Finite Group Databasebeta

← C2×#22592640 #32166972

Subgroups — C2×#22592640 #32166972

Order 18176.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D284C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic, complemented
3#22592640C2116yesmaximal, direct product factor
4C2×#17985925C212yesmaximal, complemented
5C2×#17985917C212yesmaximal
6C2×#17985897C211yesmaximal, characteristic, complemented
7C2×#17985889C212yesmaximal, complemented
8C2×#17985886C212yesmaximal, complemented
9C2×#17985856C211yesmaximal, characteristic, complemented
10C2×#17985856C211yesmaximal, characteristic, complemented
11C2×#17985856C211yesmaximal, characteristic, complemented
12C2×#389×C71C211yesmaximal, characteristic, complemented
13C23×D284C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C2×C4×D284—28no—
17C2×C4×D284C2218yes—
18C2×#14556086—28no—
19C2×#14556086C2218yescomplemented
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×C4⋊Dic71C2211yescharacteristic
23C22×Dic71⋊C4C2212yes—
24C22×(C2×D71)⋊C4C2212yes—
25C22×(C2×D71)⋊C4C2211yescharacteristic
26C22×(C2×D71)⋊C4C2211yescharacteristic
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×C4×Dic71C2212yes—
31#17985925C22116yescomplemented
32#17985917C22116yes—
33#17985897C2218yescomplemented
34#17985889C22116yescomplemented
35#17985886C22116yescomplemented
36#17985856C2218yescomplemented
37#17985856C2218yescomplemented
38#17985856C2218yescomplemented
39C2×C71⋊#95C2212yes—
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2212yes—
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2212yes—
46C22×C42×C71C2211yescharacteristic
47#389×C71C2218yescomplemented
48C2×#95×C71C2212yes—
49C24×D71C2311yescharacteristic
50C24×D71C2311yescharacteristic
51C22×D284—24no—
52C22×D284—24no—
53C22×D284—24no—
54C22×D284C2314yes—
55C23×Dic71C2311yescharacteristic
56C23×Dic71C2312yes—
57C23×Dic71C2312yes—
58C23×Dic71C2311yescharacteristic
59C22×C4×D71—24no—
60C22×C4×D71—24no—
61C22×C4×D71—24no—
62C22×C4×D71—24no—
63C22×C4×D71—24no—
64C22×C4×D71D418yes—
65C22×C4×D71C2314yes—
66C22×C4×D71C2314yes—
67C4×D284—232no—
68#14556086—232no—
69C2×C4⋊Dic71—24no—
70C2×C4⋊Dic71—24no—
71C2×C4⋊Dic71—24no—
72C2×C4⋊Dic71C2314yes—
73C2×C4⋊Dic71C2314yes—
74C2×C4⋊Dic71C2314yes—
75C2×Dic71⋊C4—28no—
76C2×Dic71⋊C4C2318yes—
77C2×(C2×D71)⋊C4—28no—
78C2×(C2×D71)⋊C4—24no—
79C2×(C2×D71)⋊C4—28no—
80C2×(C2×D71)⋊C4—24no—
81C2×(C2×D71)⋊C4—44no—
82C2×(C2×D71)⋊C4—44no—
83C2×(C2×D71)⋊C4C2318yes—
84C2×(C2×D71)⋊C4C2314yes—
85C2×(C2×D71)⋊C4C2318yes—
86C2×(C2×D71)⋊C4C2314yes—
87C2×(C2×D71)⋊C4C2318yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×C4×Dic71—28no—
90C2×C4×Dic71C2318yes—
91C71⋊#95C2318yes—
92C71⋊#95C2318yes—
93C71⋊#95C2318yes—
94C71⋊#95C2318yes—
95C23×C4×C71C2311yescharacteristic
96C23×C4×C71C2312yes—
97C23×C4×C71C2312yes—
98C23×C4×C71C2311yescharacteristic
99C23×C4×C71C2311yescharacteristic
100C2×C4⋊C4×C71—24no—
101C2×C4⋊C4×C71—24no—
102C2×C4⋊C4×C71—44no—
103C2×C4⋊C4×C71C2314yes—
104C2×C4⋊C4×C71C2314yes—
105C2×C4⋊C4×C71C2318yes—
106C2×C42×C71—24no—
107C2×C42×C71C2314yes—
108#95×C71C2318yes—
109C23×D71—22no—
110C23×D71—22no—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—22no—
121C23×D71C2×D414yes—
122C23×D71#4114yes—
123C23×D71C2412yes—
124C23×D71C2412yes—
125C2×D284—232no—
126C2×D284—28no—
127C2×D284—28no—
128C2×D284—28no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—24no—
