Finite Group Databasebeta

← C2×#22598283 #32865253

Subgroups — C2×#22598283 #32865253

Order 18176.

2326 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D71C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic
3#22598283C2116yesmaximal, direct product factor
4C2×#17986335C211yesmaximal, characteristic, complemented
5C2×#17986332C211yesmaximal, characteristic, complemented
6C2×#17986343C212yesmaximal, complemented
7C2×#17986339C212yesmaximal, complemented
8C2×#17986329C211yesmaximal, characteristic, complemented
9C2×#17986038C212yesmaximal, complemented
10C2×#17986006C212yesmaximal, complemented
11C2×#17985961C211yesmaximal, characteristic
12C2×#483×C71C211yesmaximal, characteristic, complemented
13C22×Q8×D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C22×#11765848C2212yes—
16C2×#14556108C2211yescharacteristic
17C2×#14556107C2211yescharacteristic
18C2×#14556106—21no—
19C2×#14556105—21no—
20C2×#14556104C2211yescharacteristic
21C22⋊Q8×D71—24no—
22C22⋊Q8×D71C2218yes—
23C2×#14556086—21no—
24C2×#14556086—24no—
25C2×#14556086—21no—
26C2×#14556086C2212yes—
27C2×#14556086C2212yes—
28C2×#14556086C2218yes—
29C22×C4⋊Dic71C2211yescharacteristic
30C2×C4⋊C4×D71—21no—
31C2×C4⋊C4×D71C2211yescharacteristic
32C22×(C2×D71)⋊C4—21no—
33C2×C22⋊C4×D71—21no—
34C22×C4×Dic71—21no—
35#17986343C22116yescomplemented
36#17986339C22116yescomplemented
37#17986335C2218yescomplemented
38#17986332C2218yescomplemented
39#17986329C2218yescomplemented
40#17986038C22116yescomplemented
41C2×#14556062C2212yescomplemented
42C2×#14556061C2212yescomplemented
43C2×#14556059C2212yescomplemented
44#17986006C22116yescomplemented
45C2×#14556053C2212yescomplemented
46C2×#14556052C2212yescomplemented
47C2×#14556051C2212yescomplemented
48#17985961C2218yes—
49C2×C22⋊(C71⋊C8)C2211yescharacteristic
50C2×(C2×D71)⋊C8C2212yes—
51C2×C22⋊Q8×C71C2211yescharacteristic
52C22×C4⋊C4×C71C2211yescharacteristic
53#483×C71C2218yescomplemented
54C2×#106×C71C2212yescomplemented
55C2×#103×C71C2212yescomplemented
56C2×C22⋊C8×C71C2211yescharacteristic
57C24×D71D411yescharacteristic
58C22×Dic142—21no—
59C22×Dic142C2311yescharacteristic
60C2×Q8×D71—22no—
61C2×Q8×D71—28no—
62C2×Q8×D71C2314yes—
63C23×Dic71—21no—
64C23×Dic71D411yescharacteristic
65C22×C4×D71—28no—
66C22×C4×D71—21no—
67C22×C4×D71—21no—
68C22×C4×D71D411yescharacteristic
69C22×C4×D71D414yes—
70C22×C4×D71D411yescharacteristic
71C22×C4×D71C2311yescharacteristic
72C22×C4×D71C2314yes—
73C22×C4×D71C2311yescharacteristic
74C2×#11765848—28no—
75C2×#11765848C2318yes—
76C22×C71⋊C8C2312yes—
77#14556108—22no—
78#14556108C2314yes—
79#14556107—22no—
80#14556107C2314yes—
81#14556106—28no—
82#14556105—28no—
83#14556104—22no—
84#14556104C2314yes—
85#14556086—28no—
86#14556086—24no—
87#14556086—28no—
88#14556086—24no—
89#14556086C2318yes—
90#14556086C2318yes—
91C2×C4⋊Dic71—21no—
92C2×C4⋊Dic71—21no—
93C2×C4⋊Dic71—22no—
94C2×C4⋊Dic71C2314yes—
95C2×C4⋊Dic71C2311yescharacteristic
96C2×C4⋊Dic71C2312yes—
97C2×Dic71⋊C4—21no—
