Finite Group Databasebeta

← #31934176

Subgroups — #31934176

Order 18176.

1566 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985933C212yesmaximal
2C2×#17985932C212yesmaximal
3C2×#17985896C211yesmaximal, characteristic
4C2×#17985868C212yesmaximal
5C2×#17985868C211yesmaximal, characteristic
6#22586605C211yesmaximal, characteristic
7#22586604C211yesmaximal, characteristic
8#22586603C211yesmaximal, characteristic
9Dic71⋊#95C212yesmaximal
10#95×Dic71C211yesmaximal, characteristic
11C71×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C4⋊Dic71C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×Dic71⋊C4C2212yes—
16C22×Dic71⋊C4C2212yes—
17C22×Dic71⋊C4C2211yescharacteristic
18C22×Dic71⋊C4C2211yescharacteristic
19C22×Dic71⋊C4C2212yes—
20C22×Dic71⋊C4C2212yes—
21C22×Dic71⋊C4C2211yescharacteristic
22C22×C4×Dic71C2211yescharacteristic
23C22×C4×Dic71C2212yes—
24#17985933—28no—
25#17985932—28no—
26#17985896—24no—
27#17985868—28no—
28#17985868—24no—
29C2×C71⋊#95C2211yescharacteristic
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2212yes—
33C2×C71⋊#95C2212yes—
34C2×C71⋊#95C2211yescharacteristic
35C22×C4⋊C4×C71C2212yes—
36C22×C4⋊C4×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C2×#95×C71C2212yes—
39C2×#95×C71C2211yescharacteristic
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C23×Dic71C2312yes—
45C23×Dic71C2311yescharacteristic
46C2×C4⋊Dic71—24no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—24no—
49C2×C4⋊Dic71—22no—
50C2×C4⋊Dic71—24no—
51C2×C4⋊Dic71—42no—
52C2×C4⋊Dic71—42no—
53C2×Dic71⋊C4—22no—
54C2×Dic71⋊C4—22no—
55C2×Dic71⋊C4—24no—
56C2×Dic71⋊C4—24no—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—24no—
59C2×Dic71⋊C4—22no—
60C2×Dic71⋊C4—22no—
61C2×Dic71⋊C4—24no—
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—24no—
64C2×Dic71⋊C4—24no—
65C2×Dic71⋊C4—24no—
66C2×Dic71⋊C4—24no—
67C2×Dic71⋊C4—24no—
68C2×Dic71⋊C4—24no—
69C2×Dic71⋊C4—42no—
70C2×Dic71⋊C4—42no—
71C2×Dic71⋊C4D414yes—
72C2×C4×Dic71—24no—
73C2×C4×Dic71—24no—
74C2×C4×Dic71—24no—
75C2×C4×Dic71D414yes—
76C2×C4×Dic71Q818yes—
77C71⋊#95—24no—
78C71⋊#95—24no—
79C71⋊#95—22no—
80C71⋊#95—24no—
81C71⋊#95D418yes—
82C71⋊#95Q814yes—
83C23×C4×C71C2311yescharacteristic
84C23×C4×C71C2312yes—
85C23×C4×C71C2311yescharacteristic
86C23×C4×C71C2312yes—
87C23×C4×C71C2311yescharacteristic
88C2×C4⋊C4×C71—24no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—24no—
91C2×C4⋊C4×C71—42no—
92C2×C4⋊C4×C71—42no—
93#95×C71—22no—
94#95×C71—24no—
95#95×C71Q814yes—
96C22×Dic71—22no—
97C22×Dic71—22no—
98C22×Dic71—22no—
99C22×Dic71—22no—
100C22×Dic71—21no—
101C22×Dic71—21no—
102C22×Dic71—22no—
103C22×Dic71—21no—
104C22×Dic71—22no—
105C22×Dic71—21no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—21no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—21no—
118C22×Dic71—21no—
119C22×Dic71C2×D412yes—
120C22×Dic71C2×D414yes—
121C22×Dic71C2×D412yes—
122C22×Dic71C2×D414yes—
123C22×Dic71C2×D412yes—
124C22×Dic71C2×D412yes—
