Finite Group Databasebeta

← C2×#22593816 #32167967

Subgroups — C2×#22593816 #32167967

Order 18176.

3438 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556103C211yesmaximal, characteristic
2C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#22593816C2116yesmaximal, direct product factor
4C2×#17985937C211yesmaximal, characteristic, complemented
5C2×#17985917C211yesmaximal, characteristic, complemented
6C2×#17985916C211yesmaximal, characteristic, complemented
7C2×#17985891C212yesmaximal, complemented
8C2×#17985888C212yesmaximal, complemented
9C2×#17985877C212yesmaximal, complemented
10C2×#17985877C212yesmaximal, complemented
11C2×#17985846C211yesmaximal, characteristic, complemented
12C2×#400×C71C211yesmaximal, characteristic, complemented
13C23×D284C2212yes—
14C23×D284C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C23×C4×D71C2211yescharacteristic
17C23×C4×D71C2211yescharacteristic
18C2×#14556103—28no—
19C2×#14556103C2218yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×Dic71⋊C4C2211yescharacteristic
22C22×Dic71⋊C4C2211yescharacteristic
23C2×C4⋊C4×D71—28no—
24C2×C4⋊C4×D71C2218yescomplemented
25C22×(C2×D71)⋊C4C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31C22×C4×Dic71C2212yes—
32#17985937C2218yescomplemented
33#17985917C2218yescomplemented
34#17985916C2218yescomplemented
35#17985891C22116yescomplemented
36#17985888C22116yescomplemented
37#17985877C22116yescomplemented
38#17985877C22116yescomplemented
39#17985846C2218yescomplemented
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2211yescharacteristic
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45#400×C71C2218yescomplemented
46C2×#95×C71C2212yes—
47C2×#95×C71C2212yes—
48C2×#95×C71C2211yescharacteristic
49C24×D71C2312yes—
50C24×D71C2311yescharacteristic
51C22×D284—28no—
52C22×D284—28no—
53C22×D284—28no—
54C22×D284—24no—
55C22×D284—44no—
56C22×D284C2318yes—
57C22×D284C2314yes—
58C23×Dic71C2311yescharacteristic
59C23×Dic71C2312yes—
60C23×Dic71C2311yescharacteristic
61C23×Dic71C2311yescharacteristic
62C22×C4×D71—24no—
63C22×C4×D71—24no—
64C22×C4×D71—24no—
65C22×C4×D71—24no—
66C22×C4×D71—44no—
67C22×C4×D71—44no—
68C22×C4×D71D418yes—
69C22×C4×D71C2314yes—
70C22×C4×D71C2314yes—
71C22×C4×D71C2314yes—
72#14556103—232no—
73C2×C4⋊Dic71—24no—
74C2×C4⋊Dic71C2314yes—
75C2×Dic71⋊C4—24no—
76C2×Dic71⋊C4—24no—
77C2×Dic71⋊C4C2314yes—
78C2×Dic71⋊C4C2314yes—
79C4⋊C4×D71—232no—
80C2×(C2×D71)⋊C4—28no—
81C2×(C2×D71)⋊C4—28no—
82C2×(C2×D71)⋊C4—28no—
83C2×(C2×D71)⋊C4—28no—
84C2×(C2×D71)⋊C4—44no—
85C2×(C2×D71)⋊C4—44no—
86C2×(C2×D71)⋊C4C2318yes—
87C2×(C2×D71)⋊C4C2318yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2318yes—
90C2×(C2×D71)⋊C4C2318yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×C4×Dic71—44no—
93C2×C4×Dic71C2318yes—
94C71⋊#95C2314yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2312yes—
99C23×C4×C71C2312yes—
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71—24no—
103C2×C4⋊C4×C71—24no—
104C2×C4⋊C4×C71C2314yes—
105C2×C4⋊C4×C71C2314yes—
106#95×C71C2314yes—
107#95×C71C2318yes—
108#95×C71C2318yes—
109C23×D71—22no—
110C23×D71—22no—
111C23×D71—24no—
112C23×D71—24no—
113C23×D71—24no—
114C23×D71—24no—
115C23×D71—24no—
116C23×D71—24no—
117C23×D71—24no—
