Finite Group Databasebeta

← C2×#22591940 #32165474

Subgroups — C2×#22591940 #32165474

Order 18176.

3246 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556126C211yesmaximal, characteristic, complemented
2C22×#14556108C211yesmaximal, characteristic
3C22×#14556104C211yesmaximal, characteristic
4C22×C4×Dic142C211yesmaximal, characteristic, complemented
5#22591940C2116yesmaximal, direct product factor
6C2×#17985933C212yesmaximal, complemented
7C2×#17985923C212yesmaximal
8C2×#17985896C212yesmaximal, complemented
9C2×#17985895C211yesmaximal, characteristic, complemented
10C2×#17985888C212yesmaximal
11C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
12C2×#391×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C2×#14556126—28no—
16C2×#14556126C2218yescomplemented
17C2×#14556108—28no—
18C2×#14556108C2218yes—
19C2×#14556104—28no—
20C2×#14556104C2218yes—
21C2×C4×Dic142—28no—
22C2×C4×Dic142C2218yescomplemented
23C22×C4⋊Dic71C2212yes—
24C22×C4⋊Dic71C2211yescharacteristic
25C22×Dic71⋊C4C2212yes—
26C22×Dic71⋊C4C2212yes—
27C22×Dic71⋊C4C2212yes—
28C22×Dic71⋊C4C2212yes—
29C22×Dic71⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2211yescharacteristic
33C22×(C2×D71)⋊C4C2211yescharacteristic
34C22×C4×Dic71C2212yes—
35C22×C4×Dic71C2211yescharacteristic
36#17985933C22116yescomplemented
37#17985923C22116yes—
38#17985896C22116yescomplemented
39#17985895C2218yescomplemented
40#17985888C22116yes—
41C4×(C2×D71)⋊C4C2218yes—
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2211yescharacteristic
44C2×C71⋊#95C2211yescharacteristic
45C23×Q8×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C42×C71C2211yescharacteristic
48#391×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C2×#95×C71C2212yes—
51C24×D71C2311yescharacteristic
52C22×Dic142—24no—
53C22×Dic142—24no—
54C22×Dic142D418yes—
55C22×Dic142C2314yes—
56C23×Dic71C2311yescharacteristic
57C23×Dic71C2312yes—
58C23×Dic71C2312yes—
59C23×Dic71C2311yescharacteristic
60C23×Dic71C2311yescharacteristic
61C22×C4×D71—24no—
62C22×C4×D71—24no—
63C22×C4×D71—24no—
64C22×C4×D71C2314yes—
65#14556126—232no—
66#14556108—232no—
67#14556104—232no—
68C4×Dic142—232no—
69C2×C4⋊Dic71—28no—
70C2×C4⋊Dic71D418yes—
71C2×C4⋊Dic71C2318yes—
72C2×C4⋊Dic71C2314yes—
73C2×Dic71⋊C4—28no—
74C2×Dic71⋊C4—28no—
75C2×Dic71⋊C4—28no—
76C2×Dic71⋊C4—28no—
77C2×Dic71⋊C4—28no—
78C2×Dic71⋊C4C2318yes—
79C2×Dic71⋊C4C2318yes—
80C2×Dic71⋊C4C2318yes—
81C2×Dic71⋊C4C2318yes—
82C2×Dic71⋊C4C2318yes—
83C2×(C2×D71)⋊C4—28no—
84C2×(C2×D71)⋊C4—28no—
85C2×(C2×D71)⋊C4—24no—
86C2×(C2×D71)⋊C4Q818yescomplemented
87C2×(C2×D71)⋊C4C2314yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2318yes—
90C2×(C2×D71)⋊C4C2314yes—
91C2×C4×Dic71—28no—
92C2×C4×Dic71—24no—
93C2×C4×Dic71C2314yes—
94C2×C4×Dic71C2318yes—
95C71⋊#95C2314yes—
96C71⋊#95C2314yes—
97C71⋊#95C2318yes—
98C22×Q8×C71—24no—
99C22×Q8×C71—24no—
100C22×Q8×C71—24no—
101C22×Q8×C71C2314yes—
102C23×C4×C71C2311yescharacteristic
103C23×C4×C71C2312yes—
104C23×C4×C71C2312yes—
105C23×C4×C71C2311yescharacteristic
106C23×C4×C71C2311yescharacteristic
107C2×C4⋊C4×C71—24no—
108C2×C4⋊C4×C71C2314yes—
109C2×C42×C71—24no—
110C2×C42×C71C2314yes—
111#95×C71C2318yes—
112#95×C71C2318yes—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71C2×Q814yes—
120C23×D71C2412yes—
121C2×Dic142—232no—
122C2×Dic142—28no—
123C2×Dic142—28no—
