Finite Group Databasebeta

← C2×#22591676 #32166144

Subgroups — C2×#22591676 #32166144

Order 18176.

5718 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765866C211yesmaximal, characteristic
2C22×#14556113C212yesmaximal, complemented
3C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#22591676C2116yesmaximal, direct product factor
5C2×#17985918C211yesmaximal, characteristic, complemented
6C2×#17985917C211yesmaximal, characteristic, complemented
7C2×#17985916C211yesmaximal, characteristic, complemented
8C2×#17985877C212yesmaximal, complemented
9C2×#17985873C212yesmaximal, complemented
10C2×#17985870C212yesmaximal, complemented
11C2×#17985861C211yesmaximal, characteristic, complemented
12C2×#395×C71C211yesmaximal, characteristic, complemented
13C22×#11765866—216no—
14C22×#11765866—28no—
15C22×#11765866C2218yes—
16C23×D284C2211yescharacteristic
17C23×D284C2212yes—
18C23×C4×D71C2211yescharacteristic
19C23×C4×D71C2211yescharacteristic
20C23×C4×D71C2212yes—
21C23×C4×D71C2211yescharacteristic
22C2×#14556113—216no—
23C2×#14556113C22116yescomplemented
24C22×C4⋊Dic71C2211yescharacteristic
25C22×Dic71⋊C4C2211yescharacteristic
26C22×Dic71⋊C4C2211yescharacteristic
27C22×Dic71⋊C4C2212yes—
28C2×C4⋊C4×D71—28no—
29C2×C4⋊C4×D71C2218yescomplemented
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2212yes—
33C22×(C2×D71)⋊C4C2212yes—
34C22×(C2×D71)⋊C4C2212yes—
35#17985918C2218yescomplemented
36#17985917C2218yescomplemented
37#17985916C2218yescomplemented
38#17985877C22116yescomplemented
39#17985873C22116yescomplemented
40#17985870C22116yescomplemented
41#17985861C2218yescomplemented
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2212yes—
44C2×C71⋊#95C2211yescharacteristic
45C23×Q8×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C4⋊C4×C71C2212yes—
48#395×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C2×#95×C71C2211yescharacteristic
51C24×D71C2312yes—
52C24×D71C2311yescharacteristic
53C2×#11765866—2128no—
54C2×#11765866—264no—
55C2×#11765866—232no—
56C22×D284—28no—
57C22×D284—28no—
58C22×D284—28no—
59C22×D284—24no—
60C22×D284D4116yes—
61C22×D284C2314yes—
62C22×D284C2318yes—
63C23×Dic71C2311yescharacteristic
64C23×Dic71C2312yes—
65C23×Dic71C2311yescharacteristic
66C23×Dic71C2311yescharacteristic
67C22×C4×D71—28no—
68C22×C4×D71—28no—
69C22×C4×D71—24no—
70C22×C4×D71—28no—
71C22×C4×D71—24no—
72C22×C4×D71—24no—
73C22×C4×D71—28no—
74C22×C4×D71—28no—
75C22×C4×D71—24no—
76C22×C4×D71D418yes—
77C22×C4×D71C2318yes—
78C22×C4×D71C2314yes—
79C22×C4×D71C2314yes—
80C22×C4×D71C2314yes—
81#14556113—264no—
82C2×C4⋊Dic71—24no—
83C2×C4⋊Dic71C2314yes—
84C2×Dic71⋊C4—24no—
85C2×Dic71⋊C4—24no—
86C2×Dic71⋊C4—28no—
87C2×Dic71⋊C4C2314yes—
88C2×Dic71⋊C4C2318yes—
89C2×Dic71⋊C4C2314yes—
90C4⋊C4×D71—232no—
91C2×(C2×D71)⋊C4—28no—
92C2×(C2×D71)⋊C4—28no—
93C2×(C2×D71)⋊C4—28no—
94C2×(C2×D71)⋊C4—28no—
95C2×(C2×D71)⋊C4—28no—