118C23×D71—24no—
119C23×D71C2×D414yes—
120C23×D71#4118yes—
121C23×D71C2414yes—
122C23×D71C2412yes—
123C2×D284—216no—
124C2×D284—216no—
125C2×D284—216no—
126C2×D284—28no—
127C2×D284—48no—
128C2×D284—416no—
129C2×D284—432no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—24no—
133C22×Dic71—22no—
134C22×Dic71—24no—
135C22×Dic71—24no—
136C22×Dic71—24no—
137C22×Dic71—24no—
138C22×Dic71C2×D414yes—
139C22×Dic71#4114yes—
140C22×Dic71C2412yes—
141C22×Dic71C2414yes—
142C22×Dic71C2412yes—
143C22×Dic71C2412yes—
144C2×C4×D71—232no—
145C2×C4×D71—28no—
146C2×C4×D71—28no—
147C2×C4×D71—28no—
148C2×C4×D71—28no—
149C2×C4×D71—416no—
150C2×C4×D71—48no—
151C2×C4×D71—48no—
152C2×C4×D71—416no—
153C2×C4×D71C2×D4116yes—
154C4⋊Dic71—28no—
155Dic71⋊C4—28no—
156Dic71⋊C4—28no—
157(C2×D71)⋊C4—216no—
158(C2×D71)⋊C4—216no—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—216no—
161(C2×D71)⋊C4—48no—
162(C2×D71)⋊C4—48no—
163C4×Dic71—48no—
164C24×C71C2411yescharacteristic
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—24no—
168C22×C4×C71—22no—
169C22×C4×C71—24no—
170C22×C4×C71—24no—
171C22×C4×C71—24no—
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71C2×D414yes—
177C22×C4×C71#4114yes—
178C22×C4×C71C2412yes—
179C22×C4×C71C2414yes—
180C22×C4×C71C2414yes—
181C22×C4×C71C2412yes—
182C22×C4×C71C2412yes—
183C4⋊C4×C71—28no—
184C4⋊C4×C71—28no—
185C23×C71C22×D412yes—
186C23×C71C2×#4112yes—
187C23×C71C2×#4114yes—
188C23×C71C2×#4112yes—
189C23×C71H(5,𝔽2)14yes—
190C23×C71C2511yescharacteristic
191C22×D71—28no—
192C22×D71—28no—
193C22×D71—216no—
194C22×D71—216no—
195C22×D71—22no—
196C22×D71—22no—
197C22×D71—24no—
198C22×D71—24no—
199C22×D71—24no—
200C22×D71—24no—
201C22×D71—24no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—48no—
206C22×D71—48no—
207C22×D71—48no—
208C22×D71—48no—
209C22×D71—48no—
210C22×D71—48no—
211C22×D71—48no—
212C22×D71#124116yes—
213C22×D71C22×D414yes—
214C22×D71C2×#4118yes—
215C2×C4×C71—28no—
216C2×C4×C71—28no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—48no—
229C2×C4×C71—44no—
230C2×C4×C71—48no—
231C2×C4×C71C22×D414yes—
232C2×C4×C71C2×#4114yes—
233C2×Dic71—28no—
234C2×Dic71—28no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—24no—
238C2×Dic71—22no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—44no—
244C2×Dic71—48no—
245C2×Dic71C22×D414yes—
246C2×Dic71C2×#4114yes—
247D284—416no—
248D284—432no—
249C4×D71—232no—
250C4×D71—416no—
251C4×D71—416no—
252C22×C71#52514yes—
253C22×C71C2×#12414yes—
254C22×C71#53414yes—
255C22×C71#55318yes—
256C22×C71#55318yes—
257C22×C71C23×D411yescharacteristic
258C22×C71C22×#4111yescharacteristic
259C22×C71C22×#4112yes—
260C22×C71C22×#4111yescharacteristic
261C22×C71C2×H(5,𝔽2)12yes—
262C2×D71—232no—
263C2×D71—24no—
264C2×D71—24no—
265C2×D71—28no—
266C2×D71—28no—
267C2×D71—44no—
268C2×D71—44no—
269C2×D71—44no—
270C2×D71—44no—
271C2×D71—44no—
272C2×D71—44no—
273C2×D71—44no—
274C2×D71—816no—
275C2×D71C2×#12418yes—
276C4×C71—24no—
277C4×C71—24no—
278C4×C71—44no—
279C4×C71—42no—
280C4×C71—44no—
281Dic71—24no—
282Dic71—24no—
283Dic71—42no—
284Dic71—44no—
285C2×#2591—711nomaximal, Sylow
286C2×C71#259118yescomplemented
287C2×C71C2×#52511yescharacteristic
288C2×C71C22×#12411yescharacteristic
