Finite Group Databasebeta

← C2×#22598256 #32865222

Subgroups — C2×#22598256 #32865222

Order 18176.

2378 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986512C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic, complemented
3#22598256C2116yesmaximal, direct product factor
4C2×#17986317C211yesmaximal, characteristic, complemented
5C2×#17986339C212yesmaximal, complemented
6C2×#17986326C212yesmaximal, complemented
7C2×#17986327C211yesmaximal, characteristic, complemented
8C2×#17986313C211yesmaximal, characteristic, complemented
9C2×#17986036C212yesmaximal, complemented
10C2×#17985994C212yesmaximal, complemented
11C2×C22⋊C8×D71C211yesmaximal, characteristic, complemented
12C2×#484×C71C211yesmaximal, characteristic, complemented
13C22×#11765866C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C22×C8×D71C2212yes—
16#17986512—24no—
17#17986512C2218yes—
18C2×#14556113—21no—
19C2×#14556104C2211yescharacteristic
20C2×#14556103C2211yescharacteristic
21C2×#14556102C2211yescharacteristic
22C2×#14556098—21no—
23C2×#14556090—21no—
24C2×(C22⋊D71)⋊C4—21no—
25C2×#14556086—21no—
26C2×#14556086—21no—
27C2×#14556086—24no—
28C2×#14556086C2212yes—
29C2×#14556086C2212yes—
30C2×#14556086C2211yescharacteristic
31C2×#14556086C2218yescomplemented
32C22×C4⋊Dic71C2211yescharacteristic
33C22×(C2×D71)⋊C4—21no—
34C22×C4×Dic71—21no—
35#17986339C22116yescomplemented
36#17986327C2218yescomplemented
37#17986326C22116yescomplemented
38#17986317C2218yescomplemented
39#17986313C2218yescomplemented
40#17986036C22116yescomplemented
41C2×#14556062C2212yescomplemented
42C2×#14556059C2212yescomplemented
43C2×#14556059C2212yescomplemented
44#17985994C22116yescomplemented
45C2×#14556052C2212yescomplemented
46C2×#14556050C2212yescomplemented
47C2×#14556048C2212yescomplemented
48C2×C22⋊(C71⋊C8)C2211yescharacteristic
49C2×(C2×D71)⋊C8C2212yes—
50C22⋊C8×D71C2218yescomplemented
51C2×C22⋊Q8×C71C2211yescharacteristic
52C22×C4⋊C4×C71C2211yescharacteristic
53#484×C71C2218yescomplemented
54C2×#107×C71C2212yescomplemented
55C2×#103×C71C2212yescomplemented
56C2×C22⋊C8×C71C2211yescharacteristic
57C24×D71D411yescharacteristic
58C2×#11765866—22no—
59C2×#11765866—28no—
60C2×#11765866C2314yes—
61C22×D284—21no—
62C22×D284—21no—
63C22×D284C2311yescharacteristic
64C22×C22⋊D71—21no—
65C23×Dic71—21no—
66C23×Dic71D411yescharacteristic
67C22×C4×D71—28no—
68C22×C4×D71—21no—
69C22×C4×D71D411yescharacteristic
70C22×C4×D71D411yescharacteristic
71C22×C4×D71D414yes—
72C22×C4×D71C2311yescharacteristic
73C22×C4×D71C2314yes—
74C22×C4×D71C2311yescharacteristic
75C22×C71⋊C8C2312yes—
76C2×C8×D71—28no—
77C2×C8×D71C2318yes—
78#14556113—28no—
79#14556104—22no—
80#14556104C2314yes—
81#14556103—22no—
82#14556103C2314yes—
83#14556102—22no—
84#14556102C2314yes—
85#14556098—28no—
86#14556090—28no—
87(C22⋊D71)⋊C4—28no—
88#14556086—24no—
89#14556086—24no—
90#14556086—28no—
91#14556086—28no—
92#14556086—22no—
93#14556086C2318yes—
94#14556086C2318yes—
95#14556086C2314yes—
96C2×C4⋊Dic71—21no—
97C2×C4⋊Dic71—22no—
98C2×C4⋊Dic71C2314yes—
99C2×C4⋊Dic71C2312yes—
