Finite Group Databasebeta

← C2×#22592644 #32166500

Subgroups — C2×#22592644 #32166500

Order 18176.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556110C212yesmaximal
2C22×C4×C22⋊D71C211yesmaximal, characteristic
3C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#22592644C2116yesmaximal, direct product factor
5C2×#17985933C212yesmaximal, complemented
6C2×#17985927C212yesmaximal, complemented
7C2×#17985898C211yesmaximal, characteristic, complemented
8C2×#17985895C211yesmaximal, characteristic, complemented
9C2×#17985861C212yesmaximal, complemented
10C2×#17985859C211yesmaximal, characteristic, complemented
11C2×#17985856C211yesmaximal, characteristic, complemented
12C2×#380×C71C211yesmaximal, characteristic, complemented
13C23×C22⋊D71C2211yescharacteristic
14C23×C4×D71C2212yes—
15C23×C4×D71C2211yescharacteristic
16C2×#14556110—216no—
17C2×#14556110C22116yes—
18C2×C4×C22⋊D71—28no—
19C2×C4×C22⋊D71C2218yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×C4⋊Dic71C2212yes—
23C22×Dic71⋊C4C2212yes—
24C22×Dic71⋊C4C2211yescharacteristic
25C22×Dic71⋊C4C2212yes—
26C2×C4⋊C4×D71—28no—
27C2×C4⋊C4×D71C2218yescomplemented
28C22×C22⋊Dic71C2211yescharacteristic
29C22×C22⋊Dic71C2212yes—
30C22×(C2×D71)⋊C4C2211yescharacteristic
31C22×(C2×D71)⋊C4C2212yes—
32C22×C4×Dic71C2211yescharacteristic
33#17985933C22116yescomplemented
34#17985927C22116yescomplemented
35#17985898C2218yescomplemented
36#17985895C2218yescomplemented
37#17985861C22116yescomplemented
38#17985859C2218yescomplemented
39#17985856C2218yescomplemented
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2212yes—
44C24×C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C22⋊C4×C71C2212yes—
48#380×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C24×D71C2311yescharacteristic
51C22×C22⋊D71—24no—
52C22×C22⋊D71—24no—
53C22×C22⋊D71D418yes—
54C22×C22⋊D71C2314yes—
55C23×Dic71C2312yes—
56C23×Dic71C2311yescharacteristic
57C23×Dic71C2312yes—
58C23×Dic71C2311yescharacteristic
59C23×Dic71C2311yescharacteristic
60C22×C4×D71—28no—
61C22×C4×D71—28no—
62C22×C4×D71—28no—
63C22×C4×D71—24no—
64C22×C4×D71—24no—
65C22×C4×D71D418yes—
66C22×C4×D71C2318yes—
67C22×C4×D71C2314yes—
68#14556110—264no—
69C4×C22⋊D71—232no—
70C2×C4⋊Dic71—28no—
71C2×C4⋊Dic71—24no—
72C2×C4⋊Dic71—24no—
73C2×C4⋊Dic71C2314yes—
74C2×C4⋊Dic71C2314yes—
75C2×C4⋊Dic71C2318yes—
76C2×Dic71⋊C4—28no—
77C2×Dic71⋊C4—24no—
78C2×Dic71⋊C4—28no—
79C2×Dic71⋊C4C2318yes—
80C2×Dic71⋊C4C2318yes—
81C2×Dic71⋊C4C2314yes—
82C4⋊C4×D71—232no—
83C2×C22⋊Dic71—24no—
84C2×C22⋊Dic71—28no—
85C2×C22⋊Dic71C2318yes—
86C2×C22⋊Dic71C2314yes—
87C2×(C2×D71)⋊C4—24no—
88C2×(C2×D71)⋊C4—28no—
89C2×(C2×D71)⋊C4C2318yes—
90C2×(C2×D71)⋊C4C2314yes—
91C2×C4×Dic71—24no—
92C2×C4×Dic71C2314yes—
93C71⋊#95C2318yes—
94C71⋊#95C2318yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C25×C71C2311yescharacteristic
98C23×C4×C71—24no—
