Finite Group Databasebeta

← C2×#22598153 #32864981

Subgroups — C2×#22598153 #32864981

Order 18176.

2358 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×#14556127C211yesmaximal, characteristic, complemented
2C2×#17986467C211yesmaximal, characteristic
3#22598153C2116yesmaximal, direct product factor
4C2×#17986293C212yesmaximal, complemented
5C2×#17986283C212yesmaximal, complemented
6C2×#17986288C212yesmaximal, complemented
7C2×#17986278C212yesmaximal, complemented
8C2×#17986042C211yesmaximal, characteristic, complemented
9C2×#17986022C211yesmaximal, characteristic, complemented
10C2×#17986001C211yesmaximal, characteristic
11C2×#17985998C211yesmaximal, characteristic
12C2×#428×C71C211yesmaximal, characteristic, complemented
13C22×#11765865C2211yescharacteristic
14C23×Dic142—21no—
15C22×#11765849C2211yescharacteristic
16C2×#14556127—21no—
17C2×#14556127—21no—
18C2×#14556127—24no—
19C2×#14556127C2212yes—
20C2×#14556127C2212yes—
21C2×#14556127C2218yescomplemented
22#17986467—24no—
23#17986467C2218yes—
24C2×#14556093C2211yescharacteristic
25C2×#14556093C2211yescharacteristic
26C2×#14556090C2211yescharacteristic
27C2×#14556090C2211yescharacteristic
28C2×(C22⋊D71)⋊C4—21no—
29C2×(C22⋊D71)⋊C4—21no—
30C2×#14556085—21no—
31C22×C4⋊Dic71C2211yescharacteristic
32C22×Dic71⋊C4—21no—
33C22×C4×Dic71C2211yescharacteristic
34C2×C42×D71—21no—
35#17986293C22116yescomplemented
36#17986288C22116yescomplemented
37#17986283C22116yescomplemented
38#17986278C22116yescomplemented
39#17986042C2218yescomplemented
40C2×#14556061C2212yes—
41C2×#14556059C2212yes—
42#17986022C2218yescomplemented
43C2×Dic71⋊C8C2212yes—
44C2×Dic71⋊C8C2212yes—
45#17986001C2218yes—
46#17985998C2218yes—
47C2×#14556053C2212yes—
48C2×#14556051C2212yes—
49C2×#14556050C2212yes—
50C2×#14556048C2212yes—
51C22×M16×C71C2211yescharacteristic
52C2×#117×C71C2211yescharacteristic
53C22×C4⋊C4×C71C2211yescharacteristic
54#428×C71C2218yescomplemented
55C2×#107×C71C2212yes—
56C2×#106×C71C2212yes—
57C2×#11765865—22no—
58C2×#11765865—24no—
59C2×#11765865Q818yescomplemented
60C2×#11765865C2314yes—
61C22×Dic142—22no—
62C22×Dic142—22no—
63C22×Dic142—28no—
64C22×Dic142—48no—
65C22×Dic142C2311yescharacteristic
66C22×D284C2311yescharacteristic
67C22×C22⋊D71—21no—
68C23×Dic71—21no—
69C23×Dic71D411yescharacteristic
70C23×Dic71D411yescharacteristic
71C22×C4×D71—21no—
72C22×C4×D71—21no—
73C22×C4×D71—21no—
74C2×#11765849—24no—
75C2×#11765849C2314yes—
76C22×C71⋊C8C2312yes—
77#14556127—24no—
78#14556127—24no—
79#14556127—28no—
80#14556127—28no—
81#14556127C2318yes—
82#14556127C2318yes—
83#14556093—22no—
84#14556093—22no—
85#14556093C2314yes—
86#14556093C2314yes—
87#14556090—22no—
88#14556090—22no—
89#14556090C2314yes—
90#14556090C2314yes—
91(C22⋊D71)⋊C4—28no—
92(C22⋊D71)⋊C4—28no—
93#14556085—28no—
94C2×C4⋊Dic71—21no—
95C2×C4⋊Dic71—22no—
96C2×C4⋊Dic71C2312yes—
97C2×C4⋊Dic71C2314yes—
98C2×Dic71⋊C4—22no—
