4⋊Dic71—22no—
46C2×C4⋊Dic71—22no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—21no—
49C2×C4⋊Dic71—21no—
50C2×C4⋊Dic71—22no—
51C2×C4⋊Dic71C2312yes—
52C2×C4⋊Dic71C2312yes—
53C2×C22⋊Dic71—24no—
54C2×C22⋊Dic71—44no—
55#14556059—28no—
56#14556052—28no—
57#14556052—28no—
58C22⋊(C71⋊C8)—28no—
59C71⋊#95—24no—
60C25×C71D411yescharacteristic
61C22×Q8×C71—22no—
62C22×Q8×C71—21no—
63C22×Q8×C71—22no—
64C22×Q8×C71—22no—
65C22×Q8×C71—22no—
66C22×Q8×C71—22no—
67C22×Q8×C71D414yes—
68C22×Q8×C71C2312yes—
69C23×C4×C71—21no—
70C23×C4×C71—21no—
71C23×C4×C71—22no—
72C23×C4×C71—22no—
73C23×C4×C71—22no—
74C23×C4×C71—22no—
75C23×C4×C71—22no—
76C23×C4×C71—22no—
77C23×C4×C71D411yescharacteristic
78C23×C4×C71D412yes—
79C23×C4×C71C2312yes—
80C23×C4×C71C2311yescharacteristic
81C22⋊Q8×C71—216no—
82C22⋊Q8×C71—28no—
83C22⋊Q8×C71—28no—
84C2×C4⋊C4×C71—28no—
85C2×C4⋊C4×C71—24no—
86C2×C4⋊C4×C71—21no—
87C2×C4⋊C4×C71—22no—
88C2×C4⋊C4×C71D414yes—
89C2×C4⋊C4×C71C2312yes—
90C2×C22⋊C4×C71—28no—
91C2×C22⋊C4×C71—24no—
92C22×Dic71—22no—
93C22×Dic71—22no—
94C22×Dic71—42no—
95C22×Dic71—42no—
96C22×Dic71—42no—
97C22×Dic71—42no—
98C22×Dic71—42no—
99C22×Dic71—42no—
100C2×C71⋊C8—28no—
101C2×C71⋊C8—22no—
102C2×C71⋊C8—22no—
103C2×C71⋊C8—22no—
104C2×C71⋊C8—22no—
105C2×C71⋊C8—22no—
106C2×C71⋊C8—44no—
107C2×C71⋊C8C2×D414yes—
108C4⋊Dic71—22no—
109C4⋊Dic71—22no—
110C4⋊Dic71—22no—
111C4⋊Dic71—22no—
112C4⋊Dic71—42no—
113C4⋊Dic71—42no—
114C22⋊Dic71—48no—
115C24×C71—22no—
116C24×C71—22no—
117C24×C71—22no—
118C24×C71—22no—
119C24×C71—22no—
120C24×C71—22no—
121C24×C71SD1614yes—
122C24×C71C2×D412yes—
123C24×C71C2×D411yescharacteristic
124C2×Q8×C71—24no—
125C2×Q8×C71—22no—
126C2×Q8×C71—28no—
127C2×Q8×C71—28no—
128C2×Q8×C71—22no—
129C2×Q8×C71—22no—
130C2×Q8×C71—22no—
131C2×Q8×C71—22no—
132C2×Q8×C71—44no—
133C2×Q8×C71—42no—
134C2×Q8×C71—42no—
135C2×Q8×C71—44no—
136C2×Q8×C71C2×D414yes—
137C22×C4×C71—24no—
138C22×C4×C71—24no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—24no—
145C22×C4×C71—24no—
146C22×C4×C71—24no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—42no—
154C22×C4×C71—42no—
155C22×C4×C71—42no—
156C22×C4×C71—42no—
157C22×C4×C71—44no—
158C22×C4×C71—44no—
159C22×C4×C71—44no—
160C22×C4×C71—44no—
161C22×C4×C71C2×D412yes—
162C22×C4×C71C2×D412yes—
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D412yes—
165C22×C4×C71C2×D412yes—
166C22×C4×C71#4112yes—
167C22×C4×C71#4112yes—
168C22×C4×C71#4112yes—
169C22×C4×C71C2411yescharacteristic
170C4⋊C4×C71—216no—
171C4⋊C4×C71—24no—
172C4⋊C4×C71—22no—
173C4⋊C4×C71C2×D414yes—
174C22⋊C4×C71—216no—
175C23×C71—22no—
176C23×C71—22no—
177C23×C71—22no—
178C23×C71—22no—
179C23×C71—22no—
180C23×C71—22no—
181C23×C71—28no—
182C23×C71—28no—
183C23×C71—28no—
184C23×C71—44no—
185C23×C71—44no—
186C23×C71—44no—
187C23×C71—44no—
188C23×C71#12012yes—
189C23×C71#12112yes—
190C23×C71#12312yes—
191C23×C71C2×SD1614yes—
192C23×C71C2×SD1612yes—
193C23×C71#13512yes—
194C23×C71#13712yes—
195C23×C71#13712yes—
196C23×C71C22×D411yescharacteristic
197Q8×C71—24no—
198Q8×C71—24no—
199Q8×C71—42no—
200Q8×C71—42no—
201Q8×C71—48no—
202C2×C4×C71—24no—
203C2×C4×C71—24no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—44no—
212C2×C4×C71—48no—
213C2×C4×C71—48no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—48no—
219C2×C4×C71#12012yes—
220C2×C4×C71#12114yes—
221C2×C4×C71#12114yes—
222C2×C4×C71#12112yes—
223C2×C4×C71#12314yes—