99C23×C4×C71—24no—
100C23×C4×C71—24no—
101C23×C4×C71C2312yes—
102C23×C4×C71C2312yes—
103C23×C4×C71C2311yescharacteristic
104C23×C4×C71C2314yes—
105C23×C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71—24no—
107C2×C4⋊C4×C71—24no—
108C2×C4⋊C4×C71C2314yes—
109C2×C4⋊C4×C71C2314yes—
110C2×C22⋊C4×C71—28no—
111C2×C22⋊C4×C71C2318yes—
112#95×C71C2318yes—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71C2×D414yes—
118C23×D71C2×D414yes—
119C23×D71#4114yes—
120C23×D71C2412yes—
121C2×C22⋊D71—232no—
122C2×C22⋊D71—28no—
123C2×C22⋊D71—28no—
124C2×C22⋊D71C2×D4116yes—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—24no—
130C22×Dic71—24no—
131C22×Dic71—22no—
132C22×Dic71—24no—
133C22×Dic71—24no—
134C22×Dic71—24no—
135C22×Dic71—22no—
136C22×Dic71—22no—
137C22×Dic71—24no—
138C22×Dic71C2×D414yes—
139C22×Dic71C2×D414yes—
140C22×Dic71C2414yes—
141C22×Dic71C2412yes—
142C22×Dic71C2414yes—
143C22×Dic71C2412yes—
144C22×Dic71C2412yes—
145C2×C4×D71—232no—
146C2×C4×D71—28no—
147C2×C4×D71—28no—
148C2×C4×D71—216no—
149C2×C4×D71—216no—
150C2×C4×D71—216no—
151C2×C4×D71—432no—
152C2×C4×D71C2×D4116yes—
153C4⋊Dic71—28no—
154C4⋊Dic71—28no—
155C4⋊Dic71—216no—
156Dic71⋊C4—216no—
157Dic71⋊C4—28no—
158Dic71⋊C4—216no—
159C22⋊Dic71—216no—
160C22⋊Dic71—28no—
161(C2×D71)⋊C4—28no—
162(C2×D71)⋊C4—216no—
163C4×Dic71—28no—
164C24×C71—22no—
165C24×C71—22no—
166C24×C71—22no—
167C24×C71—22no—
168C24×C71—22no—
169C24×C71—22no—
170C24×C71C2×D414yes—
171C24×C71C2412yes—
172C24×C71C2411yescharacteristic
173C22×C4×C71—24no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71—22no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—24no—
182C22×C4×C71—28no—
183C22×C4×C71—28no—
184C22×C4×C71—28no—
185C22×C4×C71—416no—
186C22×C4×C71C2×D414yes—
187C22×C4×C71#4114yes—
188C22×C4×C71#4114yes—
189C22×C4×C71C2414yes—
190C22×C4×C71C2412yes—
191C22×C4×C71C2414yes—
192C22×C4×C71C2412yes—
193C4⋊C4×C71—28no—
194C4⋊C4×C71—28no—
195C22⋊C4×C71—216no—
196C23×C71—28no—
197C23×C71—28no—
198C23×C71—28no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—22no—
204C23×C71—22no—
205C23×C71—44no—
206C23×C71—44no—
207C23×C71—44no—
208C23×C71—44no—
209C23×C71C22×D414yes—
210C23×C71C22×D412yes—
211C23×C71C22×D412yes—
212C23×C71C2×#4112yes—
213C23×C71C2×#4112yes—
214C23×C71C2×#4112yes—
215C23×C712−1+412yes—
216C23×C712−1+412yes—
217C23×C71C2511yescharacteristic
218C22×D71—28no—
219C22×D71—28no—
220C22×D71—28no—
221C22×D71—28no—
222C22×D71—28no—
223C22×D71—28no—
224C22×D71—22no—
225C22×D71—22no—
226C22×D71—22no—
227C22×D71—22no—
228C22×D71#121116yes—
229C22×D71C22×D414yes—
230C22×D71C22×D414yes—
231C22×D71C2×#4114yes—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—22no—
236C2×C4×C71—24no—
237C2×C4×C71—22no—
238C2×C4×C71—22no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—48no—
242C2×C4×C71—44no—
243C2×C4×C71—48no—
244C2×C4×C71—416no—
