Finite Group Databasebeta

← C2×#22598263 #32865230

Subgroups — C2×#22598263 #32865230

Order 18176.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2#22598263C2116yesmaximal, direct product factor
3C2×#17986334C212yesmaximal, complemented
4C2×#17986317C212yesmaximal, complemented
5C2×#17986329C212yesmaximal, complemented
6C2×#17986312C212yesmaximal, complemented
7C2×#17986039C212yesmaximal, complemented
8C2×#17985998C212yesmaximal, complemented
9C2×#17985963C211yesmaximal, characteristic, complemented
10C23×C4⋊C4×C71C211yesmaximal, characteristic, complemented
11C22×#11765865C2211yescharacteristic
12C22×#11765849C2212yes—
13#17986510—24no—
14#17986510C2218yes—
15C2×#14556111—21no—
16C2×#14556107C2211yescharacteristic
17C2×#14556107C2211yescharacteristic
18C2×#14556102C2211yescharacteristic
19C2×#14556102C2211yescharacteristic
20C2×#14556097—21no—
21C2×C4×C22⋊D71—21no—
22C2×C4×C22⋊D71—21no—
23C2×#14556087C2211yescharacteristic
24#17986334C22116yescomplemented
25#17986329C22116yescomplemented
26#17986317C22116yescomplemented
27#17986312C22116yescomplemented
28#17986039C22116yescomplemented
29C2×#14556061C2214yescomplemented
30C2×#14556059C2214yescomplemented
31#17985998C22116yescomplemented
32C2×#14556051C2214yescomplemented
33C2×#14556048C2214yescomplemented
34#17985963C2218yescomplemented
35C2×C22⋊(C71⋊C8)C2212yes—
36C2×C22⋊(C71⋊C8)C2212yes—
37C24×C4×C71—21no—
38C24×C4×C71C2211yescharacteristic
39C22×C4⋊C4×C71—24no—
40C22×C4⋊C4×C71—21no—
41C22×C4⋊C4×C71—21no—
42C22×C4⋊C4×C71—21no—
43C22×C4⋊C4×C71C2212yes—
44C22×C4⋊C4×C71C2212yes—
45C22×C4⋊C4×C71C2212yes—
46C22×C4⋊C4×C71C2218yescomplemented
47C2×#11765865—22no—
48C2×#11765865—28no—
49C2×#11765865C2314yes—
50C22×Dic142C2311yescharacteristic
51C22×D284C2311yescharacteristic
52C22×C22⋊D71—21no—
53C22×C22⋊D71—21no—
54C22×C22⋊D71—21no—
55C22×C4×D71—21no—
56C2×#11765849—28no—
57C2×#11765849C2318yes—
58C22×C71⋊C8C2314yes—
59#14556111—28no—
60#14556107—22no—
61#14556107—22no—
62#14556107C2314yes—
63#14556107C2314yes—
64#14556102—22no—
65#14556102—22no—
66#14556102C2314yes—
67#14556102C2314yes—
68#14556097—28no—
69C4×C22⋊D71—28no—
70C4×C22⋊D71—28no—
71#14556087—22no—
72#14556087C2314yes—
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×C22⋊Dic71—21no—
78C2×C22⋊Dic71—21no—
79C2×C22⋊Dic71—21no—
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82C2×C4×Dic71—21no—
83#14556061C23116yescomplemented
84#14556059C23116yescomplemented
85#14556051C23116yescomplemented
86#14556048C23116yescomplemented
87C22⋊(C71⋊C8)C2318yes—
88C22⋊(C71⋊C8)C2318yes—
89C25×C71D411yescharacteristic
90C23×C4×C71—28no—
91C23×C4×C71—22no—
92C23×C4×C71—22no—
93C23×C4×C71—22no—
94C23×C4×C71—28no—
95C23×C4×C71—48no—
96C23×C4×C71D411yescharacteristic
97C23×C4×C71D411yescharacteristic
98C23×C4×C71D411yescharacteristic
99C23×C4×C71D414yes—
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C23×C4×C71C2314yes—
103C23×C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71—22no—
105C2×C4⋊C4×C71—22no—
106C2×C4⋊C4×C71—24no—
107C2×C4⋊C4×C71—24no—
108C2×C4⋊C4×C71—22no—
109C2×C4⋊C4×C71—24no—
110C2×C4⋊C4×C71—28no—
111C2×C4⋊C4×C71—28no—
112C2×C4⋊C4×C71—28no—
113C2×C4⋊C4×C71C2318yes—
114C2×C4⋊C4×C71C2318yes—
115C2×C4⋊C4×C71C2314yes—
116C2×C4⋊C4×C71C2318yes—
