Finite Group Databasebeta

← C2×#22592806 #32166508

Subgroups — C2×#22592806 #32166508

Order 18176.

4222 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556113C212yesmaximal, complemented
2C22×#14556110C212yesmaximal, complemented
3C22×#14556101C211yesmaximal, characteristic
4C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
5#22592806C2116yesmaximal, direct product factor
6C2×#17985936C211yesmaximal, characteristic, complemented
7C2×#17985935C211yesmaximal, characteristic, complemented
8C2×#17985934C211yesmaximal, characteristic, complemented
9C2×#17985899C212yesmaximal, complemented
10C2×#17985868C212yesmaximal, complemented
11C2×#17985846C211yesmaximal, characteristic, complemented
12C2×#399×C71C211yesmaximal, characteristic, complemented
13C23×D284C2211yescharacteristic
14C23×C22⋊D71C2211yescharacteristic
15C23×C22⋊D71C2211yescharacteristic
16C23×C4×D71C2212yes—
17C23×C4×D71C2211yescharacteristic
18C2×#14556113—216no—
19C2×#14556113C22116yescomplemented
20C2×#14556110—216no—
21C2×#14556110C22116yescomplemented
22C2×#14556101—28no—
23C2×#14556101C2218yes—
24C22×C4⋊Dic71C2212yes—
25C22×C4⋊Dic71C2211yescharacteristic
26C22×Dic71⋊C4C2211yescharacteristic
27C22×Dic71⋊C4C2212yes—
28C22×Dic71⋊C4C2212yes—
29C22×Dic71⋊C4C2212yes—
30C2×C4⋊C4×D71—28no—
31C2×C4⋊C4×D71C2218yescomplemented
32C22×C22⋊Dic71C2212yes—
33C22×(C2×D71)⋊C4C2212yes—
34C22×(C2×D71)⋊C4C2212yes—
35C22×(C2×D71)⋊C4C2211yescharacteristic
36C22×(C2×D71)⋊C4C2212yes—
37C22×C4×Dic71C2212yes—
38#17985936C2218yescomplemented
39#17985935C2218yescomplemented
40#17985934C2218yescomplemented
41#17985899C22116yescomplemented
42#17985868C22116yescomplemented
43#17985846C2218yescomplemented
44C2×C71⋊#95C2211yescharacteristic
45C22×C4⋊C4×C71C2212yes—
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C22⋊C4×C71C2212yes—
48C22×C22⋊C4×C71C2211yescharacteristic
49#399×C71C2218yescomplemented
50C2×#95×C71C2211yescharacteristic
51C24×D71C2311yescharacteristic
52C24×D71C2311yescharacteristic
53C22×D284—24no—
54C22×D284—28no—
55C22×D284C2314yes—
56C22×C22⋊D71—24no—
57C22×C22⋊D71—28no—
58C22×C22⋊D71—24no—
59C22×C22⋊D71—28no—
60C22×C22⋊D71C2314yes—
61C22×C22⋊D71C2314yes—
62C23×Dic71C2312yes—
63C23×Dic71C2311yescharacteristic
64C23×Dic71C2312yes—
65C23×Dic71C2311yescharacteristic
66C22×C4×D71—28no—
67C22×C4×D71—28no—
68C22×C4×D71—28no—
69C22×C4×D71—28no—
70C22×C4×D71D418yes—
71C22×C4×D71C2318yes—
72C22×C4×D71C2314yes—
73#14556113—264no—
74#14556110—264no—
75#14556101—232no—
76C2×C4⋊Dic71—24no—
77C2×C4⋊Dic71—28no—
78C2×C4⋊Dic71C2318yes—
79C2×C4⋊Dic71C2314yes—
80C2×Dic71⋊C4—28no—
81C2×Dic71⋊C4—24no—
82C2×Dic71⋊C4—28no—
83C2×Dic71⋊C4—28no—
84C2×Dic71⋊C4C2314yes—
85C2×Dic71⋊C4C2318yes—
86C2×Dic71⋊C4C2318yes—
87C2×Dic71⋊C4C2318yes—
88C4⋊C4×D71—232no—
89C2×C22⋊Dic71—28no—
90C2×C22⋊Dic71C2318yes—
91C2×(C2×D71)⋊C4—28no—