100C2×C4⋊Dic71C2311yescharacteristic
101C2×Dic71⋊C4—21no—
102C2×(C2×D71)⋊C4—22no—
103C2×(C2×D71)⋊C4—22no—
104C2×(C2×D71)⋊C4—28no—
105C2×(C2×D71)⋊C4—21no—
106C2×(C2×D71)⋊C4—21no—
107C2×(C2×D71)⋊C4—21no—
108C2×(C2×D71)⋊C4—21no—
109C2×C4×Dic71—22no—
110C2×C4×Dic71—22no—
111C2×C4×Dic71—28no—
112C2×C4×Dic71—21no—
113#14556062C2318yescomplemented
114#14556059C2318yescomplemented
115#14556059C2318yescomplemented
116#14556052C2318yescomplemented
117#14556050C2318yescomplemented
118#14556048C2318yescomplemented
119C22⋊(C71⋊C8)C2314yes—
120(C2×D71)⋊C8C2318yes—
121C22×Q8×C71C2311yescharacteristic
122C23×C4×C71—21no—
123C23×C4×C71C2311yescharacteristic
124C22×C8×C71C2312yes—
125C22⋊Q8×C71—22no—
126C22⋊Q8×C71C2314yes—
127C2×C4⋊C4×C71—21no—
128C2×C4⋊C4×C71—22no—
129C2×C4⋊C4×C71—21no—
130C2×C4⋊C4×C71C2312yes—
131C2×C4⋊C4×C71C2314yes—
132C2×C4⋊C4×C71C2311yescharacteristic
133C2×C22⋊C4×C71—21no—
134#107×C71C2318yescomplemented
135#103×C71C2318yescomplemented
136C22⋊C8×C71C2314yes—
137C23×D71—24no—
138C23×D71—24no—
139C23×D71—21no—
140C23×D71—42no—
141C23×D71C2×D411yescharacteristic
142C23×D71C2×D414yes—
143C23×D71C2×D411yescharacteristic
144#11765866—216no—
145#11765866—48no—
146C2×D284—24no—
147C2×D284—24no—
148C2×D284—24no—
149C2×D284—21no—
150C2×D284—44no—
151C2×D284—44no—
152C2×D284C2412yes—
153C2×C22⋊D71—24no—
154C2×C22⋊D71—44no—
155C22×Dic71—24no—
156C22×Dic71—24no—
157C22×Dic71—22no—
158C22×Dic71—21no—
159C22×Dic71—44no—
160C22×Dic71C2×D411yescharacteristic
161C22×Dic71C2×D414yes—
162C22×Dic71C2×D411yescharacteristic
163C2×C4×D71—24no—
164C2×C4×D71—24no—
165C2×C4×D71—24no—
166C2×C4×D71—216no—
167C2×C4×D71—216no—
168C2×C4×D71—24no—
169C2×C4×D71—44no—
170C2×C4×D71C2×D414yes—
171C2×C4×D71C2×D412yes—
172C2×C4×D71C2×D414yes—
173C2×C4×D71C2×D412yes—
174C2×C4×D71#4118yes—
175C2×C4×D71C2412yes—
176C2×C4×D71C2412yes—
177C2×C71⋊C8—24no—
178C2×C71⋊C8C2414yes—
179C8×D71—216no—
180C4⋊Dic71—24no—
181C4⋊Dic71—22no—
182C4⋊Dic71—21no—
183C4⋊Dic71C2414yes—
184C4⋊Dic71C2412yes—
185Dic71⋊C4—24no—
186(C2×D71)⋊C4—28no—
187(C2×D71)⋊C4—28no—
188(C2×D71)⋊C4—24no—
189(C2×D71)⋊C4—24no—
190(C2×D71)⋊C4—24no—
191(C2×D71)⋊C4—24no—
192C4×Dic71—28no—
193C4×Dic71—28no—
194C4×Dic71—24no—
195C24×C71C2×D411yescharacteristic
196C2×Q8×C71—21no—
197C2×Q8×C71—24no—
198C2×Q8×C71C2412yes—
199C22×C4×C71—24no—
200C22×C4×C71—21no—
201C22×C4×C71—21no—
202C22×C4×C71—24no—
203C22×C4×C71—22no—
204C22×C4×C71—44no—
205C22×C4×C71C2×D411yescharacteristic
206C22×C4×C71C2×D412yes—
207C22×C4×C71C2412yes—
208C22×C4×C71C2411yescharacteristic
209C2×C8×C71—24no—
210C2×C8×C71C2414yes—
211C4⋊C4×C71—24no—
212C4⋊C4×C71—22no—
213C4⋊C4×C71—24no—
214C4⋊C4×C71—21no—
215C4⋊C4×C71C2414yes—
216C4⋊C4×C71C2412yes—
217C22⋊C4×C71—24no—
218C23×C71—22no—
219C23×C71—22no—
220C23×C71#13514yes—
221C23×C71C22×D412yes—
222C23×C71C22×D411yescharacteristic
223C22×D71—22no—
224C22×D71—22no—
225C22×D71—22no—
226C22×D71—28no—
227C22×D71—22no—
228C22×D71—22no—
229C22×D71—28no—
230C22×D71—22no—
231C22×D71—44no—
232C22×D71—416no—
233C22×D71—42no—
234C22×D71—42no—
