Finite Group Databasebeta

← C2×#22594083 #32163816

Subgroups — C2×#22594083 #32163816

Order 18176.

3718 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556113C212yesmaximal, complemented
2C22×#14556103C211yesmaximal, characteristic
3C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#22594083C2116yesmaximal, direct product factor
5C2×#17985930C211yesmaximal, characteristic, complemented
6C2×#17985923C212yesmaximal, complemented
7C2×#17985921C212yesmaximal, complemented
8C2×#17985917C211yesmaximal, characteristic, complemented
9C2×#17985916C211yesmaximal, characteristic, complemented
10C2×#17985877C212yesmaximal, complemented
11C2×#17985858C211yesmaximal, characteristic, complemented
12C2×#398×C71C211yesmaximal, characteristic, complemented
13C23×D284C2212yes—
14C23×D284C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C23×C4×D71C2211yescharacteristic
17C23×C4×D71C2211yescharacteristic
18C2×#14556113—216no—
19C2×#14556113C22116yescomplemented
20C2×#14556103—28no—
21C2×#14556103C2218yes—
22C22×C4⋊Dic71C2211yescharacteristic
23C22×Dic71⋊C4C2211yescharacteristic
24C22×Dic71⋊C4C2211yescharacteristic
25C22×Dic71⋊C4C2212yes—
26C22×Dic71⋊C4C2212yes—
27C2×C4⋊C4×D71—28no—
28C2×C4⋊C4×D71C2218yescomplemented
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2212yes—
33C22×(C2×D71)⋊C4C2212yes—
34C22×(C2×D71)⋊C4C2212yes—
35#17985930C2218yescomplemented
36#17985923C22116yescomplemented
37#17985921C22116yescomplemented
38#17985917C2218yescomplemented
39#17985916C2218yescomplemented
40#17985877C22116yescomplemented
41#17985858C2218yescomplemented
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2212yes—
46C22×C4⋊C4×C71C2211yescharacteristic
47#398×C71C2218yescomplemented
48C2×#95×C71C2212yes—
49C2×#95×C71C2211yescharacteristic
50C24×D71C2312yes—
51C24×D71C2311yescharacteristic
52C22×D284—28no—
53C22×D284—28no—
54C22×D284—28no—
55C22×D284—24no—
56C22×D284—44no—
57C22×D284C2318yes—
58C22×D284C2314yes—
59C23×Dic71C2311yescharacteristic
60C23×Dic71C2312yes—
61C23×Dic71C2311yescharacteristic
62C23×Dic71C2311yescharacteristic
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×D71D418yescomplemented
70C22×C4×D71C2314yes—
71C22×C4×D71C2314yes—
72C22×C4×D71C2314yes—
73#14556113—264no—
74#14556103—232no—
75C2×C4⋊Dic71—24no—
76C2×C4⋊Dic71C2314yes—
77C2×Dic71⋊C4—24no—
78C2×Dic71⋊C4—28no—
79C2×Dic71⋊C4—24no—
80C2×Dic71⋊C4—44no—
81C2×Dic71⋊C4C2314yes—
82C2×Dic71⋊C4C2314yes—
83C2×Dic71⋊C4C2318yes—
84C2×Dic71⋊C4C2318yes—
85C4⋊C4×D71—232no—
86C2×(C2×D71)⋊C4—28no—
87C2×(C2×D71)⋊C4—28no—
88C2×(C2×D71)⋊C4—28no—
89C2×(C2×D71)⋊C4—28no—
90C2×(C2×D71)⋊C4—44no—
91C2×(C2×D71)⋊C4—44no—
92C2×(C2×D71)⋊C4C2318yes—
93C2×(C2×D71)⋊C4C2318yes—
94C2×(C2×D71)⋊C4C2318yes—
95C2×(C2×D71)⋊C4C2318yes—
96C2×(C2×D71)⋊C4C2318yes—
97C2×(C2×D71)⋊C4C2318yes—
98C71⋊#95C2314yes—
99C71⋊#95C2318yes—
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2312yes—
102C23×C4×C71C2312yes—
103C23×C4×C71C2311yescharacteristic
104C23×C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71—24no—
106C2×C4⋊C4×C71—24no—
107C2×C4⋊C4×C71—28no—
108C2×C4⋊C4×C71C2314yes—
109C2×C4⋊C4×C71C2318yes—
110C2×C4⋊C4×C71C2314yes—
111#95×C71C2314yes—
112#95×C71C2318yes—
113C23×D71—24no—
114C23×D71—24no—
115C23×D71—24no—
116C23×D71—24no—
117C23×D71—24no—
118C23×D71—24no—