92C2×(C2×D71)⋊C4—24no—
93C2×(C2×D71)⋊C4—28no—
94C2×(C2×D71)⋊C4D4116yescomplemented
95C2×(C2×D71)⋊C4C2318yes—
96C2×(C2×D71)⋊C4C2318yes—
97C2×(C2×D71)⋊C4C2314yes—
98C2×(C2×D71)⋊C4C2318yes—
99C2×C4×Dic71—28no—
100C2×C4×Dic71C2318yes—
101C71⋊#95C2314yes—
102C25×C71C2311yescharacteristic
103C23×C4×C71C2312yes—
104C23×C4×C71C2311yescharacteristic
105C23×C4×C71C2311yescharacteristic
106C23×C4×C71C2312yes—
107C2×C4⋊C4×C71—24no—
108C2×C4⋊C4×C71—28no—
109C2×C4⋊C4×C71C2318yes—
110C2×C4⋊C4×C71C2314yes—
111C2×C22⋊C4×C71—24no—
112C2×C22⋊C4×C71—28no—
113C2×C22⋊C4×C71C2318yes—
114C2×C22⋊C4×C71C2314yes—
115#95×C71C2314yes—
116C23×D71—24no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—24no—
121C23×D71—24no—
122C23×D71—22no—
123C23×D71—22no—
124C23×D71C2×D414yes—
125C23×D71C2×D418yes—
126C23×D71C2412yes—
127C23×D71C2412yes—
128C2×D284—28no—
129C2×D284—216no—
130C2×D284—416no—
131C2×C22⋊D71—28no—
132C2×C22⋊D71—216no—
133C2×C22⋊D71—28no—
134C2×C22⋊D71—216no—
135C2×C22⋊D71—416no—
136C2×C22⋊D71—416no—
137C22×Dic71—24no—
138C22×Dic71—24no—
139C22×Dic71—24no—
140C22×Dic71—24no—
141C22×Dic71—24no—
142C22×Dic71—22no—
143C22×Dic71—24no—
144C22×Dic71C2×D418yes—
145C22×Dic71C2×D414yes—
146C22×Dic71#4118yes—
147C22×Dic71C2414yes—
148C22×Dic71C2412yes—
149C22×Dic71C2414yes—
150C22×Dic71C2412yes—
151C2×C4×D71—232no—
152C2×C4×D71—216no—
153C2×C4×D71—216no—
154C2×C4×D71—216no—
155C2×C4×D71—216no—
156C2×C4×D71—432no—
157C2×C4×D71C2×D4116yes—
158C4⋊Dic71—28no—
159C4⋊Dic71—216no—
160Dic71⋊C4—216no—
161Dic71⋊C4—28no—
162Dic71⋊C4—216no—
163Dic71⋊C4—216no—
164C22⋊Dic71—216no—
165(C2×D71)⋊C4—28no—
166(C2×D71)⋊C4—216no—
167(C2×D71)⋊C4—216no—
168(C2×D71)⋊C4C2×D4132yescomplemented
169C4×Dic71—216no—
170C24×C71—22no—
171C24×C71—24no—
172C24×C71—24no—
173C24×C71—22no—
174C24×C71—22no—
175C24×C71C2412yes—
176C24×C71C2411yescharacteristic
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71—24no—
181C22×C4×C71—24no—
182C22×C4×C71—24no—
183C22×C4×C71—22no—
184C22×C4×C71—24no—
185C22×C4×C71C2×D414yes—
186C22×C4×C71C2×D418yes—
187C22×C4×C71C2414yes—
188C22×C4×C71C2412yes—
189C22×C4×C71C2412yes—
190C22×C4×C71C2414yes—
191C4⋊C4×C71—28no—
192C4⋊C4×C71—216no—
193C22⋊C4×C71—28no—
194C22⋊C4×C71—216no—
195C23×C71—24no—
196C23×C71—22no—
197C23×C71—24no—
198C23×C71—22no—
199C23×C71—22no—
200C23×C71—44no—
201C23×C71—44no—
202C23×C71—44no—
203C23×C71—48no—
204C23×C71—48no—
205C23×C71C22×D412yes—
206C23×C71C22×D414yes—
207C23×C71C2×#4114yes—
208C23×C712−1+412yes—
209C23×C712−1+412yes—
210C23×C71C2511yescharacteristic
211C22×D71—28no—
212C22×D71—28no—
213C22×D71—216no—
214C22×D71—216no—
215C22×D71—24no—
216C22×D71—22no—
217C22×D71—22no—
218C22×D71—24no—
219C22×D71—22no—
220C22×D71—24no—
221C22×D71—22no—
222C22×D71—22no—
223C22×D71—44no—
224C22×D71—44no—
225C22×D71—44no—
226C22×D71—48no—
227C22×D71—48no—
