Finite Group Databasebeta

← C2×#22588574 #31998315

Subgroups — C2×#22588574 #31998315

Order 18176.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986479C211yesmaximal, characteristic
2C2×#17986042C211yesmaximal, characteristic, complemented
3C22×#14556056C211yesmaximal, characteristic
4C2×#17986001C211yesmaximal, characteristic
5C2×#17985865C211yesmaximal, characteristic, complemented
6#22588574C218yesmaximal, direct product factor
7C2×#17985767C211yesmaximal, characteristic, complemented
8C2×#339×C71C211yesmaximal, characteristic, complemented
9C22×#11765865C2211yescharacteristic
10C22×#11765849C2211yescharacteristic
11#17986479—22no—
12#17986479—22no—
13#17986479—22no—
14#17986479C2214yes—
15C2×C4×Dic142—21no—
16C2×C4×D284—21no—
17C2×(C22⋊D71)⋊C4—22no—
18C2×#14556086—21no—
19C22×C4⋊Dic71C2211yescharacteristic
20C22×C4⋊Dic71C2211yescharacteristic
21C2×C4⋊C4×D71—21no—
22#17986042—22no—
23#17986042C2214yescomplemented
24C2×#14556061—21no—
25C2×#14556059—21no—
26C2×#14556056C418yescomplemented
27C2×#14556056C2214yes—
28#17986001—22no—
29#17986001C2214yes—
30C2×#14556053—21no—
31C2×#14556050—21no—
32#17985865—22no—
33#17985865C2214yescomplemented
34C2×C71⋊#95—22no—
35#17985767C2214yescomplemented
36C22×M16×C71C2211yescharacteristic
37C2×#117×C71C2211yescharacteristic
38C22×C42×C71C2211yescharacteristic
39#339×C71C2214yescomplemented
40C2×#11765865—21no—
41C2×#11765865—21no—
42C2×#11765865—22no—
43C2×#11765865—22no—
44C2×#11765865C2×C414yes—
45C2×#11765865D412yes—
46C2×#11765865Q814yes—
47C2×#11765865C2312yes—
48C22×Dic142D411yescharacteristic
49C22×D284D411yescharacteristic
50C22×C22⋊D71—21no—
51C23×Dic71—21no—
52C23×Dic71D412yes—
53C22×C4×D71—22no—
54C22×C4×D71—21no—
55C2×#11765849—21no—
56C2×#11765849—22no—
57C2×#11765849C2×C414yescomplemented
58C2×#11765849C2312yes—
59C22×C71⋊C8—21no—
60C4×Dic142—24no—
61C4×Dic142—24no—
62C4×D284—24no—
63C4×D284—24no—
64(C22⋊D71)⋊C4—28no—
65(C22⋊D71)⋊C4—28no—
66#14556086—24no—
67#14556086—24no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71—21no—
77C2×C4⋊Dic71D412yes—
78C2×C4⋊Dic71C2312yes—
79C2×C4⋊Dic71C2312yes—
80C2×Dic71⋊C4—22no—
81C4⋊C4×D71—24no—
82C4⋊C4×D71—24no—
83C2×(C2×D71)⋊C4—22no—
84C2×C4×Dic71—22no—
85#14556061—24no—
86#14556059—24no—
87#14556056C2×C4116yescomplemented
88#14556053—24no—
89#14556050—24no—
90C71⋊#95—28no—
91C23×C4×C71—21no—
92C23×C4×C71C2311yescharacteristic
93C2×M16×C71—21no—
94C2×M16×C71—22no—
95C2×M16×C71—22no—
96C2×M16×C71C2312yes—
97C22×C8×C71—21no—
98#117×C71—21no—
99#117×C71—21no—
100#117×C71D412yes—
101#117×C71C2312yes—
102C2×C4⋊C4×C71D411yescharacteristic
103C2×C4⋊C4×C71D411yescharacteristic
104C2×C22⋊C4×C71—21no—
105C2×C42×C71—21no—
106C2×C42×C71—21no—
107C2×C42×C71—22no—
108C2×C42×C71C2×C414yescomplemented
109C2×C42×C71C2312yes—
110C23×D71—21no—
111#11765865—24no—
112#11765865—24no—
113#11765865—24no—
114#11765865—24no—
115#11765865—24no—
116#11765865—22no—
117#11765865—22no—
118#11765865C4⋊C418yes—
119#11765865C22×C414yes—
120#11765865C2×Q814yes—
121C2×Dic142—21no—
122C2×Dic142—22no—
123C2×Dic142C4⋊C414yes—
124C2×Dic142C2×D412yes—
125C2×D284—21no—
126C2×D284—22no—
127C2×D284C4⋊C414yes—
128C2×D284C2×D412yes—
129C2×C22⋊D71—24no—
130C2×C22⋊D71—22no—
131C2×C22⋊D71—24no—
132C2×C22⋊D71—22no—
133C22×Dic71—21no—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71—22no—
137C22×Dic71—44no—