96C2×(C2×D71)⋊C4C2318yes—
97C2×(C2×D71)⋊C4C2318yes—
98C2×(C2×D71)⋊C4C2318yes—
99C2×(C2×D71)⋊C4C2318yes—
100C2×(C2×D71)⋊C4C2318yes—
101C71⋊#95C2318yes—
102C71⋊#95C2314yes—
103C71⋊#95C2318yes—
104C22×Q8×C71—28no—
105C22×Q8×C71—24no—
106C22×Q8×C71C2314yes—
107C23×C4×C71C2311yescharacteristic
108C23×C4×C71C2312yes—
109C23×C4×C71C2312yes—
110C23×C4×C71C2311yescharacteristic
111C23×C4×C71C2311yescharacteristic
112C2×C4⋊C4×C71—24no—
113C2×C4⋊C4×C71—28no—
114C2×C4⋊C4×C71C2318yes—
115C2×C4⋊C4×C71C2314yes—
116#95×C71C2314yes—
117#95×C71C2318yes—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—24no—
121C23×D71—24no—
122C23×D71—24no—
123C23×D71—24no—
124C23×D71—24no—
125C23×D71—24no—
126C23×D71—24no—
127C23×D71C2×D418yes—
128C23×D71C2×D414yes—
129C23×D71C2×D418yes—
130C23×D71C2414yes—
131C23×D71C2412yes—
132#11765866—2256no—
133C2×D284—232no—
134C2×D284—264no—
135C2×D284—216no—
136C2×D284—216no—
137C2×D284—216no—
138C2×D284—28no—
139C2×D284C2×D4132yes—
140C22×Dic71—24no—
141C22×Dic71—22no—
142C22×Dic71—22no—
143C22×Dic71—24no—
144C22×Dic71—22no—
145C22×Dic71—24no—
146C22×Dic71—24no—
147C22×Dic71—24no—
148C22×Dic71C2×D414yes—
149C22×Dic71#4114yescomplemented
150C22×Dic71C2412yes—
151C22×Dic71C2414yes—
152C22×Dic71C2412yes—
153C22×Dic71C2412yes—
154C2×C4×D71—232no—
155C2×C4×D71—264no—
156C2×C4×D71—232no—
157C2×C4×D71—216no—
158C2×C4×D71—216no—
159C2×C4×D71—28no—
160C2×C4×D71—216no—
161C2×C4×D71—28no—
162C2×C4×D71—216no—
163C2×C4×D71—28no—
164C2×C4×D71—28no—
165C2×C4×D71—216no—
166C2×C4×D71—416no—
167C2×C4×D71C2×D4116yes—
168C4⋊Dic71—28no—
169Dic71⋊C4—28no—
170Dic71⋊C4—216no—
171Dic71⋊C4—28no—
172(C2×D71)⋊C4—216no—
173(C2×D71)⋊C4—216no—
174(C2×D71)⋊C4—216no—
175(C2×D71)⋊C4—216no—
176(C2×D71)⋊C4—216no—
177C24×C71C2411yescharacteristic
178C2×Q8×C71—232no—
179C2×Q8×C71—216no—
180C2×Q8×C71—28no—
181C22×C4×C71—24no—
182C22×C4×C71—24no—
183C22×C4×C71—24no—
184C22×C4×C71—22no—
185C22×C4×C71—22no—
186C22×C4×C71—24no—
187C22×C4×C71—22no—
188C22×C4×C71—24no—
189C22×C4×C71—24no—
190C22×C4×C71—24no—
191C22×C4×C71C2×D418yes—
192C22×C4×C71C2×D414yes—
193C22×C4×C71#4114yescomplemented
194C22×C4×C71C2412yes—
195C22×C4×C71C2414yes—
196C22×C4×C71C2414yes—
197C22×C4×C71C2412yes—
198C22×C4×C71C2412yes—
199C4⋊C4×C71—28no—
200C4⋊C4×C71—216no—
201C23×C71C22×D412yes—
202C23×C71C22×D414yes—
203C23×C71C2×#4112yes—
204C23×C71C2×#4112yes—
205C23×C712−1+414yes—
206C23×C71C2511yescharacteristic
207C22×D71—28no—
208C22×D71—28no—
209C22×D71—216no—
210C22×D71—216no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—24no—
214C22×D71—216no—
215C22×D71—216no—
216C22×D71—216no—
217C22×D71—24no—
218C22×D71—24no—
219C22×D71—24no—
220C22×D71—24no—
221C22×D71—24no—
222C22×D71—24no—
223C22×D71—48no—
224C22×D71—48no—
225C22×D71—48no—
226C22×D71—48no—
227C22×D71#120116yes—
228C22×D71C22×D418yes—
229C22×D71C22×D418yes—
230C22×D71C22×D414yes—
231Q8×C71—232no—