235C22×D71—42no—
236C22×D71#12314yes—
237C22×D71C22×D412yes—
238C22×D71C22×D412yes—
239Q8×C71—24no—
240Q8×C71—42no—
241C2×C4×C71—22no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—22no—
245C2×C4×C71—22no—
246C2×C4×C71—24no—
247C2×C4×C71—48no—
248C2×C4×C71—42no—
249C2×C4×C71—44no—
250C2×C4×C71—42no—
251C2×C4×C71C22×D412yes—
252C2×C4×C71C22×D411yescharacteristic
253C2×C4×C71C2×#4114yes—
254C2×C4×C71C2511yescharacteristic
255C22⋊D71—48no—
256C2×Dic71—22no—
257C2×Dic71—28no—
258C2×Dic71—24no—
259C2×Dic71—22no—
260C2×Dic71—48no—
261C2×Dic71—44no—
262C2×Dic71—42no—
263C2×Dic71#12314yes—
264C2×Dic71C22×D412yes—
265C2×Dic71C22×D412yes—
266D284—24no—
267D284—48no—
268D284—48no—
269D284—42no—
270C4×D71—28no—
271C4×D71—24no—
272C4×D71—24no—
273C4×D71—28no—
274C4×D71—48no—
275C4×D71#12318yes—
276C4×D71C2×#4118yes—
277C8×C71—24no—
278C71⋊C8—24no—
279C22×C71—28no—
280C22×C71—22no—
281C22×C71—22no—
282C22×C71C2×#12312yes—
283C22×C71C2×#13514yes—
284C22×C71C2×#13512yes—
285C22×C71#57512yes—
286C22×C71C23×D411yescharacteristic
287C2×D71—28no—
288C2×D71—22no—
289C2×D71—22no—
290C2×D71—22no—
291C2×D71—22no—
292C2×D71—22no—
293C2×D71—416no—
294C2×D71—44no—
295C2×D71—42no—
296C2×D71—42no—
297C2×D71—42no—
298C2×D71—84no—
299C2×D71C2×#12314yes—
300C4×C71—22no—
301C4×C71—42no—
302C4×C71—42no—
303C4×C71—44no—
304C4×C71C2×#12312yes—
305C4×C71C22×#4112yes—
306Dic71—22no—
307Dic71—44no—
308Dic71—42no—
309Dic71C2×#12314yes—
310C2×#3091—711nomaximal, Sylow
311C2×C71—24no—
312C2×C71#309114yescomplemented
313C2×C71C22×#12311yescharacteristic
314C2×C71C22×#13511yescharacteristic
315C2×C71C2×#57511yescharacteristic
316D71—24no—
317D71—42no—
318D71—82no—
319C2×#525—711no—
320C22×#117—711no—
321#3091—7116no—
322C2×#484D7111yescharacteristic, complemented
323C2×#481—711no—
324C2×#485—712no—
325C2×#482—712no—
326C2×#483—711no—
327C2×#480—711no—
328C2×#427—712no—
329C2×#416—712no—
330C22×C22⋊C8—711no—
331C71C2×#309111yescharacteristic, complemented, Sylow
332C22×#41—711no—
333C24×C4—711no—
334C23×C8—712no—
335#525—718no—
336#525—1424no—
337C2×#123—1421no—
338C2×C22⋊Q8—711no—
339C2×C22⋊Q8C2×D7111yescharacteristic
340C2×#121—711no—
341C2×#121—711no—
342C2×#120—1421no—
343C2×C4×D4—1421no—
344C2×C4×D4—1421no—
345C2×#117—712no—
346C2×#117—718no—
347C2×#117—712no—
348C2×#117—711no—
349C2×#117—1421no—
350C2×#117—1421no—
351C2×#117—1424no—
352C22×C4⋊C4—711no—
353C22×C4⋊C4C2×D7111yescharacteristic
354C22×C22⋊C4—1421no—
355C22×C42—1421no—
356#485—7116no—
357#484C2×D7118yescomplemented
358#483—718no—
359#482—7116no—
360#481—718no—
361#480—718no—
362#427—7116no—
363C2×#107—712no—
364C2×#107C2×D7112yescomplemented
365C2×#106—712no—
366C2×#106—712no—
367#416—7116no—
368C2×#103—712no—
369C2×#103C2×D7112yescomplemented
370C2×#102—712no—
371C2×#102—712no—
372C2×C22⋊C8—711no—
373C2×C22⋊C8—712no—
374C2×C22⋊C8—718no—
375C2×C22⋊C8C2×D7111yescharacteristic
376C25—711no—
377C2×#41—714no—
378C2×#41—1428no—
379C2×#41—1422no—
380C22×Q8C22×D7111yescharacteristic
381C22×D4—711no—
382C22×D4—1421no—
383C22×D4—1421no—