228C22×D71#127116yes—
229C22×D71C22×D418yes—
230C22×D71C22×D414yes—
231C2×C4×C71—28no—
232C2×C4×C71—216no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—48no—
242C2×C4×C71—44no—
243C2×C4×C71C22×D418yes—
244C2×C4×C71C22×D414yes—
245C22⋊D71—416no—
246C22⋊D71—416no—
247C2×Dic71—216no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—22no—
252C2×Dic71—24no—
253C2×Dic71—24no—
254C2×Dic71—24no—
255C2×Dic71—48no—
256C2×Dic71—44no—
257C2×Dic71#123116yescomplemented
258C2×Dic71C22×D418yes—
259C2×Dic71C22×D414yes—
260C2×Dic71C2×#4118yes—
261D284—416no—
262C4×D71—232no—
263C4×D71—432no—
264C22×C71—42no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71—44no—
268C22×C71—44no—
269C22×C71—88no—
270C22×C71C2×#12314yes—
271C22×C71C2×#12714yes—
272C22×C71#53614yes—
273C22×C71#54718yes—
274C22×C71#54718yes—
275C22×C71C23×D411yescharacteristic
276C22×C71C23×D412yes—
277C22×C71C22×#4112yes—
278C22×C71C2×2−1+411yescharacteristic
279C22×C71C2×2−1+411yescharacteristic
280C2×D71—24no—
281C2×D71—24no—
282C2×D71—28no—
283C2×D71—28no—
284C2×D71—232no—
285C2×D71—42no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71—44no—
289C2×D71—44no—
290C2×D71—88no—
291C2×D71C2×#12718yes—
292C4×C71—24no—
293C4×C71—28no—
294C4×C71—44no—
295C4×C71—42no—
296Dic71—28no—
297Dic71—44no—
298Dic71—42no—
299Dic71C2×#12318yescomplemented
300C2×#2501—711nomaximal, Sylow
301C2×C71—82no—
302C2×C71#250118yescomplemented
303C2×C71C22×#12311yescharacteristic
304C2×C71C22×#12711yescharacteristic
305C2×C71C2×#53611yescharacteristic
306C2×C71C2×#54712yes—
307C2×C71C2×#54712yes—
308D71—28no—
309D71—82no—
310C22×#123—712no—
311C22×#123—712no—
312C22×#121—711no—
313C23×C4⋊C4—711no—
314#2501—7116no—
315C2×#399—711no—
316C2×#399—711no—
317C2×#399—711no—
318C2×#399D7111yescharacteristic, complemented
319C2×#390—712no—
320C2×#384—712no—
321C2×C4×C22⋊C4—711no—
322C71C2×#250111yescharacteristic, complemented, Sylow
323C23×D4—711no—
324C23×D4—711no—
325C23×D4—711no—
326C24×C4—711no—
327C24×C4—712no—
328C2×#123—7116no—
329C2×#123—7116no—
330C2×#123—14216no—
331C2×#123—14216no—
332C2×#121—718no—
333C2×#121—1428no—
334C22×C4⋊C4—711no—
335C22×C4⋊C4—712no—
336C22×C4⋊C4—712no—
337C22×C4⋊C4—712no—
338C22×C4⋊C4—712no—
339C22×C4⋊C4—711no—
340C22×C4⋊C4—718no—
341C22×C4⋊C4—1428no—
342C22×C4⋊C4C2×D7111yescharacteristic
343C22×C4⋊C4C2×D7112yes—
344C22×C22⋊C4—712no—
345C22×C22⋊C4—712no—
346C22×C22⋊C4—711no—
347C22×C22⋊C4—712no—
348C22×C22⋊C4—712no—
349C22×C22⋊C4C2×D7111yescharacteristic
350C22×C22⋊C4C2×D7112yes—
351C22×C42—712no—
352#399—718no—
353#399—718no—
354#399—718no—
355#399C2×D7118yescomplemented
356#390—7116no—
357#384—7116no—
358C4×C22⋊C4—718no—
359C2×#95—711no—
360C2×#95C2×D7111yescharacteristic
361C25—711no—
362C25—711no—
363C25C22×D7111yescharacteristic
364C22×D4—714no—
365C22×D4—714no—
366C22×D4—714no—
