Finite Group Databasebeta

← C2×#22592609 #32167214

Subgroups — C2×#22592609 #32167214

Order 18176.

4166 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556086C211yesmaximal, characteristic, complemented
2C22×C22⋊C4×D71C211yesmaximal, characteristic
3#22592609C2116yesmaximal, direct product factor
4C2×#17985936C211yesmaximal, characteristic, complemented
5C2×#17985935C211yesmaximal, characteristic, complemented
6C2×#17985925C211yesmaximal, characteristic, complemented
7C2×#17985892C212yesmaximal, complemented
8C2×#17985888C212yesmaximal, complemented
9C2×#17985873C212yesmaximal, complemented
10C2×#17985861C211yesmaximal, characteristic, complemented
11C2×#17985854C212yesmaximal
12C2×#397×C71C211yesmaximal, characteristic, complemented
13C25×D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C2×#14556086—28no—
17C2×#14556086C2218yescomplemented
18C22×C4⋊Dic71C2211yescharacteristic, complemented
19C22×C4⋊Dic71C2212yescomplemented
20C22×Dic71⋊C4C2212yescomplemented
21C22×C22⋊Dic71C2211yescharacteristic
22C22×C22⋊Dic71C2212yes—
23C22×(C2×D71)⋊C4C2211yescharacteristic
24C22×(C2×D71)⋊C4C2211yescharacteristic
25C22×(C2×D71)⋊C4C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C2×C22⋊C4×D71—28no—
30C2×C22⋊C4×D71C2218yes—
31C22×C4×Dic71C2212yescomplemented
32#17985936C2218yescomplemented
33#17985935C2218yescomplemented
34#17985925C2218yescomplemented
35#17985892C22116yescomplemented
36#17985888C22116yescomplemented
37#17985873C22116yescomplemented
38#17985861C2218yescomplemented
39#17985854C22116yes—
40C2×C71⋊#95C2212yescomplemented
41C2×C71⋊#95C2212yescomplemented
42C2×C71⋊#95C2211yescharacteristic, complemented
43C22×C4⋊C4×C71C2211yescharacteristic, complemented
44C22×C22⋊C4×C71C2211yescharacteristic
45C22×C22⋊C4×C71C2212yes—
46#397×C71C2218yescomplemented
47C2×#95×C71C2212yescomplemented
48C2×#95×C71C2211yescharacteristic, complemented
49C24×D71—24no—
50C24×D71—24no—
51C24×D71—28no—
52C24×D71—24no—
53C24×D71—44no—
54C24×D71C2311yescharacteristic
55C24×D71C2314yes—
56C24×D71C2311yescharacteristic
57C23×Dic71C2311yescharacteristic
58C23×Dic71C2312yes—
59C23×Dic71C2312yes—
60C23×Dic71C2311yescharacteristic
61C22×C4×D71—28no—
62C22×C4×D71—24no—
63C22×C4×D71—44no—
64C22×C4×D71D418yes—
65C22×C4×D71C2314yes—
66C22×C4×D71C2314yes—
67#14556086—232no—
68C2×C4⋊Dic71—24no—
69C2×C4⋊Dic71—44no—
70C2×C4⋊Dic71C2314yescomplemented
71C2×C4⋊Dic71C2318yescomplemented
72C2×Dic71⋊C4—44no—
73C2×Dic71⋊C4C2318yescomplemented
74C2×C22⋊Dic71—24no—
75C2×C22⋊Dic71—44no—
76C2×C22⋊Dic71C2314yes—
77C2×C22⋊Dic71C2318yes—
78C2×(C2×D71)⋊C4—28no—
79C2×(C2×D71)⋊C4—28no—
80C2×(C2×D71)⋊C4—28no—
81C2×(C2×D71)⋊C4—24no—
82C2×(C2×D71)⋊C4—24no—
83C2×(C2×D71)⋊C4—44no—
84C2×(C2×D71)⋊C4C2318yes—
85C2×(C2×D71)⋊C4C2318yes—
86C2×(C2×D71)⋊C4C2318yes—
87C2×(C2×D71)⋊C4C2314yes—
88C2×(C2×D71)⋊C4C2314yes—
89C2×(C2×D71)⋊C4C2318yes—
90C22⋊C4×D71—232no—