124C2×Dic142C2×D4116yes—
125C22×Dic71—24no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—24no—
129C22×Dic71—24no—
130C22×Dic71—22no—
131C22×Dic71—24no—
132C22×Dic71—22no—
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71C2×D418yes—
137C22×Dic71C2×D418yes—
138C22×Dic71C2×Q814yes—
139C22×Dic71#4114yes—
140C22×Dic71C2412yes—
141C22×Dic71C2414yes—
142C22×Dic71C2414yes—
143C22×Dic71C2412yes—
144C22×Dic71C2412yes—
145C2×C4×D71—28no—
146C2×C4×D71—28no—
147C2×C4×D71—28no—
148C2×C4×D71—416no—
149C4⋊Dic71—216no—
150C4⋊Dic71C2×D4116yes—
151Dic71⋊C4—216no—
152Dic71⋊C4—216no—
153Dic71⋊C4—216no—
154Dic71⋊C4—216no—
155Dic71⋊C4—216no—
156(C2×D71)⋊C4—216no—
157(C2×D71)⋊C4—216no—
158(C2×D71)⋊C4—28no—
159(C2×D71)⋊C4C2×Q8116yescomplemented
160C4×Dic71—28no—
161C4×Dic71—216no—
162C24×C71C2411yescharacteristic
163C2×Q8×C71—28no—
164C2×Q8×C71—28no—
165C2×Q8×C71—28no—
166C2×Q8×C71—416no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71—24no—
171C22×C4×C71—24no—
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—22no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—22no—
178C22×C4×C71C2×D414yes—
179C22×C4×C71C2×D414yes—
180C22×C4×C71C2×Q814yes—
181C22×C4×C71#4114yes—
182C22×C4×C71C2412yes—
183C22×C4×C71C2414yes—
184C22×C4×C71C2414yes—
185C22×C4×C71C2412yes—
186C22×C4×C71C2412yes—
187C4⋊C4×C71—28no—
188C42×C71—28no—
189C23×C71C22×D412yes—
190C23×C71C22×D412yes—
191C23×C71C22×Q812yes—
192C23×C71C2×#4112yes—
193C23×C71C2×#4112yes—
194C23×C712−1+412yes—
195C23×C712−1+412yes—
196C23×C71C2511yescharacteristic
197C22×D71—28no—
198C22×D71—28no—
199C22×D71—28no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—44no—
207C22×D71—44no—
208C22×D71—44no—
209C22×D71—44no—
210C22×D71C22×Q814yes—
211Q8×C71—416no—
212C2×C4×C71—28no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—22no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—22no—
224C2×C4×C71—48no—
225C2×C4×C71—44no—
226C2×C4×C71—48no—
227C2×C4×C71#121116yes—
228C2×C4×C71C22×D414yes—
229C2×C4×C71C22×D414yes—
230C2×C4×C71C22×Q814yes—
231C2×C4×C71C2×#4114yes—
232C2×Dic71—216no—
233C2×Dic71—216no—
234C2×Dic71—28no—
235C2×Dic71—28no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—22no—
242C2×Dic71—24no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—44no—
248C2×Dic71C22×D418yes—
249C2×Dic71C22×D418yes—
250C2×Dic71C22×Q814yes—
251C2×Dic71C2×#4114yes—
252C4×D71—416no—
253Dic142—232no—
254C22×C71C2×#12114yes—
255C22×C71#53314yes—
256C22×C71#54114yes—
257C22×C71#54714yes—
258C22×C71#54814yes—
259C22×C71D4×Q814yes—
260C22×C71D4×Q814yes—
261C22×C71C23×D411yescharacteristic
262C22×C71C23×D411yescharacteristic
263C22×C71C23×Q811yescharacteristic
264C22×C71C22×#4111yescharacteristic
265C22×C71C22×#4111yescharacteristic
266C22×C71C2×2−1+411yescharacteristic
267C22×C71C2×2−1+411yescharacteristic
268C2×D71—24no—
269C2×D71—24no—
270C2×D71—24no—
271C2×D71—42no—
272C2×D71—416no—
273C2×D71—42no—
274C2×D71—42no—
275C2×D71—42no—
276C4×C71—24no—
277C4×C71—44no—
278C4×C71—42no—
279C4×C71—44no—
280C4×C71C2×#12118yes—
281Dic71—28no—
282Dic71—28no—
283Dic71—24no—
284Dic71—24no—
285Dic71—42no—
286C2×#2429—711nomaximal, Sylow
287C2×C71#242918yescomplemented
288C2×C71C22×#12111yescharacteristic
289C2×C71C2×#53311yescharacteristic
290C2×C71C2×#54111yescharacteristic
291C2×C71C2×#54711yescharacteristic