119C23×D71—24no—
120C23×D71—24no—
121C23×D71—24no—
122C23×D71C2×D414yes—
123C23×D71C2×D414yes—
124C23×D71#4114yes—
125C23×D71C2414yes—
126C23×D71C2412yes—
127C2×D284—216no—
128C2×D284—216no—
129C2×D284—216no—
130C2×D284—28no—
131C2×D284—48no—
132C2×D284—416no—
133C2×D284—432no—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71—24no—
137C22×Dic71—22no—
138C22×Dic71—24no—
139C22×Dic71—24no—
140C22×Dic71—24no—
141C22×Dic71—24no—
142C22×Dic71C2×D414yes—
143C22×Dic71C2×D414yes—
144C22×Dic71C2412yes—
145C22×Dic71C2414yes—
146C22×Dic71C2412yes—
147C22×Dic71C2412yes—
148C2×C4×D71—232no—
149C2×C4×D71—28no—
150C2×C4×D71—28no—
151C2×C4×D71—28no—
152C2×C4×D71—416no—
153C2×C4×D71—48no—
154C2×C4×D71—48no—
155C2×C4×D71—416no—
156C2×C4×D71C2×D4116yes—
157C2×C4×D71C2×D4116yescomplemented
158C4⋊Dic71—28no—
159Dic71⋊C4—28no—
160Dic71⋊C4—28no—
161Dic71⋊C4—216no—
162Dic71⋊C4—48no—
163(C2×D71)⋊C4—216no—
164(C2×D71)⋊C4—216no—
165(C2×D71)⋊C4—216no—
166(C2×D71)⋊C4—216no—
167(C2×D71)⋊C4—48no—
168(C2×D71)⋊C4—48no—
169C24×C71C2411yescharacteristic
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—24no—
173C22×C4×C71—22no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71—24no—
181C22×C4×C71C2×D414yes—
182C22×C4×C71C2×D414yes—
183C22×C4×C71C2412yes—
184C22×C4×C71C2414yes—
185C22×C4×C71C2414yes—
186C22×C4×C71C2412yes—
187C22×C4×C71C2412yes—
188C4⋊C4×C71—28no—
189C4⋊C4×C71—28no—
190C4⋊C4×C71—216no—
191C23×C71C22×D412yes—
192C23×C71C22×D412yes—
193C23×C71C2×#4112yes—
194C23×C71H(5,𝔽2)14yes—
195C23×C712−1+414yes—
196C23×C71C2511yescharacteristic
197C22×D71—28no—
198C22×D71—28no—
199C22×D71—216no—
200C22×D71—216no—
201C22×D71—24no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—48no—
211C22×D71—48no—
212C22×D71—48no—
213C22×D71—48no—
214C22×D71—48no—
215C22×D71—48no—
216C22×D71—48no—
217C22×D71#121116yes—
218C22×D71C22×D414yes—
219C22×D71C22×D414yes—
220C22×D71C2×#4114yes—
221C2×C4×C71—28no—
222C2×C4×C71—28no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—48no—
235C2×C4×C71—44no—
236C2×C4×C71—48no—
237C2×C4×C71C22×D414yes—
238C2×C4×C71C22×D414yes—
239C2×Dic71—28no—
240C2×Dic71—28no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—22no—
246C2×Dic71—24no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—44no—
250C2×Dic71—48no—
251C2×Dic71C22×D414yes—
252C2×Dic71C22×D414yes—
253D284—416no—
254D284—432no—
255C4×D71—232no—
256C4×D71—416no—
257C4×D71—416no—
258C22×C71C2×#12114yes—
259C22×C71#53414yes—
260C22×C71#53614yes—
261C22×C71#53718yes—
262C22×C71#53918yes—
263C22×C71C23×D411yescharacteristic
264C22×C71C23×D411yescharacteristic
265C22×C71C22×#4111yescharacteristic
266C22×C71C2×H(5,𝔽2)12yes—
267C22×C71C2×2−1+412yes—
268C2×D71—232no—
269C2×D71—24no—
270C2×D71—24no—
271C2×D71—28no—
272C2×D71—28no—
273C2×D71—44no—
274C2×D71—44no—
275C2×D71—44no—
276C2×D71—44no—
277C2×D71—44no—
278C2×D71—44no—
279C2×D71—44no—
280C2×D71—816no—
281C2×D71C2×#12118yes—
282C4×C71—24no—
283C4×C71—24no—
284C4×C71—44no—
285C4×C71—42no—
286C4×C71—44no—
287Dic71—24no—
288Dic71—24no—
289Dic71—42no—
290Dic71—44no—
291C2×#2618—711nomaximal, Sylow
292C2×C71#261818yescomplemented
293C2×C71C22×#12111yescharacteristic
294C2×C71C2×#53411yescharacteristic
295C2×C71C2×#53611yescharacteristic
296C2×C71C2×#53712yes—
297C2×C71C2×#53912yes—
