Finite Group Databasebeta

← C2×#22592827 #32166851

Subgroups — C2×#22592827 #32166851

Order 18176.

3438 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556101C211yesmaximal, characteristic
2C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#22592827C2116yesmaximal, direct product factor
4C2×#17985926C211yesmaximal, characteristic, complemented
5C2×#17985925C211yesmaximal, characteristic, complemented
6C2×#17985924C211yesmaximal, characteristic, complemented
7C2×#17985899C212yesmaximal, complemented
8C2×#17985877C212yesmaximal, complemented
9C2×#17985875C212yesmaximal, complemented
10C2×#17985863C212yesmaximal, complemented
11C2×#17985846C211yesmaximal, characteristic, complemented
12C2×#397×C71C211yesmaximal, characteristic, complemented
13C23×D284C2211yescharacteristic
14C23×C22⋊D71C2211yescharacteristic
15C23×C22⋊D71C2211yescharacteristic
16C23×C4×D71C2211yescharacteristic
17C23×C4×D71C2212yes—
18C2×#14556101—28no—
19C2×#14556101C2218yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×Dic71⋊C4C2211yescharacteristic
22C22×Dic71⋊C4C2212yes—
23C2×C4⋊C4×D71—28no—
24C2×C4⋊C4×D71C2218yescomplemented
25C22×C22⋊Dic71C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2211yescharacteristic
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×C4×Dic71C2212yes—
31#17985926C2218yescomplemented
32#17985925C2218yescomplemented
33#17985924C2218yescomplemented
34#17985899C22116yescomplemented
35#17985877C22116yescomplemented
36#17985875C22116yescomplemented
37#17985863C22116yescomplemented
38#17985846C2218yescomplemented
39C2×C71⋊#95C2212yes—
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2211yescharacteristic
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C22⋊C4×C71C2211yescharacteristic
45C22×C22⋊C4×C71C2212yes—
46#397×C71C2218yescomplemented
47C2×#95×C71C2212yes—
48C2×#95×C71C2211yescharacteristic
49C24×D71C2311yescharacteristic
50C24×D71C2311yescharacteristic
51C22×D284—24no—
52C22×D284—28no—
53C22×D284C2314yes—
54C22×C22⋊D71—24no—
55C22×C22⋊D71—28no—
56C22×C22⋊D71—24no—
57C22×C22⋊D71—28no—
58C22×C22⋊D71C2314yes—
59C22×C22⋊D71C2314yes—
60C23×Dic71C2312yes—
61C23×Dic71C2311yescharacteristic
62C23×Dic71C2312yes—
63C23×Dic71C2311yescharacteristic
64C22×C4×D71—28no—
65C22×C4×D71—28no—
66C22×C4×D71—44no—
67C22×C4×D71—44no—
68C22×C4×D71D418yes—
69C22×C4×D71C2314yes—
70C22×C4×D71C2318yes—
71#14556101—232no—
72C2×C4⋊Dic71—24no—
73C2×C4⋊Dic71C2314yes—
74C2×Dic71⋊C4—28no—
75C2×Dic71⋊C4—24no—
76C2×Dic71⋊C4C2314yes—
77C2×Dic71⋊C4C2318yes—
78C4⋊C4×D71—232no—
79C2×C22⋊Dic71—44no—
80C2×C22⋊Dic71C2318yes—
81C2×(C2×D71)⋊C4—28no—
82C2×(C2×D71)⋊C4—24no—
83C2×(C2×D71)⋊C4—44no—
84C2×(C2×D71)⋊C4—44no—
85C2×(C2×D71)⋊C4C2318yes—
86C2×(C2×D71)⋊C4C2314yes—
87C2×(C2×D71)⋊C4C2318yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×C4×Dic71—28no—
90C2×C4×Dic71C2318yes—
91C71⋊#95C2314yes—
92C71⋊#95C2318yes—
93C71⋊#95C2318yes—
94C71⋊#95C2318yes—
95C25×C71C2311yescharacteristic
96C23×C4×C71C2312yes—
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2312yes—
99C23×C4×C71C2311yescharacteristic
100C2×C4⋊C4×C71—24no—
101C2×C4⋊C4×C71C2314yes—
102C2×C22⋊C4×C71—24no—
103C2×C22⋊C4×C71—44no—
104C2×C22⋊C4×C71C2314yes—
105C2×C22⋊C4×C71C2318yes—
106#95×C71C2314yes—
107#95×C71C2318yes—
108C23×D71—24no—
109C23×D71—22no—
110C23×D71—22no—
111C23×D71—22no—