367C22×D4—1424no—
368C22×D4—1428no—
369C22×D4—1424no—
370C22×D4—1428no—
371C22×D4—1424no—
372C22×D4—1428no—
373C23×C4—711no—
374C23×C4—712no—
375C23×C4—718no—
376C23×C4—714no—
377C23×C4—712no—
378C23×C4—711no—
379C23×C4—718no—
380C23×C4—1428no—
381C23×C4—1428no—
382C23×C4—1428no—
383C23×C4—1428no—
384C23×C4C22×D7112yes—
385C23×C4C22×D7111yescharacteristic
386C23×C4C22×D7112yes—
387C23×C4C22×D7111yescharacteristic
388#123—14264no—
389#123—14264no—
390#121—14232no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—718no—
394C2×C4⋊C4—714no—
395C2×C4⋊C4—714no—
396C2×C4⋊C4—718no—
397C2×C4⋊C4—718no—
398C2×C4⋊C4—718no—
399C2×C4⋊C4—14232no—
400C2×C4⋊C4—1428no—
401C2×C4⋊C4—1424no—
402C2×C4⋊C4—1428no—
403C2×C4⋊C4—1428no—
404C2×C4⋊C4—1424no—
405C2×C4⋊C4—1428no—
406C2×C4⋊C4C22×D7114yes—
407C2×C4⋊C4C22×D7118yes—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—28no—
410C2×C22⋊C4—7116no—
411C2×C22⋊C4—718no—
412C2×C22⋊C4—718no—
413C2×C22⋊C4—714no—
414C2×C22⋊C4—718no—
415C2×C22⋊C4—718no—
416C2×C22⋊C4—1428no—
417C2×C22⋊C4—1428no—
418C2×C22⋊C4—1424no—
419C2×C22⋊C4—1428no—
420C2×C22⋊C4C22×D7114yes—
421C2×C22⋊C4C22×D7118yes—
422C2×C42—718no—
423C2×C42—1428no—
424#95—714no—
425#95C22×D7114yes—
426C24—24no—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—22no—
431C24—718no—
432C24—714no—
433C24—712no—
434C24—712no—
435C24—1424no—
436C24—1422no—
437C24—1424no—
438C24—1422no—
439C24—1422no—
440C24—1424no—
441C24—1422no—
442C24—1422no—
443C24C23×D7112yes—
444C24C23×D7111yescharacteristic
445C2×D4—28416no—
446C2×D4—28416no—
447C2×D4—28416no—
448C2×D4—1428no—
449C2×D4—14216no—
450C2×D4—1428no—
451C2×D4—14216no—
452C2×D4—1428no—
453C2×D4—14216no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—28432no—
463C22×C4—714no—
464C22×C4—718no—
465C22×C4—712no—
466C22×C4—718no—
467C22×C4—714no—
468C22×C4—7116no—
469C22×C4—714no—
470C22×C4—712no—
471C22×C4—14216no—
472C22×C4—14216no—
473C22×C4—14216no—
474C22×C4—14216no—
475C22×C4—14232no—
476C22×C4—1424no—
477C22×C4—1424no—
478C22×C4—1422no—
479C22×C4—1424no—
480C22×C4—1424no—
481C22×C4—1424no—
482C22×C4—1424no—
483C22×C4D4×D7118yes—
484C22×C4D4×D7114yes—
485C22×C4C23×D7114yes—
486C22×C4C23×D7112yes—
487C22×C4C23×D7112yes—
488C22×C4C23×D7114yes—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—14216no—
492C4⋊C4—1428no—
493C4⋊C4—14216no—
494C4⋊C4—14216no—
495C4⋊C4—1428no—
496C4⋊C4—14216no—
497C22⋊C4—28no—
498C22⋊C4—216no—
499C22⋊C4—7132no—
500C22⋊C4—14216no—
501C22⋊C4—14216no—
502C22⋊C4—1428no—
503C22⋊C4—14216no—
504C42—14216no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—48no—
512C23—48no—
513C23—44no—
514C23—44no—
515C23—2844no—
516C23—2848no—
517C23—2848no—
518C23—2844no—
519C23—2844no—
520C23—718no—
521C23—714no—
522C23—7116no—
523C23—1424no—
524C23—1422no—
525C23—1424no—
526C23—1422no—
527C23—1422no—
528C23—1424no—