245C2×C4×C71#123116yescomplemented
246C2×C4×C71C22×D414yes—
247C2×C4×C71C2×#4114yes—
248C2×C4×C71C2×#4114yes—
249C22⋊D71—232no—
250C2×Dic71—28no—
251C2×Dic71—28no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—22no—
263C2×Dic71—22no—
264C2×Dic71—24no—
265C2×Dic71—48no—
266C2×Dic71—44no—
267C2×Dic71—48no—
268C2×Dic71C22×D414yes—
269C2×Dic71C22×D414yes—
270C4×D71—232no—
271C4×D71—432no—
272C22×C71—24no—
273C22×C71—24no—
274C22×C71—24no—
275C22×C71—42no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71—42no—
279C22×C71—416no—
280C22×C71C2×#12114yes—
281C22×C71C2×#12314yes—
282C22×C71#53614yes—
283C22×C71#54714yes—
284C22×C71#54714yes—
285C22×C71#55214yes—
286C22×C71#55214yes—
287C22×C71C23×D411yescharacteristic
288C22×C71C23×D411yescharacteristic
289C22×C71C22×#4111yescharacteristic
290C22×C71C22×#4111yescharacteristic
291C22×C71C22×#4111yescharacteristic
292C22×C71C2×2−1+411yescharacteristic
293C22×C71C2×2−1+411yescharacteristic
294C2×D71—24no—
295C2×D71—24no—
296C2×D71—24no—
297C2×D71—24no—
298C2×D71—24no—
299C2×D71—24no—
300C2×D71—232no—
301C2×D71C2×#12118yes—
302C4×C71—44no—
303C4×C71—42no—
304C4×C71—44no—
305C4×C71C2×#12318yescomplemented
306Dic71—24no—
307Dic71—24no—
308Dic71—44no—
309Dic71—42no—
310Dic71—44no—
311C2×#2486—711nomaximal, Sylow
312C2×C71—44no—
313C2×C71#248618yescomplemented
314C2×C71C22×#12111yescharacteristic
315C2×C71C22×#12311yescharacteristic
316C2×C71C2×#53611yescharacteristic
317C2×C71C2×#54711yescharacteristic
318C2×C71C2×#54711yescharacteristic
319C2×C71C2×#55211yescharacteristic
320C2×C71C2×#55211yescharacteristic
321D71—28no—
322C22×#123—712no—
323C22×C4×D4—711no—
324C23×C4⋊C4—711no—
325#2486—7116no—
326C2×#398—712no—
327C2×#397—712no—
328C2×#389—711no—
329C2×#389—711no—
330C2×#381—712no—
331C2×#380—711no—
332C2×#380—711no—
333C2×#380D7111yescharacteristic, complemented
334C71C2×#248611yescharacteristic, complemented, Sylow
335C23×D4—711no—
336C24×C4—711no—
337C24×C4—712no—
338C24×C4C2×D7111yescharacteristic
339C2×#123—7116no—
340C2×#123—14216no—
341C2×C4×D4—718no—
342C2×C4×D4—1428no—
343C22×C4⋊C4—712no—
344C22×C4⋊C4—711no—
345C22×C4⋊C4—712no—
346C22×C4⋊C4—712no—
347C22×C4⋊C4—711no—
348C22×C4⋊C4—711no—
349C22×C4⋊C4—718no—
350C22×C4⋊C4—1428no—
351C22×C4⋊C4C2×D7111yescharacteristic
352C22×C4⋊C4C2×D7111yescharacteristic
353C22×C22⋊C4—711no—
354C22×C22⋊C4—712no—
355C22×C22⋊C4—711no—
356C22×C22⋊C4—712no—
357C22×C22⋊C4C2×D7112yes—
358C22×C42—711no—
359#398—7116no—
360#397—7116no—
361#389—718no—
362#389—718no—
363#381—7116no—
364#380—718no—
365#380—718no—
366#380C2×D7118yescomplemented
367C2×#95—712no—
368C2×#95—712no—
369C2×#95—712no—
370C2×#95—712no—
371C2×#95C2×D7112yes—
372C25—711no—
373C25C22×D7111yescharacteristic
374C22×D4—718no—
375C22×D4—714no—
376C22×D4—1424no—
377C22×D4—1424no—
378C23×C4—24no—
379C23×C4—24no—
380C23×C4—24no—
