Finite Group Databasebeta

← C2×#22589650 #32025380

Subgroups — C2×#22589650 #32025380

Order 18176.

1918 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765850C211yesmaximal, characteristic
2C22×C4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3C22×#14556049C211yesmaximal, characteristic, complemented
4C22×#14556048C211yesmaximal, characteristic, complemented
5C2×#17985901C211yesmaximal, characteristic, complemented
6#22589650C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C2×#390×C71C211yesmaximal, characteristic, complemented
9C23×D284C2211yescharacteristic
10C22×#11765850—24no—
11C22×#11765850—24no—
12C22×#11765850—24no—
13C22×#11765850C2214yes—
14C23×C71⋊C8C2211yescharacteristic
15C23×C71⋊C8C2211yescharacteristic
16C22×C4⋊Dic71C2211yescharacteristic
17C22×(C2×D71)⋊C4—21no—
18C22×(C2×D71)⋊C4—21no—
19C2×C4⋊(C71⋊C8)—24no—
20C2×C4⋊(C71⋊C8)C2214yescomplemented
21C2×#14556049—24no—
22C2×#14556049C2214yescomplemented
23C2×#14556048—24no—
24C2×#14556048C2214yescomplemented
25#17985901—22no—
26#17985901C2214yescomplemented
27#17985761C2214yescomplemented
28C23×D4×C71C2211yescharacteristic
29C22×C4⋊C4×C71C2211yescharacteristic
30C22×C22⋊C4×C71—21no—
31C22×C22⋊C4×C71—21no—
32C22×C42×C71C2211yescharacteristic
33#390×C71—22no—
34#390×C71C2214yescomplemented
35C24×D71—21no—
36C22×D284—22no—
37C22×D284—22no—
38C22×D284—21no—
39C22×D284—22no—
40C22×D284—22no—
41C22×D284—22no—
42C22×D284D414yes—
43C22×D284C2312yes—
44C23×Dic71—21no—
45C2×#11765850—232no—
46C2×#11765850—28no—
47C2×#11765850—28no—
48C2×#11765850—28no—
49C22×C71⋊C8—22no—
50C22×C71⋊C8—22no—
51C22×C71⋊C8—22no—
52C22×C71⋊C8—22no—
53C22×C71⋊C8—22no—
54C22×C71⋊C8D414yes—
55C22×C71⋊C8C2312yes—
56C22×C71⋊C8C2312yes—
57C2×C4⋊Dic71—22no—
58C2×C4⋊Dic71—21no—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71C2312yes—
61C2×(C2×D71)⋊C4—24no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—44no—
64C2×(C2×D71)⋊C4—44no—
65C4⋊(C71⋊C8)—28no—
66#14556049—28no—
67#14556048—28no—
68C25×C71—21no—
69C22×D4×C71—22no—
70C22×D4×C71—22no—
71C22×D4×C71—21no—
72C22×D4×C71—22no—
73C22×D4×C71—22no—
74C22×D4×C71—22no—
75C22×D4×C71D414yes—
76C22×D4×C71C2312yes—
77C23×C4×C71—21no—
78C23×C4×C71D411yescharacteristic
79C23×C4×C71D411yescharacteristic
80C23×C4×C71C2311yescharacteristic
81C2×C4⋊C4×C71—22no—
82C2×C4⋊C4×C71—22no—
83C2×C4⋊C4×C71—21no—
84C2×C4⋊C4×C71C2312yes—
85C2×C22⋊C4×C71—24no—
86C2×C22⋊C4×C71—24no—
87C2×C22⋊C4×C71—44no—
88C2×C22⋊C4×C71—44no—
89C2×C42×C71—22no—
90C2×C42×C71—22no—
91C2×C42×C71D412yes—
92C2×C42×C71C2312yes—
93C23×D71—24no—
94C23×D71—22no—
95C23×D71—42no—
96C23×D71—42no—
97C23×D71—42no—
98C23×D71—42no—
99C23×D71—42no—
100C23×D71—42no—
101C2×D284—28no—
102C2×D284—28no—
103C2×D284—22no—
104C2×D284—22no—
105C2×D284—22no—
106C2×D284—22no—
107C2×D284—24no—
108C2×D284—22no—
109C2×D284—42no—
110C2×D284—42no—
111C2×D284—44no—
112C2×D284—44no—
113C2×D284C2×D414yes—
114C22×Dic71—22no—
115C22×Dic71—42no—
116C22×Dic71—42no—
117C22×Dic71—42no—
118#11765850—232no—
119C2×C71⋊C8—28no—
120C2×C71⋊C8—22no—
121C2×C71⋊C8—22no—
122C2×C71⋊C8—22no—
123C2×C71⋊C8—22no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8—44no—
126C2×C71⋊C8C2×D414yes—
127C4⋊Dic71—22no—
128C4⋊Dic71—22no—
129C4⋊Dic71—42no—
130(C2×D71)⋊C4—48no—
131(C2×D71)⋊C4—48no—
132C24×C71—24no—
133C24×C71—22no—
134C24×C71—42no—
