Finite Group Databasebeta

← C2×#22588772 #31999133

Subgroups — C2×#22588772 #31999133

Order 18176.

1966 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765848C211yesmaximal, characteristic
2C22×#14556108C211yesmaximal, characteristic
3C22×#14556053C211yesmaximal, characteristic, complemented
4C22×#14556051C211yesmaximal, characteristic, complemented
5C2×#17985856C211yesmaximal, characteristic
6#22588772C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C2×#340×C71C211yesmaximal, characteristic, complemented
9C23×Dic142C2211yescharacteristic
10C23×C4×D71C2211yescharacteristic
11C22×#11765848—24no—
12C22×#11765848—24no—
13C22×#11765848—24no—
14C22×#11765848C2214yes—
15C23×C71⋊C8C2211yescharacteristic
16C2×#14556108—22no—
17C2×#14556108—24no—
18C2×#14556108C418yescomplemented
19C2×#14556108C2214yes—
20C22×C4⋊Dic71—21no—
21C22×C4⋊Dic71C2211yescharacteristic
22C22×(C2×D71)⋊C4—21no—
23C22×(C2×D71)⋊C4—21no—
24C2×#14556053—24no—
25C2×#14556053C2214yescomplemented
26C2×#14556051—24no—
27C2×#14556051C2214yescomplemented
28#17985856—22no—
29#17985856C2214yes—
30C2×C71⋊#95—21no—
31#17985761C2214yescomplemented
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C4⋊C4×C71C2211yescharacteristic
35#340×C71C2214yescomplemented
36C24×D71D411yescharacteristic
37C22×Dic142—21no—
38C22×Dic142—22no—
39C22×Dic142—22no—
40C22×Dic142—22no—
41C22×Dic142—22no—
42C22×Dic142—22no—
43C22×Dic142C2×C414yes—
44C22×Dic142C2312yes—
45C23×Dic71—21no—
46C23×Dic71—21no—
47C23×Dic71D411yescharacteristic
48C22×C4×D71—22no—
49C22×C4×D71—22no—
50C22×C4×D71—22no—
51C22×C4×D71—22no—
52C22×C4×D71C2×C414yes—
53C22×C4×D71D412yes—
54C22×C4×D71D414yes—
55C22×C4×D71C2312yes—
56C2×#11765848—232no—
57C2×#11765848—28no—
58C2×#11765848—28no—
59C2×#11765848—28no—
60C22×C71⋊C8—22no—
61C22×C71⋊C8—22no—
62C22×C71⋊C8D414yes—
63C22×C71⋊C8C2312yes—
64#14556108—216no—
65#14556108—28no—
66#14556108C2×C4116yescomplemented
67C2×C4⋊Dic71—28no—
68C2×C4⋊Dic71—24no—
69C2×C4⋊Dic71—22no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71C2312yes—
73C2×(C2×D71)⋊C4—28no—
74C2×(C2×D71)⋊C4—24no—
75C2×(C2×D71)⋊C4—24no—
76C2×(C2×D71)⋊C4—44no—
77#14556053—28no—
78#14556051—28no—
79C71⋊#95—24no—
80C23×C4×C71—21no—
81C23×C4×C71—21no—
82C23×C4×C71C2311yescharacteristic
83C22×C8×C71—22no—
84C22×C8×C71—22no—
85C22×C8×C71D414yes—
86C22×C8×C71C2312yes—
87C2×C4⋊C4×C71—21no—
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71C2×C414yes—
93C2×C4⋊C4×C71C2312yes—
94C2×C4⋊C4×C71C2312yes—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71C22⋊C414yes—
102C23×D71C2×D412yes—
103C2×Dic142—28no—
104C2×Dic142—28no—
105C2×Dic142—22no—
106C2×Dic142—22no—
107C2×Dic142—22no—
108C2×Dic142—22no—
109C2×Dic142—24no—
110C2×Dic142—22no—
111C2×Dic142—44no—
112C2×Dic142—42no—
113C2×Dic142—42no—
114C2×Dic142—44no—
115C2×Dic142C22×C414yes—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—24no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—42no—
122C22×Dic71—42no—
123C22×Dic71—42no—
124C22×Dic71—42no—
125C22×Dic71—42no—
126C22×Dic71C22⋊C414yes—
127C22×Dic71C2×D412yes—
128C2×C4×D71—28no—
129C2×C4×D71—28no—
130C2×C4×D71—28no—
131C2×C4×D71—24no—
132C2×C4×D71—24no—
133C2×C4×D71—28no—
134C2×C4×D71—22no—
135C2×C4×D71—22no—
136C2×C4×D71—22no—
137C2×C4×D71—22no—