138C22×Dic71—44no—
139C22×Dic71—41no—
140C22×Dic71—41no—
141C22×Dic71C2×D414yes—
142C2×C4×D71—24no—
143C2×C4×D71—22no—
144C2×C4×D71—24no—
145C2×C4×D71—22no—
146C2×C4×D71—28no—
147C2×C4×D71—84no—
148#11765849—24no—
149#11765849—22no—
150#11765849C22×C414yescomplemented
151C2×C71⋊C8—24no—
152C2×C71⋊C8—22no—
153C4⋊Dic71—22no—
154C4⋊Dic71—22no—
155C4⋊Dic71—22no—
156C4⋊Dic71—22no—
157C4⋊Dic71—22no—
158C4⋊Dic71—22no—
159C4⋊Dic71—41no—
160C4⋊Dic71—41no—
161Dic71⋊C4—28no—
162(C2×D71)⋊C4—28no—
163C4×Dic71—28no—
164C24×C71C2×D411yescharacteristic
165C22×C4×C71—24no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—22no—
169C22×C4×C71—42no—
170C22×C4×C71—42no—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×Q812yes—
178C22×C4×C71#4112yes—
179C22×C4×C71#4112yes—
180C22×C4×C71C2411yescharacteristic
181M16×C71—22no—
182M16×C71—22no—
183M16×C71—42no—
184C2×C8×C71—22no—
185C2×C8×C71—42no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71C2×D412yes—
189C4⋊C4×C71C2×D412yes—
190C22⋊C4×C71—22no—
191C22⋊C4×C71—22no—
192C42×C71—24no—
193C42×C71—22no—
194C42×C71—22no—
195C42×C71—22no—
196C42×C71C22×C414yescomplemented
197C23×C71—22no—
198C23×C71—22no—
199C23×C71C4×D412yes—
200C23×C71#12011yescharacteristic
201C23×C71#12011yescharacteristic
202C23×C71C22⋊Q812yes—
203C23×C71C22×D411yescharacteristic
204C22×D71—22no—
205C22×D71—24no—
206C22×D71—42no—
207C22×D71—42no—
208C2×C4×C71—24no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—48no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71C2×C4⋊C412yes—
217C2×C4×C71C2×C4⋊C412yes—
218C2×C4×C71C4×D412yes—
219C2×C4×C71C4×D412yes—
220C2×C4×C71C4×D412yes—
221C2×C4×C71#12011yescharacteristic
222C2×C4×C71#12011yescharacteristic
223C2×C4×C71C22⋊Q812yes—
224C2×C4×C71C22⋊Q812yes—
225C2×C4×C71C22⋊Q812yes—
226C2×C4×C71#12312yes—
227C2×C4×C71#12312yes—
228C2×C4×C71C23×C411yescharacteristic
229C2×C4×C71C22×D411yescharacteristic
230C2×C4×C71C22×D411yescharacteristic
231C2×C4×C71C22×Q811yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C22⋊D71—24no—
235C22⋊D71—24no—
236C22⋊D71—28no—
237C2×Dic71—22no—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—44no—
241C2×Dic71—44no—
242C2×Dic71—44no—
243C2×Dic71—42no—
244C2×Dic71—82no—
245C2×Dic71—82no—
246C2×Dic71—82no—
247D284—22no—
248D284—24no—
249D284C2×C4⋊C414yes—
250C4×D71—24no—
251C4×D71—24no—
252C4×D71—28no—
253C4×D71—88no—
254Dic142—22no—
255Dic142—24no—
256Dic142C2×C4⋊C414yes—
257C8×C71—42no—
258C71⋊C8—24no—
259C22×C71—24no—
260C22×C71—24no—
261C22×C71—21no—
262C22×C71—21no—
263C22×C71—21no—
264C22×C71—21no—
265C22×C71#38512yes—
266C22×C71#38512yes—
267C22×C71#45412yes—
268C22×C71#45512yes—
269C22×C71C2×C4×D411yescharacteristic
270C22×C71C2×#12011yescharacteristic
271C22×C71C2×C22⋊Q811yescharacteristic
272C2×D71—24no—
273C2×D71—48no—
274C2×D71—42no—
275C2×D71—42no—
276C4×C71—44no—
277C4×C71—42no—
278C4×C71—42no—
279C4×C71#38512yes—
280C4×C71#38512yes—
281C4×C71C22×C4⋊C411yescharacteristic
282C4×C71C2×C4×D411yescharacteristic
283C4×C71C2×C22⋊Q811yescharacteristic
284C4×C71C2×#12311yescharacteristic
285Dic71—24no—
286Dic71—44no—
287Dic71—82no—
288C2×#1860—711nomaximal, Sylow
289C2×C71—21no—
290C2×C71—21no—
291C2×C71—22no—
292C2×C71#186014yescomplemented
293C2×C71C2×#38511yescharacteristic
294C2×C71C2×#45511yescharacteristic
295C2×C71C2×#45411yescharacteristic
296D71—44no—
297C2×#520—711no—