99C2×Dic71⋊C4—22no—
100C2×Dic71⋊C4—28no—
101C2×Dic71⋊C4—21no—
102C2×Dic71⋊C4—21no—
103C2×(C2×D71)⋊C4—21no—
104C2×(C2×D71)⋊C4—21no—
105C2×C4×Dic71—21no—
106C2×C4×Dic71—21no—
107C2×C4×Dic71D411yescharacteristic
108C2×C4×Dic71D411yescharacteristic
109C2×C4×Dic71D414yes—
110C2×C4×Dic71C2311yescharacteristic
111C2×C4×Dic71C2311yescharacteristic
112C2×C4×Dic71C2314yes—
113C42×D71—28no—
114#14556061C2318yes—
115#14556059C2318yes—
116Dic71⋊C8C2318yes—
117Dic71⋊C8C2318yes—
118#14556053C2318yes—
119#14556051C2318yes—
120#14556050C2318yes—
121#14556048C2318yes—
122C23×C4×C71—21no—
123C23×C4×C71C2311yescharacteristic
124C2×M16×C71—24no—
125C2×M16×C71C2314yes—
126C22×C8×C71C2312yes—
127#117×C71—22no—
128#117×C71C2314yes—
129C2×C4⋊C4×C71—21no—
130C2×C4⋊C4×C71—22no—
131C2×C4⋊C4×C71C2311yescharacteristic
132C2×C4⋊C4×C71C2312yes—
133C2×C4⋊C4×C71C2311yescharacteristic
134C2×C4⋊C4×C71C2314yes—
135C2×C22⋊C4×C71—21no—
136C2×C42×C71—21no—
137#107×C71C2318yes—
138#106×C71C2318yes—
139C23×D71—21no—
140#11765865—216no—
141#11765865—28no—
142#11765865C2×Q8116yescomplemented
143C2×Dic142—21no—
144C2×Dic142—22no—
145C2×Dic142—28no—
146C2×Dic142—28no—
147C2×Dic142—48no—
148C2×Dic142—432no—
149C2×Dic142—48no—
150C2×Dic142C2×Q814yescomplemented
151C2×Dic142C2412yes—
152C2×D284—21no—
153C2×D284—22no—
154C2×D284C2×Q814yescomplemented
155C2×D284C2412yes—
156C2×C22⋊D71—28no—
157C2×C22⋊D71—24no—
158C22×Dic71—21no—
159C22×Dic71—24no—
160C22×Dic71—24no—
161C22×Dic71—24no—
162C22×Dic71—22no—
163C22×Dic71—44no—
164C22×Dic71C2×D411yescharacteristic
165C22×Dic71C2×D414yes—
166C22×Dic71C2×D411yescharacteristic
167C22×Dic71C2×D414yes—
168C22×Dic71C2×D411yescharacteristic
169C22×Dic71C2×D411yescharacteristic
170C2×C4×D71—28no—
171C2×C4×D71—24no—
172C2×C4×D71—24no—
173C2×C4×D71—24no—
174C2×C4×D71—44no—
175C2×C4×D71—44no—
176#11765849—28no—
177C2×C71⋊C8—24no—
178C2×C71⋊C8C2414yes—
179C4⋊Dic71—24no—
180C4⋊Dic71—22no—
181C4⋊Dic71C2414yes—
182Dic71⋊C4—28no—
183Dic71⋊C4—28no—
184Dic71⋊C4—24no—
185Dic71⋊C4—24no—
186(C2×D71)⋊C4—24no—
187(C2×D71)⋊C4—24no—
188C4×Dic71—24no—
189C4×Dic71—24no—
190C4×Dic71C2×D414yes—
191C4×Dic71C2×D414yes—
192C4×Dic71C2×D412yes—
193C4×Dic71C2×D412yes—
194C4×Dic71C2412yes—
195C4×Dic71C2412yes—
196C24×C71C2×D411yescharacteristic
197C22×C4×C71—21no—
198C22×C4×C71—21no—
199C22×C4×C71—24no—
200C22×C4×C71—22no—
201C22×C4×C71—44no—
202C22×C4×C71C2×D411yescharacteristic
203C22×C4×C71C2×D412yes—
204C22×C4×C71C2×Q814yes—
205C22×C4×C71#4114yes—
206C22×C4×C71C2411yescharacteristic
207C22×C4×C71C2412yes—
208M16×C71—28no—
209C2×C8×C71—24no—
210C2×C8×C71C2414yes—
211C4⋊C4×C71—24no—
212C4⋊C4×C71—22no—
213C4⋊C4×C71—21no—
214C4⋊C4×C71—21no—
215C4⋊C4×C71C2412yes—
216C4⋊C4×C71C2414yes—
217C4⋊C4×C71C2412yes—
218C22⋊C4×C71—24no—
219C42×C71—24no—
220C23×C71—22no—
221C23×C71—22no—
222C23×C71C22⋊Q814yes—
223C23×C71C22×D412yes—
224C23×C71C22×D411yescharacteristic