138C2×C4×D71C22⋊C418yes—
139C2×C4×D71C22×C414yes—
140C2×C4×D71C2×D414yes—
141#11765848—232no—
142C2×C71⋊C8—28no—
143C2×C71⋊C8—22no—
144C2×C71⋊C8—22no—
145C2×C71⋊C8C2×D414yes—
146C4⋊Dic71—216no—
147C4⋊Dic71—22no—
148C4⋊Dic71—22no—
149C4⋊Dic71—42no—
150(C2×D71)⋊C4—216no—
151(C2×D71)⋊C4—48no—
152C24×C71C2×D411yescharacteristic
153C22×C4×C71—24no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71—42no—
160C22×C4×C71—42no—
161C22×C4×C71C22×C412yes—
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D412yes—
164C22×C4×C71C2×D412yes—
165C22×C4×C71C2×D412yes—
166C22×C4×C71#4112yes—
167C22×C4×C71#4112yes—
168C22×C4×C71C2411yescharacteristic
169C2×C8×C71—28no—
170C2×C8×C71—22no—
171C2×C8×C71—22no—
172C2×C8×C71C2×D414yes—
173C4⋊C4×C71—24no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—22no—
177C4⋊C4×C71—42no—
178C4⋊C4×C71C22×C414yes—
179C23×C71C2×C22⋊C412yes—
180C23×C71#12112yes—
181C23×C71#12112yes—
182C23×C71AGL(1,ℤ/8)12yes—
183C23×C71#13712yes—
184C23×C71#13712yes—
185C23×C71#13712yes—
186C23×C71C22×D411yescharacteristic
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—22no—
190C22×D71—22no—
191C22×D71—22no—
192C22×D71—22no—
193C22×D71—28no—
194C22×D71—28no—
195C22×D71—28no—
196C22×D71—44no—
197C22×D71—44no—
198C22×D71—44no—
199C22×D71—44no—
200C22×D71C2×C22⋊C414yes—
201C2×C4×C71—24no—
202C2×C4×C71—42no—
203C2×C4×C71—42no—
204C2×C4×C71—48no—
205C2×C4×C71—42no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—84no—
209C2×C4×C71C2×C22⋊C412yes—
210C2×C4×C71C4×D414yes—
211C2×C4×C71C4×D414yes—
212C2×C4×C71#12014yes—
213C2×C4×C71#12112yes—
214C2×C4×C71#12112yes—
215C2×C4×C71#12314yes—
216C2×C4×C71C23×C411yescharacteristic
217C2×C4×C71C22×D411yescharacteristic
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C22×D411yescharacteristic
220C2×C4×C71C2×#4111yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C2×Dic71—22no—
223C2×Dic71—22no—
224C2×Dic71—28no—
225C2×Dic71—24no—
226C2×Dic71—42no—
227C2×Dic71—42no—
228C2×Dic71—48no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71—84no—
233C2×Dic71C2×C22⋊C414yes—
234C4×D71—24no—
235C4×D71—24no—
236C4×D71—24no—
237C4×D71—24no—
238C4×D71—216no—
239Dic142—24no—
240Dic142—24no—
241Dic142—42no—
242Dic142—42no—
243Dic142—48no—
244C8×C71—24no—
245C71⋊C8—24no—
246C22×C71#38614yes—
247C22×C71#41714yes—
248C22×C71#41914yes—
249C22×C71#46814yes—
250C22×C71#46814yes—
251C22×C71#47014yes—
252C22×C71#47014yes—
253C22×C71C22×C22⋊C411yescharacteristic
254C22×C71C2×#12111yescharacteristic
255C22×C71C2×#12111yescharacteristic
256C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C71C2×#13711yescharacteristic
258C22×C71C2×#13711yescharacteristic
259C22×C71C2×#13711yescharacteristic
260C2×D71—24no—
261C2×D71—24no—
262C2×D71—24no—
263C2×D71—416no—
264C2×D71—42no—
265C2×D71—42no—
266C2×D71—42no—
267C2×D71—42no—
268C4×C71—44no—
269C4×C71—82no—
270C4×C71#38614yes—
271C4×C71C2×C4×D411yescharacteristic
272C4×C71C2×C4×D411yescharacteristic
273C4×C71C2×#12011yescharacteristic
274C4×C71C2×#12311yescharacteristic
275Dic71—24no—
276Dic71—44no—
277Dic71—82no—
278C2×#1894—711nomaximal, Sylow
279C2×C71#189418yescomplemented
280C2×C71C2×#38611yescharacteristic
281C2×C71C2×#41911yescharacteristic
282C2×C71C2×#41711yescharacteristic