117C23×D71—21no—
118#11765865—216no—
119#11765865—48no—
120C2×Dic142—24no—
121C2×Dic142—21no—
122C2×Dic142C2412yes—
123C2×D284—24no—
124C2×D284—21no—
125C2×D284C2412yes—
126C2×C22⋊D71—24no—
127C2×C22⋊D71—24no—
128C2×C22⋊D71—24no—
129C2×C22⋊D71—44no—
130C2×C22⋊D71—44no—
131C2×C22⋊D71—44no—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C2×C4×D71—24no—
136C2×C4×D71—44no—
137#11765849—216no—
138C2×C71⋊C8—28no—
139C2×C71⋊C8C2418yes—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71C2412yes—
143C4⋊Dic71C2412yes—
144Dic71⋊C4—24no—
145Dic71⋊C4—24no—
146C22⋊Dic71—24no—
147C22⋊Dic71—24no—
148C22⋊Dic71—24no—
149(C2×D71)⋊C4—24no—
150(C2×D71)⋊C4—24no—
151C4×Dic71—24no—
152C24×C71—24no—
153C24×C71—24no—
154C24×C71—24no—
155C24×C71C2×D414yes—
156C24×C71C2×D411yescharacteristic
157C24×C71C2×D411yescharacteristic
158C24×C71C2×D411yescharacteristic
159C22×C4×C71—24no—
160C22×C4×C71—24no—
161C22×C4×C71—24no—
162C22×C4×C71—216no—
163C22×C4×C71—24no—
164C22×C4×C71—24no—
165C22×C4×C71—216no—
166C22×C4×C71—28no—
167C22×C4×C71—28no—
168C22×C4×C71—28no—
169C22×C4×C71—24no—
170C22×C4×C71—48no—
171C22×C4×C71—48no—
172C22×C4×C71—432no—
173C22×C4×C71—48no—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D414yes—
177C22×C4×C71C2×D414yes—
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2×D412yes—
180C22×C4×C71C2×D414yes—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D412yes—
183C22×C4×C71#4118yes—
184C22×C4×C71C2412yes—
185C22×C4×C71C2412yes—
186C22×C4×C71C2411yescharacteristic
187C22×C4×C71C2412yes—
188C4⋊C4×C71—28no—
189C4⋊C4×C71—28no—
190C4⋊C4×C71—28no—
191C4⋊C4×C71—24no—
192C4⋊C4×C71C2418yes—
193C23×C71—232no—
194C23×C71—28no—
195C23×C71—22no—
196C23×C71—22no—
197C23×C71—28no—
198C23×C71—22no—
199C23×C71—22no—
200C23×C71—28no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71#12314yes—
204C23×C71AGL(1,ℤ/8)14yes—
205C23×C71#13714yes—
206C23×C71C22×D412yes—
207C23×C71C22×D412yes—
208C23×C71C22×D412yes—
209C23×C71C22×D411yescharacteristic
210C22×D71—22no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C2×C4×C71—28no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—28no—
220C2×C4×C71—28no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—28no—
224C2×C4×C71—416no—
225C2×C4×C71—416no—
226C2×C4×C71—416no—
227C2×C4×C71—48no—
228C2×C4×C71#12314yes—
229C2×C4×C71#12318yes—
230C2×C4×C71C22×D412yes—
231C2×C4×C71C22×D412yes—
232C2×C4×C71C22×D412yes—
233C2×C4×C71C22×D411yescharacteristic
234C2×C4×C71C2×#4114yes—
235C2×C4×C71C2×#4118yes—
236C2×C4×C71C2511yescharacteristic
237C22⋊D71—48no—
238C22⋊D71—48no—
239C22⋊D71—48no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246D284—24no—
247D284—42no—
248C4×D71—48no—
249Dic142—24no—
250Dic142—42no—
251C71⋊C8—28no—
252C22×C71—232no—
253C22×C71—28no—
254C22×C71—28no—
255C22×C71—28no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71C2×#12314yes—
263C22×C71C2×#12312yes—
264C22×C71C2×AGL(1,ℤ/8)14yes—
265C22×C71C2×AGL(1,ℤ/8)12yes—
266C22×C71C2×#13714yes—
267C22×C71C2×#13712yes—
268C22×C71C23×D411yescharacteristic
269C2×D71—42no—
270C2×D71—42no—
271C2×D71—42no—
272C2×D71—84no—
273C4×C71—22no—
274C4×C71—22no—
275C4×C71—48no—
276C4×C71C2×#12312yes—
277C4×C71C2×#12314yes—
278C4×C71C22×#4112yes—
279Dic71—42no—
280Dic71—42no—