232C2×C4×C71—216no—
233C2×C4×C71—28no—
234C2×C4×C71—28no—
235C2×C4×C71—22no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—22no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71—22no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—44no—
246C2×C4×C71—48no—
247C2×C4×C71C22×D418yes—
248C2×C4×C71C22×D414yes—
249C2×C4×C71C2×#4114yescomplemented
250C2×Dic71—28no—
251C2×Dic71—28no—
252C2×Dic71—22no—
253C2×Dic71—24no—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—22no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—44no—
261C2×Dic71—48no—
262C2×Dic71C22×D414yes—
263C2×Dic71C2×#4114yescomplemented
264D284—232no—
265D284—264no—
266C4×D71—232no—
267C4×D71—264no—
268C4×D71—232no—
269C4×D71—416no—
270C22×C71C2×#12014yes—
271C22×C71#52514yes—
272C22×C71#53614yes—
273C22×C71#54718yes—
274C22×C71#54718yes—
275C22×C71C23×D411yescharacteristic
276C22×C71C23×D412yes—
277C22×C71C22×#4111yescharacteristic
278C22×C71C22×#4111yescharacteristic
279C22×C71C2×2−1+412yes—
280C2×D71—232no—
281C2×D71—24no—
282C2×D71—24no—
283C2×D71—28no—
284C2×D71—28no—
285C2×D71—28no—
286C2×D71—28no—
287C2×D71—28no—
288C2×D71—432no—
289C2×D71—44no—
290C2×D71—44no—
291C2×D71—44no—
292C2×D71—44no—
293C2×D71C2×#12018yes—
294C4×C71—28no—
295C4×C71—24no—
296C4×C71—24no—
297C4×C71—42no—
298C4×C71—44no—
299Dic71—24no—
300Dic71—24no—
301Dic71—42no—
302Dic71—44no—
303C2×#2406—711nomaximal, Sylow
304C2×C71#240618yescomplemented
305C2×C71C22×#12011yescharacteristic
306C2×C71C2×#52511yescharacteristic
307C2×C71C2×#53611yescharacteristic
308C2×C71C2×#54712yes—
309C2×C71C2×#54712yes—
310D71—28no—
311D71—48no—
312C23×#41—711no—
313C22×#123—712no—
314C23×C4⋊C4—711no—
315#2406—7116no—
316C2×#395—711no—
317C2×#395D7111yescharacteristic, complemented
318C2×#394—711no—
319C2×#394—711no—
320C2×#386—712no—
321C2×#385—712no—
322C2×#385—712no—
323C2×#381—711no—
324C71C2×#240611yescharacteristic, complemented, Sylow
325C22×#41—718no—
326C22×#41—1428no—
327C22×#41—14216no—
328C23×Q8C2×D7111yescharacteristic
329C23×D4—712no—
330C23×D4—711no—
331C24×C4—711no—
332C24×C4—712no—
333C24×C4—711no—
334C24×C4—711no—
335C2×#123—7116no—
336C2×#123—14216no—
337C22×C4⋊C4—711no—
338C22×C4⋊C4—712no—
339C22×C4⋊C4—711no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4—718no—
342C22×C4⋊C4—1428no—
343C22×C4⋊C4C2×D7111yescharacteristic
344C22×C4⋊C4C2×D7112yes—
345C22×C22⋊C4—712no—
346C22×C22⋊C4—712no—
347C22×C22⋊C4—712no—
348C22×C22⋊C4—712no—
349C22×C22⋊C4—712no—
350#395—718no—
351#395C2×D7118yescomplemented
352#394—718no—
353#394—718no—
354#386—7116no—
355#385—7116no—
356#385—7116no—
357#381—718no—
358C2×#95—711no—
359C2×#95—712no—
360C2×#95—712no—
361C2×#95C2×D7111yescharacteristic
362C2×#95C2×D7112yes—
363C25—711no—
364C25—712no—
365C2×#41—14232no—
366C2×#41—14264no—
367C2×#41—142128no—
368C22×Q8—24no—
369C22×Q8—28no—
370C22×Q8C22×D7114yes—
371C22×D4—7116no—