298D71—28no—
299D71—84no—
300C22×#123—712no—
301C22×#121—711no—
302C23×C4⋊C4—711no—
303#2618—7116no—
304C2×#398—711no—
305C2×#398D7111yescharacteristic, complemented
306C2×#396—712no—
307C2×#396—712no—
308C2×#394—711no—
309C2×#394—711no—
310C2×#386—712no—
311C2×#380—711no—
312C71C2×#261811yescharacteristic, complemented, Sylow
313C23×D4—711no—
314C23×D4—712no—
315C24×C4—711no—
316C24×C4—711no—
317C24×C4—711no—
318C2×#123—7116no—
319C2×#123—14216no—
320C2×#121—718no—
321C2×#121—1428no—
322C22×C4⋊C4—712no—
323C22×C4⋊C4—711no—
324C22×C4⋊C4—711no—
325C22×C4⋊C4—711no—
326C22×C4⋊C4—712no—
327C22×C4⋊C4—718no—
328C22×C4⋊C4—1428no—
329C22×C4⋊C4C2×D7111yescharacteristic
330C22×C4⋊C4C2×D7112yes—
331C22×C4⋊C4C2×D7111yescharacteristic
332C22×C22⋊C4—712no—
333C22×C22⋊C4—712no—
334C22×C22⋊C4—712no—
335C22×C22⋊C4—712no—
336C22×C22⋊C4—712no—
337C22×C22⋊C4—712no—
338#398—718no—
339#398C2×D7118yescomplemented
340#396—7116no—
341#396—7116no—
342#394—718no—
343#394—718no—
344#386—7116no—
345#380—718no—
346C2×#95—711no—
347C2×#95—712no—
348C2×#95C2×D7111yescharacteristic
349C2×#95C2×D7112yes—
350C25—711no—
351C25—712no—
352C22×D4—2844no—
353C22×D4—714no—
354C22×D4—718no—
355C22×D4—1428no—
356C22×D4—1428no—
357C22×D4—1428no—
358C22×D4—1424no—
359C23×C4—2844no—
360C23×C4—2844no—
361C23×C4—718no—
362C23×C4—714no—
363C23×C4—714no—
364C23×C4—711no—
365C23×C4—711no—
366C23×C4—718no—
367C23×C4—714no—
368C23×C4—712no—
369C23×C4—711no—
370C23×C4—1424no—
371C23×C4—1424no—
372C23×C4—1424no—
373C23×C4C22×D7112yes—
374C23×C4C22×D7111yescharacteristic
375C23×C4C22×D7112yes—
376C23×C4C22×D7111yescharacteristic
377C23×C4C22×D7111yescharacteristic
378#123—14264no—
379#121—14232no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—28no—
383C2×C4⋊C4—2844no—
384C2×C4⋊C4—718no—
385C2×C4⋊C4—714no—
386C2×C4⋊C4—714no—
387C2×C4⋊C4—714no—
388C2×C4⋊C4—718no—
389C2×C4⋊C4—14232no—
390C2×C4⋊C4—1424no—
391C2×C4⋊C4—1424no—
392C2×C4⋊C4—1428no—
393C2×C4⋊C4—1424no—
394C2×C4⋊C4C22×D7114yes—
395C2×C4⋊C4C22×D7114yes—
396C2×C4⋊C4C22×D7118yes—
397C2×C22⋊C4—2844no—
398C2×C22⋊C4—2844no—
399C2×C22⋊C4—718no—
400C2×C22⋊C4—718no—
401C2×C22⋊C4—718no—
402C2×C22⋊C4—718no—
403C2×C22⋊C4—718no—
404C2×C22⋊C4—718no—
405C2×C22⋊C4—1428no—
406C2×C22⋊C4—1428no—
407C2×C22⋊C4—1428no—
408C2×C22⋊C4—1428no—
409#95—714no—
410#95—718no—
411#95C22×D7114yes—
412#95C22×D7118yes—
413C24—714no—
414C24—714no—
415C24—714no—
416C24—712no—
417C24—714no—
418C24—1424no—
419C24—1424no—
420C24—1424no—
421C24—1424no—
422C24—1424no—
423C24—1424no—
424C24—1424no—
425C24—1424no—
426C24—1424no—
427C24C23×D7111yescharacteristic
428C2×D4—28416no—
429C2×D4—2848no—
430C2×D4—28432no—
431C2×D4—14216no—
432C2×D4—14216no—
433C2×D4—14216no—
434C2×D4—1428no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28416no—
447C22×C4—2848no—
448C22×C4—2848no—
449C22×C4—28416no—
450C22×C4—7116no—
451C22×C4—714no—
452C22×C4—714no—
453C22×C4—712no—
454C22×C4—712no—
455C22×C4—7116no—
456C22×C4—714no—
457C22×C4—712no—
458C22×C4—1428no—
459C22×C4—1428no—
460C22×C4—1428no—
461C22×C4—14232no—
462C22×C4—1424no—
463C22×C4—1422no—
464C22×C4—1424no—
465C22×C4—1424no—
466C22×C4—1424no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1424no—
470C22×C4D4×D7114yes—
471C22×C4D4×D7114yes—
472C22×C4C23×D7114yes—
473C22×C4C23×D7114yes—