133C22×Dic71—22no—
134C22×Dic71—22no—
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—28no—
147C2×C4×D71—28no—
148C2×C4×D71—28no—
149C2×C4×D71—28no—
150C2×C4×D71—28no—
151C2×C4×D71—416no—
152C2×C4×D71C2×D4116yes—
153C4⋊Dic71—28no—
154C4⋊Dic71—28no—
155C4⋊Dic71—28no—
156Dic71⋊C4—216no—
157(C2×D71)⋊C4—216no—
158(C2×D71)⋊C4—28no—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—28no—
161(C2×D71)⋊C4—48no—
162(C2×D71)⋊C4—48no—
163C4×Dic71—216no—
164C24×C71C2411yescharacteristic
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—24no—
169C22×C4×C71—24no—
170C22×C4×C71—24no—
171C22×C4×C71—22no—
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—22no—
175C22×C4×C71—24no—
176C22×C4×C71C2×D414yes—
177C22×C4×C71#4114yes—
178C22×C4×C71#4114yes—
179C22×C4×C71#4114yes—
180C22×C4×C71C2414yes—
181C22×C4×C71C2414yes—
182C22×C4×C71C2412yes—
183C22×C4×C71C2412yes—
184C22×C4×C71C2412yes—
185C4⋊C4×C71—28no—
186C4⋊C4×C71—28no—
187C4⋊C4×C71—48no—
188C42×C71—28no—
189C23×C71C22×D412yes—
190C23×C71C2×#4112yes—
191C23×C71C2×#4112yes—
192C23×C71C2×#4112yes—
193C23×C71C2×#4112yes—
194C23×C71H(5,𝔽2)12yes—
195C23×C712−1+412yes—
196C23×C71C2511yescharacteristic
197C22×D71—28no—
198C22×D71—28no—
199C22×D71—28no—
200C22×D71—28no—
201C22×D71—28no—
202C22×D71—28no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—22no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—44no—
216C22×D71—44no—
217C22×D71—44no—
218C22×D71—44no—
219C22×D71—44no—
220C22×D71—44no—
221C22×D71—44no—
222C22×D71—44no—
223C22×D71C22×D414yes—
224C22×D71C2×#4114yes—
225C2×C4×C71—28no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—22no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—22no—
237C2×C4×C71—44no—
238C2×C4×C71—48no—
239C2×C4×C71—48no—
240C2×C4×C71#123116yes—
241C2×C4×C71C22×D414yes—
242C2×C4×C71C2×#4114yes—
243C2×C4×C71C2×#4114yes—
244C2×C4×C71C2×#4114yes—
245C2×Dic71—28no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—24no—
249C2×Dic71—22no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—24no—
254C2×Dic71—22no—
255C2×Dic71—22no—
256C2×Dic71—24no—
257C2×Dic71—44no—
258C2×Dic71—48no—
259C2×Dic71—48no—
260C2×Dic71C22×D414yes—
261D284—232no—
262C4×D71—232no—
263C4×D71—416no—
264C22×C71C2×#12314yes—
265C22×C71#52514yes—
266C22×C71#53714yes—
267C22×C71#55214yes—
268C22×C71#55314yes—
269C22×C71#55514yes—
270C22×C71#55614yes—
271C22×C71C23×D411yescharacteristic
272C22×C71C22×#4111yescharacteristic
273C22×C71C22×#4111yescharacteristic
274C22×C71C22×#4111yescharacteristic
275C22×C71C22×#4111yescharacteristic
276C22×C71C2×H(5,𝔽2)11yescharacteristic
277C22×C71C2×2−1+411yescharacteristic
278C2×D71—24no—
279C2×D71—24no—
280C2×D71—24no—
281C2×D71—24no—
282C2×D71—24no—
283C2×D71—24no—
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71—42no—
289C2×D71—42no—
290C2×D71—42no—
291C2×D71—42no—
292C2×D71—416no—
293C2×D71—416no—
294C4×C71—24no—
295C4×C71—42no—
296C4×C71—44no—
297C4×C71—44no—
298C4×C71C2×#12318yes—
299Dic71—24no—
300Dic71—42no—
301Dic71—44no—
302Dic71—44no—
303C2×#2485—711nomaximal, Sylow
304C2×C71#248518yescomplemented
305C2×C71C22×#12311yescharacteristic
306C2×C71C2×#52511yescharacteristic
307C2×C71C2×#53711yescharacteristic
308C2×C71C2×#55211yescharacteristic
309C2×C71C2×#55311yescharacteristic
310C2×C71C2×#55511yescharacteristic
311C2×C71C2×#55611yescharacteristic
312D71—44no—
313D71—44no—
314C22×C4×D4—711no—
315C22×#117—711no—