225C22×D71—24no—
226C22×D71—22no—
227C22×D71—42no—
228C22×D71—42no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—24no—
232C2×C4×C71—48no—
233C2×C4×C71—44no—
234C2×C4×C71—42no—
235C2×C4×C71—42no—
236C2×C4×C71C22⋊Q814yes—
237C2×C4×C71C22⋊Q818yes—
238C2×C4×C71C22×D412yes—
239C2×C4×C71C22×D411yescharacteristic
240C2×C4×C71C22×Q812yes—
241C2×C4×C71C22×Q814yes—
242C2×C4×C71C2×#4112yes—
243C2×C4×C71C2×#4114yes—
244C2×C4×C71C2511yescharacteristic
245C22⋊D71—216no—
246C2×Dic71—24no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—22no—
250C2×Dic71—28no—
251C2×Dic71—28no—
252C2×Dic71—24no—
253C2×Dic71—48no—
254C2×Dic71—44no—
255C2×Dic71AGL(1,ℤ/8)14yes—
256C2×Dic71#13714yes—
257C2×Dic71C22×D412yes—
258C2×Dic71C22×D412yes—
259C2×Dic71C22×D412yes—
260C2×Dic71C22×D412yes—
261D284—24no—
262D284—22no—
263D284C22×Q814yescomplemented
264C4×D71—216no—
265C4×D71—48no—
266C4×D71—48no—
267Dic142—24no—
268Dic142—22no—
269Dic142—416no—
270Dic142—416no—
271Dic142C22×Q814yescomplemented
272C8×C71—24no—
273C71⋊C8—24no—
274C22×C71—28no—
275C22×C71—22no—
276C22×C71—22no—
277C22×C71C2×C22⋊Q814yes—
278C22×C71C2×C22⋊Q812yes—
279C22×C71C2×AGL(1,ℤ/8)12yes—
280C22×C71C2×#13712yes—
281C22×C71C23×D411yescharacteristic
282C2×D71—24no—
283C2×D71—48no—
284C2×D71—42no—
285C2×D71—42no—
286C4×C71—44no—
287C4×C71—42no—
288C4×C71—42no—
289C4×C71C2×C22⋊Q812yes—
290C4×C71C2×C22⋊Q814yes—
291C4×C71C23×Q811yescharacteristic
292C4×C71C22×#4111yescharacteristic
293Dic71—24no—
294Dic71—22no—
295Dic71—22no—
296Dic71—44no—
297Dic71C2×AGL(1,ℤ/8)14yes—
298Dic71C2×#13714yes—
299C2×#3079—711nomaximal, Sylow
300C2×C71—24no—
301C2×C71#307914yescomplemented
302C2×C71C22×C22⋊Q811yescharacteristic
303C2×C71C22×AGL(1,ℤ/8)11yescharacteristic
304C2×C71C22×#13711yescharacteristic
305D71—44no—
306C22×C4⋊Q8—711no—
307C2×C4×#41—711no—
308#3079—7116no—
309C2×#477—712no—
310C2×#475—712no—
311C2×#476—712no—
312C2×#474—712no—
313C2×#428—711no—
314C2×#428D7111yescharacteristic, complemented
315C2×#423—711no—
316C2×#417—711no—
317C2×#417—711no—
318C71C2×#307911yescharacteristic, complemented, Sylow
319C22×#41—711no—
320C23×Q8—1421no—
321C22×M16—711no—
322C22×M16C2×D7111yescharacteristic
323C2×C4⋊Q8—712no—
324C2×C4⋊Q8—712no—
325C2×C4⋊Q8—718no—
326C2×C4⋊Q8—1421no—
327C2×C4⋊Q8—1424no—
328C2×C4⋊Q8—1421no—
329C4×#41—718no—
330C4×#41—1424no—
331C2×C4×Q8—711no—
332C2×C4×Q8—711no—
333C2×C4×D4—711no—
334C2×C4×D4—711no—
335C2×C4×D4—1421no—
336C2×C4×D4—1421no—
337C2×#117—1421no—
338C2×#117C2×D7111yescharacteristic
339C22×C4⋊C4—711no—
340C22×C4⋊C4—1421no—
341C22×C4⋊C4C2×D7111yescharacteristic
342C22×C42—711no—
343C22×C42—1421no—
344#477—7116no—
345#476—7116no—
346#475—7116no—
347#474—7116no—
348#428—718no—
349#428C2×D7118yescomplemented
350C2×#107—712no—
351C2×#107C2×D7112yes—
352C2×#106—712no—
353C2×#106C2×D7112yes—
354#423—718no—
355C2×C4⋊C8—712no—
356C2×C4⋊C8—712no—
357#417—718no—