381C23×C4—711no—
382C23×C4—711no—
383C23×C4—718no—
384C23×C4—714no—
385C23×C4—712no—
386C23×C4—711no—
387C23×C4—712no—
388C23×C4—718no—
389C23×C4—1428no—
390C23×C4—1428no—
391C23×C4—1428no—
392C23×C4—1424no—
393C23×C4—1424no—
394C23×C4C22×D7114yes—
395C23×C4C22×D7112yes—
396C23×C4C22×D7111yescharacteristic
397C23×C4C22×D7112yes—
398C23×C4C22×D7111yescharacteristic
399#123—14264no—
400C4×D4—14232no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—714no—
404C2×C4⋊C4—718no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—718no—
407C2×C4⋊C4—718no—
408C2×C4⋊C4—714no—
409C2×C4⋊C4—14232no—
410C2×C4⋊C4—1428no—
411C2×C4⋊C4—1424no—
412C2×C4⋊C4—1428no—
413C2×C4⋊C4—1428no—
414C2×C4⋊C4—1424no—
415C2×C4⋊C4—1424no—
416C2×C4⋊C4C22×D7114yes—
417C2×C4⋊C4C22×D7114yes—
418C2×C22⋊C4—28no—
419C2×C22⋊C4—714no—
420C2×C22⋊C4—718no—
421C2×C22⋊C4—714no—
422C2×C22⋊C4—718no—
423C2×C22⋊C4—1424no—
424C2×C22⋊C4—1428no—
425C2×C22⋊C4—1424no—
426C2×C22⋊C4—1428no—
427C2×C22⋊C4C22×D7118yes—
428C2×C42—714no—
429C2×C42—1424no—
430#95—718no—
431#95—718no—
432#95—718no—
433#95—718no—
434#95C22×D7118yes—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—714no—
442C24—714no—
443C24—714no—
444C24—712no—
445C24—1422no—
446C24—1422no—
447C24—1422no—
448C24—1422no—
449C24D4×D7114yes—
450C24C23×D7112yes—
451C24C23×D7111yescharacteristic
452C2×D4—7116no—
453C2×D4—1428no—
454C2×D4—1428no—
455C2×D4—14232no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—416no—
469C22×C4—28432no—
470C22×C4—714no—
471C22×C4—714no—
472C22×C4—712no—
473C22×C4—712no—
474C22×C4—7116no—
475C22×C4—712no—
476C22×C4—714no—
477C22×C4—714no—
478C22×C4—14216no—
479C22×C4—14216no—
480C22×C4—14216no—
481C22×C4—1422no—
482C22×C4—1424no—
483C22×C4—1424no—
484C22×C4—1424no—
485C22×C4—1424no—
486C22×C4—1424no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1424no—
490C22×C4—1422no—
491C22×C4—1422no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1428no—
495C22×C4—1428no—
496C22×C4—14232no—
497C22×C4D4×D7114yes—
498C22×C4#1176586414yes—
499C22×C4#1176586514yes—
500C22×C4C23×D7112yes—
501C22×C4C23×D7114yes—
502C22×C4C23×D7114yes—
503C22×C4C23×D7112yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—14216no—
507C4⋊C4—1428no—
508C4⋊C4—14216no—
509C4⋊C4—14216no—
510C4⋊C4—1428no—
511C4⋊C4—1428no—
512C22⋊C4—216no—
513C22⋊C4—1428no—
514C22⋊C4—14216no—
515C22⋊C4—1428no—
516C22⋊C4—14216no—
517C42—1428no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—714no—
532C23—714no—
533C23—714no—
534C23—7116no—
535C23—1422no—
536C23—1422no—
537C23—1428no—
538C23—1422no—
539C23—1422no—
540C23—1428no—
541C23—1428no—
542C23—1428no—
543C23—1428no—
544C23—1428no—
545C23C2×D4×D7114yes—
546C23C2×D4×D7112yes—
547C23C2×D4×D7112yes—
548C23#41×D7112yes—
549C23C2×#1176586412yes—
550C23C2×#1176586512yes—
551C23#1455617512yes—
552C23#1455617612yes—
553C23C24×D7111yescharacteristic
