Finite Group Databasebeta

← #31923828

Subgroups — #31923828

Order 18176.

2142 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556104C211yesmaximal, characteristic
2C22×#14556085C211yesmaximal, characteristic
3C2×#17985906C211yesmaximal, characteristic, complemented
4C2×#17985856C211yesmaximal, characteristic
5#22586591C214yesmaximal, complemented
6#22586588C214yesmaximal
7#22586570C211yesmaximal, characteristic, complemented
8#22586569C211yesmaximal, characteristic, complemented
9C2×#391×C71C211yesmaximal, characteristic, complemented
10C23×C4×D71C2211yescharacteristic
11C2×#14556104—24no—
12C2×#14556104—24no—
13C2×#14556104C418yescomplemented
14C2×#14556085—24no—
15C2×#14556085—24no—
16C2×#14556085—24no—
17C22×C4⋊Dic71C2211yescharacteristic
18C22×C4⋊Dic71C2211yescharacteristic
19C22×Dic71⋊C4C2212yes—
20C22×Dic71⋊C4C2212yes—
21C22×(C2×D71)⋊C4C2212yes—
22C22×(C2×D71)⋊C4C2212yes—
23C22×(C2×D71)⋊C4C2212yes—
24C22×C4×Dic71C2211yescharacteristic
25#17985906—24no—
26#17985856—24no—
27C2×C71⋊#95C2214yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2214yes—
30C2×C71⋊#95C2214yes—
31C23×Q8×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C42×C71C2211yescharacteristic
34#391×C71—24no—
35C2×#95×C71C2214yes—
36C24×D71C2311yescharacteristic
37C23×Dic71C2312yes—
38C23×Dic71C2312yes—
39C23×Dic71C2312yes—
40C23×Dic71C2311yescharacteristic
41C22×C4×D71—22no—
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71C2×C414yes—
46C22×C4×D71D414yes—
47C22×C4×D71D414yes—
48#14556104—232no—
49#14556085—232no—
50C2×C4⋊Dic71—22no—
51C2×C4⋊Dic71—22no—
52C2×C4⋊Dic71—22no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71C2×C414yes—
56C2×Dic71⋊C4—24no—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—24no—
59C2×Dic71⋊C4—24no—
60C2×Dic71⋊C4—24no—
61C2×Dic71⋊C4C2×C418yes—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4—24no—
65C2×(C2×D71)⋊C4—24no—
66C2×(C2×D71)⋊C4—24no—
67C2×(C2×D71)⋊C4—24no—
68C2×(C2×D71)⋊C4—44no—
69C2×(C2×D71)⋊C4C2×C418yes—
70C2×C4×Dic71—22no—
71C2×C4×Dic71—22no—
72C2×C4×Dic71D414yes—
73C71⋊#95—24no—
74C71⋊#95—28no—
75C71⋊#95—44no—
76C71⋊#95—44no—
77C22×Q8×C71—22no—
78C22×Q8×C71—22no—
79C22×Q8×C71—22no—
80C22×Q8×C71—22no—
81C22×Q8×C71—22no—
82C22×Q8×C71—22no—
83C22×Q8×C71C2×C414yes—
84C23×C4×C71C2312yes—
85C23×C4×C71C2312yes—
86C23×C4×C71C2312yes—
87C23×C4×C71C2311yescharacteristic
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—22no—
91C2×C42×C71—22no—
92C2×C42×C71—22no—
93C2×C42×C71D414yes—
94#95×C71—28no—
95C23×D71—21no—
96C23×D71—21no—
97C23×D71—21no—
98C23×D71—21no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71—22no—
103C23×D71C22×C412yes—
104C23×D71C2×D412yes—
105C23×D71C2×D412yes—
106C22×Dic71—21no—
107C22×Dic71—21no—
108C22×Dic71—21no—
109C22×Dic71—21no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—24no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—24no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—24no—
124C22×Dic71—24no—
125C22×Dic71C22×C414yes—
126C22×Dic71C22×C412yes—
127C22×Dic71C2×D414yes—
128C22×Dic71C2×D412yes—
129C22×Dic71C2×D412yes—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71—28no—
133C2×C4×D71—28no—
134C2×C4×D71—28no—