135C24×C71—42no—
136C24×C71—42no—
137C24×C71—42no—
138C24×C71—42no—
139C24×C71—42no—
140C24×C71C2×D411yescharacteristic
141C2×D4×C71—28no—
142C2×D4×C71—24no—
143C2×D4×C71—28no—
144C2×D4×C71—22no—
145C2×D4×C71—22no—
146C2×D4×C71—22no—
147C2×D4×C71—22no—
148C2×D4×C71—22no—
149C2×D4×C71—42no—
150C2×D4×C71—42no—
151C2×D4×C71—44no—
152C2×D4×C71—44no—
153C2×D4×C71C2×D414yes—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—42no—
161C22×C4×C71—42no—
162C22×C4×C71—42no—
163C22×C4×C71D814yescomplemented
164C22×C4×C71C2×D412yes—
165C22×C4×C71C2×D412yes—
166C22×C4×C71C2×D412yes—
167C22×C4×C71C2×D412yes—
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D412yes—
170C22×C4×C71#4112yes—
171C22×C4×C71#4112yes—
172C22×C4×C71#4112yes—
173C22×C4×C71C2411yescharacteristic
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—42no—
177C22⋊C4×C71—48no—
178C22⋊C4×C71—48no—
179C42×C71—24no—
180C42×C71—22no—
181C42×C71—22no—
182C23×C71—24no—
183C23×C71—42no—
184C23×C71—42no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—48no—
190C23×C71—48no—
191C23×C71—48no—
192C23×C71—84no—
193C23×C71—84no—
194C23×C71—84no—
195C23×C71—84no—
196C23×C71#12112yes—
197C23×C71#12112yes—
198C23×C71#12112yes—
199C23×C71C2×D812yes—
200C23×C71#13512yes—
201C23×C71AGL(1,ℤ/8)12yes—
202C23×C71AGL(1,ℤ/8)12yes—
203C23×C71C22×D411yescharacteristic
204C22×D71—24no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C22×D71—48no—
212C22×D71—48no—
213C22×D71—48no—
214C22×D71—84no—
215C22×D71—84no—
216C22×D71—84no—
217C22×D71—84no—
218D4×C71—24no—
219D4×C71—24no—
220D4×C71—42no—
221D4×C71—42no—
222D4×C71—48no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—28no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—44no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—84no—
234C2×C4×C71#12014yes—
235C2×C4×C71#12112yes—
236C2×C4×C71#12112yes—
237C2×C4×C71#12112yes—
238C2×C4×C71#12314yes—
239C2×C4×C71#12314yes—
240C2×C4×C71#12414yes—
241C2×C4×C71C2×D814yescomplemented
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C22×D411yescharacteristic
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—84no—
252D284—24no—
253D284—24no—
254D284—42no—
255D284—42no—
256D284—48no—
257C71⋊C8—24no—
258C71⋊C8—42no—
259C22×C71—44no—
260C22×C71—44no—
261C22×C71—44no—
262C22×C71—816no—
263C22×C71—82no—
264C22×C71—82no—
265C22×C71—82no—
266C22×C71—82no—
267C22×C71#39414yes—
268C22×C71#45914yes—
269C22×C71#45914yes—
270C22×C71#46314yes—
271C22×C71#46314yes—
272C22×C71#46614yes—
273C22×C71#46914yes—
274C22×C71C2×#12111yescharacteristic
275C22×C71C2×#12111yescharacteristic
276C22×C71C2×#12111yescharacteristic
277C22×C71C22×D811yescharacteristic
278C22×C71C2×#13511yescharacteristic
279C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
281C2×D71—44no—
282C2×D71—44no—
283C2×D71—44no—
284C2×D71—816no—
285C2×D71—82no—
286C2×D71—82no—
287C2×D71—82no—
288C2×D71—82no—
289C4×C71—24no—
290C4×C71—42no—
291C4×C71—82no—
292C4×C71#39414yes—
293C4×C71C2×#12011yescharacteristic
294C4×C71C2×#12311yescharacteristic
295C4×C71C2×#12311yescharacteristic
296C4×C71C2×#12411yescharacteristic
297Dic71—82no—
298C2×#2054—711nomaximal, Sylow
299C2×C71—84no—
300C2×C71#205418yescomplemented
301C2×C71C2×#39411yescharacteristic
302C2×C71C2×#45911yescharacteristic
303C2×C71C2×#45911yescharacteristic
304C2×C71C2×#46311yescharacteristic
