Finite Group Databasebeta

← #32026327

Subgroups — #32026327

Order 18176.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986042C211yesmaximal, characteristic, complemented
2C2×#17985963C211yesmaximal, characteristic
3C2×#17985859C211yesmaximal, characteristic
4#22589777C212yesmaximal, complemented
5#22589776C212yesmaximal, complemented
6#22589771C212yesmaximal, complemented
7#22589766C212yesmaximal, complemented
8#22588061C211yesmaximal, characteristic, complemented
9#22588061C211yesmaximal, characteristic, complemented
10#22588061C211yesmaximal, characteristic, complemented
11C71×#2280C211yesmaximal, characteristic
12C22×#11765849C2211yescharacteristic
13C22×#11765849C2211yescharacteristic
14C22×C4⋊Dic71C2211yescharacteristic
15C22×C4⋊Dic71C2211yescharacteristic
16C22×C22⋊Dic71—21no—
17#17986042—24no—
18C2×#14556061C2212yescomplemented
19C2×#14556059C2212yescomplemented
20#17985963—24no—
21C2×C22⋊(C71⋊C8)C2212yes—
22C2×C22⋊(C71⋊C8)C2212yes—
23#17985859—21no—
24#17985859—21no—
25#17985859C2212yes—
26#17985859C2212yes—
27C2×C71⋊#95—21no—
28#17985761C2212yescomplemented
29#17985761C2212yescomplemented
30#17985761C2212yescomplemented
31#17985761C2212yescomplemented
32#17985761C2212yescomplemented
33#17985761C2212yescomplemented
34C24×C4×C71C2211yescharacteristic
35C2×#117×C71C2211yescharacteristic
36C2×#117×C71C2211yescharacteristic
37#381×C71—21no—
38#380×C71C2211yescharacteristic
39#380×C71C2211yescharacteristic
40#380×C71C2211yescharacteristic
41#380×C71C2211yescharacteristic
42#379×C71—21no—
43C4×C22⋊C4×C71—21no—
44C4×C22⋊C4×C71—21no—
45C23×Dic71—21no—
46C23×Dic71—21no—
47C2×#11765849—22no—
48C2×#11765849—22no—
49C2×#11765849—22no—
50C2×#11765849—22no—
51C2×#11765849—22no—
52C2×#11765849D414yes—
53C22×C71⋊C8C2312yes—
54C22×C71⋊C8C2312yes—
55C2×C4⋊Dic71—22no—
56C2×C4⋊Dic71—21no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71C2312yes—
62C2×C4⋊Dic71C2312yes—
63C2×C22⋊Dic71—22no—
64C2×C22⋊Dic71—22no—
65C2×C22⋊Dic71—44no—
66#14556061—24no—
67#14556059—24no—
68C22⋊(C71⋊C8)—24no—
69C22⋊(C71⋊C8)—24no—
70C71⋊#95—22no—
71C71⋊#95—22no—
72C25×C71D411yescharacteristic
73C23×C4×C71—22no—
74C23×C4×C71—22no—
75C23×C4×C71—22no—
76C23×C4×C71—22no—
77C23×C4×C71—22no—
78C23×C4×C71—22no—
79C23×C4×C71D411yescharacteristic
80C23×C4×C71D411yescharacteristic
81C23×C4×C71D411yescharacteristic
82C23×C4×C71C2311yescharacteristic
83C23×C4×C71C2311yescharacteristic
84C23×C4×C71C2311yescharacteristic
85#117×C71—22no—
86#117×C71—22no—
87#117×C71—22no—
88#117×C71—22no—
89C2×C4⋊C4×C71C2311yescharacteristic
90C2×C4⋊C4×C71C2311yescharacteristic
91C2×C4⋊C4×C71C2311yescharacteristic
92C2×C4⋊C4×C71C2311yescharacteristic
93C2×C22⋊C4×C71—21no—
94C2×C22⋊C4×C71—21no—
95C2×C22⋊C4×C71—21no—
96C2×C22⋊C4×C71—21no—
97C2×C22⋊C4×C71—21no—
98C2×C22⋊C4×C71—21no—
99C2×C42×C71—21no—
100C2×C42×C71—21no—
101#95×C71—21no—
102#95×C71—21no—
103#95×C71—21no—
104#95×C71—21no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—42no—
108C22×Dic71—42no—
109C22×Dic71—42no—
110C22×Dic71—42no—
111C22×Dic71—42no—
112C22×Dic71—42no—
113#11765849—28no—
114#11765849—44no—