224C2×C4×C71#12312yes—
225C2×C4×C71#12414yes—
226C2×C4×C71C22×D411yescharacteristic
227C2×C4×C71C22×D411yescharacteristic
228C2×C4×C71C22×D411yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—84no—
239C2×Dic71—84no—
240C71⋊C8—24no—
241C71⋊C8—42no—
242C22×C71—24no—
243C22×C71—24no—
244C22×C71—24no—
245C22×C71—42no—
246C22×C71—42no—
247C22×C71—42no—
248C22×C71—42no—
249C22×C71—416no—
250C22×C71#39414yes—
251C22×C71#44814yes—
252C22×C71#45214yes—
253C22×C71#46514yes—
254C22×C71#47014yes—
255C22×C71#48314yes—
256C22×C71#48514yes—
257C22×C71C2×#12011yescharacteristic
258C22×C71C2×#12111yescharacteristic
259C22×C71C2×#12311yescharacteristic
260C22×C71C22×SD1611yescharacteristic
261C22×C71C2×#13511yescharacteristic
262C22×C71C2×#13711yescharacteristic
263C22×C71C2×#13711yescharacteristic
264C4×C71—42no—
265C4×C71—44no—
266C4×C71—44no—
267C4×C71#39414yes—
268C4×C71C2×#12111yescharacteristic
269C4×C71C2×#12111yescharacteristic
270C4×C71C2×#12311yescharacteristic
271C4×C71C2×#12411yescharacteristic
272Dic71—82no—
273Dic71—82no—
274C2×#2035—711nomaximal, Sylow
275C2×C71—44no—
276C2×C71#203518yescomplemented
277C2×C71C2×#39411yescharacteristic
278C2×C71C2×#44811yescharacteristic
279C2×C71C2×#45211yescharacteristic
280C2×C71C2×#46511yescharacteristic
281C2×C71C2×#47011yescharacteristic
282C2×C71C2×#48311yescharacteristic
283C2×C71C2×#48511yescharacteristic
284C22×C22⋊Q8D7111yescharacteristic
285C22×#106—711no—
286C22×#103—711no—
287C22×#103—711no—
288C22×C22⋊C8—711no—
289C2×#380—711no—
290#2035—718no—
291C2×#340—711no—
292C71C2×#203511yescharacteristic, complemented, Sylow
293C23×Q8C2×D7111yescharacteristic
294C24×C4C2×D7111yescharacteristic
295C23×C8—711no—
296C23×C8—711no—
297C2×C22⋊Q8—24no—
298C2×C22⋊Q8—22no—
299C2×C22⋊Q8—24no—
300C2×C22⋊Q8C2×D7114yes—
301C22×C4⋊C4—21no—
302C22×C4⋊C4—711no—
303C22×C4⋊C4—711no—
304C22×C4⋊C4C2×D7111yescharacteristic
305C22×C22⋊C4—21no—
306C22×C22⋊C4—1421no—
307C2×#106—714no—
308C2×#106—1424no—
309C2×#103—714no—
310C2×#103—714no—
311C2×#103—1424no—
312C2×#103—1424no—
313C2×C22⋊C8—714no—
314C2×C22⋊C8—1424no—
315#380—714no—
316#380—1422no—
317C2×#95—1421no—
318#340—714no—
319C25C22⋊D7111yescharacteristic
320C22×Q8—21no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8C22⋊D7114yes—
327C22×Q8C22×D7112yes—
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—1421no—
337C23×C4—1421no—
338C23×C4C22⋊D7112yes—
339C23×C4C22⋊D7111yescharacteristic
340C23×C4C22×D7112yes—
341C23×C4C22×D7111yescharacteristic
342C22×C8—714no—
343C22×C8—712no—
344C22×C8—712no—
345C22×C8—1422no—
346C22×C8—1422no—
347C22×C8—1422no—
348C22×C8—1422no—
349C22×C8—1422no—
350C22⋊Q8—216no—
351C22⋊Q8—28no—
352C22⋊Q8—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—712no—
358C2×C4⋊C4—712no—
359C2×C4⋊C4—1421no—
360C2×C4⋊C4—1421no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1422no—
363C2×C4⋊C4—1422no—
364C2×C4⋊C4—1422no—
365C2×C4⋊C4C22⋊D7114yes—
366C2×C4⋊C4C22×D7112yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—2844no—
370C2×C22⋊C4—1424no—
371#106—1428no—
372#103—1428no—
373#103—1428no—
374C22⋊C8—1428no—
375#95—1424no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#1176585214yes—
383C24C2×C22⋊D7112yes—
384C24C2×C22⋊D7111yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D7114yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
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—2842no—
423C22×C4—2842no—