125C22×Dic71C2×Q814yes—
126C22×Dic71C2×Q814yes—
127C22×Dic71C2×Q814yes—
128C22×Dic71#4112yes—
129C22×Dic71#4114yes—
130C4⋊Dic71—48no—
131C4⋊Dic71—44no—
132C4⋊Dic71—44no—
133Dic71⋊C4—28no—
134Dic71⋊C4—48no—
135Dic71⋊C4—48no—
136Dic71⋊C4—48no—
137Dic71⋊C4—44no—
138Dic71⋊C4—44no—
139Dic71⋊C4—48no—
140C4×Dic71—216no—
141C4×Dic71—28no—
142C24×C71C2411yescharacteristic
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—21no—
150C22×C4×C71—22no—
151C22×C4×C71—21no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—21no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71C2×D412yes—
166C22×C4×C71C2×D414yes—
167C22×C4×C71C2×D412yes—
168C22×C4×C71C2×Q812yes—
169C22×C4×C71C2×Q814yes—
170C22×C4×C71C2×Q814yes—
171C22×C4×C71C2×Q812yes—
172C4⋊C4×C71—48no—
173C4⋊C4×C71—44no—
174C23×C71C22×D411yescharacteristic
175C23×C71C22×D411yescharacteristic
176C23×C71C22×D412yes—
177C23×C71C22×Q812yes—
178C23×C71C22×Q811yescharacteristic
179C23×C71C22×Q811yescharacteristic
180C23×C71C2×#4111yescharacteristic
181C23×C71C2×#4112yes—
182C23×C71H(5,𝔽2)12yes—
183C23×C712−1+411yescharacteristic
184C23×C712−1+411yescharacteristic
185C2×C4×C71—24no—
186C2×C4×C71—22no—
187C2×C4×C71—22no—
188C2×C4×C71—24no—
189C2×C4×C71—24no—
190C2×C4×C71—24no—
191C2×C4×C71—24no—
192C2×C4×C71—24no—
193C2×C4×C71—24no—
194C2×C4×C71—24no—
195C2×C4×C71—22no—
196C2×C4×C71—24no—
197C2×C4×C71—41no—
198C2×C4×C71—41no—
199C2×C4×C71—42no—
200C2×C4×C71—42no—
201C2×C4×C71—41no—
202C2×C4×C71—41no—
203C2×C4×C71—42no—
204C2×C4×C71—42no—
205C2×C4×C71—41no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71C4⋊Q818yes—
213C2×C4×C71C4⋊Q814yes—
214C2×Dic71—24no—
215C2×Dic71—22no—
216C2×Dic71—22no—
217C2×Dic71—24no—
218C2×Dic71—24no—
219C2×Dic71—24no—
220C2×Dic71—24no—
221C2×Dic71—24no—
222C2×Dic71—24no—
223C2×Dic71—24no—
224C2×Dic71—22no—
225C2×Dic71—24no—
226C2×Dic71—41no—
227C2×Dic71—41no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—41no—
231C2×Dic71—42no—
232C2×Dic71—41no—
233C2×Dic71—42no—
234C2×Dic71—41no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241C2×Dic71#12114yes—
242C2×Dic71C22⋊Q818yes—
243C2×Dic71#12314yes—
244C2×Dic71#12518yes—
245C2×Dic71#12714yes—
246C2×Dic71C4⋊Q818yes—
247C2×Dic71C4⋊Q814yes—
248C22×C71C2×#12111yescharacteristic
249C22×C71C2×C22⋊Q812yes—
250C22×C71C2×#12311yescharacteristic
251C22×C71C2×#12512yes—
252C22×C71C2×#12711yescharacteristic
253C22×C71C2×C4⋊Q811yescharacteristic
254C22×C71C2×C4⋊Q812yes—
255C22×C71C2×C4⋊Q812yes—
256C22×C71C2×C4⋊Q811yescharacteristic
257C22×C71#53511yescharacteristic
258C22×C71#53611yescharacteristic
259C22×C71#54412yes—
260C22×C71#54412yes—
261C22×C71D4212yes—
262C22×C71#54712yes—
263C22×C71#54712yes—
264C22×C71D4×Q811yescharacteristic
265C22×C71D4×Q811yescharacteristic
266C22×C71#55012yes—
267C22×C71#55412yes—
268C22×C71#55412yes—
269C22×C71#55711yescharacteristic
270C22×C71#55711yescharacteristic
271C4×C71—28no—
272C4×C71—24no—
273C4×C71—42no—
274C4×C71—81no—