474C22×C4C23×D7112yes—
475C22×C4C23×D7112yes—
476C22×C4C23×D7112yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—2848no—
481C4⋊C4—1428no—
482C4⋊C4—14216no—
483C4⋊C4—1428no—
484C4⋊C4—1428no—
485C22⋊C4—2848no—
486C22⋊C4—2848no—
487C22⋊C4—14216no—
488C22⋊C4—14216no—
489C22⋊C4—14216no—
490C22⋊C4—14216no—
491C23—2848no—
492C23—2848no—
493C23—2848no—
494C23—2848no—
495C23—2848no—
496C23—2848no—
497C23—2848no—
498C23—714no—
499C23—714no—
500C23—714no—
501C23—7116no—
502C23—1424no—
503C23—1424no—
504C23—1424no—
505C23—1424no—
506C23—1424no—
507C23—1424no—
508C23—1424no—
509C23—1428no—
510C23—1428no—
511C23—1424no—
512C23—1424no—
513C23—14216no—
514C23—14216no—
515C23C2×D4×D7112yes—
516C23C2×D4×D7112yes—
517C23#41×D7112yes—
518C23#1455617414yes—
519C23#1455617614yes—
520C23C24×D7111yescharacteristic
521D4—28416no—
522D4—28432no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—28416no—
540C2×C4—28416no—
541C2×C4—2844no—
542C2×C4—2848no—
543C2×C4—714no—
544C2×C4—714no—
545C2×C4—14232no—
546C2×C4—1424no—
547C2×C4—1422no—
548C2×C4—1424no—
549C2×C4—1424no—
550C2×C4—1424no—
551C2×C4—1428no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4—1424no—
555C2×C4—1428no—
556C2×C4C2×D4×D7114yes—
557C2×C4C2×D4×D7114yes—
558C22—2844no—
559C22—2844no—
560C22—2844no—
561C22—2844no—
562C22—2844no—
563C22—2844no—
564C22—2844no—
565C22—56816no—
566C22—718no—
567C22—14232no—
568C22—1424no—
569C22—1424no—
570C22—1428no—
571C22—1428no—
572C22#121×D7114yes—
573C22#1798656014yes—
574C22#1798657414yes—
575C22#1798657918yes—
576C22#1798659718yes—
577C22C22×D4×D7111yescharacteristic
578C22C22×D4×D7111yescharacteristic
579C22C2×#41×D7111yescharacteristic
580C22C2×#1455617412yes—
581C22C2×#1455617612yes—
582C4—24no—
583C4—24no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—2842no—
588C4—2844no—
589C4—1424no—
590C4—1424no—
591C2—5684no—
592C2—1428no—
593C2#2259408318yesdirect product factor
594C2C2×#121×D7111yescharacteristic
595C2C2×#1798656011yescharacteristic
596C2C2×#1798657411yescharacteristic
597C2C2×#1798657912yes—
598C2C2×#1798659712yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#398×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×#398D71
hypocenterC71C2×#2618
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
1. C22×#14556113C2semidirect
3. C22×C4⋊C4×D71C2semidirect
4. #22594083C2direct (N × H)
5. C2×#17985930C2semidirect
6. C2×#17985923C2semidirect
7. C2×#17985921C2semidirect
8. C2×#17985917C2semidirect
9. C2×#17985916C2semidirect
10. C2×#17985877C2semidirect
11. C2×#17985858C2semidirect
12. C2×#398×C71C2semidirect
19. C2×#14556113C22semidirect
28. C2×C4⋊C4×D71C22semidirect
35. #17985930C22semidirect
36. #17985923C22semidirect
37. #17985921C22semidirect
38. #17985917C22semidirect
39. #17985916C22semidirect
40. #17985877C22semidirect
41. #17985858C22semidirect
47. #398×C71C22semidirect
69. C22×C4×D71D4semidirect
157. C2×C4×D71C2×D4semidirect
292. C2×C71#2618semidirect
305. C2×#398D71semidirect
312. C71C2×#2618semidirect
339. #398C2×D71semidirect
593. C2#22594083direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561132
C22×#145561032
C22×C4⋊C4×D712
#225940832
C2×#179859302
C2×#179859232
C2×#179859212
C2×#179859172
C2×#179859162
C2×#179858772
C2×#179858582
C2×#398×C712
C2×#261871