135C2×C4×D71—28no—
136C2×C4×D71C22⋊C4116yes—
137C4⋊Dic71—28no—
138C4⋊Dic71—44no—
139Dic71⋊C4—216no—
140Dic71⋊C4—216no—
141(C2×D71)⋊C4—216no—
142(C2×D71)⋊C4—216no—
143(C2×D71)⋊C4—48no—
144C4×Dic71—28no—
145C24×C71C2411yescharacteristic
146C2×Q8×C71—28no—
147C2×Q8×C71—28no—
148C2×Q8×C71—28no—
149C2×Q8×C71—44no—
150C2×Q8×C71—44no—
151C2×Q8×C71—44no—
152C2×Q8×C71—44no—
153C22×C4×C71—22no—
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—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—24no—
169C22×C4×C71—24no—
170C22×C4×C71C22×C414yes—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×D414yes—
176C22×C4×C71C2×D412yes—
177C22×C4×C71#4112yes—
178C22×C4×C71#4112yes—
179C4⋊C4×C71—44no—
180C42×C71—28no—
181C23×C71C23×C411yescharacteristic
182C23×C71C22×D411yescharacteristic
183C23×C71C22×D411yescharacteristic
184C23×C71C22×D411yescharacteristic
185C23×C71C22×D411yescharacteristic
186C23×C71C2×#4111yescharacteristic
187C23×C71C2×#4111yescharacteristic
188C23×C71H(5,𝔽2)12yes—
189C23×C712−1+412yes—
190C23×C712−1+412yes—
191C23×C712−1+412yes—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—24no—
195C22×D71—24no—
196C22×D71—22no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—24no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—24no—
206C22×D71—42no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71C2×C22⋊C414yes—
215Q8×C71—416no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—44no—
234C2×C4×C71—44no—
235C2×C4×C71C2×C22⋊C414yes—
236C2×C4×C71C4×D414yes—
237C2×C4×C71C4×D414yes—
238C2×C4×C71#12014yes—
239C2×C4×C71#12114yes—
240C2×C4×C71#12114yes—
241C2×C4×C71#12314yes—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—24no—
249C2×Dic71—28no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—44no—
259C2×Dic71—42no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—44no—
264C2×Dic71—44no—
265C2×Dic71C2×C22⋊C414yes—
266C4×D71—232no—
267C22×C71C22×C22⋊C411yescharacteristic
268C22×C71C2×C4×D411yescharacteristic
269C22×C71C2×C4×D411yescharacteristic
270C22×C71#51912yes—
271C22×C71#52012yes—
272C22×C71C2×#12011yescharacteristic
273C22×C71C2×#12111yescharacteristic
274C22×C71C2×#12111yescharacteristic
275C22×C71C2×#12311yescharacteristic
276C22×C71#53511yescharacteristic
277C22×C71#53611yescharacteristic
278C22×C71#53612yes—
279C22×C71#53612yes—
280C22×C71#53612yes—
281C22×C71#53611yescharacteristic
282C22×C71#53611yescharacteristic
283C22×C71#53712yes—
284C22×C71#53712yes—
285C22×C71#53712yes—
286C22×C71#54112yes—
287C22×C71#54112yes—
288C22×C71#54212yes—
289C22×C71#54212yes—
290C2×D71—24no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71—44no—
294C2×D71—44no—
295C2×D71—44no—
296C2×D71—42no—
297C2×D71—42no—
298C2×D71—42no—
299C2×D71—42no—
300C2×D71—44no—
301C4×C71—44no—
302C4×C71—44no—
303C4×C71—82no—
304C4×C71#38618yes—
305Dic71—24no—
306Dic71—44no—
307Dic71—44no—
308Dic71—82no—
309#14258—711nomaximal, Sylow
310C2×C71C2×#38611yescharacteristic
311C2×C71#231011yescharacteristic
312C2×C71#231111yescharacteristic
313C2×C71#232312yes—
314C2×C71#232312yes—
315C2×C71#261612yes—
316C2×C71#262011yescharacteristic
317C2×C71#262011yescharacteristic
318C2×C71#262111yescharacteristic