275C4×C71—82no—
276Dic71—28no—
277Dic71—24no—
278Dic71—42no—
279Dic71—81no—
280Dic71—82no—
281Dic71#39818yescomplemented
282#15145—711nomaximal, Sylow
283C2×C71C2×#39811yescharacteristic
284C2×C71#244012yes—
285C2×C71#244012yes—
286C2×C71#250111yescharacteristic
287C2×C71#250211yescharacteristic
288C2×C71#254111yescharacteristic
289C2×C71#254611yescharacteristic
290C2×C71#255212yes—
291C2×C71#255212yes—
292C2×C71#266011yescharacteristic
293C2×C71#286811yescharacteristic
294C2×#398—712no—
295C2×#398—712no—
296C2×#389—711no—
297C2×#384—711no—
298C2×#384—712no—
299#1444—711no—
300#1444—711no—
301#1444—711no—
302#1444D7111yescharacteristic
303C4⋊#95—712no—
304C4×#95—711no—
305C71#1514511yescharacteristic, complemented, Sylow
306C22×C4⋊C4—711no—
307C22×C4⋊C4—712no—
308C22×C4⋊C4—712no—
309C22×C4⋊C4—712no—
310C22×C4⋊C4—711no—
311C22×C4⋊C4—712no—
312C22×C4⋊C4—711no—
313C22×C4⋊C4—711no—
314C22×C4⋊C4—711no—
315C22×C4⋊C4—712no—
316C22×C4⋊C4C2×D7112yes—
317C22×C4⋊C4C2×D7111yescharacteristic
318C22×C42—712no—
319C22×C42—711no—
320#398—1428no—
321#398—1428no—
322#389—1424no—
323#384—1424no—
324#384—1428no—
325C2×#95—712no—
326C2×#95—711no—
327C2×#95—712no—
328C2×#95—712no—
329C2×#95—712no—
330C2×#95—711no—
331C2×#95C2×D7112yes—
332C2×#95C2×D7111yescharacteristic
333C2×#95C2×D7111yescharacteristic
334C23×C4—711no—
335C23×C4—711no—
336C23×C4—712no—
337C23×C4—711no—
338C23×C4—712no—
339C23×C4—711no—
340C23×C4C22×D7112yes—
341C23×C4C22×D7111yescharacteristic
342C23×C4C22×D7111yescharacteristic
343C23×C4C22×D7112yes—
344C23×C4C22×D7111yescharacteristic
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—2842no—
351C2×C4⋊C4—2842no—
352C2×C4⋊C4—2842no—
353C2×C4⋊C4—2842no—
354C2×C4⋊C4—714no—
355C2×C4⋊C4—1424no—
356C2×C4⋊C4—1424no—
357C2×C4⋊C4—1424no—
358C2×C4⋊C4—1424no—
359C2×C4⋊C4—1424no—
360C2×C4⋊C4—1422no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1422no—
363C2×C4⋊C4—1424no—
364C2×C4⋊C4—1424no—
365C2×C4⋊C4—1424no—
366C2×C4⋊C4—1424no—
367C2×C4⋊C4—1424no—
368C2×C4⋊C4—1424no—
369C2×C4⋊C4—1424no—
370C2×C4⋊C4—1424no—
371C2×C4⋊C4—1424no—
372C2×C4⋊C4—1422no—
373C2×C4⋊C4—1424no—
374C2×C4⋊C4—1422no—
375C2×C4⋊C4—1422no—
376C2×C42—714no—
377C2×C42—718no—
378C2×C42—1424no—
379C2×C42—1424no—
380C2×C42—1424no—
381#95—22no—
382#95—24no—
383#95—718no—
384#95—714no—
385#95—1424no—
386#95—1424no—
387#95—1422no—
388#95—1424no—
389#95Dic14214yes—
390C24C23×D7111yescharacteristic
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—712no—
414C22×C4—712no—
415C22×C4—714no—
416C22×C4—714no—
417C22×C4—712no—
418C22×C4—714no—
419C22×C4—714no—
420C22×C4—712no—
421C22×C4—714no—
422C22×C4—714no—
423C22×C4—712no—
424C22×C4—1422no—
425C22×C4—1421no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1422no—
433C22×C4—1422no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1422no—
438C22×C4—1421no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4—1422no—
447C22×C4D4×D7112yes—
448C22×C4D4×D7114yes—
449C22×C4D4×D7112yes—