289C2×C71C2×#53411yescharacteristic
290C2×C71C2×#55312yes—
291C2×C71C2×#55312yes—
292D71—28no—
293D71—84no—
294C22×#121—711no—
295C23×C4⋊C4—711no—
296#2591—7116no—
297C2×#400—711no—
298C2×#400D7111yescharacteristic, complemented
299C2×#394—711no—
300C2×#394—711no—
301C2×#388—712no—
302C2×#388—712no—
303C2×#386—712no—
304C2×#386—712no—
305C2×C4×C22⋊C4—711no—
306C71C2×#259111yescharacteristic, complemented, Sylow
307C23×D4—711no—
308C23×D4—712no—
309C24×C4—711no—
310C24×C4—711no—
311C24×C4—711no—
312C2×#121—718no—
313C2×#121—1428no—
314C22×C4⋊C4—711no—
315C22×C4⋊C4—711no—
316C22×C4⋊C4—711no—
317C22×C4⋊C4—718no—
318C22×C4⋊C4—1428no—
319C22×C4⋊C4C2×D7111yescharacteristic
320C22×C4⋊C4C2×D7111yescharacteristic
321C22×C22⋊C4—712no—
322C22×C22⋊C4—712no—
323C22×C22⋊C4—712no—
324C22×C22⋊C4—712no—
325C22×C22⋊C4—712no—
326C22×C22⋊C4—712no—
327C22×C42—712no—
328#400—718no—
329#400C2×D7118yescomplemented
330#394—718no—
331#394—718no—
332#388—7116no—
333#388—7116no—
334#386—7116no—
335#386—7116no—
336C4×C22⋊C4—718no—
337C2×#95—711no—
338C2×#95—712no—
339C2×#95—712no—
340C2×#95C2×D7111yescharacteristic
341C2×#95C2×D7112yes—
342C2×#95C2×D7112yes—
343C25—711no—
344C25—712no—
345C22×D4—2844no—
346C22×D4—714no—
347C22×D4—718no—
348C22×D4—1428no—
349C22×D4—1428no—
350C22×D4—1428no—
351C22×D4—1424no—
352C23×C4—2844no—
353C23×C4—2844no—
354C23×C4—718no—
355C23×C4—714no—
356C23×C4—714no—
357C23×C4—711no—
358C23×C4—711no—
359C23×C4—712no—
360C23×C4—711no—
361C23×C4—714no—
362C23×C4—1424no—
363C23×C4—1424no—
364C23×C4—1424no—
365C23×C4—1424no—
366C23×C4C22×D7112yes—
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7112yes—
369C23×C4C22×D7111yescharacteristic
370C23×C4C22×D7111yescharacteristic
371#121—14232no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—714no—
375C2×C4⋊C4—714no—
376C2×C4⋊C4—714no—
377C2×C4⋊C4—14232no—
378C2×C4⋊C4—1424no—
379C2×C4⋊C4—1424no—
380C2×C4⋊C4—1424no—
381C2×C4⋊C4C22×D7114yes—
382C2×C4⋊C4C22×D7114yes—
383C2×C22⋊C4—2844no—
384C2×C22⋊C4—2844no—
385C2×C22⋊C4—718no—
386C2×C22⋊C4—718no—
387C2×C22⋊C4—718no—
388C2×C22⋊C4—718no—
389C2×C22⋊C4—718no—
390C2×C22⋊C4—718no—
391C2×C22⋊C4—1428no—
392C2×C22⋊C4—1428no—
393C2×C22⋊C4—1428no—
394C2×C22⋊C4—1428no—
395C2×C42—2844no—
396C2×C42—718no—
397#95—714no—
398#95—718no—
399#95—718no—
400#95C22×D7114yes—
401#95C22×D7118yes—
402#95C22×D7118yes—
403C24—714no—
404C24—718no—
405C24—712no—
406C24—714no—
407C24—1424no—
408C24—1424no—
409C24—1424no—
410C24—1424no—
411C24—1424no—
412C24—1424no—
413C24—1424no—
414C24—1422no—
415C24—1422no—
416C24—1424no—
417C24C23×D7111yescharacteristic
418C2×D4—28416no—
419C2×D4—2848no—
420C2×D4—28432no—
421C2×D4—14216no—
422C2×D4—14216no—
423C2×D4—14216no—
424C2×D4—1428no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28416no—
437C22×C4—2848no—
438C22×C4—2848no—
439C22×C4—28416no—
440C22×C4—7116no—
441C22×C4—714no—
442C22×C4—714no—
443C22×C4—712no—
444C22×C4—712no—
445C22×C4—714no—
446C22×C4—712no—
447C22×C4—1428no—
448C22×C4—1428no—
449C22×C4—1428no—
450C22×C4—14232no—
451C22×C4—1422no—
452C22×C4—1424no—
453C22×C4—1424no—
454C22×C4—1424no—
455C22×C4—1424no—
456C22×C4—1422no—
457C22×C4—1422no—
458C22×C4—1424no—
459C22×C4—1428no—
460C22×C4D4×D7114yes—