298C2×#428—711no—
299C22×#104—711no—
300C2×#417—711no—
301C2×#382—711no—
302#1860—718no—
303C2×#343—711no—
304C2×#339D7111yescharacteristic, complemented
305C71C2×#186011yescharacteristic, complemented, Sylow
306C22×#41—711no—
307C22×M16—711no—
308C22×M16C2×D7111yescharacteristic
309#520—714no—
310#520—1422no—
311#520—1422no—
312#520—1422no—
313C2×C4×Q8—1421no—
314C2×C4×D4—1421no—
315C2×C4×D4—1422no—
316C2×#117—1421no—
317C2×#117C2×D7111yescharacteristic
318C22×C4⋊C4—711no—
319C22×C4⋊C4—711no—
320C22×C4⋊C4—1421no—
321C22×C42C2×D7111yescharacteristic
322#428—714no—
323#428—1422no—
324C2×#107—1421no—
325C2×#106—1421no—
326C2×#104—718no—
327C2×#104—714no—
328#417—714no—
329#417—1422no—
330C2×#103—1421no—
331C2×#102—1421no—
332#382—714no—
333#382—1422no—
334C2×#95—1422no—
335#343—714no—
336#339C2×D7114yescomplemented
337C2×#41—714no—
338C2×#41—712no—
339C2×#41—714no—
340C2×#41—712no—
341C2×#41—1421no—
342C2×#41—1421no—
343C2×#41—1422no—
344C2×#41—1422no—
345C22×Q8—711no—
346C22×D4—711no—
347C22×D4—1421no—
348C23×C4—21no—
349C23×C4—712no—
350C23×C4—1422no—
351C23×C4—1421no—
352C23×C4—1421no—
353C23×C4C22×D7111yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—714no—
358C2×M16—712no—
359C2×M16—1421no—
360C2×M16—1422no—
361C2×M16C22×D7112yes—
362C22×C8—21no—
363C22×C8—1421no—
364C4×Q8—1424no—
365C4×Q8—1424no—
366C4×D4—1424no—
367C4×D4—1424no—
368C4×D4—1428no—
369C4×D4—1428no—
370#117—21no—
371#117—21no—
372#117—1424no—
373#117—1424no—
374#117D28412yes—
375#117C22×D7112yes—
376C2×C4⋊C4—712no—
377C2×C4⋊C4—712no—
378C2×C4⋊C4—712no—
379C2×C4⋊C4—1424no—
380C2×C4⋊C4—1421no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1421no—
384C2×C4⋊C4—1421no—
385C2×C4⋊C4—1422no—
386C2×C4⋊C4—1421no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1424no—
391C2×C4⋊C4D28411yescharacteristic
392C2×C4⋊C4D28411yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1422no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—1422no—
399C2×C42C4×D7114yescomplemented
400C2×C42C22×D7112yes—
401#107—1424no—
402#106—1424no—
403#104—7116no—
404#103—1424no—
405#102—1424no—
406#95—1428no—
407C24—1421no—
408C24D4×D7111yescharacteristic
409#41—714no—
410#41—714no—
411#41—718no—
412#41—1422no—
413#41—1422no—
414#41—1424no—
415#41—1424no—
416#41—1424no—
417#41—1424no—
418#41—1424no—
419C2×Q8—712no—
420C2×Q8—714no—
421C2×Q8—1421no—
422C2×Q8—1422no—
423C2×D4—712no—
424C2×D4—714no—
425C2×D4—1422no—
426C2×D4—1424no—
427C2×D4—1422no—
428C2×D4—1424no—
429C2×D4—1421no—
430C2×D4—1422no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—2844no—
438C22×C4—2844no—
439C22×C4—2841no—
440C22×C4—2841no—
441C22×C4—5684no—
442C22×C4—714no—
443C22×C4—1428no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1421no—
448C22×C4—1422no—
449C22×C4—1424no—
450C22×C4—1422no—
451C22×C4—1424no—
452C22×C4C2×C4×D7112yes—
453C22×C4D4×D7111yescharacteristic
454C22×C4D4×D7111yescharacteristic
455C22×C4D4×D7111yescharacteristic
456C22×C4C2×D28411yescharacteristic
457C22×C4C2×D28411yescharacteristic
458C22×C4Q8×D7112yes—
459C22×C4#1176586412yes—
460C22×C4#1176586512yes—
461C22×C4C23×D7111yescharacteristic
462M16—22no—
463M16—22no—
464M16—42no—
465M16—714no—
466M16—1422no—
467M16—1424no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—1422no—
471C2×C8—1424no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—2841no—
475C4⋊C4—2841no—