450C22×C4Q8×D7112yes—
451C22×C4Q8×D7114yes—
452C22×C4C2×Dic14214yes—
453C22×C4C2×Dic14212yes—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—2844no—
457C4⋊C4—2848no—
458C4⋊C4—2848no—
459C4⋊C4—2848no—
460C4⋊C4—2844no—
461C4⋊C4—2844no—
462C4⋊C4—2848no—
463C4⋊C4—2844no—
464C4⋊C4—2848no—
465C4⋊C4—1428no—
466C42—14216no—
467C42—1428no—
468C23C2×D4×D7111yescharacteristic
469C23C2×D4×D7111yescharacteristic
470C23C2×D4×D7112yes—
471C23C2×Q8×D7112yes—
472C23C2×Q8×D7111yescharacteristic
473C23C22×Dic14211yescharacteristic
474C23#41×D7112yes—
475C23#41×D7111yescharacteristic
476C23#1455617312yes—
477C23#1455617511yescharacteristic
478C23#1455617611yescharacteristic
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—2841no—
507C2×C4—2842no—
508C2×C4—2841no—
509C2×C4—2842no—
510C2×C4—2841no—
511C2×C4—2842no—
512C2×C4—2842no—
513C2×C4—2842no—
514C2×C4—2842no—
515C2×C4—2842no—
516C2×C4—2842no—
517C2×C4—2842no—
518C2×C4—2841no—
519C2×C4—2842no—
520C2×C4—2841no—
521C2×C4—714no—
522C2×C4—718no—
523C2×C4—714no—
524C2×C4—718no—
525C2×C4—714no—
526C2×C4—714no—
527C2×C4—718no—
528C2×C4—1424no—
529C2×C4—1422no—
530C2×C4—1422no—
531C2×C4—1424no—
532C2×C4—1424no—
533C2×C4—1424no—
534C2×C4—1424no—
535C2×C4—1424no—
536C2×C4—1424no—
537C2×C4—1424no—
538C2×C4—1424no—
539C2×C4—1422no—
540C2×C4#1455612718yes—
541C2×C4#1455612714yes—
542C22#121×D7111yescharacteristic
543C22C22⋊Q8×D7112yes—
544C22#123×D7111yescharacteristic
545C22#125×D7112yes—
546C22#127×D7111yescharacteristic
547C22C4⋊Q8×D7112yes—
548C22C4⋊Q8×D7111yescharacteristic
549C22C2×#1455612712yes—
550C22C2×#1455612711yescharacteristic
551C22#1798656411yescharacteristic
552C22#1798657211yescharacteristic
553C22#1798662612yes—
554C22#1798662812yes—
555C22#1798663112yes—
556C22#1798664712yes—
557C22#1798665112yes—
558C22#1798666411yescharacteristic
559C22D4×Dic14211yescharacteristic
560C22#1798666712yes—
561C22#1798670312yes—
562C22#1798670812yes—
563C22#1798672511yescharacteristic
564C22#1798672611yescharacteristic
565C4—28no—
566C4—24no—
567C4—42no—
568C4—82no—
569C4—81no—
570C4—2842no—
571C4—5682no—
572C4—5681no—
573C4—718no—
574C4—1428no—
575C4—1424no—
576C2#398×D7111yescharacteristic
577C2#2259203512yes—
578C2#2259204112yes—
579C2#2259280611yescharacteristic
580C2#2259281611yescharacteristic
581C2#2259328111yescharacteristic
582C2#2259332711yescharacteristic
583C2#2259337012yes—
584C2#2259337712yes—
585C2#2259445811yescharacteristic
586C2#2259673711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutator, socle, prime radical, cosocleC24×C71C24
FittingC71×#1444C2
hypercenter#1444D71
hypocenterC71#15145
LCSG ▷ C24×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
281. Dic71#398semidirect
305. C71#15145semidirect

Maximal subgroups

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

subgroupindex
C2×#179859332
C2×#179859322
C2×#179858962
C2×#179858682
#225866052
#225866042
#225866032
Dic71⋊#952
#95×Dic712
C71×#14442
#1514571