112C23×D71—24no—
113C23×D71—24no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71C2×D414yes—
117C23×D71#4118yes—
118C23×D71C2412yes—
119C23×D71C2412yes—
120C2×D284—28no—
121C2×D284—216no—
122C2×D284—416no—
123C2×C22⋊D71—28no—
124C2×C22⋊D71—216no—
125C2×C22⋊D71—28no—
126C2×C22⋊D71—216no—
127C2×C22⋊D71—416no—
128C2×C22⋊D71—416no—
129C22×Dic71—24no—
130C22×Dic71—22no—
131C22×Dic71—24no—
132C22×Dic71—24no—
133C22×Dic71—24no—
134C22×Dic71—24no—
135C22×Dic71—24no—
136C22×Dic71—24no—
137C22×Dic71C2×D414yes—
138C22×Dic71#4118yes—
139C22×Dic71C2414yes—
140C22×Dic71C2412yes—
141C22×Dic71C2414yes—
142C22×Dic71C2412yes—
143C2×C4×D71—232no—
144C2×C4×D71—216no—
145C2×C4×D71—216no—
146C2×C4×D71—48no—
147C2×C4×D71—48no—
148C2×C4×D71—432no—
149C2×C4×D71C2×D4116yes—
150C4⋊Dic71—28no—
151Dic71⋊C4—216no—
152Dic71⋊C4—28no—
153C22⋊Dic71—48no—
154(C2×D71)⋊C4—216no—
155(C2×D71)⋊C4—28no—
156(C2×D71)⋊C4—48no—
157(C2×D71)⋊C4—48no—
158C4×Dic71—216no—
159C24×C71—22no—
160C24×C71—24no—
161C24×C71—24no—
162C24×C71—22no—
163C24×C71—22no—
164C24×C71C2412yes—
165C24×C71C2411yescharacteristic
166C22×C4×C71—24no—
167C22×C4×C71—24no—
168C22×C4×C71—22no—
169C22×C4×C71—24no—
170C22×C4×C71—24no—
171C22×C4×C71—24no—
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—24no—
175C22×C4×C71C2×D414yes—
176C22×C4×C71C2414yes—
177C22×C4×C71C2412yes—
178C22×C4×C71C2414yes—
179C22×C4×C71C2412yes—
180C4⋊C4×C71—28no—
181C22⋊C4×C71—28no—
182C22⋊C4×C71—48no—
183C23×C71—22no—
184C23×C71—24no—
185C23×C71—24no—
186C23×C71—22no—
187C23×C71—22no—
188C23×C71—44no—
189C23×C71—44no—
190C23×C71—44no—
191C23×C71—48no—
192C23×C71—48no—
193C23×C71C22×D412yes—
194C23×C71C2×#4114yes—
195C23×C71C2×#4114yes—
196C23×C71H(5,𝔽2)12yes—
197C23×C71H(5,𝔽2)12yes—
198C23×C71C2511yescharacteristic
199C22×D71—28no—
200C22×D71—28no—
201C22×D71—216no—
202C22×D71—216no—
203C22×D71—24no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—22no—
210C22×D71—22no—
211C22×D71—44no—
212C22×D71—44no—
213C22×D71—44no—
214C22×D71—48no—
215C22×D71—48no—
216C22×D71#124116yes—
217C22×D71C22×D414yes—
218C22×D71C2×#4118yes—
219C2×C4×C71—28no—
220C2×C4×C71—24no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—44no—
230C2×C4×C71—48no—
231C2×C4×C71—48no—
232C2×C4×C71C22×D414yes—
233C22⋊D71—416no—
234C22⋊D71—416no—
235C2×Dic71—22no—
236C2×Dic71—24no—
237C2×Dic71—24no—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—44no—
244C2×Dic71—48no—
245C2×Dic71—48no—
246C2×Dic71#123116yes—
247C2×Dic71C22×D414yes—
248C2×Dic71C2×#4118yes—
249D284—416no—
250C4×D71—232no—
251C4×D71—432no—
252C22×C71—42no—
253C22×C71—42no—
254C22×C71—42no—
255C22×C71—44no—
256C22×C71—44no—
257C22×C71—88no—
258C22×C71C2×#12314yes—
259C22×C71C2×#12414yes—
260C22×C71#53514yes—
261C22×C71#55318yes—
262C22×C71#55318yes—
263C22×C71C23×D411yescharacteristic
264C22×C71C22×#4112yes—
265C22×C71C22×#4112yes—
266C22×C71C2×H(5,𝔽2)11yescharacteristic
267C22×C71C2×H(5,𝔽2)11yescharacteristic
268C2×D71—24no—
269C2×D71—24no—
270C2×D71—28no—
271C2×D71—28no—
272C2×D71—232no—
273C2×D71—42no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—44no—
277C2×D71—44no—
278C2×D71—88no—
279C2×D71C2×#12418yes—
280C4×C71—24no—
281C4×C71—42no—
282C4×C71—44no—
283C4×C71—44no—
284Dic71—42no—
285Dic71—44no—
286Dic71—44no—
287Dic71C2×#12318yes—
288C2×#2504—711nomaximal, Sylow