91C2×C4×Dic71—28no—
92C2×C4×Dic71C2318yescomplemented
93C71⋊#95C2318yescomplemented
94C71⋊#95C2314yescomplemented
95C71⋊#95C2318yescomplemented
96C25×C71C2311yescharacteristic
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2312yes—
99C23×C4×C71C2312yes—
100C23×C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71—24no—
102C2×C4⋊C4×C71C2314yescomplemented
103C2×C22⋊C4×C71—28no—
104C2×C22⋊C4×C71—24no—
105C2×C22⋊C4×C71C2318yes—
106C2×C22⋊C4×C71C2314yes—
107#95×C71C2314yescomplemented
108#95×C71C2318yescomplemented
109C23×D71—22no—
110C23×D71—22no—
111C23×D71—24no—
112C23×D71—232no—
113C23×D71—22no—
114C23×D71—24no—
115C23×D71—28no—
116C23×D71—22no—
117C23×D71—24no—
118C23×D71—28no—
119C23×D71—28no—
120C23×D71—216no—
121C23×D71—48no—
122C23×D71—432no—
123C23×D71—432no—
124C23×D71—416no—
125C23×D71—416no—
126C23×D71C2×D414yes—
127C23×D71#4114yes—
128C23×D71#4118yes—
129C23×D71C2412yes—
130C23×D71C2412yes—
131C22×Dic71—24no—
132C22×Dic71—22no—
133C22×Dic71—24no—
134C22×Dic71—24no—
135C22×Dic71—24no—
136C22×Dic71—24no—
137C22×Dic71—24no—
138C22×Dic71—24no—
139C22×Dic71—24no—
140C22×Dic71C2×D414yes—
141C22×Dic71C2412yes—
142C22×Dic71C2414yes—
143C22×Dic71C2414yes—
144C22×Dic71C2412yes—
145C2×C4×D71—232no—
146C2×C4×D71—216no—
147C2×C4×D71—28no—
148C2×C4×D71—48no—
149C2×C4×D71—416no—
150C2×C4×D71C2×D4116yes—
151C4⋊Dic71—28no—
152C4⋊Dic71—48no—
153Dic71⋊C4—48no—
154C22⋊Dic71—28no—
155C22⋊Dic71—48no—
156(C2×D71)⋊C4—216no—
157(C2×D71)⋊C4—216no—
158(C2×D71)⋊C4—216no—
159(C2×D71)⋊C4—28no—
160(C2×D71)⋊C4—28no—
161(C2×D71)⋊C4—48no—
162C4×Dic71—216no—
163C24×C71—22no—
164C24×C71—22no—
165C24×C71—24no—
166C24×C71—24no—
167C24×C71#4114yes—
168C24×C71C2412yes—
169C24×C71C2411yescharacteristic
170C22×C4×C71—24no—
171C22×C4×C71—22no—
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71C2×D414yes—
180C22×C4×C71C2412yes—
181C22×C4×C71C2414yes—
182C22×C4×C71C2414yes—
183C22×C4×C71C2412yes—
184C4⋊C4×C71—28no—
185C22⋊C4×C71—216no—
186C22⋊C4×C71—28no—
187C23×C71—28no—
188C23×C71—216no—
189C23×C71—22no—
190C23×C71—22no—
191C23×C71—24no—
192C23×C71—24no—
193C23×C71—44no—
194C23×C71—44no—
195C23×C71—48no—
196C23×C71C22×D412yes—
197C23×C71C2×#4114yes—
198C23×C71C2×#4112yes—
199C23×C71C2×#4114yes—
200C23×C71C2×#4112yes—
201C23×C71H(5,𝔽2)14yes—
202C23×C71C2511yescharacteristic
203C22×D71—28no—
204C22×D71—28no—
205C22×D71—216no—
206C22×D71—216no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—24no—
210C22×D71—232no—
211C22×D71—28no—
212C22×D71—216no—
213C22×D71—22no—
214C22×D71—24no—
215C22×D71—22no—
216C22×D71—24no—
217C22×D71—432no—
218C22×D71—432no—
219C22×D71—416no—
220C22×D71—4128no—
221C22×D71—44no—
222C22×D71—44no—
223C22×D71—48no—
224C22×D71—416no—
225C22×D71#123116yes—
226C22×D71C22×D414yes—
227C22×D71C2×#4114yes—
228C22×D71C2×#4118yes—
229C2×C4×C71—28no—
230C2×C4×C71—24no—
231C2×C4×C71—22no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—48no—