384C22×D4—1421no—
385C23×C4—21no—
386C23×C4—711no—
387C23×C4—711no—
388C23×C4—711no—
389C23×C4—711no—
390C23×C4—714no—
391C23×C4—711no—
392C23×C4—714no—
393C23×C4—1421no—
394C23×C4—1428no—
395C23×C4—1421no—
396C23×C4C22×D7111yescharacteristic
397C22×C8—712no—
398C22×C8—718no—
399C22×C8—1428no—
400C22×C8C22×D7112yes—
401#123—1428no—
402C22⋊Q8—22no—
403C22⋊Q8—714no—
404C22⋊Q8—1422no—
405C22⋊Q8C22×D7114yes—
406#121—714no—
407#121—714no—
408#121—1422no—
409#121—1422no—
410#120—1428no—
411C4×D4—1428no—
412C4×D4—1428no—
413#117—718no—
414#117—718no—
415#117—714no—
416#117—1424no—
417#117—1428no—
418#117—1424no—
419#117—1428no—
420#117—1422no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—712no—
425C2×C4⋊C4—714no—
426C2×C4⋊C4—711no—
427C2×C4⋊C4—1421no—
428C2×C4⋊C4—1422no—
429C2×C4⋊C4—1421no—
430C2×C4⋊C4C22×D7112yes—
431C2×C4⋊C4C22×D7114yes—
432C2×C4⋊C4C22×D7111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1421no—
435C2×C22⋊C4—1428no—
436C2×C22⋊C4—1422no—
437C2×C22⋊C4—1422no—
438C2×C22⋊C4—1421no—
439C2×C22⋊C4—1421no—
440C2×C22⋊C4—1421no—
441C2×C42—1428no—
442C2×C42—1422no—
443C2×C42—1422no—
444C2×C42—1421no—
445#107—718no—
446#107C22×D7118yescomplemented
447#106—718no—
448#106—718no—
449#103—718no—
450#103C22×D7118yescomplemented
451#102—718no—
452#102—718no—
453C22⋊C8—714no—
454C22⋊C8—718no—
455C22⋊C8C22×D7114yes—
456C24—2842no—
457C24—714no—
458C24—711no—
459C24—711no—
460C24—1424no—
461C24—1421no—
462C24—1424no—
463C24D4×D7111yescharacteristic
464#41—2848no—
465#41—14216no—
466C2×Q8—21no—
467C2×Q8—24no—
468C2×Q8C23×D7112yes—
469C2×D4—2844no—
470C2×D4—2844no—
471C2×D4—2844no—
472C2×D4—712no—
473C2×D4—1424no—
474C2×D4—1424no—
475C2×D4—1424no—
476C2×D4—1421no—
477C2×D4—1424no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—44no—
484C22×C4—2844no—
485C22×C4—2844no—
486C22×C4—711no—
487C22×C4—712no—
488C22×C4—718no—
489C22×C4—712no—
490C22×C4—714no—
491C22×C4—711no—
492C22×C4—714no—
493C22×C4—712no—
494C22×C4—714no—
495C22×C4—712no—
496C22×C4—1424no—
497C22×C4—1424no—
498C22×C4—14216no—
499C22×C4—14216no—
500C22×C4—1424no—
501C22×C4—1424no—
502C22×C4—1422no—
503C22×C4—1421no—
504C22×C4—1424no—
505C22×C4—1424no—
506C22×C4D4×D7112yes—
507C22×C4D4×D7111yescharacteristic
508C22×C4C23×D7112yes—
509C22×C4C23×D7111yescharacteristic
510C2×C8—24no—
511C2×C8—714no—
512C2×C8—14216no—
513C2×C8—1424no—
514C2×C8C23×D7114yes—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—714no—
520C4⋊C4—712no—
521C4⋊C4—1424no—
522C4⋊C4—1424no—
523C4⋊C4—1422no—
524C4⋊C4—1421no—
525C4⋊C4C23×D7114yes—
526C4⋊C4C23×D7112yes—
527C22⋊C4—24no—
528C22⋊C4—1424no—
529C22⋊C4—1428no—
530C22⋊C4—1428no—
531C22⋊C4—1424no—
532C22⋊C4—1424no—
533C22⋊C4—1424no—
534C42—1428no—
535C42—1428no—
536C42—1424no—
537C23—22no—
538C23—22no—
539C23—28416no—
540C23—2844no—
541C23—2842no—
542C23—2842no—
543C23—2842no—
544C23—712no—
545C23—714no—
546C23—712no—
547C23—1428no—
548C23—1422no—
549C23—1422no—
550C23—1428no—
551C23—1422no—
552C23—1422no—
553C23—1422no—
554C23—1422no—
555C23#1455615114yes—
556C23C2×D4×D7112yes—