319C2×C71#262111yescharacteristic
320C2×C71#262512yes—
321D71—44no—
322C22×C22⋊Q8—711no—
323C22×#117—711no—
324C2×#391—711no—
325C2×#391D7111yescharacteristic, complemented
326C2×#380—711no—
327#1441—714no—
328#1440—714no—
329C4⋊#95—711no—
330C4⋊#95—711no—
331C71#1425811yescharacteristic, complemented, Sylow
332C23×Q8C2×D7111yescharacteristic
333C24×C4—711no—
334C2×C22⋊Q8—718no—
335C2×C22⋊Q8—1424no—
336C2×C22⋊Q8—1424no—
337C2×#117—1424no—
338C2×#117—1424no—
339C2×#117—1424no—
340C22×C4⋊C4—712no—
341C22×C4⋊C4—711no—
342C22×C4⋊C4—712no—
343C22×C4⋊C4—711no—
344C22×C4⋊C4C2×D7111yescharacteristic
345C22×C22⋊C4—712no—
346C22×C22⋊C4—712no—
347C22×C22⋊C4—712no—
348C22×C42—711no—
349C22×C42C2×D7111yescharacteristic
350#391—24no—
351#391—1424no—
352#380—1424no—
353C2×#95—714no—
354C2×#95—714no—
355C2×#95—714no—
356C2×#95—712no—
357C2×#95C2×D7114yes—
358C25—711no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—22no—
365C22×Q8C4×D7114yes—
366C23×C4—711no—
367C23×C4—714no—
368C23×C4—714no—
369C23×C4—714no—
370C23×C4—712no—
371C23×C4—712no—
372C23×C4—712no—
373C23×C4—1422no—
374C23×C4—1422no—
375C23×C4—1422no—
376C23×C4—1422no—
377C23×C4C22×D7112yes—
378C23×C4C22×D7112yes—
379C23×C4C22×D7112yes—
380C23×C4C22×D7111yescharacteristic
381C22⋊Q8—14232no—
382#117—14232no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—718no—
387C2×C4⋊C4—714no—
388C2×C4⋊C4—1422no—
389C2×C4⋊C4—1422no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4—1424no—
392C2×C4⋊C4—1424no—
393C2×C4⋊C4—1424no—
394C2×C4⋊C4—1424no—
395C2×C4⋊C4—1422no—
396C2×C4⋊C4—1422no—
397C2×C4⋊C4—1424no—
398C2×C22⋊C4—2844no—
399C2×C22⋊C4—718no—
400C2×C22⋊C4—1424no—
401C2×C22⋊C4—1424no—
402C2×C22⋊C4—1424no—
403C2×C22⋊C4—1424no—
404C2×C22⋊C4—1424no—
405C2×C22⋊C4—1424no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—714no—
409C2×C42—1422no—
410C2×C42—1422no—
411C2×C42C22⋊D7114yes—
412#95—28no—
413#95—2844no—
414#95—2844no—
415#95—1424no—
416#95—1428no—
417C24—712no—
418C24—712no—
419C24—712no—
420C24—1421no—
421C24—1421no—
422C24—1421no—
423C24—1421no—
424C24—1422no—
425C24—1422no—
426C24—1422no—
427C24—1422no—
428C24C23×D7111yescharacteristic
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—712no—
454C22×C4—712no—
455C22×C4—712no—
456C22×C4—7116no—
457C22×C4—714no—
458C22×C4—714no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1424no—
463C22×C4—1424no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1424no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1422no—
472C22×C4—1422no—
473C22×C4—1424no—
474C22×C4—1421no—
475C22×C4—1421no—
476C22×C4—1421no—
477C22×C4—1421no—
478C22×C4—1428no—
479C22×C4—1428no—
480C22×C4—1428no—
481C22×C4—1428no—
482C22×C4—1428no—
483C22×C4—1428no—
484C22×C4C2×C4×D7114yes—
485C22×C4C2×C4×D7112yes—
486C22×C4D4×D7112yes—
487C22×C4D4×D7112yes—
488C22×C4D4×D7112yes—
489C22×C4C2×C22⋊D7114yes—
490C22×C4C2×C22⋊D7112yes—
491C22×C4#1176586412yes—
492C22×C4#1176586512yes—
493C4⋊C4—44no—
494C4⋊C4—2844no—
495C4⋊C4—14216no—
496C4⋊C4—14216no—
497C4⋊C4—1428no—
498C22⋊C4—2848no—
499C22⋊C4—14216no—
500C22⋊C4—14216no—
501C42—28no—
502C42—1428no—
503C23—2842no—
504C23—2842no—
505C23—2842no—
506C23—2842no—