240C2×C4×C71—44no—
241C2×C4×C71—48no—
242C2×C4×C71C22×D414yes—
243C2×Dic71—28no—
244C2×Dic71—24no—
245C2×Dic71—24no—
246C2×Dic71—22no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—44no—
254C2×Dic71—48no—
255C2×Dic71—48no—
256C2×Dic71C22×D414yes—
257C4×D71—232no—
258C4×D71—416no—
259C22×C71—24no—
260C22×C71—28no—
261C22×C71—42no—
262C22×C71—42no—
263C22×C71—44no—
264C22×C71—416no—
265C22×C71C2×#12314yes—
266C22×C71#52514yes—
267C22×C71#53514yes—
268C22×C71#55118yes—
269C22×C71#55118yes—
270C22×C71C23×D411yescharacteristic
271C22×C71C22×#4112yes—
272C22×C71C22×#4111yescharacteristic
273C22×C71C22×#4111yescharacteristic
274C22×C71C2×H(5,𝔽2)12yes—
275C2×D71—24no—
276C2×D71—24no—
277C2×D71—28no—
278C2×D71—28no—
279C2×D71—232no—
280C2×D71—24no—
281C2×D71—28no—
282C2×D71—464no—
283C2×D71—42no—
284C2×D71—42no—
285C2×D71—44no—
286C2×D71—416no—
287C2×D71C2×#12318yes—
288C4×C71—24no—
289C4×C71—44no—
290C4×C71—42no—
291C4×C71—44no—
292Dic71—24no—
293Dic71—42no—
294Dic71—44no—
295Dic71—44no—
296C2×#2482—711nomaximal, Sylow
297C2×C71—44no—
298C2×C71#248218yescomplemented
299C2×C71C22×#12311yescharacteristic
300C2×C71C2×#52511yescharacteristic
301C2×C71C2×#53511yescharacteristic
302C2×C71C2×#55112yes—
303C2×C71C2×#55112yes—
304D71—28no—
305D71—44no—
306C22×#117—711no—
307C23×C22⋊C4—711no—
308#2482—7116no—
309C2×#399—711no—
310C2×#399—711no—
311C2×#397—711no—
312C2×#397D7111yescharacteristic, complemented
313C2×#388—712no—
314C2×#388—712no—
315C2×#385—712no—
316C2×#381—711no—
317C2×#379—712no—
318C71C2×#248211yescharacteristic, complemented, Sylow
319C26—711no—
320C24×C4—711no—
321C24×C4—711no—
322C2×#117—718no—
323C2×#117—1428no—
324C22×C4⋊C4—712no—
325C22×C4⋊C4—712no—
326C22×C4⋊C4—711no—
327C22×C4⋊C4C2×D7111yescharacteristic, complemented
328C22×C22⋊C4—712no—
329C22×C22⋊C4—712no—
330C22×C22⋊C4—711no—
331C22×C22⋊C4—711no—
332C22×C22⋊C4—712no—
333C22×C22⋊C4—711no—
334C22×C22⋊C4—712no—
335C22×C22⋊C4—712no—
336C22×C22⋊C4—718no—
337C22×C22⋊C4—1428no—
338C22×C22⋊C4C2×D7111yescharacteristic
339C22×C22⋊C4C2×D7112yes—
340C22×C42—712no—
341#399—718no—
342#399—718no—
343#397—718no—
344#397C2×D7118yescomplemented
345#388—7116no—
346#388—7116no—
347#385—7116no—
348#381—718no—
349#379—7116no—
350C2×#95—711no—
351C2×#95—712no—
352C2×#95—712no—
353C2×#95C2×D7111yescharacteristic, complemented
354C2×#95C2×D7112yescomplemented
355C25—2844no—
356C25—714no—
357C25—711no—
358C25—711no—
359C25—1428no—
360C25—1424no—
361C25—1424no—
362C25—1424no—
363C25C22×D7111yescharacteristic
364C23×C4—2844no—
365C23×C4—714no—
366C23×C4—711no—
367C23×C4—718no—
368C23×C4—714no—
369C23×C4—712no—
370C23×C4—712no—
371C23×C4—711no—
372C23×C4—1424no—
373C23×C4—1428no—
374C23×C4C22×D7112yes—
375C23×C4C22×D7112yes—
376C23×C4C22×D7111yescharacteristic
377C23×C4C22×D7111yescharacteristic
378#117—14232no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—2844no—