557C23C2×D4×D7111yescharacteristic
558Q8—24no—
559Q8—42no—
560D4—2848no—
561D4—2848no—
562D4—2848no—
563D4—2842no—
564D4—1424no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—2848no—
576C2×C4—2848no—
577C2×C4—2844no—
578C2×C4—2842no—
579C2×C4—712no—
580C2×C4—714no—
581C2×C4—718no—
582C2×C4—718no—
583C2×C4—712no—
584C2×C4—1428no—
585C2×C4—1428no—
586C2×C4—1424no—
587C2×C4—1422no—
588C2×C4—1428no—
589C2×C4—1422no—
590C2×C4—1424no—
591C2×C4—1424no—
592C2×C4C2×D4×D7112yes—
593C2×C4C2×D4×D7111yescharacteristic
594C2×C4#41×D7114yes—
595C2×C4C24×D7111yescharacteristic
596C8—24no—
597C8—1424no—
598C22—22no—
599C22—22no—
600C22—28no—
601C22—28416no—
602C22—2844no—
603C22—2842no—
604C22—2842no—
605C22—2842no—
606C22—5684no—
607C22—714no—
608C22—1422no—
609C22—1422no—
610C22—1422no—
611C22—1422no—
612C22—1422no—
613C22—1428no—
614C22#123×D7112yes—
615C22C2×#1455615114yes—
616C22C2×#1455615112yes—
617C22#1798679512yes—
618C22C22×D4×D7111yescharacteristic
619C4—22no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—2844no—
624C4—2842no—
625C4—714no—
626C4—1422no—
627C4#123×D7112yes—
628C4C2×#41×D7112yes—
629C2—24no—
630C2—2842no—
631C2—5682no—
632C2—1424no—
633C2#2259825614yesdirect product factor
634C2C2×#123×D7111yescharacteristic
635C2C22×#1455615111yescharacteristic
636C2C2×#1798679511yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC2×#484×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986512C2
hypercenterC2×#484D71
hypocenterC71C2×#3091
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#14556086C2semidirect
3. #22598256C2direct (N × H)
4. C2×#17986317C2semidirect
5. C2×#17986339C2semidirect
6. C2×#17986326C2semidirect
7. C2×#17986327C2semidirect
8. C2×#17986313C2semidirect
9. C2×#17986036C2semidirect
10. C2×#17985994C2semidirect
11. C2×C22⋊C8×D71C2semidirect
12. C2×#484×C71C2semidirect
31. C2×#14556086C22semidirect
35. #17986339C22semidirect
36. #17986327C22semidirect
37. #17986326C22semidirect
38. #17986317C22semidirect
39. #17986313C22semidirect
40. #17986036C22semidirect
41. C2×#14556062C22semidirect
42. C2×#14556059C22semidirect
43. C2×#14556059C22semidirect
44. #17985994C22semidirect
45. C2×#14556052C22semidirect
46. C2×#14556050C22semidirect
47. C2×#14556048C22semidirect
50. C22⋊C8×D71C22semidirect
53. #484×C71C22semidirect
54. C2×#107×C71C22semidirect
55. C2×#103×C71C22semidirect
113. #14556062C23semidirect
114. #14556059C23semidirect
115. #14556059C23semidirect
116. #14556052C23semidirect
117. #14556050C23semidirect
118. #14556048C23semidirect
134. #107×C71C23semidirect
135. #103×C71C23semidirect
312. C2×C71#3091semidirect
322. C2×#484D71semidirect
331. C71C2×#3091semidirect
357. #484C2×D71semidirect
364. C2×#107C2×D71semidirect
369. C2×#103C2×D71semidirect
446. #107C22×D71semidirect
450. #103C22×D71semidirect
633. C2#22598256direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179865122
C22×#145560862
#225982562
C2×#179863172
C2×#179863392
C2×#179863262
C2×#179863272
C2×#179863132
C2×#179860362
C2×#179859942
C2×C22⋊C8×D712
C2×#484×C712
C2×#309171