507C23—2842no—
508C23—2842no—
509C23—2842no—
510C23—2842no—
511C23—714no—
512C23—1422no—
513C23—1422no—
514C23—1424no—
515C23—1422no—
516C23—1422no—
517C23—1424no—
518C23—1422no—
519C23—1422no—
520C23—1424no—
521C23—1424no—
522C23—1422no—
523C23—1422no—
524C23—1422no—
525C23—1422no—
526C23C22×C4×D7111yescharacteristic
527C23C2×D4×D7111yescharacteristic
528C23C2×D4×D7111yescharacteristic
529C23C2×D4×D7111yescharacteristic
530C23C22×C22⋊D7111yescharacteristic
531C23C2×#1176586411yescharacteristic
532C23C2×#1176586511yescharacteristic
533C23#1455617412yes—
534C23#1455617512yes—
535C23#1455617612yes—
536C23#1455617612yes—
537Q8—416no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—2842no—
558C2×C4—2842no—
559C2×C4—2842no—
560C2×C4—2842no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—2844no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2844no—
567C2×C4—2844no—
568C2×C4—2842no—
569C2×C4—714no—
570C2×C4—1424no—
571C2×C4—1424no—
572C2×C4—1424no—
573C2×C4—1428no—
574C2×C4—1424no—
575C2×C4—1422no—
576C2×C4—1422no—
577C2×C4—1422no—
578C2×C4—1422no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—14232no—
582C2×C4C22⋊C4×D7114yes—
583C2×C4(C22⋊D71)⋊C414yes—
584C2×C4C4×C22⋊D7114yes—
585C2×C4#1455609614yes—
586C2×C4#1455609914yes—
587C2×C4#1455610114yes—
588C2×C4#1455611014yes—
589C22—2844no—
590C22—2844no—
591C22—2844no—
592C22—2842no—
593C22—2842no—
594C22—2842no—
595C22—2842no—
596C22—2844no—
597C22—1424no—
598C22—1424no—
599C22—1424no—
600C22C2×C22⋊C4×D7111yescharacteristic
601C22C2×(C22⋊D71)⋊C411yescharacteristic
602C22C2×C4×C22⋊D7111yescharacteristic
603C22#1798647512yes—
604C22#1798648112yes—
605C22C2×#1455609611yescharacteristic
606C22C2×#1455609911yescharacteristic
607C22C2×#1455610111yescharacteristic
608C22C2×#1455611011yescharacteristic
609C22#1798656711yescharacteristic
610C22#1798656911yescharacteristic
611C22#1798657212yes—
612C22#1798657212yes—
613C22#1798657312yes—
614C22#1798657411yescharacteristic
615C22#1798657411yescharacteristic
616C22#1798657712yes—
617C22#1798657912yes—
618C22#1798657912yes—
619C22#1798660712yes—
620C22#1798661012yes—
621C22#1798661812yes—
622C22#1798662112yes—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—2844no—
627C4—2844no—
628C4—5682no—
629C4—1424no—
630C4#1798587518yes—
631C2—2844no—
632C2C2×#1798587511yescharacteristic
633C2#2259092311yescharacteristic
634C2#2259093111yescharacteristic
635C2#2259101612yes—
636C2#2259102512yes—
637C2#2259407012yes—
638C2#2259409811yescharacteristic
639C2#2259410411yescharacteristic
640C2#2259410711yescharacteristic
641C2#2259411111yescharacteristic
642C2#2259414012yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#391×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#391D71
hypocenterC71#14258
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#17985906C2semidirect
5. #22586591C2semidirect
7. #22586570C2semidirect
8. #22586569C2semidirect
9. C2×#391×C71C2semidirect
13. C2×#14556104C4semidirect
325. C2×#391D71semidirect
331. C71#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#145561042
C22×#145560852
C2×#179859062
C2×#179858562
#225865912
#225865882
#225865702
#225865692
C2×#391×C712
#1425871