98C2×Dic71⋊C4—21no—
99C2×Dic71⋊C4—21no—
100C4⋊C4×D71—28no—
101C4⋊C4×D71—22no—
102C4⋊C4×D71C2314yes—
103C2×C22⋊Dic71—21no—
104C2×(C2×D71)⋊C4—21no—
105C2×(C2×D71)⋊C4—21no—
106C2×(C2×D71)⋊C4—22no—
107C2×(C2×D71)⋊C4—28no—
108C2×(C2×D71)⋊C4—22no—
109C22⋊C4×D71—28no—
110C2×C4×Dic71—22no—
111C2×C4×Dic71—28no—
112C2×C4×Dic71—22no—
113#14556062C2318yescomplemented
114#14556061C2318yescomplemented
115#14556059C2318yescomplemented
116#14556053C2318yescomplemented
117#14556052C2318yescomplemented
118#14556051C2318yescomplemented
119C22⋊(C71⋊C8)C2314yes—
120(C2×D71)⋊C8C2318yes—
121C22×Q8×C71C2311yescharacteristic
122C23×C4×C71—21no—
123C23×C4×C71C2311yescharacteristic
124C22×C8×C71C2312yes—
125C22⋊Q8×C71—22no—
126C22⋊Q8×C71C2314yes—
127C2×C4⋊C4×C71—21no—
128C2×C4⋊C4×C71—22no—
129C2×C4⋊C4×C71—21no—
130C2×C4⋊C4×C71C2312yes—
131C2×C4⋊C4×C71C2311yescharacteristic
132C2×C4⋊C4×C71C2314yes—
133C2×C22⋊C4×C71—21no—
134#106×C71C2318yescomplemented
135#103×C71C2318yescomplemented
136C22⋊C8×C71C2314yes—
137C23×D71—24no—
138C23×D71—24no—
139C23×D71—42no—
140C23×D71C2×D411yescharacteristic
141C23×D71C2×D414yes—
142C23×D71C2×D411yescharacteristic
143C2×Dic142—24no—
144C2×Dic142—21no—
145C2×Dic142—24no—
146C2×Dic142—44no—
147C2×Dic142C2412yes—
148Q8×D71—216no—
149Q8×D71—48no—
150C22×Dic71—24no—
151C22×Dic71—21no—
152C22×Dic71—21no—
153C22×Dic71—24no—
154C22×Dic71—22no—
155C22×Dic71—44no—
156C22×Dic71C2×D411yescharacteristic
157C22×Dic71C2×D414yes—
158C22×Dic71C2×D411yescharacteristic
159C2×C4×D71—24no—
160C2×C4×D71—24no—
161C2×C4×D71—24no—
162C2×C4×D71—216no—
163C2×C4×D71—216no—
164C2×C4×D71—24no—
165C2×C4×D71—24no—
166C2×C4×D71—44no—
167C2×C4×D71—44no—
168C2×C4×D71C2×D414yes—
169C2×C4×D71C2×D412yes—
170C2×C4×D71C2×D414yes—
171C2×C4×D71C2×D412yes—
172C2×C4×D71#4118yes—
173C2×C4×D71C2412yes—
174C2×C4×D71C2412yes—
175#11765848—216no—
176C2×C71⋊C8—24no—
177C2×C71⋊C8C2414yes—
178C4⋊Dic71—21no—
179C4⋊Dic71—24no—
180C4⋊Dic71—22no—
181C4⋊Dic71—24no—
182C4⋊Dic71C2414yes—
183C4⋊Dic71C2412yes—
184Dic71⋊C4—24no—
185Dic71⋊C4—24no—
186Dic71⋊C4—24no—
187C22⋊Dic71—24no—
188(C2×D71)⋊C4—24no—
189(C2×D71)⋊C4—24no—
190(C2×D71)⋊C4—28no—
191(C2×D71)⋊C4—28no—
192C4×Dic71—28no—
193C4×Dic71—28no—
194C24×C71C2×D411yescharacteristic
195C2×Q8×C71—21no—
196C2×Q8×C71—24no—
197C2×Q8×C71C2412yes—
198C22×C4×C71—24no—
199C22×C4×C71—21no—
200C22×C4×C71—21no—
201C22×C4×C71—24no—
202C22×C4×C71—22no—
203C22×C4×C71—44no—
204C22×C4×C71C2×D411yescharacteristic
205C22×C4×C71C2×D412yes—
206C22×C4×C71C2412yes—
207C22×C4×C71C2411yescharacteristic
208C2×C8×C71—24no—
209C2×C8×C71C2414yes—
210C4⋊C4×C71—24no—
211C4⋊C4×C71—22no—
212C4⋊C4×C71—24no—
213C4⋊C4×C71—21no—
214C4⋊C4×C71C2414yes—
215C4⋊C4×C71C2412yes—
216C22⋊C4×C71—24no—
217C23×C71—22no—
218C23×C71—22no—
219C23×C71#13714yes—
220C23×C71C22×D412yes—
221C23×C71C22×D411yescharacteristic
222C22×D71—22no—
223C22×D71—22no—
224C22×D71—22no—
225C22×D71—28no—
226C22×D71—22no—
227C22×D71—22no—
228C22×D71—28no—
229C22×D71—44no—