372C22×D4—718no—
373C22×D4—714no—
374C22×D4—1428no—
375C22×D4—1428no—
376C22×D4—1428no—
377C22×D4—1424no—
378C23×C4—711no—
379C23×C4—711no—
380C23×C4—718no—
381C23×C4—714no—
382C23×C4—712no—
383C23×C4—711no—
384C23×C4—714no—
385C23×C4—714no—
386C23×C4—718no—
387C23×C4—1424no—
388C23×C4—1428no—
389C23×C4—1424no—
390C23×C4—1428no—
391C23×C4—1424no—
392C23×C4—1428no—
393C23×C4—1428no—
394C23×C4—1428no—
395C23×C4—1424no—
396C23×C4C22×D7112yes—
397C23×C4C22×D7111yescharacteristic
398C23×C4C22×D7111yescharacteristic
399C23×C4C22×D7112yes—
400C23×C4C22×D7111yescharacteristic
401#123—14264no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—28no—
404C2×C4⋊C4—714no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—714no—
407C2×C4⋊C4—718no—
408C2×C4⋊C4—14232no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1424no—
411C2×C4⋊C4—1424no—
412C2×C4⋊C4—1428no—
413C2×C4⋊C4C22×D7114yes—
414C2×C4⋊C4C22×D7118yes—
415C2×C22⋊C4—718no—
416C2×C22⋊C4—718no—
417C2×C22⋊C4—718no—
418C2×C22⋊C4—718no—
419C2×C22⋊C4—718no—
420C2×C22⋊C4—1428no—
421C2×C22⋊C4—1428no—
422C2×C22⋊C4—1428no—
423C2×C22⋊C4—1428no—
424C2×C22⋊C4—1428no—
425#95—718no—
426#95—714no—
427#95—718no—
428#95C22×D7114yes—
429#95C22×D7118yes—
430C24—718no—
431C24—714no—
432C24—712no—
433C24—718no—
434C24—714no—
435C24—1424no—
436C24—1424no—
437C24—1424no—
438C24—1424no—
439C24—1424no—
440C24—1424no—
441C24—1422no—
442C24—1422no—
443C24—1424no—
444C24C23×D7111yescharacteristic
445#41—142256no—
446C2×Q8—232no—
447C2×Q8—28no—
448C2×Q8—216no—
449C2×D4—7132no—
450C2×D4—14216no—
451C2×D4—14216no—
452C2×D4—14216no—
453C2×D4—14264no—
454C2×D4—14232no—
455C2×D4—1428no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—28416no—
467C22×C4—714no—
468C22×C4—714no—
469C22×C4—712no—
470C22×C4—712no—
471C22×C4—7116no—
472C22×C4—714no—
473C22×C4—712no—
474C22×C4—1428no—
475C22×C4—14216no—
476C22×C4—1428no—
477C22×C4—14216no—
478C22×C4—1428no—
479C22×C4—14216no—
480C22×C4—14216no—
481C22×C4—14232no—
482C22×C4—14232no—
483C22×C4—14264no—
484C22×C4—1424no—
485C22×C4—1424no—
486C22×C4—1422no—
487C22×C4—1424no—
488C22×C4—1424no—
489C22×C4—1422no—
490C22×C4—1424no—
491C22×C4—1422no—
492C22×C4—14216no—
493C22×C4—1428no—
494C22×C4D4×D7114yes—
495C22×C4D4×D7118yes—
496C22×C4#1176586614yescomplemented
497C22×C4C23×D7114yes—
498C22×C4C23×D7112yes—
499C22×C4C23×D7112yes—
500C22×C4C23×D7114yes—
501C22×C4C23×D7112yes—
502C4⋊C4—28no—
503C4⋊C4—216no—
504C4⋊C4—1428no—
505C4⋊C4—1428no—
506C4⋊C4—1428no—
507C4⋊C4—14216no—
508C22⋊C4—14216no—
509C22⋊C4—14216no—
510C22⋊C4—14216no—
511C22⋊C4—14216no—
512C22⋊C4—14216no—
513C23—2848no—
514C23—2848no—
515C23—2848no—
516C23—2848no—
517C23—718no—
518C23—714no—
519C23—7116no—
520C23—718no—
521C23—1424no—
522C23—1424no—
523C23—1424no—
524C23—1424no—
525C23—1424no—
526C23—1424no—
527C23—14216no—
528C23—14216no—
529C23—14216no—
530C23—1422no—