381C2×C4⋊C4—2844no—
382C2×C4⋊C4—718no—
383C2×C4⋊C4—718no—
384C2×C4⋊C4—714no—
385C2×C4⋊C4—1424no—
386C2×C4⋊C4C22×D7114yescomplemented
387C2×C22⋊C4—28no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—2844no—
390C2×C22⋊C4—2844no—
391C2×C22⋊C4—718no—
392C2×C22⋊C4—718no—
393C2×C22⋊C4—714no—
394C2×C22⋊C4—714no—
395C2×C22⋊C4—714no—
396C2×C22⋊C4—718no—
397C2×C22⋊C4—718no—
398C2×C22⋊C4—718no—
399C2×C22⋊C4—14232no—
400C2×C22⋊C4—1424no—
401C2×C22⋊C4—1428no—
402C2×C22⋊C4—1428no—
403C2×C22⋊C4—1428no—
404C2×C22⋊C4—1424no—
405C2×C22⋊C4—1424no—
406C2×C22⋊C4C22×D7114yes—
407C2×C22⋊C4C22×D7118yes—
408C2×C42—718no—
409C2×C42—1428no—
410#95—718no—
411#95—714no—
412#95—718no—
413#95C22×D7114yescomplemented
414#95C22×D7118yescomplemented
415C24—22no—
416C24—24no—
417C24—22no—
418C24—24no—
419C24—28432no—
420C24—2848no—
421C24—28416no—
422C24—28416no—
423C24—28432no—
424C24—714no—
425C24—718no—
426C24—712no—
427C24—714no—
428C24—712no—
429C24—14216no—
430C24—1428no—
431C24—1428no—
432C24—14232no—
433C24—1422no—
434C24—1424no—
435C24—1428no—
436C24—1422no—
437C24—1424no—
438C24—1422no—
439C24—1422no—
440C24—1424no—
441C24#1176586414yes—
442C24C23×D7112yes—
443C24C23×D7111yescharacteristic
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—2848no—
454C22×C4—28416no—
455C22×C4—714no—
456C22×C4—712no—
457C22×C4—7116no—
458C22×C4—714no—
459C22×C4—712no—
460C22×C4—714no—
461C22×C4—1428no—
462C22×C4—14216no—
463C22×C4—14232no—
464C22×C4—1424no—
465C22×C4—1422no—
466C22×C4—1424no—
467C22×C4—1424no—
468C22×C4—1424no—
469C22×C4—1424no—
470C22×C4—1424no—
471C22×C4—1424no—
472C22×C4—1424no—
473C22×C4D4×D7114yes—
474C22×C4C23×D7114yes—
475C22×C4C23×D7114yes—
476C22×C4C23×D7112yes—
477C22×C4C23×D7112yes—
478C4⋊C4—28no—
479C4⋊C4—2848no—
480C4⋊C4—2848no—
481C4⋊C4—1428no—
482C22⋊C4—216no—
483C22⋊C4—28no—
484C22⋊C4—2848no—
485C22⋊C4—2848no—
486C22⋊C4—1428no—
487C22⋊C4—14216no—
488C22⋊C4—14216no—
489C22⋊C4—14216no—
490C22⋊C4—1428no—
491C22⋊C4—1428no—
492C42—14216no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—216no—
498C23—28no—
499C23—44no—
500C23—48no—
501C23—44no—
502C23—28432no—
503C23—28416no—
504C23—284128no—
505C23—28416no—
506C23—28432no—
507C23—2844no—
508C23—2848no—
509C23—2844no—
510C23—714no—
511C23—718no—
512C23—7116no—
513C23—714no—
514C23—14232no—
515C23—1422no—
516C23—1424no—
517C23—1422no—
518C23—1424no—
519C23—14216no—
520C23—1428no—
521C23—1428no—
522C23—1422no—
523C23—1422no—
524C23—1424no—
525C23—14216no—
526C23—14216no—
527C23—1428no—
528C23C2×D4×D7112yes—
529C23#41×D7114yes—
530C23#41×D7112yes—
531C23C2×#1176586414yes—
532C23C2×#1176586412yes—
533C23#1455617414yes—
534C23C24×D7111yescharacteristic
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—28416no—
549C2×C4—2844no—
550C2×C4—2848no—
551C2×C4—2848no—
552C2×C4—714no—
553C2×C4—14232no—
554C2×C4—1424no—
555C2×C4—1422no—