230C22×D71—416no—
231C22×D71#12314yes—
232C22×D71C22×D412yes—
233C22×D71C22×D412yes—
234Q8×C71—24no—
235Q8×C71—42no—
236C2×C4×C71—22no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—22no—
240C2×C4×C71—22no—
241C2×C4×C71—24no—
242C2×C4×C71—42no—
243C2×C4×C71—48no—
244C2×C4×C71—44no—
245C2×C4×C71—42no—
246C2×C4×C71C22×D412yes—
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C2×#4114yes—
249C2×C4×C71C2511yescharacteristic
250C2×Dic71—22no—
251C2×Dic71—28no—
252C2×Dic71—24no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—48no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—44no—
259C2×Dic71#12314yes—
260C2×Dic71C22×D412yes—
261C2×Dic71C22×D412yes—
262C4×D71—28no—
263C4×D71—24no—
264C4×D71—24no—
265C4×D71—28no—
266C4×D71—48no—
267C4×D71—48no—
268C4×D71#12318yes—
269C4×D71C2×#4118yes—
270Dic142—24no—
271Dic142—42no—
272Dic142—48no—
273C8×C71—24no—
274C71⋊C8—24no—
275C22×C71—28no—
276C22×C71—22no—
277C22×C71—22no—
278C22×C71C2×#12312yes—
279C22×C71C2×#13714yes—
280C22×C71C2×#13712yes—
281C22×C71#57512yes—
282C22×C71C23×D411yescharacteristic
283C2×D71—28no—
284C2×D71—22no—
285C2×D71—22no—
286C2×D71—22no—
287C2×D71—22no—
288C2×D71—22no—
289C2×D71—416no—
290C2×D71—44no—
291C2×D71C2×#12314yes—
292C4×C71—22no—
293C4×C71—42no—
294C4×C71—44no—
295C4×C71—42no—
296C4×C71C2×#12312yes—
297C4×C71C22×#4112yes—
298Dic71—22no—
299Dic71—42no—
300Dic71—42no—
301Dic71—44no—
302Dic71C2×#12314yes—
303C2×#3095—711nomaximal, Sylow
304C2×C71—24no—
305C2×C71#309514yescomplemented
306C2×C71C22×#12311yescharacteristic
307C2×C71C22×#13711yescharacteristic
308C2×C71C2×#57511yescharacteristic
309D71—24no—
310D71—42no—
311C22×C22⋊Q8—711no—
312C22×#117—711no—
313#3095—7116no—
314C2×#484—711no—
315C2×#484—711no—
316C2×#485—712no—
317C2×#485—712no—
318C2×#483—711no—
319C2×#483D7111yescharacteristic, complemented
320C2×#428—712no—
321C2×#419—712no—
322C2×#407—711no—
323C71C2×#309511yescharacteristic, complemented, Sylow
324C23×Q8—711no—
325C24×C4—711no—
326C22×M16—712no—
327C2×C22⋊Q8—711no—
328C2×C22⋊Q8—711no—
329C2×C22⋊Q8—718no—
330C2×C22⋊Q8—711no—
331C2×C22⋊Q8—1421no—
332C2×C22⋊Q8—1421no—
333C2×C22⋊Q8—1424no—
334C2×C22⋊Q8C2×D7111yescharacteristic
335C2×#117—712no—
336C2×#117—718no—
337C2×#117—712no—
338C2×#117—1421no—
339C2×#117—1424no—
340C2×#117—1421no—
341C22×C4⋊C4—711no—
342C22×C4⋊C4—711no—
343C22×C4⋊C4—1421no—
344C22×C4⋊C4C2×D7111yescharacteristic
345C22×C22⋊C4—1421no—
346C22×C22⋊C4—1421no—
347C22×C42—1421no—
348#485—7116no—
349#485—7116no—
350#484—718no—
351#484—718no—
352#483—718no—
353#483C2×D7118yescomplemented
354#428—7116no—
355C2×#107—712no—
356C2×#107—712no—
357C2×#106—712no—
358C2×#106C2×D7112yescomplemented
359#419—7116no—
360C2×#103—712no—
361C2×#103—712no—
362C2×#103—712no—
363C2×#103C2×D7112yescomplemented
364#407—718no—
365C2×C22⋊C8—711no—
366C2×C22⋊C8—712no—
367C2×C22⋊C8C2×D7111yescharacteristic
368C25—711no—
369C22×Q8—714no—
370C22×Q8—711no—
371C22×Q8—1421no—
372C22×Q8—1428no—
373C22×Q8—1422no—