281Dic71—42no—
282C2×#3092—711nomaximal, Sylow
283C2×C71—24no—
284C2×C71—24no—
285C2×C71—24no—
286C2×C71#309214yescomplemented
287C2×C71C22×#12311yescharacteristic
288C2×C71C22×AGL(1,ℤ/8)11yescharacteristic
289C2×C71C22×#13711yescharacteristic
290D71—82no—
291C2×#525—711no—
292C23×C4⋊C4D7111yescharacteristic, complemented
293#3092—7116no—
294C2×#484—712no—
295C2×#481—712no—
296C2×#483—712no—
297C2×#480—712no—
298C2×#428—712no—
299C2×#417—712no—
300C2×#407—711no—
301C71C2×#309211yescharacteristic, complemented, Sylow
302C22×#41—711no—
303C24×C4—21no—
304C24×C4C2×D7111yescharacteristic
305C22×M16—712no—
306#525—718no—
307#525—1424no—
308C2×#123—1421no—
309C2×C22⋊Q8—711no—
310C2×C22⋊Q8—711no—
311C2×#121—711no—
312C2×#121—711no—
313C2×#120—1421no—
314C2×C4×D4—1421no—
315C2×C4×D4—1421no—
316C2×#117—711no—
317C22×C4⋊C4—24no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4—21no—
321C22×C4⋊C4C2×D7112yes—
322C22×C4⋊C4C2×D7112yes—
323C22×C4⋊C4C2×D7112yes—
324C22×C4⋊C4C2×D7118yescomplemented
325#484—7116no—
326#483—7116no—
327#481—7116no—
328#480—7116no—
329#428—7116no—
330C2×#107—714no—
331C2×#106—714no—
332#417—7116no—
333C2×#103—714no—
334C2×#102—714no—
335#407—718no—
336C2×C22⋊C8—712no—
337C2×C22⋊C8—712no—
338C25C22⋊D7111yescharacteristic
339C2×#41—714no—
340C2×#41—1428no—
341C2×#41—1422no—
342C22×Q8—711no—
343C22×D4—711no—
344C22×D4—1421no—
345C22×D4—1421no—
346C22×D4—1421no—
347C23×C4—28no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—28no—
352C23×C4—48no—
353C23×C4—1421no—
354C23×C4C22⋊D7111yescharacteristic
355C23×C4C22⋊D7111yescharacteristic
356C23×C4C22⋊D7111yescharacteristic
357C23×C4C22⋊D7114yes—
358C23×C4C22×D7111yescharacteristic
359C23×C4C22×D7111yescharacteristic
360C23×C4C22×D7114yes—
361C23×C4C22×D7111yescharacteristic
362C2×M16—718no—
363C2×M16—1428no—
364C22×C8—714no—
365#123—1428no—
366C22⋊Q8—714no—
367C22⋊Q8—714no—
368C22⋊Q8—1422no—
369C22⋊Q8—1422no—
370#121—714no—
371#121—714no—
372#121—1422no—
373#121—1422no—
374#120—1428no—
375C4×D4—1428no—
376C4×D4—1428no—
377#117—714no—
378#117—1422no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4C22×D7114yes—
393C2×C4⋊C4C22×D7118yes—
394C2×C4⋊C4C22×D7118yes—
395C2×C4⋊C4C22×D7118yes—
396C2×C22⋊C4—1421no—
397C2×C22⋊C4—1421no—
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1421no—
400C2×C22⋊C4—1421no—
401C2×C42—1421no—
402#107—7116no—
403#106—7116no—
404#103—7116no—
405#102—7116no—
406C22⋊C8—718no—
407C22⋊C8—718no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—1421no—
412C24C2×C22⋊D7114yes—
413C24C2×C22⋊D7111yescharacteristic
414C24C2×C22⋊D7111yescharacteristic
415C24C2×C22⋊D7111yescharacteristic
416#41—2848no—
417#41—14216no—
418C2×Q8—712no—
419C2×Q8—1424no—
420C2×Q8—1421no—
421C2×D4—2844no—
422C2×D4—2844no—
423C2×D4—2844no—
424C2×D4—712no—
425C2×D4—1424no—
426C2×D4—1424no—
427C2×D4—1424no—
428C2×D4—1421no—
429C2×D4—1424no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—2844no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4—1424no—
450C22×C4C2×C22⋊D7114yes—
451C22×C4C2×C22⋊D7114yes—
452C22×C4C2×C22⋊D7112yes—
453C22×C4C2×C22⋊D7112yes—
454C22×C4C2×C22⋊D7114yes—
455C22×C4C2×C22⋊D7111yescharacteristic
456C22×C4C2×C22⋊D7111yescharacteristic
457C22×C4C2×C22⋊D7112yes—