529C23—1428no—
530C23—1422no—
531C23—1422no—
532C23—14216no—
533C23—14216no—
534C23—1428no—
535C23C2×D4×D7112yes—
536C23C2×D4×D7114yes—
537C23#41×D7114yes—
538C23#1455617512yes—
539C23#1455617612yes—
540C23C24×D7111yescharacteristic
541D4—28416no—
542D4—28416no—
543D4—28416no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—216no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—28432no—
557C2×C4—2848no—
558C2×C4—2844no—
559C2×C4—714no—
560C2×C4—718no—
561C2×C4—7116no—
562C2×C4—718no—
563C2×C4—14232no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1422no—
567C2×C4—1424no—
568C2×C4—1424no—
569C2×C4—1424no—
570C2×C4—1424no—
571C2×C4—14216no—
572C2×C4C2×D4×D7118yes—
573C2×C4C2×D4×D7114yes—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—88no—
580C22—2844no—
581C22—2844no—
582C22—2842no—
583C22—2842no—
584C22—2842no—
585C22—5688no—
586C22—718no—
587C22—14232no—
588C22—1428no—
589C22—1428no—
590C22—1424no—
591C22—1424no—
592C22#123×D7114yes—
593C22#127×D7114yes—
594C22#1798657214yes—
595C22#1798664718yes—
596C22#1798665118yes—
597C22C22×D4×D7112yes—
598C22C22×D4×D7111yescharacteristic
599C22C2×#41×D7112yes—
600C22C2×#1455617511yescharacteristic
601C22C2×#1455617611yescharacteristic
602C4—28no—
603C4—24no—
604C4—44no—
605C4—42no—
606C4—2844no—
607C4—2842no—
608C4—718no—
609C4—1428no—
610C2—82no—
611C2—5682no—
612C2—1428no—
613C2#2259280618yesdirect product factor
614C2C2×#123×D7111yescharacteristic
615C2C2×#127×D7111yescharacteristic
616C2C2×#1798657211yescharacteristic
617C2C2×#1798664712yes—
618C2C2×#1798665112yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#399×C71C2
socleC24×C71C24
prime radicalC22×#14556101C2
hypercenterC2×#399D71
hypocenterC71C2×#2501
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#14556113C2semidirect
2. C22×#14556110C2semidirect
4. C22×C4⋊C4×D71C2semidirect
5. #22592806C2direct (N × H)
6. C2×#17985936C2semidirect
7. C2×#17985935C2semidirect
8. C2×#17985934C2semidirect
9. C2×#17985899C2semidirect
10. C2×#17985868C2semidirect
11. C2×#17985846C2semidirect
12. C2×#399×C71C2semidirect
19. C2×#14556113C22semidirect
21. C2×#14556110C22semidirect
31. C2×C4⋊C4×D71C22semidirect
38. #17985936C22semidirect
39. #17985935C22semidirect
40. #17985934C22semidirect
41. #17985899C22semidirect
42. #17985868C22semidirect
43. #17985846C22semidirect
49. #399×C71C22semidirect
94. C2×(C2×D71)⋊C4D4semidirect
168. (C2×D71)⋊C4C2×D4semidirect
257. C2×Dic71#123semidirect
299. Dic71C2×#123semidirect
302. C2×C71#2501semidirect
318. C2×#399D71semidirect
322. C71C2×#2501semidirect
355. #399C2×D71semidirect
613. C2#22592806direct (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×#145561102
C22×#145561012
C22×C4⋊C4×D712
#225928062
C2×#179859362
C2×#179859352
C2×#179859342
C2×#179858992
C2×#179858682
C2×#179858462
C2×#399×C712
C2×#250171