374C22×Q8C22×D7111yescharacteristic
375C23×C4—21no—
376C23×C4—711no—
377C23×C4—711no—
378C23×C4—711no—
379C23×C4—711no—
380C23×C4—714no—
381C23×C4—711no—
382C23×C4—714no—
383C23×C4—1421no—
384C23×C4—1421no—
385C23×C4—1428no—
386C23×C4—1421no—
387C23×C4C22×D7111yescharacteristic
388C2×M16—718no—
389C2×M16—1428no—
390C22×C8—712no—
391C22×C8C22×D7112yes—
392C22⋊Q8—22no—
393C22⋊Q8—714no—
394C22⋊Q8—714no—
395C22⋊Q8—714no—
396C22⋊Q8—1428no—
397C22⋊Q8—1428no—
398C22⋊Q8—1422no—
399C22⋊Q8—1422no—
400C22⋊Q8—1422no—
401C22⋊Q8C22×D7114yes—
402#117—718no—
403#117—718no—
404#117—1424no—
405#117—1428no—
406#117—1424no—
407#117—1428no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—712no—
412C2×C4⋊C4—714no—
413C2×C4⋊C4—711no—
414C2×C4⋊C4—714no—
415C2×C4⋊C4—1422no—
416C2×C4⋊C4—1428no—
417C2×C4⋊C4—1421no—
418C2×C4⋊C4—1421no—
419C2×C4⋊C4—1421no—
420C2×C4⋊C4—1422no—
421C2×C4⋊C4—1421no—
422C2×C4⋊C4—1421no—
423C2×C4⋊C4C22×D7111yescharacteristic
424C2×C4⋊C4C22×D7112yes—
425C2×C4⋊C4C22×D7114yes—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1428no—
428C2×C22⋊C4—1428no—
429C2×C22⋊C4—1422no—
430C2×C22⋊C4—1422no—
431C2×C22⋊C4—1421no—
432C2×C22⋊C4—1421no—
433C2×C22⋊C4—1421no—
434C2×C42—1428no—
435C2×C42—1422no—
436C2×C42—1422no—
437#107—718no—
438#107—718no—
439#106—718no—
440#106C22×D7118yescomplemented
441#103—718no—
442#103—718no—
443#103—718no—
444#103C22×D7118yescomplemented
445C22⋊C8—714no—
446C22⋊C8—718no—
447C22⋊C8C22×D7114yes—
448C24—2842no—
449C24—714no—
450C24—711no—
451C24—711no—
452C24—1424no—
453C24—1424no—
454C24D4×D7111yescharacteristic
455C2×Q8—21no—
456C2×Q8—24no—
457C2×Q8—2844no—
458C2×Q8—2848no—
459C2×Q8—712no—
460C2×Q8—1424no—
461C2×Q8—14216no—
462C2×Q8—1424no—
463C2×Q8—1421no—
464C2×Q8C23×D7112yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—2844no—
472C22×C4—2844no—
473C22×C4—2844no—
474C22×C4—711no—
475C22×C4—712no—
476C22×C4—718no—
477C22×C4—712no—
478C22×C4—714no—
479C22×C4—711no—
480C22×C4—714no—
481C22×C4—714no—
482C22×C4—712no—
483C22×C4—712no—
484C22×C4—1424no—
485C22×C4—1424no—
486C22×C4—1424no—
487C22×C4—14216no—
488C22×C4—14216no—
489C22×C4—1424no—
490C22×C4—1424no—
491C22×C4—1422no—
492C22×C4—1421no—
493C22×C4—1421no—
494C22×C4—1424no—
495C22×C4—1424no—
496C22×C4D4×D7112yes—
497C22×C4D4×D7111yescharacteristic
498C22×C4C23×D7112yes—
499C22×C4C23×D7111yescharacteristic
500M16—14216no—
501C2×C8—24no—
502C2×C8—714no—
503C2×C8—1424no—
504C2×C8C23×D7114yes—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—714no—
510C4⋊C4—712no—
511C4⋊C4—1424no—
512C4⋊C4—1424no—
513C4⋊C4—1424no—
514C4⋊C4—1424no—
515C4⋊C4—1424no—
516C4⋊C4—1422no—
517C4⋊C4—1421no—
518C4⋊C4C23×D7114yes—
519C4⋊C4C23×D7112yes—
520C22⋊C4—24no—
521C22⋊C4—1428no—
522C22⋊C4—1428no—
523C22⋊C4—1424no—
524C22⋊C4—1424no—
525C22⋊C4—1424no—
526C42—1428no—
527C42—1428no—
528C23—22no—
529C23—22no—
530C23—28416no—
531C23—2844no—
532C23—712no—
533C23—714no—
534C23—712no—
535C23—1428no—
536C23—1422no—
537C23—1428no—
538C23—1422no—