115C2×C71⋊C8—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8C2×D414yes—
121C4⋊Dic71—22no—
122C4⋊Dic71—22no—
123C4⋊Dic71—22no—
124C4⋊Dic71—22no—
125C4⋊Dic71—42no—
126C4⋊Dic71—42no—
127C22⋊Dic71—44no—
128C22⋊Dic71—44no—
129C24×C71—22no—
130C24×C71—22no—
131C24×C71—22no—
132C24×C71—22no—
133C24×C71—22no—
134C24×C71—41no—
135C24×C71—41no—
136C24×C71C2×D411yescharacteristic
137C24×C71C2×D411yescharacteristic
138C24×C71C2×D411yescharacteristic
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—44no—
165C22×C4×C71—44no—
166C22×C4×C71—44no—
167C22×C4×C71—44no—
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71#4112yes—
173C22×C4×C71#4112yes—
174C22×C4×C71#4112yes—
175C22×C4×C71#4112yes—
176C22×C4×C71#4112yes—
177C22×C4×C71C2411yescharacteristic
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C22⋊C4×C71—42no—
187C22⋊C4×C71—42no—
188C22⋊C4×C71—42no—
189C22⋊C4×C71—42no—
190C22⋊C4×C71—42no—
191C22⋊C4×C71—42no—
192C42×C71—42no—
193C42×C71—42no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—28no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71—21no—
202C23×C71—21no—
203C23×C71—21no—
204C23×C71—21no—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71—21no—
209C23×C71—21no—
210C23×C71—21no—
211C23×C71—44no—
212C23×C71—44no—
213C23×C71—44no—
214C23×C71—44no—
215C23×C71—44no—
216C23×C71—44no—
217C23×C71#12112yes—
218C23×C71#12312yes—
219C23×C71#12312yes—
220C23×C71AGL(1,ℤ/8)12yes—
221C23×C71#13712yes—
222C23×C71C22×D411yescharacteristic
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—44no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—44no—
239C2×C4×C71—44no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—41no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—41no—
247C2×C4×C71—41no—
248C2×C4×C71—41no—
249C2×C4×C71—41no—
250C2×C4×C71—41no—
251C2×C4×C71—41no—
252C2×C4×C71#12112yes—
253C2×C4×C71#12312yes—
254C2×C4×C71#12314yes—
255C2×C4×C71#12312yes—
256C2×C4×C71#12414yes—
257C2×C4×C71#12614yes—
258C2×C4×C71#12614yes—
259C2×C4×C71C22×D411yescharacteristic
260C2×C4×C71C2×#4111yescharacteristic
261C2×C4×C71C2×#4111yescharacteristic
262C2×C4×C71C2×#4111yescharacteristic
263C2×C4×C71C2×#4111yescharacteristic
264C2×C4×C71C2×#4111yescharacteristic
265C2×Dic71—42no—
266C2×Dic71—42no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—42no—
270C2×Dic71—42no—
271C2×Dic71—84no—
272C2×Dic71—84no—
273C71⋊C8—24no—
274C71⋊C8—42no—
275C22×C71—22no—
276C22×C71—22no—
277C22×C71—22no—
278C22×C71—22no—
279C22×C71—22no—
280C22×C71—22no—
281C22×C71—22no—
282C22×C71—22no—
283C22×C71—22no—
284C22×C71—22no—
285C22×C71—22no—
286C22×C71—22no—
287C22×C71—416no—
288C22×C71—41no—
289C22×C71—41no—
290C22×C71—41no—
291C22×C71—41no—
292C22×C71—41no—
293C22×C71—41no—
294C22×C71—41no—
295C22×C71—41no—
296C22×C71#39714yes—
297C22×C71C2×#12111yescharacteristic
298C22×C71C2×#12311yescharacteristic
299C22×C71C2×#12311yescharacteristic
300C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C71C2×#13711yescharacteristic
302C22×C71#57511yescharacteristic
303C22×C71#57511yescharacteristic
304C4×C71—41no—