316#2485—7116no—
317C2×#397—712no—
318C2×#394—712no—
319C2×#389—711no—
320C2×#389D7111yescharacteristic, complemented
321C2×#388—712no—
322C2×#387—712no—
323C2×#380—711no—
324C2×#380—711no—
325C2×#380—711no—
326C71C2×#248511yescharacteristic, complemented, Sylow
327C23×D4—711no—
328C24×C4—711no—
329C24×C4—711no—
330C2×C4×D4—718no—
331C2×C4×D4—1428no—
332C2×#117—718no—
333C2×#117—1428no—
334C22×C4⋊C4—712no—
335C22×C4⋊C4—711no—
336C22×C4⋊C4—711no—
337C22×C4⋊C4—711no—
338C22×C4⋊C4C2×D7111yescharacteristic
339C22×C4⋊C4C2×D7111yescharacteristic
340C22×C4⋊C4C2×D7112yes—
341C22×C22⋊C4—712no—
342C22×C22⋊C4—712no—
343C22×C22⋊C4—712no—
344C22×C22⋊C4—712no—
345C22×C22⋊C4—711no—
346C22×C22⋊C4—711no—
347C22×C42—712no—
348C22×C42C2×D7111yescharacteristic
349#397—7116no—
350#394—7116no—
351#389—718no—
352#389C2×D7118yescomplemented
353#388—7116no—
354#387—7116no—
355#380—718no—
356#380—718no—
357#380—718no—
358C2×#95—712no—
359C2×#95—712no—
360C2×#95—712no—
361C2×#95—712no—
362C2×#95C2×D7112yes—
363C25—711no—
364C25—711no—
365C22×D4—714no—
366C22×D4—1424no—
367C22×D4—1424no—
368C22×D4—1424no—
369C23×C4—711no—
370C23×C4—718no—
371C23×C4—714no—
372C23×C4—712no—
373C23×C4—712no—
374C23×C4—711no—
375C23×C4—714no—
376C23×C4—1424no—
377C23×C4—1424no—
378C23×C4—1424no—
379C23×C4—1424no—
380C23×C4—1424no—
381C23×C4C22×D7112yes—
382C23×C4C22×D7111yescharacteristic
383C23×C4C22×D7112yes—
384C23×C4C22×D7111yescharacteristic
385C23×C4C22×D7111yescharacteristic
386C4×D4—14232no—
387#117—14232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—44no—
391C2×C4⋊C4—714no—
392C2×C4⋊C4—714no—
393C2×C4⋊C4—714no—
394C2×C4⋊C4—718no—
395C2×C4⋊C4—1424no—
396C2×C4⋊C4—1424no—
397C2×C4⋊C4—1428no—
398C2×C4⋊C4—1424no—
399C2×C4⋊C4C22×D7114yes—
400C2×C4⋊C4C22×D7114yes—
401C2×C4⋊C4C22×D7118yes—
402C2×C22⋊C4—2844no—
403C2×C22⋊C4—2844no—
404C2×C22⋊C4—718no—
405C2×C22⋊C4—718no—
406C2×C22⋊C4—718no—
407C2×C22⋊C4—714no—
408C2×C22⋊C4—714no—
409C2×C22⋊C4—718no—
410C2×C22⋊C4—1424no—
411C2×C22⋊C4—1428no—
412C2×C22⋊C4—1424no—
413C2×C22⋊C4—1428no—
414C2×C42—24no—
415C2×C42—718no—
416C2×C42—1428no—
417C2×C42C22×D7114yes—
418#95—718no—
419#95—718no—
420#95—718no—
421#95—718no—
422#95C22×D7118yes—
423C24—714no—
424C24—714no—
425C24—712no—
426C24—712no—
427C24—1422no—
428C24—1422no—
429C24—1422no—
430C24—1422no—
431C24—1422no—
432C24—1422no—
433C24—1422no—
434C24—1422no—
435C24—1422no—
436C24—1422no—
437C24—1422no—
438C24—1422no—
439C24C23×D7111yescharacteristic
440C2×D4—1428no—
441C2×D4—1428no—
442C2×D4—14232no—
443C2×D4—1428no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—28416no—
456C22×C4—714no—
457C22×C4—712no—
458C22×C4—7116no—
459C22×C4—712no—
460C22×C4—714no—
461C22×C4—714no—
462C22×C4—1424no—
463C22×C4—1422no—
464C22×C4—1424no—
465C22×C4—1422no—
466C22×C4—1424no—
467C22×C4—1424no—
468C22×C4—1424no—
469C22×C4—1422no—
470C22×C4—1424no—
471C22×C4—1428no—
472C22×C4—1428no—
473C22×C4—1428no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1428no—
477C22×C4—1428no—
478C22×C4—14232no—
479C22×C4D4×D7114yes—
480C22×C4#1176586514yes—
481C22×C4#1176586614yes—
482C22×C4#1176586614yes—
483C22×C4C23×D7114yes—
484C22×C4C23×D7112yes—
485C22×C4C23×D7114yes—
486C22×C4C23×D7112yes—
487C22×C4C23×D7112yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—1428no—
492C4⋊C4—1428no—
493C4⋊C4—14216no—
494C4⋊C4—1428no—
495C22⋊C4—2848no—
496C22⋊C4—2848no—
497C22⋊C4—1428no—