424C22×C4—2842no—
425C22×C4—2842no—
426C22×C4—2842no—
427C22×C4—2842no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4D4×D7112yes—
431C22×C4C2×C22⋊D7112yes—
432C22×C4C2×C22⋊D7112yes—
433C22×C4C2×C22⋊D7112yes—
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4#1176586412yes—
436C22×C4#1176586412yes—
437C22×C4#1176586412yes—
438C22×C4C23×D7111yescharacteristic
439C2×C8—2844no—
440C2×C8—714no—
441C2×C8—1422no—
442C2×C8—1422no—
443C2×C8—1422no—
444C2×C8—1422no—
445C2×C8—1422no—
446C2×C8—1428no—
447C4⋊C4—216no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—2842no—
451C4⋊C4—2842no—
452C4⋊C4—1422no—
453C4⋊C4—1422no—
454C4⋊C4—1422no—
455C4⋊C4—1422no—
456C4⋊C4C2×C22⋊D7114yes—
457C22⋊C4—216no—
458C22⋊C4—2848no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1455609712yes—
473C23#1455610012yes—
474C23#1455610912yes—
475C23C2×#1176585214yes—
476C23C2×#1176585212yes—
477C23#1455614812yes—
478C23#1455615812yes—
479C23#1455615912yes—
480C23C22×C22⋊D7111yescharacteristic
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—2842no—
504C2×C4—2842no—
505C2×C4—2842no—
506C2×C4—2842no—
507C2×C4—2842no—
508C2×C4—2842no—
509C2×C4—5684no—
510C2×C4—5684no—
511C2×C4#1455609712yes—
512C2×C4#1455609914yes—
513C2×C4#1455610012yes—
514C2×C4#1455610014yes—
515C2×C4#1455610912yes—
516C2×C4#1455610914yes—
517C2×C4#1455611514yes—
518C2×C4C2×D4×D7111yescharacteristic
519C2×C4C22×C22⋊D7111yescharacteristic
520C2×C4C22×C22⋊D7111yescharacteristic
521C2×C4C2×#1176586411yescharacteristic
522C2×C4C2×#1176586411yescharacteristic
523C2×C4C2×#1176586411yescharacteristic
524C8—2842no—
525C8—1424no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—416no—
533C22—42no—
534C22#1798591314yes—
535C22#1798613814yes—
536C22#1798616114yes—
537C22#1798623014yes—
538C22#1798625514yes—
539C22#1798632714yes—
540C22#1798633714yes—
541C22C2×#1455609711yescharacteristic
542C22C2×#1455610011yescharacteristic
543C22C2×#1455610911yescharacteristic
544C22C22×#1176585211yescharacteristic
545C22C2×#1455614811yescharacteristic
546C22C2×#1455615811yescharacteristic
547C22C2×#1455615911yescharacteristic
548C4—42no—
549C4—44no—
550C4—44no—
551C4—5682no—
552C4—5682no—
553C4#1798591314yes—
554C4C2×#1455609911yescharacteristic
555C4C2×#1455610011yescharacteristic
556C4C2×#1455610911yescharacteristic
557C4C2×#1455611511yescharacteristic
558C2—44no—
559C2#2258953318yesdirect product factor
560C2C2×#1798591311yescharacteristic
561C2C2×#1798613811yescharacteristic
562C2C2×#1798616111yescharacteristic
563C2C2×#1798623011yescharacteristic
564C2C2×#1798625511yescharacteristic
565C2C2×#1798632711yescharacteristic
566C2C2×#1798633711yescharacteristic
subgroupG/H
centerC24C2×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC22×C22⋊Q8×C71C2
socleC24×C71C2×D4
prime radicalC25×C71D4
hypercenterC22×C22⋊Q8D71
hypocenterC71C2×#2035
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14556059C2semidirect
2. C22×#14556052C2semidirect
3. C22×#14556052C2semidirect
6. #22589533C2direct (N × H)
7. C2×#17985761C2semidirect
15. C2×#14556059C22semidirect
18. C2×#14556052C22semidirect
19. C2×#14556052C22semidirect
25. #17985761C22semidirect
276. C2×C71#2035semidirect
292. C71C2×#2035semidirect
559. C2#22589533direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560592
C22×#145560522
C22×C22⋊(C71⋊C8)2
C2×#179858592
#225895332
C2×#179857612
C22×C22⋊Q8×C712
C2×#203571