461C22×C4#1176586614yes—
462C22×C4C23×D7114yes—
463C22×C4C23×D7114yes—
464C22×C4C23×D7112yes—
465C22×C4C23×D7112yes—
466C22×C4C23×D7112yes—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—1428no—
470C4⋊C4—1428no—
471C4⋊C4—1428no—
472C22⋊C4—2848no—
473C22⋊C4—2848no—
474C22⋊C4—14216no—
475C22⋊C4—14216no—
476C22⋊C4—14216no—
477C22⋊C4—14216no—
478C42—2848no—
479C23—2848no—
480C23—2848no—
481C23—2848no—
482C23—2848no—
483C23—2848no—
484C23—2848no—
485C23—2848no—
486C23—714no—
487C23—718no—
488C23—7116no—
489C23—1424no—
490C23—1424no—
491C23—1424no—
492C23—1424no—
493C23—1424no—
494C23—1424no—
495C23—1424no—
496C23—1428no—
497C23—1422no—
498C23—1422no—
499C23—1424no—
500C23—14216no—
501C23—14216no—
502C23—1428no—
503C23C2×D4×D7112yes—
504C23#41×D7112yes—
505C23#41×D7114yes—
506C23C2×#1176586612yes—
507C23#1455617414yes—
508C23C24×D7111yescharacteristic
509D4—28416no—
510D4—28432no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—28416no—
528C2×C4—28416no—
529C2×C4—2844no—
530C2×C4—2848no—
531C2×C4—714no—
532C2×C4—714no—
533C2×C4—14232no—
534C2×C4—1422no—
535C2×C4—1424no—
536C2×C4—1424no—
537C2×C4—1424no—
538C2×C4—1424no—
539C2×C4—1428no—
540C2×C4—1422no—
541C2×C4—1422no—
542C2×C4—1424no—
543C2×C4—1428no—
544C2×C4C2×D4×D7114yes—
545C2×C4C2×#1176586614yes—
546C22—2844no—
547C22—2844no—
548C22—2844no—
549C22—2844no—
550C22—2844no—
551C22—2844no—
552C22—2844no—
553C22—56816no—
554C22—718no—
555C22—14232no—
556C22—1424no—
557C22—1428no—
558C22—1428no—
559C22—1424no—
560C22#1798651214yes—
561C22#124×D7114yes—
562C22#1798656014yes—
563C22#1798670018yes—
564C22#1798670218yes—
565C22C22×D4×D7111yescharacteristic
566C22C2×#41×D7111yescharacteristic
567C22C2×#41×D7112yes—
568C22C22×#1176586611yescharacteristic
569C22C2×#1455617412yes—
570C4—24no—
571C4—24no—
572C4—44no—
573C4—44no—
574C4—42no—
575C4—2844no—
576C4—2842no—
577C4—1424no—
578C4—1424no—
579C2—5684no—
580C2—1428no—
581C2#2259381618yesdirect product factor
582C2C2×#1798651211yescharacteristic
583C2C2×#124×D7111yescharacteristic
584C2C2×#1798656011yescharacteristic
585C2C2×#1798670012yes—
586C2C2×#1798670212yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#400×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×#400D71
hypocenterC71C2×#2591
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D71C2semidirect
3. #22593816C2direct (N × H)
4. C2×#17985937C2semidirect
5. C2×#17985917C2semidirect
6. C2×#17985916C2semidirect
7. C2×#17985891C2semidirect
8. C2×#17985888C2semidirect
9. C2×#17985877C2semidirect
10. C2×#17985877C2semidirect
11. C2×#17985846C2semidirect
12. C2×#400×C71C2semidirect
24. C2×C4⋊C4×D71C22semidirect
32. #17985937C22semidirect
33. #17985917C22semidirect
34. #17985916C22semidirect
35. #17985891C22semidirect
36. #17985888C22semidirect
37. #17985877C22semidirect
38. #17985877C22semidirect
39. #17985846C22semidirect
45. #400×C71C22semidirect
286. C2×C71#2591semidirect
298. C2×#400D71semidirect
306. C71C2×#2591semidirect
329. #400C2×D71semidirect
581. C2#22593816direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561032
C22×C4⋊C4×D712
#225938162
C2×#179859372
C2×#179859172
C2×#179859162
C2×#179858912
C2×#179858882
C2×#179858772
C2×#179858462
C2×#400×C712
C2×#259171