305C4×C71—41no—
306C4×C71—81no—
307C4×C71—81no—
308C4×C71—81no—
309C4×C71—81no—
310C4×C71#39714yes—
311C4×C71C2×#12311yescharacteristic
312C4×C71C2×#12411yescharacteristic
313C4×C71C2×#12611yescharacteristic
314C4×C71C2×#12611yescharacteristic
315Dic71—82no—
316Dic71—82no—
317#23642—711nomaximal, Sylow
318C2×C71—24no—
319C2×C71—42no—
320C2×C71—42no—
321C2×C71C2×#39711yescharacteristic
322C2×C71#305511yescharacteristic
323C2×C71#305611yescharacteristic
324C2×C71#309211yescharacteristic
325C2×C71#309311yescharacteristic
326C2×C71#309411yescharacteristic
327C2×C71#309511yescharacteristic
328C2×#428—711no—
329C2×#407—711no—
330#2280D7111yescharacteristic
331C2×#380—711no—
332#2077—712no—
333#2077—712no—
334#2076—712no—
335#2075—712no—
336#1736—711no—
337#1736—711no—
338#1736—711no—
339C71#2364211yescharacteristic, complemented, Sylow
340C24×C4C2×D7111yescharacteristic
341C22×M16—711no—
342C22×M16—711no—
343C2×#117C2×D7111yescharacteristic
344C2×#117C2×D7111yescharacteristic
345C22×C4⋊C4—711no—
346C22×C4⋊C4—711no—
347C22×C22⋊C4—1421no—
348#428—1424no—
349C2×#107—712no—
350C2×#106—712no—
351#407—1424no—
352C2×C22⋊C8—712no—
353C2×C22⋊C8—712no—
354#381—21no—
355#380—712no—
356#380—712no—
357#380—1421no—
358#380—1421no—
359#380C2×D7111yescharacteristic
360#380C2×D7111yescharacteristic
361#380C2×D7111yescharacteristic
362#380C2×D7111yescharacteristic
363#379—21no—
364C4×C22⋊C4—21no—
365C4×C22⋊C4—21no—
366C2×#95—1421no—
367#340—712no—
368#340—712no—
369#340—712no—
370#340—712no—
371#340—712no—
372#340—712no—
373C25C22⋊D7111yescharacteristic
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1421no—
381C23×C4—1421no—
382C23×C4C22⋊D7111yescharacteristic
383C23×C4C22⋊D7111yescharacteristic
384C23×C4C22⋊D7111yescharacteristic
385C23×C4C22×D7111yescharacteristic
386C23×C4C22×D7111yescharacteristic
387C23×C4C22×D7111yescharacteristic
388C2×M16—714no—
389C2×M16—1422no—
390C2×M16—1422no—
391C2×M16—1422no—
392C2×M16—1422no—
393C2×M16—1422no—
394C22×C8—712no—
395C22×C8—712no—
396#117—22no—
397#117—22no—
398#117—22no—
399#117—22no—
400C2×C4⋊C4—712no—
401C2×C4⋊C4—712no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4—1421no—
404C2×C4⋊C4—1422no—
405C2×C4⋊C4—1422no—
406C2×C4⋊C4—1422no—
407C2×C4⋊C4—1422no—
408C2×C4⋊C4C22×D7111yescharacteristic
409C2×C4⋊C4C22×D7111yescharacteristic
410C2×C4⋊C4C22×D7111yescharacteristic
411C2×C4⋊C4C22×D7111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—2844no—
419C2×C22⋊C4—1422no—
420C2×C22⋊C4—1422no—
421C2×C42—21no—
422C2×C42—21no—
423#107—1424no—
424#106—1424no—
425C22⋊C8—1424no—
426C22⋊C8—1424no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1422no—
432#95—1422no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—41no—
439C24—41no—
440C24C2×C22⋊D7111yescharacteristic
441C24C2×C22⋊D7111yescharacteristic
442C24C2×C22⋊D7111yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—2842no—
473C22×C4—2842no—
474C22×C4—2842no—
475C22×C4—2842no—
476C22×C4—2842no—
477C22×C4—2842no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4D4×D7112yes—
481C22×C4C2×C22⋊D7111yescharacteristic