283C2×C71C2×#46811yescharacteristic
284C2×C71C2×#46811yescharacteristic
285C2×C71C2×#47011yescharacteristic
286C2×C71C2×#47011yescharacteristic
287D71—44no—
288C23×M16—711no—
289C22×C22⋊Q8—711no—
290C22×#103—711no—
291C22×#103—711no—
292C2×#380—711no—
293#1894—718no—
294C2×#340—711no—
295C2×#340D7111yescharacteristic, complemented
296C71C2×#189411yescharacteristic, complemented, Sylow
297C23×Q8—711no—
298C24×C4—711no—
299C22×M16—714no—
300C22×M16—1424no—
301C22×M16—1424no—
302C22×M16—1424no—
303C23×C8—711no—
304C23×C8C2×D7111yescharacteristic
305C2×C22⋊Q8—718no—
306C2×C22⋊Q8—714no—
307C2×C22⋊Q8—1424no—
308C2×C22⋊Q8—1422no—
309C22×C4⋊C4—711no—
310C22×C4⋊C4—1421no—
311C22×C4⋊C4C2×D7111yescharacteristic
312C22×C4⋊C4C2×D7111yescharacteristic
313C22×C22⋊C4—1421no—
314C22×C22⋊C4—1421no—
315C2×#103—714no—
316C2×#103—714no—
317C2×#103—1424no—
318C2×#103—1424no—
319#380—714no—
320#380—1422no—
321C2×#95—1421no—
322#340—714no—
323#340C2×D7114yescomplemented
324C25—711no—
325C22×Q8—714no—
326C22×Q8—712no—
327C22×Q8—1422no—
328C22×Q8—1422no—
329C22×Q8—1422no—
330C22×Q8—1422no—
331C22×Q8—1421no—
332C22×Q8—1422no—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—711no—
336C23×C4—714no—
337C23×C4—714no—
338C23×C4—712no—
339C23×C4—712no—
340C23×C4—1421no—
341C23×C4—1421no—
342C23×C4—1422no—
343C23×C4—1422no—
344C23×C4—1422no—
345C23×C4—1422no—
346C23×C4C22×D7111yescharacteristic
347C2×M16—1428no—
348C2×M16—1428no—
349C2×M16—14232no—
350C2×M16—1428no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—714no—
354C22×C8—712no—
355C22×C8—1422no—
356C22×C8—1422no—
357C22×C8D28414yes—
358C22×C8C22×D7112yes—
359C22⋊Q8—7116no—
360C22⋊Q8—1428no—
361C22⋊Q8—14216no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—712no—
368C2×C4⋊C4—1421no—
369C2×C4⋊C4—1422no—
370C2×C4⋊C4—1422no—
371C2×C4⋊C4—1424no—
372C2×C4⋊C4—1428no—
373C2×C4⋊C4C4×D7114yes—
374C2×C4⋊C4C22×D7112yes—
375C2×C4⋊C4C22×D7112yes—
376C2×C22⋊C4—2844no—
377C2×C22⋊C4—1424no—
378C2×C22⋊C4—1424no—
379C2×C22⋊C4—1428no—
380#103—1428no—
381#103—1428no—
382#95—1424no—
383C24—712no—
384C24—714no—
385C24—1422no—
386C24—1422no—
387C24—1422no—
388C24—1422no—
389C24—1422no—
390C24—1422no—
391C24D4×D7111yescharacteristic
392C2×Q8—2844no—
393C2×Q8—2842no—
394C2×Q8—2842no—
395C2×Q8—2844no—
396C2×Q8—714no—
397C2×Q8—1422no—
398C2×Q8—1422no—
399C2×Q8—1422no—
400C2×Q8—1422no—
401C2×Q8—1428no—
402C2×Q8—1428no—
403C2×Q8—1422no—
404C2×Q8—1424no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—2842no—
414C22×C4—2842no—
415C22×C4—2842no—
416C22×C4—2842no—
417C22×C4—2842no—
418C22×C4—712no—
419C22×C4—714no—
420C22×C4—714no—
421C22×C4—714no—
422C22×C4—718no—
423C22×C4—1422no—
424C22×C4—1422no—
425C22×C4—1424no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1428no—
433C22×C4—1428no—
434C22×C4—1428no—
435C22×C4—1424no—
436C22×C4—1424no—
437C22×C4—1428no—
438C22×C4C2×C4×D7112yes—
439C22×C4D4×D7112yes—
440C22×C4D4×D7112yes—
441C22×C4D4×D7111yescharacteristic
442C22×C4C2×D28412yes—
443C22×C4#1176586512yes—
444C22×C4#1176586612yes—
445C22×C4C23×D7111yescharacteristic
446M16—14232no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—28no—
450C2×C8—714no—
451C2×C8—1422no—
452C2×C8—1422no—
453C2×C8—1428no—
454C2×C8C2×D28414yes—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—2842no—