539C23—1422no—
540C23—1422no—
541C23—1422no—
542C23#1455615714yes—
543C23C2×D4×D7112yes—
544C23C2×D4×D7111yescharacteristic
545Q8—24no—
546Q8—42no—
547Q8—2848no—
548Q8—2842no—
549Q8—1424no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—2848no—
561C2×C4—2848no—
562C2×C4—2848no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2844no—
566C2×C4—712no—
567C2×C4—714no—
568C2×C4—718no—
569C2×C4—718no—
570C2×C4—712no—
571C2×C4—1428no—
572C2×C4—1428no—
573C2×C4—1424no—
574C2×C4—1422no—
575C2×C4—1422no—
576C2×C4—1428no—
577C2×C4—1422no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4C2×D4×D7112yes—
581C2×C4C2×D4×D7111yescharacteristic
582C2×C4#41×D7114yes—
583C2×C4C24×D7111yescharacteristic
584C8—24no—
585C8—1424no—
586C22—22no—
587C22—22no—
588C22—28no—
589C22—28416no—
590C22—2844no—
591C22—714no—
592C22—1422no—
593C22—1422no—
594C22—1422no—
595C22—1422no—
596C22—1422no—
597C22—1428no—
598C22#123×D7112yes—
599C22C2×#1455615714yes—
600C22C2×#1455615712yes—
601C22#1798679812yes—
602C22C22×D4×D7111yescharacteristic
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—2842no—
608C4—2844no—
609C4—2842no—
610C4—714no—
611C4—1422no—
612C4#123×D7112yes—
613C4C2×#41×D7112yes—
614C2—24no—
615C2—2842no—
616C2—1424no—
617C2#2259828314yesdirect product factor
618C2C2×#123×D7111yescharacteristic
619C2C22×#1455615711yescharacteristic
620C2C2×#1798679811yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC2×#483×C71C2
socleC23×C71C22×D4
prime radicalC24×D71D4
hypercenterC2×#483D71
hypocenterC71C2×#3095
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #22598283C2direct (N × H)
4. C2×#17986335C2semidirect
5. C2×#17986332C2semidirect
6. C2×#17986343C2semidirect
7. C2×#17986339C2semidirect
8. C2×#17986329C2semidirect
9. C2×#17986038C2semidirect
10. C2×#17986006C2semidirect
12. C2×#483×C71C2semidirect
35. #17986343C22semidirect
36. #17986339C22semidirect
37. #17986335C22semidirect
38. #17986332C22semidirect
39. #17986329C22semidirect
40. #17986038C22semidirect
41. C2×#14556062C22semidirect
42. C2×#14556061C22semidirect
43. C2×#14556059C22semidirect
44. #17986006C22semidirect
45. C2×#14556053C22semidirect
46. C2×#14556052C22semidirect
47. C2×#14556051C22semidirect
53. #483×C71C22semidirect
54. C2×#106×C71C22semidirect
55. C2×#103×C71C22semidirect
113. #14556062C23semidirect
114. #14556061C23semidirect
115. #14556059C23semidirect
116. #14556053C23semidirect
117. #14556052C23semidirect
118. #14556051C23semidirect
134. #106×C71C23semidirect
135. #103×C71C23semidirect
305. C2×C71#3095semidirect
319. C2×#483D71semidirect
323. C71C2×#3095semidirect
353. #483C2×D71semidirect
358. C2×#106C2×D71semidirect
363. C2×#103C2×D71semidirect
440. #106C22×D71semidirect
444. #103C22×D71semidirect
617. C2#22598283direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D712
C22×#145560862
#225982832
C2×#179863352
C2×#179863322
C2×#179863432
C2×#179863392
C2×#179863292
C2×#179860382
C2×#179860062
C2×#179859612
C2×#483×C712
C2×#309571