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4#1176586518yes—
460C22×C4C23×D7112yes—
461C22×C4C23×D7112yes—
462C22×C4C23×D7112yes—
463C22×C4C23×D7111yescharacteristic
464M16—14216no—
465C2×C8—718no—
466C2×C8—1428no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—712no—
472C4⋊C4—712no—
473C4⋊C4—1424no—
474C4⋊C4—1424no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C4⋊C4C23×D7118yes—
478C22⋊C4—1424no—
479C22⋊C4—1424no—
480C22⋊C4—1424no—
481C22⋊C4—1424no—
482C22⋊C4—1424no—
483C42—1424no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—2842no—
495C23—2842no—
496C23—2842no—
497C23—1422no—
498C23#1455611114yes—
499C23#1455615414yes—
500C23#1455615914yes—
501C23C22×C22⋊D7112yes—
502C23C22×C22⋊D7112yes—
503C23C22×C22⋊D7112yes—
504C23C22×C22⋊D7111yescharacteristic
505Q8—2842no—
506Q8—1424no—
507D4—2848no—
508D4—2848no—
509D4—2848no—
510D4—2842no—
511D4—1424no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—48no—
526C2×C4—2842no—
527C2×C4—2842no—
528C2×C4—2842no—
529C2×C4—2848no—
530C2×C4—1422no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4#1455611114yes—
534C2×C4#1455611118yes—
535C2×C4C22×C22⋊D7112yes—
536C2×C4C22×C22⋊D7112yes—
537C2×C4C22×C22⋊D7112yes—
538C2×C4C22×C22⋊D7111yescharacteristic
539C2×C4C2×#1176586514yes—
540C2×C4C2×#1176586518yes—
541C2×C4C24×D7111yescharacteristic
542C8—1428no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—2842no—
554C22—2842no—
555C22—2842no—
556C22—5684no—
557C22C2×#1455611114yes—
558C22C2×#1455611112yes—
559C22C2×#1455615414yes—
560C22C2×#1455615412yes—
561C22C2×#1455615914yes—
562C22C2×#1455615912yes—
563C22C23×C22⋊D7111yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—2842no—
568C4—2842no—
569C4—2842no—
570C4C2×#1455611114yes—
571C4C2×#1455611112yes—
572C4C22×#1176586512yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—5682no—
577C2#2259826314yesdirect product factor
578C2C22×#1455611111yescharacteristic
579C2C22×#1455615411yescharacteristic
580C2C22×#1455615911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC23×C4⋊C4×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986510C2
hypercenterC23×C4⋊C4D71
hypocenterC71C2×#3092
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22598263C2direct (N × H)
3. C2×#17986334C2semidirect
4. C2×#17986317C2semidirect
5. C2×#17986329C2semidirect
6. C2×#17986312C2semidirect
7. C2×#17986039C2semidirect
8. C2×#17985998C2semidirect
9. C2×#17985963C2semidirect
10. C23×C4⋊C4×C71C2semidirect
24. #17986334C22semidirect
25. #17986329C22semidirect
26. #17986317C22semidirect
27. #17986312C22semidirect
28. #17986039C22semidirect
29. C2×#14556061C22semidirect
30. C2×#14556059C22semidirect
31. #17985998C22semidirect
32. C2×#14556051C22semidirect
33. C2×#14556048C22semidirect
34. #17985963C22semidirect
46. C22×C4⋊C4×C71C22semidirect
83. #14556061C23semidirect
84. #14556059C23semidirect
85. #14556051C23semidirect
86. #14556048C23semidirect
286. C2×C71#3092semidirect
292. C23×C4⋊C4D71semidirect
301. C71C2×#3092semidirect
324. C22×C4⋊C4C2×D71semidirect
577. C2#22598263direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179865102
#225982632
C2×#179863342
C2×#179863172
C2×#179863292
C2×#179863122
C2×#179860392
C2×#179859982
C2×#179859632
C23×C4⋊C4×C712
C2×#309271