289C2×C71—82no—
290C2×C71#250418yescomplemented
291C2×C71C22×#12311yescharacteristic
292C2×C71C22×#12411yescharacteristic
293C2×C71C2×#53511yescharacteristic
294C2×C71C2×#55312yes—
295C2×C71C2×#55312yes—
296D71—28no—
297D71—82no—
298C22×#121—711no—
299C23×C4⋊C4—711no—
300#2504—7116no—
301C2×#397—711no—
302C2×#397—711no—
303C2×#397—711no—
304C2×#397D7111yescharacteristic, complemented
305C2×#390—712no—
306C2×#386—712no—
307C2×#386—712no—
308C2×#382—712no—
309C2×C4×C22⋊C4—711no—
310C71C2×#250411yescharacteristic, complemented, Sylow
311C23×D4—711no—
312C23×D4—711no—
313C23×D4—711no—
314C24×C4—712no—
315C24×C4—711no—
316C2×#121—718no—
317C2×#121—1428no—
318C22×C4⋊C4—711no—
319C22×C4⋊C4—712no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—718no—
322C22×C4⋊C4—1428no—
323C22×C4⋊C4C2×D7111yescharacteristic
324C22×C22⋊C4—712no—
325C22×C22⋊C4—712no—
326C22×C22⋊C4—712no—
327C22×C22⋊C4—711no—
328C22×C22⋊C4—712no—
329C22×C22⋊C4C2×D7111yescharacteristic
330C22×C22⋊C4C2×D7112yes—
331C22×C42—712no—
332#397—718no—
333#397—718no—
334#397—718no—
335#397C2×D7118yescomplemented
336#390—7116no—
337#386—7116no—
338#386—7116no—
339#382—7116no—
340C4×C22⋊C4—718no—
341C2×#95—711no—
342C2×#95—712no—
343C2×#95—712no—
344C2×#95—712no—
345C2×#95C2×D7111yescharacteristic
346C2×#95C2×D7112yes—
347C25—711no—
348C25—711no—
349C25C22×D7111yescharacteristic
350C22×D4—714no—
351C22×D4—714no—
352C22×D4—714no—
353C22×D4—1424no—
354C22×D4—1428no—
355C22×D4—1424no—
356C22×D4—1428no—
357C22×D4—1424no—
358C22×D4—1428no—
359C23×C4—2844no—
360C23×C4—2844no—
361C23×C4—718no—
362C23×C4—711no—
363C23×C4—718no—
364C23×C4—714no—
365C23×C4—712no—
366C23×C4—712no—
367C23×C4—711no—
368C23×C4—1428no—
369C23×C4—1428no—
370C23×C4C22×D7112yes—
371C23×C4C22×D7112yes—
372C23×C4C22×D7111yescharacteristic
373C23×C4C22×D7111yescharacteristic
374#121—14232no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—718no—
377C2×C4⋊C4—714no—
378C2×C4⋊C4—714no—
379C2×C4⋊C4—14232no—
380C2×C4⋊C4—1428no—
381C2×C4⋊C4—1424no—
382C2×C4⋊C4—1424no—
383C2×C4⋊C4C22×D7114yes—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—44no—
386C2×C22⋊C4—2844no—
387C2×C22⋊C4—2844no—
388C2×C22⋊C4—2844no—
389C2×C22⋊C4—718no—
390C2×C22⋊C4—718no—
391C2×C22⋊C4—718no—
392C2×C22⋊C4—714no—
393C2×C22⋊C4—718no—
394C2×C22⋊C4—1428no—
395C2×C22⋊C4—1424no—
396C2×C22⋊C4C22×D7114yes—
397C2×C22⋊C4C22×D7118yes—
398C2×C42—718no—
399C2×C42—1428no—
400#95—714no—
401#95—718no—
402#95—718no—
403#95—718no—
404#95C22×D7114yes—
405#95C22×D7118yes—
406C24—22no—
407C24—24no—
408C24—24no—
409C24—22no—
410C24—22no—
411C24—714no—
412C24—718no—
413C24—712no—
414C24—712no—
415C24—1422no—
416C24—1424no—
417C24—1424no—
418C24—1422no—
419C24—1422no—
420C24—1424no—
421C24—1422no—
422C24—1422no—
423C24C23×D7112yes—
424C24C23×D7111yescharacteristic
425C2×D4—28416no—
426C2×D4—28416no—
427C2×D4—28416no—
428C2×D4—1428no—
429C2×D4—14216no—
430C2×D4—1428no—
431C2×D4—14216no—
432C2×D4—1428no—
433C2×D4—14216no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—2848no—
444C22×C4—2848no—
445C22×C4—28432no—
446C22×C4—714no—
447C22×C4—718no—
448C22×C4—712no—
449C22×C4—7116no—
450C22×C4—712no—
451C22×C4—714no—
452C22×C4—714no—
453C22×C4—14216no—
454C22×C4—14216no—
455C22×C4—14232no—
456C22×C4—1422no—
457C22×C4—1424no—
458C22×C4—1424no—
459C22×C4—1424no—