461C4⋊C4—14216no—
462C4⋊C4—1422no—
463C4⋊C4—1422no—
464C4⋊C4C2×C4×D7114yes—
465C22⋊C4—2848no—
466C22⋊C4—14216no—
467C23—2844no—
468C23—2844no—
469C23—2844no—
470C23—2844no—
471C23—714no—
472C23—1428no—
473C23—1422no—
474C23—1422no—
475C23—1428no—
476C23—1428no—
477C23—1422no—
478C23—1422no—
479C23—1422no—
480C23—1422no—
481C23C22⋊C4×D7112yes—
482C23#1455610112yes—
483C23#1455610312yes—
484C23#1455615212yes—
485C23#1455615712yes—
486C23#1455615712yes—
487C23#1455616012yes—
488C23C2×D4×D7111yescharacteristic
489Q8—2848no—
490Q8—2842no—
491Q8—2842no—
492Q8—1424no—
493Q8—1424no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—2848no—
503C2×C4—2842no—
504C2×C4—2842no—
505C2×C4—2842no—
506C2×C4—2842no—
507C2×C4—2842no—
508C2×C4—5684no—
509C2×C4—714no—
510C2×C4—1424no—
511C2×C4—1428no—
512C2×C4—1422no—
513C2×C4—1422no—
514C2×C4—14216no—
515C2×C4—1424no—
516C2×C4—1424no—
517C2×C4—1424no—
518C2×C4—1424no—
519C2×C4C22⋊C4×D7112yes—
520C2×C4#1455609014yes—
521C2×C4C4×D28414yes—
522C2×C4#1455609814yes—
523C2×C4#1455610112yes—
524C2×C4#1455610312yes—
525C2×C4#1455611314yes—
526C2×C4C22×C4×D7111yescharacteristic
527C2×C4C2×D4×D7111yescharacteristic
528C2×C4C2×D4×D7111yescharacteristic
529C2×C4C22×D28411yescharacteristic
530C2×C4C2×#1176586511yescharacteristic
531C2×C4C2×#1176586611yescharacteristic
532C8—24no—
533C8—1424no—
534C22—2842no—
535C22—2842no—
536C22—2842no—
537C22—2842no—
538C22—28416no—
539C22—1424no—
540C22—1424no—
541C22—1424no—
542C22#1798587714yes—
543C22#1798599714yes—
544C22#1798600614yes—
545C22#1798624414yes—
546C22#1798624614yes—
547C22#1798625614yes—
548C22#1798625814yes—
549C22C2×C22⋊C4×D7111yescharacteristic
550C22C2×#1455610111yescharacteristic
551C22C2×#1455610311yescharacteristic
552C22C2×#1455615211yescharacteristic
553C22C2×#1455615711yescharacteristic
554C22C2×#1455615711yescharacteristic
555C22C2×#1455616011yescharacteristic
556C4—44no—
557C4—82no—
558C4—2844no—
559C4—5682no—
560C4—1424no—
561C4#1798587714yes—
562C4C2×#1455609011yescharacteristic
563C4C2×C4×D28411yescharacteristic
564C4C2×#1455609811yescharacteristic
565C4C2×#1455611311yescharacteristic
566C2—2844no—
567C2#2258877218yesdirect product factor
568C2C2×#1798587711yescharacteristic
569C2C2×#1798600611yescharacteristic
570C2C2×#1798599711yescharacteristic
571C2C2×#1798624411yescharacteristic
572C2C2×#1798624611yescharacteristic
573C2C2×#1798625611yescharacteristic
574C2C2×#1798625811yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#340×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenterC2×#340D71
hypocenterC71C2×#1894
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#14556053C2semidirect
4. C22×#14556051C2semidirect
6. #22588772C2direct (N × H)
7. C2×#17985761C2semidirect
8. C2×#340×C71C2semidirect
18. C2×#14556108C4semidirect
25. C2×#14556053C22semidirect
27. C2×#14556051C22semidirect
31. #17985761C22semidirect
35. #340×C71C22semidirect
66. #14556108C2×C4semidirect
279. C2×C71#1894semidirect
295. C2×#340D71semidirect
296. C71C2×#1894semidirect
323. #340C2×D71semidirect
567. C2#22588772direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658482
C22×#145561082
C22×#145560532
C22×#145560512
C2×#179858562
#225887722
C2×#179857612
C2×#340×C712
C2×#189471