482C22×C4C2×C22⋊D7111yescharacteristic
483C22×C4C2×C22⋊D7111yescharacteristic
484C22×C4#1176586412yes—
485C22×C4#1176586412yes—
486C22×C4#1176586412yes—
487C22×C4#1176586512yes—
488C22×C4#1176586512yes—
489C22×C4C23×D7111yescharacteristic
490M16—2844no—
491M16—1428no—
492C2×C8—714no—
493C2×C8—1422no—
494C2×C8—1422no—
495C2×C8—1422no—
496C2×C8—1422no—
497C2×C8—1422no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—2842no—
507C4⋊C4—2842no—
508C4⋊C4—1422no—
509C4⋊C4—1422no—
510C4⋊C4—1422no—
511C4⋊C4—1422no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—2844no—
519C22⋊C4—2844no—
520C42—42no—
521C42—42no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#1455610012yes—
546C23#1455610912yes—
547C23#1455611112yes—
548C23#1455615312yes—
549C23#1455615812yes—
550C23C22×C22⋊D7111yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—2842no—
581C2×C4—2842no—
582C2×C4—2842no—
583C2×C4—2842no—
584C2×C4—2842no—
585C2×C4—2842no—
586C2×C4—5684no—
587C2×C4—5684no—
588C2×C4#1455610012yes—
589C2×C4#1455610912yes—
590C2×C4#1455611014yes—
591C2×C4#1455611112yes—
592C2×C4#1455611414yes—
593C2×C4#1455612114yes—
594C2×C4#1455612114yes—
595C2×C4C2×D4×D7111yescharacteristic
596C2×C4C2×#1176586411yescharacteristic
597C2×C4C2×#1176586411yescharacteristic
598C2×C4C2×#1176586411yescharacteristic
599C2×C4C2×#1176586511yescharacteristic
600C2×C4C2×#1176586511yescharacteristic
601C8—2842no—
602C8—1424no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#1798592714yes—
625C22C2×#1455610011yescharacteristic
626C22C2×#1455610911yescharacteristic
627C22C2×#1455611111yescharacteristic
628C22C2×#1455615311yescharacteristic
629C22C2×#1455615811yescharacteristic
630C22#1798679711yescharacteristic
631C22#1798679711yescharacteristic
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—5682no—
639C4—5682no—
640C4#1798592714yes—
641C4C2×#1455611011yescharacteristic
642C4C2×#1455611411yescharacteristic
643C4C2×#1455612111yescharacteristic
644C4C2×#1455612111yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×#1798592711yescharacteristic
649C2#2259796111yescharacteristic
650C2#2259797211yescharacteristic
651C2#2259826011yescharacteristic
652C2#2259826911yescharacteristic
653C2#2259828011yescharacteristic
654C2#2259828811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2280C2
socleC23×C71C22×D4
prime radicalC25×C71D4
hypercenter#2280D71
hypocenterC71#23642
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
1. C2×#17986042C2semidirect
4. #22589777C2semidirect
5. #22589776C2semidirect
6. #22589771C2semidirect
7. #22589766C2semidirect
8. #22588061C2semidirect
9. #22588061C2semidirect
10. #22588061C2semidirect
18. C2×#14556061C22semidirect
19. C2×#14556059C22semidirect
28. #17985761C22semidirect
29. #17985761C22semidirect
30. #17985761C22semidirect
31. #17985761C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
339. C71#23642semidirect

Maximal subgroups

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

subgroupindex
C2×#179860422
C2×#179859632
C2×#179858592
#225897772
#225897762
#225897712
#225897662
#225880612
C71×#22802
#2364271