292C2×C71C2×#54811yescharacteristic
293C2×C71C2×D4×Q811yescharacteristic
294C2×C71C2×D4×Q811yescharacteristic
295D71—44no—
296C22×C4⋊Q8—711no—
297C22×C22⋊Q8—711no—
298C22×C22⋊Q8—711no—
299C22×C4×Q8—711no—
300#2429—7116no—
301C2×#398—712no—
302C2×#396—712no—
303C2×#391D7111yescharacteristic, complemented
304C2×#389—712no—
305C2×#389—711no—
306C2×#388—712no—
307C2×C4×C22⋊C4—711no—
308C71C2×#242911yescharacteristic, complemented, Sylow
309C23×Q8—711no—
310C23×Q8C2×D7111yescharacteristic
311C24×C4—711no—
312C2×C4⋊Q8—718no—
313C2×C4⋊Q8—1428no—
314C2×C22⋊Q8—718no—
315C2×C22⋊Q8—718no—
316C2×C22⋊Q8—1428no—
317C2×C22⋊Q8—1428no—
318C2×C4×Q8—718no—
319C2×C4×Q8—1428no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—712no—
322C22×C4⋊C4—712no—
323C22×C4⋊C4—712no—
324C22×C4⋊C4—712no—
325C22×C4⋊C4—712no—
326C22×C4⋊C4—712no—
327C22×C4⋊C4C2×D7111yescharacteristic
328C22×C22⋊C4—711no—
329C22×C22⋊C4—711no—
330C22×C22⋊C4—712no—
331C22×C22⋊C4—712no—
332C22×C42—712no—
333C22×C42—711no—
334C22×C42C2×D7111yescharacteristic
335#398—7116no—
336#396—7116no—
337#391C2×D7118yescomplemented
338#389—718no—
339#389—7116no—
340#388—7116no—
341C4×C22⋊C4—718no—
342C2×#95—711no—
343C2×#95—712no—
344C2×#95—711no—
345C2×#95C2×D7112yes—
346C2×#95C2×D7112yes—
347C25—711no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—24no—
351C22×Q8—718no—
352C22×Q8—714no—
353C22×Q8—1424no—
354C22×Q8—1424no—
355C22×Q8C22×D7114yes—
356C23×C4—712no—
357C23×C4—712no—
358C23×C4—711no—
359C23×C4—711no—
360C23×C4—711no—
361C23×C4—714no—
362C23×C4—1424no—
363C23×C4—1424no—
364C23×C4—1424no—
365C23×C4C22×D7112yes—
366C23×C4C22×D7112yes—
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7111yescharacteristic
369C23×C4C22×D7111yescharacteristic
370C4⋊Q8—14232no—
371C22⋊Q8—14232no—
372C22⋊Q8—14232no—
373C4×Q8—14232no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—718no—
376C2×C4⋊C4—714no—
377C2×C4⋊C4—718no—
378C2×C4⋊C4—718no—
379C2×C4⋊C4—718no—
380C2×C4⋊C4—718no—
381C2×C4⋊C4—718no—
382C2×C4⋊C4—718no—
383C2×C4⋊C4—1428no—
384C2×C4⋊C4—1428no—
385C2×C4⋊C4—1428no—
386C2×C4⋊C4—1428no—
387C2×C4⋊C4—1428no—
388C2×C4⋊C4—1428no—
389C2×C4⋊C4C22×D7114yes—
390C2×C22⋊C4—718no—
391C2×C22⋊C4—714no—
392C2×C22⋊C4—718no—
393C2×C22⋊C4—718no—
394C2×C22⋊C4—714no—
395C2×C22⋊C4—1428no—
396C2×C22⋊C4—1428no—
397C2×C22⋊C4—1424no—
398C2×C42—24no—
399C2×C42—718no—
400C2×C42—714no—
401C2×C42—1428no—
402C2×C42—1424no—
403C2×C42C22×D7114yes—
404#95—714no—
405#95—714no—
406#95—718no—
407#95C22×D7118yes—
408#95C22×D7118yes—
409C24—714no—
410C24—712no—
411C24—1422no—
412C24—1422no—
413C24—1422no—
414C24—1422no—
415C24—1422no—
416C24—1422no—
417C24C23×D7111yescharacteristic
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—416no—
422C2×Q8—7116no—
423C2×Q8—1428no—
424C2×Q8—1428no—
425C2×Q8—14232no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28416no—
438C22×C4—718no—
439C22×C4—718no—
440C22×C4—714no—
441C22×C4—714no—
442C22×C4—712no—
443C22×C4—714no—
444C22×C4—712no—
445C22×C4—714no—
446C22×C4—712no—
447C22×C4—1422no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1428no—
451C22×C4—1428no—
452C22×C4—1428no—
453C22×C4—1422no—
454C22×C4—1422no—
455C22×C4—1424no—
456C22×C4—1424no—
457C22×C4—1424no—
458C22×C4—1422no—
459C22×C4—1424no—
460C22×C4—1422no—
461C22×C4D4×D7114yes—
462C22×C4D4×D7114yes—
463C22×C4Q8×D7114yes—