554D4—14232no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—416no—
568C2×C4—28432no—
569C2×C4—2844no—
570C2×C4—2848no—
571C2×C4—2848no—
572C2×C4—714no—
573C2×C4—714no—
574C2×C4—1422no—
575C2×C4—1424no—
576C2×C4—1424no—
577C2×C4—1424no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4—1422no—
581C2×C4—1424no—
582C2×C4—1422no—
583C2×C4—1422no—
584C2×C4—1422no—
585C2×C4—1422no—
586C2×C4—1422no—
587C2×C4—1428no—
588C2×C4—1428no—
589C2×C4—14232no—
590C2×C4#14556110116yescomplemented
591C2×C4C2×D4×D7114yes—
592C2×C4C2×#1176586414yes—
593C2×C4C2×#1176586514yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—416no—
601C22—42no—
602C22—718no—
603C22—1424no—
604C22—1424no—
605C22—1424no—
606C22—1424no—
607C22—1424no—
608C22—14232no—
609C22—1424no—
610C22#121×D7114yes—
611C22C2×#1455611014yes—
612C22#1798657214yes—
613C22#1798664914yes—
614C22#1798665014yes—
615C22#1798668314yes—
616C22#1798668814yes—
617C22C22×D4×D7111yescharacteristic
618C22C22×D4×D7111yescharacteristic
619C22C2×#41×D7111yescharacteristic
620C22C22×#1176586411yescharacteristic
621C22C22×#1176586511yescharacteristic
622C22C2×#1455617511yescharacteristic
623C22C2×#1455617611yescharacteristic
624C4—44no—
625C4—44no—
626C4—42no—
627C4—2844no—
628C4—2844no—
629C4—2842no—
630C4—1424no—
631C4—1424no—
632C4C2×#1455611018yescomplemented
633C2—44no—
634C2—1428no—
635C2#2259264418yesdirect product factor
636C2C2×#121×D7111yescharacteristic
637C2C22×#1455611011yescharacteristic
638C2C2×#1798657211yescharacteristic
639C2C2×#1798664911yescharacteristic
640C2C2×#1798665011yescharacteristic
641C2C2×#1798668311yescharacteristic
642C2C2×#1798668811yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#380×C71C2
socleC24×C71C24
prime radicalC23×C22⋊D71C22
hypercenterC2×#380D71
hypocenterC71C2×#2486
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D71C2semidirect
4. #22592644C2direct (N × H)
5. C2×#17985933C2semidirect
6. C2×#17985927C2semidirect
7. C2×#17985898C2semidirect
8. C2×#17985895C2semidirect
9. C2×#17985861C2semidirect
10. C2×#17985859C2semidirect
11. C2×#17985856C2semidirect
12. C2×#380×C71C2semidirect
27. C2×C4⋊C4×D71C22semidirect
33. #17985933C22semidirect
34. #17985927C22semidirect
35. #17985898C22semidirect
36. #17985895C22semidirect
37. #17985861C22semidirect
38. #17985859C22semidirect
39. #17985856C22semidirect
48. #380×C71C22semidirect
245. C2×C4×C71#123semidirect
305. C4×C71C2×#123semidirect
313. C2×C71#2486semidirect
333. C2×#380D71semidirect
334. C71C2×#2486semidirect
366. #380C2×D71semidirect
590. C2×C4#14556110semidirect
632. C4C2×#14556110semidirect
635. C2#22592644direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561102
C22×C4×C22⋊D712
C22×C4⋊C4×D712
#225926442
C2×#179859332
C2×#179859272
C2×#179858982
C2×#179858952
C2×#179858612
C2×#179858592
C2×#179858562
C2×#380×C712
C2×#248671