556C2×C4—1424no—
557C2×C4—1424no—
558C2×C4—1424no—
559C2×C4—1424no—
560C2×C4—1424no—
561C2×C4—1424no—
562C2×C4—1424no—
563C2×C4—1428no—
564C2×C4C2×D4×D7114yes—
565C22—28no—
566C22—24no—
567C22—416no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—28464no—
572C22—28416no—
573C22—2842no—
574C22—2844no—
575C22—2842no—
576C22—718no—
577C22—1428no—
578C22—1424no—
579C22—14232no—
580C22—1428no—
581C22—1428no—
582C22—1424no—
583C22—1424no—
584C22#123×D7114yes—
585C22#1798650914yes—
586C22#1798656714yes—
587C22#1798667218yes—
588C22#1798667418yes—
589C22C22×D4×D7111yescharacteristic
590C22C2×#41×D7112yes—
591C22C2×#41×D7111yescharacteristic
592C22C22×#1176586411yescharacteristic
593C22C2×#1455617412yes—
594C4—24no—
595C4—44no—
596C4—44no—
597C4—42no—
598C4—2844no—
599C4—2842no—
600C4—2844no—
601C4—1424no—
602C2—44no—
603C2—2844no—
604C2—1428no—
605C2#2259260918yesdirect product factor
606C2C2×#123×D7111yescharacteristic
607C2C2×#1798650911yescharacteristic
608C2C2×#1798656711yescharacteristic
609C2C2×#1798667212yes—
610C2C2×#1798667412yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#397×C71C2
socleC24×C71C24
prime radicalC25×D71C22
hypercenterC2×#397D71
hypocenterC71C2×#2482
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×#14556086C2semidirect
3. #22592609C2direct (N × H)
4. C2×#17985936C2semidirect
5. C2×#17985935C2semidirect
6. C2×#17985925C2semidirect
7. C2×#17985892C2semidirect
8. C2×#17985888C2semidirect
9. C2×#17985873C2semidirect
10. C2×#17985861C2semidirect
12. C2×#397×C71C2semidirect
17. C2×#14556086C22semidirect
18. C22×C4⋊Dic71C22semidirect
19. C22×C4⋊Dic71C22semidirect
20. C22×Dic71⋊C4C22semidirect
31. C22×C4×Dic71C22semidirect
32. #17985936C22semidirect
33. #17985935C22semidirect
34. #17985925C22semidirect
35. #17985892C22semidirect
36. #17985888C22semidirect
37. #17985873C22semidirect
38. #17985861C22semidirect
40. C2×C71⋊#95C22semidirect
41. C2×C71⋊#95C22semidirect
42. C2×C71⋊#95C22semidirect
43. C22×C4⋊C4×C71C22semidirect
46. #397×C71C22semidirect
47. C2×#95×C71C22semidirect
48. C2×#95×C71C22semidirect
70. C2×C4⋊Dic71C23semidirect
71. C2×C4⋊Dic71C23semidirect
73. C2×Dic71⋊C4C23semidirect
92. C2×C4×Dic71C23semidirect
93. C71⋊#95C23semidirect
94. C71⋊#95C23semidirect
95. C71⋊#95C23semidirect
102. C2×C4⋊C4×C71C23semidirect
107. #95×C71C23semidirect
108. #95×C71C23semidirect
298. C2×C71#2482semidirect
312. C2×#397D71semidirect
318. C71C2×#2482semidirect
327. C22×C4⋊C4C2×D71semidirect
344. #397C2×D71semidirect
353. C2×#95C2×D71semidirect
354. C2×#95C2×D71semidirect
386. C2×C4⋊C4C22×D71semidirect
413. #95C22×D71semidirect
414. #95C22×D71semidirect
605. C2#22592609direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560862
C22×C22⋊C4×D712
#225926092
C2×#179859362
C2×#179859352
C2×#179859252
C2×#179858922
C2×#179858882
C2×#179858732
C2×#179858612
C2×#179858542
C2×#397×C712
C2×#248271