305C2×C71C2×#46311yescharacteristic
306C2×C71C2×#46611yescharacteristic
307C2×C71C2×#46911yescharacteristic
308D71—84no—
309C23×D8—711no—
310C22×C4⋊C8—711no—
311C22×#102—711no—
312C22×#102—711no—
313C2×#390—711no—
314C2×#390D7111yescharacteristic, complemented
315#2054—718no—
316C2×#340—711no—
317C71C2×#205411yescharacteristic, complemented, Sylow
318C23×D4—711no—
319C23×D4C2×D7111yescharacteristic
320C22×D8—714no—
321C22×D8—1424no—
322C22×D8—1424no—
323C22×D8—1424no—
324C23×C8—711no—
325C23×C8—711no—
326C22×C4⋊C4—711no—
327C22×C4⋊C4C2×D7111yescharacteristic
328C22×C22⋊C4—21no—
329C22×C22⋊C4—21no—
330C22×C22⋊C4—1421no—
331C22×C22⋊C4—1421no—
332C22×C42C2×D7111yescharacteristic
333C2×C4⋊C8—714no—
334C2×C4⋊C8—1424no—
335C2×#102—714no—
336C2×#102—714no—
337C2×#102—1424no—
338C2×#102—1424no—
339#390—22no—
340#390—714no—
341#390—1422no—
342#390C2×D7114yescomplemented
343#340—714no—
344C25—21no—
345C25—1421no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—21no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—714no—
353C22×D4—712no—
354C22×D4—1422no—
355C22×D4—1421no—
356C22×D4—1422no—
357C22×D4—1422no—
358C22×D4—1422no—
359C22×D4—1422no—
360C22×D4D28414yes—
361C22×D4C22×D7112yes—
362C23×C4—21no—
363C23×C4—1421no—
364C23×C4C22⋊D7111yescharacteristic
365C23×C4C22⋊D7111yescharacteristic
366C23×C4C22×D7111yescharacteristic
367C2×D8—1428no—
368C2×D8—1428no—
369C2×D8—14232no—
370C2×D8—1428no—
371C22×C8—714no—
372C22×C8—712no—
373C22×C8—712no—
374C22×C8—1422no—
375C22×C8—1422no—
376C22×C8—1422no—
377C22×C8—1422no—
378C22×C8—1422no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—712no—
383C2×C4⋊C4—1421no—
384C2×C4⋊C4—1422no—
385C2×C4⋊C4—1422no—
386C2×C4⋊C4C22×D7112yes—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—44no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—2844no—
392C2×C22⋊C4—2844no—
393C2×C22⋊C4—1424no—
394C2×C22⋊C4—1424no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42C22⋊D7112yes—
398C2×C42C22×D7112yes—
399C4⋊C8—1428no—
400#102—1428no—
401#102—1428no—
402C24—22no—
403C24—24no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—2842no—
411C24—2842no—
412C24—2842no—
413C24—2842no—
414C24—2842no—
415C24—2842no—
416C24—1422no—
417C24—1424no—
418C24C2×C22⋊D7111yescharacteristic
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—2842no—
432C2×D4—2842no—
433C2×D4—2844no—
434C2×D4—2844no—
435C2×D4—714no—
436C2×D4—1422no—
437C2×D4—1422no—
438C2×D4—1422no—
439C2×D4—1422no—
440C2×D4—1422no—
441C2×D4—1428no—
442C2×D4—1424no—
443C2×D4—1428no—
444C2×D4C2×D28414yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—2842no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—1422no—
458C22×C4#1176585014yescomplemented
459C22×C4D4×D7112yes—
460C22×C4D4×D7112yes—
461C22×C4C2×C22⋊D7112yes—
462C22×C4C2×C22⋊D7112yes—
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×D28412yes—
465C22×C4#1176586412yes—
466C22×C4#1176586412yes—
467C22×C4#1176586512yes—
468C22×C4C23×D7111yescharacteristic
469D8—14232no—
470C2×C8—2844no—
471C2×C8—714no—
472C2×C8—1422no—
473C2×C8—1422no—
474C2×C8—1422no—
475C2×C8—1422no—
476C2×C8—1422no—
477C2×C8—1428no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—2842no—
482C4⋊C4—1422no—
483C4⋊C4—1422no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—2848no—
487C22⋊C4—2848no—
488C42—22no—
489C42—22no—