358#417—718no—
359C2×#103—712no—
360C2×#103—712no—
361C2×#102—712no—
362C2×#102—712no—
363C2×#41—718no—
364C2×#41—714no—
365C2×#41—1422no—
366C2×#41—1424no—
367C22×Q8—2848no—
368C22×Q8—711no—
369C22×Q8—1428no—
370C22×Q8—1422no—
371C22×Q8—1422no—
372C22×D4—711no—
373C22×D4—1421no—
374C23×C4—21no—
375C23×C4—711no—
376C23×C4—711no—
377C23×C4—1421no—
378C23×C4—1421no—
379C23×C4—1421no—
380C23×C4—1421no—
381C23×C4C22×D7111yescharacteristic
382C2×M16—24no—
383C2×M16—714no—
384C2×M16—1424no—
385C2×M16C22×D7114yes—
386C22×C8—712no—
387C22×C8C22×D7112yes—
388C4⋊Q8—718no—
389C4⋊Q8—718no—
390C4⋊Q8—1428no—
391C4⋊Q8—1428no—
392C4⋊Q8—1424no—
393C4⋊Q8—1424no—
394C4×Q8—714no—
395C4×Q8—714no—
396C4×Q8—1422no—
397C4×Q8—1422no—
398C4×D4—714no—
399C4×D4—714no—
400C4×D4—1422no—
401C4×D4—1422no—
402C4×D4—1428no—
403C4×D4—1428no—
404#117—22no—
405#117—1428no—
406#117C22×D7114yes—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—712no—
410C2×C4⋊C4—714no—
411C2×C4⋊C4—1428no—
412C2×C4⋊C4—1422no—
413C2×C4⋊C4—1422no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1422no—
417C2×C4⋊C4—1421no—
418C2×C4⋊C4C22×D7112yes—
419C2×C4⋊C4C22×D7114yes—
420C2×C4⋊C4C22×D7111yescharacteristic
421C2×C4⋊C4C22×D7111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1421no—
424C2×C22⋊C4—1421no—
425C2×C42—21no—
426C2×C42—711no—
427C2×C42—714no—
428C2×C42—711no—
429C2×C42—711no—
430C2×C42—711no—
431C2×C42—714no—
432C2×C42—1428no—
433C2×C42—1421no—
434C2×C42—1421no—
435#107—718no—
436#107C22×D7118yes—
437#106—718no—
438#106C22×D7118yes—
439C4⋊C8—718no—
440C4⋊C8—718no—
441#103—718no—
442#103—718no—
443#102—718no—
444#102—718no—
445C24—1421no—
446C24D4×D7111yescharacteristic
447#41—7116no—
448#41—1428no—
449#41—14216no—
450C2×Q8—2848no—
451C2×Q8—28432no—
452C2×Q8—2848no—
453C2×Q8—714no—
454C2×Q8—712no—
455C2×Q8—1428no—
456C2×Q8—1428no—
457C2×Q8—1421no—
458C2×Q8—1422no—
459C2×D4—714no—
460C2×D4—712no—
461C2×D4—1424no—
462C2×D4—1428no—
463C2×D4—1421no—
464C2×D4—1422no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—2844no—
471C22×C4—2844no—
472C22×C4—2844no—
473C22×C4—711no—
474C22×C4—711no—
475C22×C4—714no—
476C22×C4—714no—
477C22×C4—711no—
478C22×C4—711no—
479C22×C4—1424no—
480C22×C4—1424no—
481C22×C4—1424no—
482C22×C4—1422no—
483C22×C4—1424no—
484C22×C4—1424no—
485C22×C4—1421no—
486C22×C4—1424no—
487C22×C4—1428no—
488C22×C4D4×D7112yes—
489C22×C4D4×D7111yescharacteristic
490C22×C4Q8×D7114yes—
491C22×C4#1176586514yes—
492C22×C4C23×D7112yes—
493C22×C4C23×D7111yescharacteristic
494M16—28no—
495M16—1428no—
496C2×C8—24no—
497C2×C8—714no—
498C2×C8—1424no—
499C2×C8C23×D7114yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—714no—
505C4⋊C4—1428no—
506C4⋊C4—1428no—
507C4⋊C4—1424no—
508C4⋊C4—1424no—
509C4⋊C4—1424no—
510C4⋊C4—1422no—
511C4⋊C4C23×D7114yes—
512C4⋊C4C23×D7112yes—
513C4⋊C4C23×D7112yes—
514C22⋊C4—24no—
515C22⋊C4—1424no—
516C22⋊C4—1424no—
517C42—24no—