460C22×C4—1424no—
461C22×C4—1424no—
462C22×C4—1424no—
463C22×C4—1424no—
464C22×C4D4×D7114yes—
465C22×C4C23×D7112yes—
466C22×C4C23×D7114yes—
467C22×C4C23×D7114yes—
468C22×C4C23×D7112yes—
469C4⋊C4—28no—
470C4⋊C4—14216no—
471C4⋊C4—1428no—
472C4⋊C4—1428no—
473C22⋊C4—28no—
474C22⋊C4—48no—
475C22⋊C4—2848no—
476C22⋊C4—2848no—
477C22⋊C4—2848no—
478C22⋊C4—14216no—
479C22⋊C4—1428no—
480C42—14216no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—44no—
487C23—48no—
488C23—48no—
489C23—44no—
490C23—44no—
491C23—2844no—
492C23—2848no—
493C23—2848no—
494C23—2844no—
495C23—2844no—
496C23—714no—
497C23—718no—
498C23—7116no—
499C23—1422no—
500C23—1424no—
501C23—1424no—
502C23—1422no—
503C23—1422no—
504C23—1428no—
505C23—1424no—
506C23—1422no—
507C23—1422no—
508C23—14216no—
509C23—14216no—
510C23—1428no—
511C23C2×D4×D7112yes—
512C23#41×D7114yes—
513C23#41×D7114yes—
514C23#1455617312yes—
515C23#1455617412yes—
516C23C24×D7111yescharacteristic
517D4—28416no—
518D4—28416no—
519D4—28416no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—28432no—
534C2×C4—2848no—
535C2×C4—2848no—
536C2×C4—2844no—
537C2×C4—714no—
538C2×C4—718no—
539C2×C4—7116no—
540C2×C4—14232no—
541C2×C4—1422no—
542C2×C4—1424no—
543C2×C4—1424no—
544C2×C4—1424no—
545C2×C4—1424no—
546C2×C4—1424no—
547C2×C4—1424no—
548C2×C4—1424no—
549C2×C4C2×D4×D7114yes—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—88no—
556C22—2842no—
557C22—2842no—
558C22—2844no—
559C22—2844no—
560C22—2842no—
561C22—5688no—
562C22—718no—
563C22—14232no—
564C22—1424no—
565C22—1428no—
566C22—1428no—
567C22—1424no—
568C22#123×D7114yes—
569C22#124×D7114yes—
570C22#1798656514yes—
571C22#1798669418yes—
572C22#1798670018yes—
573C22C22×D4×D7111yescharacteristic
574C22C2×#41×D7112yes—
575C22C2×#41×D7112yes—
576C22C2×#1455617311yescharacteristic
577C22C2×#1455617411yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—44no—
582C4—2842no—
583C4—2844no—
584C4—2844no—
585C4—718no—
586C2—82no—
587C2—5682no—
588C2—1428no—
589C2#2259282718yesdirect product factor
590C2C2×#123×D7111yescharacteristic
591C2C2×#124×D7111yescharacteristic
592C2C2×#1798656511yescharacteristic
593C2C2×#1798669412yes—
594C2C2×#1798670012yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#397×C71C2
socleC24×C71C24
prime radicalC22×#14556101C2
hypercenterC2×#397D71
hypocenterC71C2×#2504
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D71C2semidirect
3. #22592827C2direct (N × H)
4. C2×#17985926C2semidirect
5. C2×#17985925C2semidirect
6. C2×#17985924C2semidirect
7. C2×#17985899C2semidirect
8. C2×#17985877C2semidirect
9. C2×#17985875C2semidirect
10. C2×#17985863C2semidirect
11. C2×#17985846C2semidirect
12. C2×#397×C71C2semidirect
24. C2×C4⋊C4×D71C22semidirect
31. #17985926C22semidirect
32. #17985925C22semidirect
33. #17985924C22semidirect
34. #17985899C22semidirect
35. #17985877C22semidirect
36. #17985875C22semidirect
37. #17985863C22semidirect
38. #17985846C22semidirect
46. #397×C71C22semidirect
290. C2×C71#2504semidirect
304. C2×#397D71semidirect
310. C71C2×#2504semidirect
335. #397C2×D71semidirect
589. C2#22592827direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561012
C22×C4⋊C4×D712
#225928272
C2×#179859262
C2×#179859252
C2×#179859242
C2×#179858992
C2×#179858772
C2×#179858752
C2×#179858632
C2×#179858462
C2×#397×C712
C2×#250471