476C4⋊C4—1428no—
477C4⋊C4—1422no—
478C4⋊C4—1422no—
479C4⋊C4—1422no—
480C4⋊C4—1422no—
481C4⋊C4—1422no—
482C4⋊C4—1422no—
483C4⋊C4C2×D28412yes—
484C4⋊C4C2×D28412yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—1428no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—1428no—
493C42C2×C4×D7114yescomplemented
494C23—22no—
495C23—22no—
496C23—2842no—
497C23—2842no—
498C23—1424no—
499C23—1422no—
500C23(C22⋊D71)⋊C412yes—
501C23#1455609811yescharacteristic
502C23#1455609811yescharacteristic
503C23#1455610612yes—
504C23C2×D4×D7111yescharacteristic
505Q8—714no—
506Q8—1424no—
507Q8—1422no—
508D4—714no—
509D4—1428no—
510D4—1424no—
511D4—1424no—
512D4—1424no—
513D4—1422no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—2844no—
523C2×C4—2844no—
524C2×C4—2844no—
525C2×C4—2842no—
526C2×C4—5688no—
527C2×C4—5682no—
528C2×C4—5682no—
529C2×C4—5682no—
530C2×C4—1424no—
531C2×C4—1424no—
532C2×C4—1422no—
533C2×C4—1428no—
534C2×C4—1424no—
535C2×C4—1424no—
536C2×C4C4⋊C4×D7112yes—
537C2×C4C4⋊C4×D7112yes—
538C2×C4(C22⋊D71)⋊C412yes—
539C2×C4C4×D28412yes—
540C2×C4C4×D28412yes—
541C2×C4#1455609811yescharacteristic
542C2×C4#1455609811yescharacteristic
543C2×C4#1455610612yes—
544C2×C4#1455610812yes—
545C2×C4#1455610812yes—
546C2×C4#1455611012yes—
547C2×C4#1455611012yes—
548C2×C4C22×C4×D7111yescharacteristic
549C2×C4C2×D4×D7111yescharacteristic
550C2×C4C22×D28411yescharacteristic
551C2×C4C2×Q8×D7111yescharacteristic
552C2×C4C2×#1176586411yescharacteristic
553C2×C4C2×#1176586511yescharacteristic
554C8—42no—
555C8—1424no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—2842no—
563C22—2842no—
564C22—2848no—
565C22—1424no—
566C22#1798587312yes—
567C22#1798587312yes—
568C22#1798617312yes—
569C22#1798617812yes—
570C22C2×(C22⋊D71)⋊C411yescharacteristic
571C22C2×#1455609811yescharacteristic
572C22C2×#1455610611yescharacteristic
573C4—44no—
574C4—42no—
575C4—42no—
576C4—2844no—
577C4—5682no—
578C4—1424no—
579C4#1798587312yes—
580C4#1798587312yes—
581C4C2×C4⋊C4×D7111yescharacteristic
582C4C2×C4×D28411yescharacteristic
583C4C2×#1455610811yescharacteristic
584C4C2×#1455611011yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—2844no—
589C2#2258857414yesdirect product factor
590C2C2×#1798587311yescharacteristic
591C2C2×#1798617811yescharacteristic
592C2C2×#1798617311yescharacteristic
subgroupG/H
centerC23#14556098
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#339×C71C2
socleC23×C71#120
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenterC2×#339D71
hypocenterC71C2×#1860
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. C2×#17986042C2semidirect
5. C2×#17985865C2semidirect
6. #22588574C2direct (N × H)
7. C2×#17985767C2semidirect
8. C2×#339×C71C2semidirect
23. #17986042C22semidirect
26. C2×#14556056C4semidirect
33. #17985865C22semidirect
35. #17985767C22semidirect
39. #339×C71C22semidirect
57. C2×#11765849C2×C4semidirect
87. #14556056C2×C4semidirect
108. C2×C42×C71C2×C4semidirect
150. #11765849C22×C4semidirect
196. C42×C71C22×C4semidirect
292. C2×C71#1860semidirect
304. C2×#339D71semidirect
305. C71C2×#1860semidirect
336. #339C2×D71semidirect
399. C2×C42C4×D71semidirect
493. C42C2×C4×D71semidirect
589. C2#22588574direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179864792
C2×#179860422
C22×#145560562
C2×#179860012
C2×#179858652
#225885742
C2×#179857672
C2×#339×C712
C2×#186071