498C22⋊C4—14216no—
499C22⋊C4—14216no—
500C22⋊C4—1428no—
501C42—28no—
502C42—14216no—
503C23—2844no—
504C23—2844no—
505C23—2844no—
506C23—2844no—
507C23—2844no—
508C23—2844no—
509C23—2844no—
510C23—2844no—
511C23—714no—
512C23—714no—
513C23—1422no—
514C23—1422no—
515C23—1422no—
516C23—1422no—
517C23—1422no—
518C23—1422no—
519C23—1422no—
520C23—1422no—
521C23—1422no—
522C23—1422no—
523C23—1422no—
524C23—1422no—
525C23—1428no—
526C23—1428no—
527C23—1428no—
528C23—1428no—
529C23—1428no—
530C23—1428no—
531C23C2×D4×D7112yes—
532C23#41×D7112yes—
533C23C2×#1176586512yes—
534C23C2×#1176586612yes—
535C23C2×#1176586612yes—
536C23#1455617312yes—
537C23#1455617512yes—
538C23C24×D7111yescharacteristic
539D4—14232no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—2848no—
556C2×C4—2844no—
557C2×C4—2848no—
558C2×C4—28416no—
559C2×C4—714no—
560C2×C4—1424no—
561C2×C4—1422no—
562C2×C4—1424no—
563C2×C4—1422no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1424no—
567C2×C4—1422no—
568C2×C4—1424no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1428no—
572C2×C4—14232no—
573C2×C4#14556113116yes—
574C2×C4C2×D4×D7114yes—
575C2×C4C2×#1176586514yes—
576C2×C4C2×#1176586614yes—
577C2×C4C2×#1176586614yes—
578C22—2842no—
579C22—2842no—
580C22—28416no—
581C22—28416no—
582C22—2842no—
583C22—2842no—
584C22—2842no—
585C22—2842no—
586C22—2842no—
587C22—2842no—
588C22—1424no—
589C22—1424no—
590C22—1424no—
591C22—1424no—
592C22—1424no—
593C22—1424no—
594C22C2×#1455611314yes—
595C22#1798651214yes—
596C22#1798657814yes—
597C22#1798668714yes—
598C22#1798669914yes—
599C22#1798671514yes—
600C22#1798672214yes—
601C22C22×D4×D7111yescharacteristic
602C22C2×#41×D7111yescharacteristic
603C22C22×#1176586511yescharacteristic
604C22C22×#1176586611yescharacteristic
605C22C22×#1176586611yescharacteristic
606C22C2×#1455617311yescharacteristic
607C22C2×#1455617511yescharacteristic
608C4—24no—
609C4—44no—
610C4—42no—
611C4—44no—
612C4—2842no—
613C4—2844no—
614C4—2844no—
615C4—1424no—
616C4C2×#1455611318yes—
617C2—2844no—
618C2—2844no—
619C2#2259264018yesdirect product factor
620C2C22×#1455611311yescharacteristic
621C2C2×#1798651211yescharacteristic
622C2C2×#1798657811yescharacteristic
623C2C2×#1798668711yescharacteristic
624C2C2×#1798669911yescharacteristic
625C2C2×#1798671511yescharacteristic
626C2C2×#1798672211yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#389×C71C2
socleC24×C71C24
prime radicalC23×D284C22
hypercenterC2×#389D71
hypocenterC71C2×#2485
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×#14556086C2semidirect
3. #22592640C2direct (N × H)
4. C2×#17985925C2semidirect
6. C2×#17985897C2semidirect
7. C2×#17985889C2semidirect
8. C2×#17985886C2semidirect
9. C2×#17985856C2semidirect
10. C2×#17985856C2semidirect
11. C2×#17985856C2semidirect
12. C2×#389×C71C2semidirect
19. C2×#14556086C22semidirect
31. #17985925C22semidirect
33. #17985897C22semidirect
34. #17985889C22semidirect
35. #17985886C22semidirect
36. #17985856C22semidirect
37. #17985856C22semidirect
38. #17985856C22semidirect
47. #389×C71C22semidirect
304. C2×C71#2485semidirect
320. C2×#389D71semidirect
326. C71C2×#2485semidirect
352. #389C2×D71semidirect
619. C2#22592640direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D2842
C22×#145560862
#225926402
C2×#179859252
C2×#179859172
C2×#179858972
C2×#179858892
C2×#179858862
C2×#179858562
C2×#389×C712
C2×#248571