518C42—714no—
519C42—714no—
520C42—712no—
521C42—712no—
522C42—712no—
523C42—712no—
524C42—1424no—
525C42—1424no—
526C23—22no—
527C23—22no—
528C23—2842no—
529C23—2842no—
530C23—1422no—
531C23—1424no—
532C23#1455610614yes—
533C23C2×D4×D7112yes—
534C23C2×D4×D7111yescharacteristic
535Q8—28416no—
536Q8—28416no—
537Q8—714no—
538Q8—1422no—
539Q8—1424no—
540D4—714no—
541D4—14216no—
542D4—1422no—
543D4—1424no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—2848no—
552C2×C4—2848no—
553C2×C4—2848no—
554C2×C4—2844no—
555C2×C4—712no—
556C2×C4—712no—
557C2×C4—714no—
558C2×C4—714no—
559C2×C4—712no—
560C2×C4—712no—
561C2×C4—1424no—
562C2×C4—1428no—
563C2×C4—1428no—
564C2×C4—1422no—
565C2×C4—1424no—
566C2×C4—1422no—
567C2×C4—1422no—
568C2×C4—14216no—
569C2×C4#1455610614yes—
570C2×C4#1455610618yes—
571C2×C4C2×D4×D7112yes—
572C2×C4C2×D4×D7111yescharacteristic
573C2×C4C2×Q8×D7112yes—
574C2×C4C2×Q8×D7114yes—
575C2×C4C2×#1176586512yes—
576C2×C4C2×#1176586514yes—
577C2×C4C24×D7111yescharacteristic
578C8—24no—
579C8—1424no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—2848no—
584C22—2842no—
585C22—2842no—
586C22—1424no—
587C22C2×#1455610614yes—
588C22C2×#1455610612yes—
589C22AGL(1,ℤ/8)×D7112yes—
590C22#137×D7112yes—
591C22C22×D4×D7111yescharacteristic
592C4—44no—
593C4—42no—
594C4—42no—
595C4—2844no—
596C4—714no—
597C4—714no—
598C4—1422no—
599C4—1422no—
600C4—1424no—
601C4C2×#1455610614yes—
602C4C2×#1455610612yes—
603C4C22×Q8×D7111yescharacteristic
604C4C22×#1176586511yescharacteristic
605C2—24no—
606C2—2844no—
607C2#2259815314yesdirect product factor
608C2C22×#1455610611yescharacteristic
609C2C2×AGL(1,ℤ/8)×D7111yescharacteristic
610C2C2×#137×D7111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC2×#428×C71C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
hypercenterC2×#428D71
hypocenterC71C2×#3079
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C22×#14556127C2semidirect
3. #22598153C2direct (N × H)
4. C2×#17986293C2semidirect
5. C2×#17986283C2semidirect
6. C2×#17986288C2semidirect
7. C2×#17986278C2semidirect
8. C2×#17986042C2semidirect
9. C2×#17986022C2semidirect
12. C2×#428×C71C2semidirect
21. C2×#14556127C22semidirect
35. #17986293C22semidirect
36. #17986288C22semidirect
37. #17986283C22semidirect
38. #17986278C22semidirect
39. #17986042C22semidirect
42. #17986022C22semidirect
54. #428×C71C22semidirect
59. C2×#11765865Q8semidirect
142. #11765865C2×Q8semidirect
150. C2×Dic142C2×Q8semidirect
154. C2×D284C2×Q8semidirect
263. D284C22×Q8semidirect
271. Dic142C22×Q8semidirect
301. C2×C71#3079semidirect
314. C2×#428D71semidirect
318. C71C2×#3079semidirect
349. #428C2×D71semidirect
607. C2#22598153direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561272
C2×#179864672
#225981532
C2×#179862932
C2×#179862832
C2×#179862882
C2×#179862782
C2×#179860422
C2×#179860222
C2×#179860012
C2×#179859982
C2×#428×C712
C2×#307971