531C23—1422no—
532C23—1428no—
533C23—1424no—
534C23—14216no—
535C23—14216no—
536C23—1428no—
537C23C2×D4×D7112yes—
538C23C2×D4×D7114yes—
539C23#41×D7112yes—
540C23C2×#1176586612yes—
541C23#1455617614yes—
542C23C24×D7111yescharacteristic
543Q8—232no—
544D4—14264no—
545D4—14232no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—28416no—
562C2×C4—2848no—
563C2×C4—2844no—
564C2×C4—714no—
565C2×C4—714no—
566C2×C4—14232no—
567C2×C4—14264no—
568C2×C4—14232no—
569C2×C4—1424no—
570C2×C4—1424no—
571C2×C4—1422no—
572C2×C4—1424no—
573C2×C4—1424no—
574C2×C4—1422no—
575C2×C4—1424no—
576C2×C4—1422no—
577C2×C4—1428no—
578C2×C4—1428no—
579C2×C4C2×D4×D7114yes—
580C2×C4C2×D4×D7118yes—
581C2×C4C2×#1176586614yescomplemented
582C22—2844no—
583C22—2844no—
584C22—2844no—
585C22—2844no—
586C22—28432no—
587C22—718no—
588C22—1428no—
589C22—1428no—
590C22—1428no—
591C22—1428no—
592C22—1428no—
593C22—1424no—
594C22—1424no—
595C22—14232no—
596C22#120×D7114yes—
597C22#1798651214yes—
598C22#1798657414yes—
599C22#1798665118yes—
600C22#1798665218yes—
601C22C22×D4×D7112yes—
602C22C22×D4×D7111yescharacteristic
603C22C2×#41×D7111yescharacteristic
604C22C22×#1176586611yescharacteristic
605C22C2×#1455617612yes—
606C4—24no—
607C4—28no—
608C4—24no—
609C4—44no—
610C4—42no—
611C4—2844no—
612C4—2842no—
613C4—1424no—
614C4—1424no—
615C2—2848no—
616C2—1428no—
617C2#2259167618yesdirect product factor
618C2C2×#120×D7111yescharacteristic
619C2C2×#1798651211yescharacteristic
620C2C2×#1798657411yescharacteristic
621C2C2×#1798665112yes—
622C2C2×#1798665212yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#395×C71C2
socleC24×C71C24
prime radicalC23×#11765866C2
hypercenterC2×#395D71
hypocenterC71C2×#2406
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C22×#14556113C2semidirect
3. C22×C4⋊C4×D71C2semidirect
4. #22591676C2direct (N × H)
5. C2×#17985918C2semidirect
6. C2×#17985917C2semidirect
7. C2×#17985916C2semidirect
8. C2×#17985877C2semidirect
9. C2×#17985873C2semidirect
10. C2×#17985870C2semidirect
11. C2×#17985861C2semidirect
12. C2×#395×C71C2semidirect
23. C2×#14556113C22semidirect
29. C2×C4⋊C4×D71C22semidirect
35. #17985918C22semidirect
36. #17985917C22semidirect
37. #17985916C22semidirect
38. #17985877C22semidirect
39. #17985873C22semidirect
40. #17985870C22semidirect
41. #17985861C22semidirect
48. #395×C71C22semidirect
149. C22×Dic71#41semidirect
193. C22×C4×C71#41semidirect
249. C2×C4×C71C2×#41semidirect
263. C2×Dic71C2×#41semidirect
304. C2×C71#2406semidirect
317. C2×#395D71semidirect
324. C71C2×#2406semidirect
351. #395C2×D71semidirect
496. C22×C4#11765866semidirect
581. C2×C4C2×#11765866semidirect
617. C2#22591676direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658662
C22×#145561132
C22×C4⋊C4×D712
#225916762
C2×#179859182
C2×#179859172
C2×#179859162
C2×#179858772
C2×#179858732
C2×#179858702
C2×#179858612
C2×#395×C712
C2×#240671