490C42—24no—
491C23—24no—
492C23—48no—
493C23—48no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—2848no—
506C23—2848no—
507C23—2848no—
508C23—2842no—
509C23—2842no—
510C23—2842no—
511C23—2842no—
512C23—2842no—
513C23—2842no—
514C23—5684no—
515C23—5684no—
516C23—5684no—
517C23—5684no—
518C23—1424no—
519C23#1455609912yes—
520C23#1455610012yes—
521C23#1455610212yes—
522C23C2×#1176585012yes—
523C23#1455614812yes—
524C23#1455615312yes—
525C23#1455615412yes—
526C23C22×C22⋊D7111yescharacteristic
527D4—24no—
528D4—24no—
529D4—48no—
530D4—42no—
531D4—42no—
532D4—2848no—
533D4—2842no—
534D4—2842no—
535D4—1424no—
536D4—1424no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—5684no—
552C2×C4#1455609814yes—
553C2×C4#1455609912yes—
554C2×C4#1455610012yes—
555C2×C4#1455610212yes—
556C2×C4#1455611014yes—
557C2×C4#1455611214yes—
558C2×C4#1455611414yes—
559C2×C4C2×#1176585014yescomplemented
560C2×C4C2×D4×D7111yescharacteristic
561C2×C4C2×D4×D7111yescharacteristic
562C2×C4C22×D28411yescharacteristic
563C2×C4C2×#1176586411yescharacteristic
564C2×C4C2×#1176586411yescharacteristic
565C2×C4C2×#1176586511yescharacteristic
566C8—2842no—
567C8—1424no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—816no—
576C22—2844no—
577C22—2844no—
578C22—2844no—
579C22—5682no—
580C22—5682no—
581C22—5682no—
582C22—5682no—
583C22—56816no—
584C22#1798591514yes—
585C22#1798619114yes—
586C22#1798619414yes—
587C22#1798621114yes—
588C22#1798621414yes—
589C22#1798623314yes—
590C22#1798625014yes—
591C22C2×#1455609911yescharacteristic
592C22C2×#1455610011yescharacteristic
593C22C2×#1455610211yescharacteristic
594C22C22×#1176585011yescharacteristic
595C22C2×#1455614811yescharacteristic
596C22C2×#1455615311yescharacteristic
597C22C2×#1455615411yescharacteristic
598C4—24no—
599C4—42no—
600C4—82no—
601C4—5682no—
602C4#1798591514yes—
603C4C2×#1455609811yescharacteristic
604C4C2×#1455611011yescharacteristic
605C4C2×#1455611211yescharacteristic
606C4C2×#1455611411yescharacteristic
607C2—84no—
608C2—5684no—
609C2#2258965018yesdirect product factor
610C2C2×#1798591511yescharacteristic
611C2C2×#1798619111yescharacteristic
612C2C2×#1798619411yescharacteristic
613C2C2×#1798621111yescharacteristic
614C2C2×#1798621411yescharacteristic
615C2C2×#1798623311yescharacteristic
616C2C2×#1798625011yescharacteristic
subgroupG/H
centerC24C2×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#390×C71C2
socleC24×C71C2×D4
prime radicalC23×#11765850C2
hypercenterC2×#390D71
hypocenterC71C2×#2054
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊(C71⋊C8)C2semidirect
3. C22×#14556049C2semidirect
4. C22×#14556048C2semidirect
5. C2×#17985901C2semidirect
6. #22589650C2direct (N × H)
7. C2×#17985761C2semidirect
8. C2×#390×C71C2semidirect
20. C2×C4⋊(C71⋊C8)C22semidirect
22. C2×#14556049C22semidirect
24. C2×#14556048C22semidirect
26. #17985901C22semidirect
27. #17985761C22semidirect
34. #390×C71C22semidirect
163. C22×C4×C71D8semidirect
241. C2×C4×C71C2×D8semidirect
300. C2×C71#2054semidirect
314. C2×#390D71semidirect
317. C71C2×#2054semidirect
342. #390C2×D71semidirect
458. C22×C4#11765850semidirect
559. C2×C4C2×#11765850semidirect
609. C2#22589650direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658502
C22×C4⋊(C71⋊C8)2
C22×#145560492
C22×#145560482
C2×#179859012
#225896502
C2×#179857612
C2×#390×C712
C2×#205471