464C22×C4#1176586514yes—
465C22×C4C23×D7112yes—
466C22×C4C23×D7114yes—
467C22×C4C23×D7114yes—
468C22×C4C23×D7112yes—
469C22×C4C23×D7112yes—
470C4⋊C4—28no—
471C4⋊C4—7116no—
472C4⋊C4—14216no—
473C4⋊C4—14216no—
474C4⋊C4—14216no—
475C4⋊C4—14216no—
476C4⋊C4—14216no—
477C4⋊C4—14216no—
478C22⋊C4—7116no—
479C22⋊C4—1428no—
480C22⋊C4—14216no—
481C22⋊C4—14216no—
482C42—28no—
483C42—14216no—
484C42—1428no—
485C23—2844no—
486C23—2844no—
487C23—2844no—
488C23—2844no—
489C23—714no—
490C23—1422no—
491C23—1422no—
492C23—1422no—
493C23—1422no—
494C23—1422no—
495C23—1422no—
496C23—1428no—
497C23—1428no—
498C23—1428no—
499C23C2×D4×D7112yes—
500C23C2×D4×D7112yes—
501C23C2×Q8×D7112yes—
502C23#41×D7112yes—
503C23C2×#1176586512yes—
504C23#1455617512yes—
505C23#1455617612yes—
506C23C24×D7111yescharacteristic
507Q8—416no—
508Q8—14232no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—2844no—
525C2×C4—28416no—
526C2×C4—718no—
527C2×C4—718no—
528C2×C4—714no—
529C2×C4—714no—
530C2×C4—1422no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4—1422no—
535C2×C4—1424no—
536C2×C4—1424no—
537C2×C4—1424no—
538C2×C4—1422no—
539C2×C4—1424no—
540C2×C4—1422no—
541C2×C4—14216no—
542C2×C4—1428no—
543C2×C4—1428no—
544C2×C4—14216no—
545C2×C4#14556101116yes—
546C2×C4C2×D4×D7114yes—
547C2×C4C2×D4×D7114yes—
548C2×C4C2×Q8×D7114yes—
549C2×C4C2×#1176586514yes—
550C22—2842no—
551C22—2842no—
552C22—2842no—
553C22—28416no—
554C22—2842no—
555C22—1424no—
556C22—1424no—
557C22—1424no—
558C22C2×#1455610114yes—
559C22#1798655314yes—
560C22#1798660714yes—
561C22#1798665114yes—
562C22#1798665714yes—
563C22#1798666214yes—
564C22#1798666414yes—
565C22C22×D4×D7111yescharacteristic
566C22C22×D4×D7111yescharacteristic
567C22C22×Q8×D7111yescharacteristic
568C22C2×#41×D7111yescharacteristic
569C22C22×#1176586511yescharacteristic
570C22C2×#1455617511yescharacteristic
571C22C2×#1455617611yescharacteristic
572C4—24no—
573C4—42no—
574C4—44no—
575C4—44no—
576C4—2842no—
577C4—1428no—
578C4—1424no—
579C4—1424no—
580C4—1428no—
581C4C2×#1455610118yes—
582C2—2844no—
583C2#2259194018yesdirect product factor
584C2C22×#1455610111yescharacteristic
585C2C2×#1798655311yescharacteristic
586C2C2×#1798660711yescharacteristic
587C2C2×#1798665111yescharacteristic
588C2C2×#1798665711yescharacteristic
589C2C2×#1798666211yescharacteristic
590C2C2×#1798666411yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#391×C71C2
socleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#391D71
hypocenterC71C2×#2429
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
1. C22×#14556126C2semidirect
4. C22×C4×Dic142C2semidirect
5. #22591940C2direct (N × H)
6. C2×#17985933C2semidirect
8. C2×#17985896C2semidirect
9. C2×#17985895C2semidirect
12. C2×#391×C71C2semidirect
16. C2×#14556126C22semidirect
22. C2×C4×Dic142C22semidirect
36. #17985933C22semidirect
38. #17985896C22semidirect
39. #17985895C22semidirect
48. #391×C71C22semidirect
86. C2×(C2×D71)⋊C4Q8semidirect
159. (C2×D71)⋊C4C2×Q8semidirect
287. C2×C71#2429semidirect
303. C2×#391D71semidirect
308. C71C2×#2429semidirect
337. #391C2×D71semidirect
583. C2#22591940direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561262
C22×#145561082
C22×#145561042
C22×C4×Dic1422
#225919402
C2×#179859332
C2×#179859232
C2×#179858962
C2×#179858952
C2×#179858882
C2×C4×(C2×D71)⋊C42
C2×#391×C712
C2×#242971