Finite Group Databasebeta

← #32000507

Subgroups — #32000507

Order 18176.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986099C211yesmaximal, characteristic
2C42×Dic142C211yesmaximal, characteristic, complemented
3#22588964C211yesmaximal, characteristic
4#22588957C211yesmaximal, characteristic
5#22588929C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588698C211yesmaximal, characteristic
8#22588698C211yesmaximal, characteristic
9#22588607C211yesmaximal, characteristic, complemented
10#22588593C211yesmaximal, characteristic, complemented
11#22588238C211yesmaximal, characteristic, complemented
12#22588173C211yesmaximal, characteristic, complemented
13#22588165C211yesmaximal, characteristic
14#22588089C211yesmaximal, characteristic, complemented
15C71×#2293C211yesmaximal, characteristic
16C22×#11765853C2211yescharacteristic
17C2×C4×Dic142C2211yescharacteristic
18C2×C4×Dic142C2211yescharacteristic
19C2×C4×Dic142C2211yescharacteristic
20#17986099C418yescomplemented
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556051C2211yescharacteristic
25C2×#14556051C2211yescharacteristic
26C2×#14556051C2211yescharacteristic
27C2×#14556049C2211yescharacteristic
28C2×#14556049C2211yescharacteristic
29C2×#14556049C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×#14556039C2211yescharacteristic
32C4×Dic71⋊C4—21no—
33C4×Dic71⋊C4—21no—
34C4×Dic71⋊C4—21no—
35C4×C4⋊Dic71C2211yescharacteristic
36C4×C4⋊Dic71C2211yescharacteristic
37C4×C4⋊Dic71C2211yescharacteristic
38C42×Dic71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985754C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×D4×C71C2211yescharacteristic
50C2×#117×C71—21no—
51#390×C71C2211yescharacteristic
52#390×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#388×C71—21no—
56#388×C71—21no—
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C43×C71C2211yescharacteristic
60C22×Dic142C2311yescharacteristic
61C2×#11765853—22no—
62C2×#11765853—22no—
63C2×#11765853C2×C414yes—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C4×Dic142—22no—
69C4×Dic142—22no—
70C4×Dic142—22no—
71C4×Dic142—22no—
72C4×Dic142—22no—
73C4×Dic142C2×C414yes—
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4—21no—
80C2×C4×Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86#14556061—22no—
87#14556061C2×C414yes—
88#14556059—22no—
89#14556051—22no—
90#14556051—22no—
91#14556051C2×C414yes—
92#14556049—22no—
93#14556049—22no—
94#14556049C2×C414yes—
95#14556039—22no—
96#14556039C2×C414yes—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C22×D4×C71C2311yescharacteristic
101C23×C4×C71—21no—
102C4×D4×C71—22no—
103C4×D4×C71C2×C414yes—
104#117×C71—44no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C22⋊C4×C71—21no—
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C22⋊C4×C71—21no—
115C2×C42×C71—21no—
116C2×C42×C71D411yescharacteristic
117C2×C42×C71D411yescharacteristic
118C2×C42×C71D411yescharacteristic
119C2×C42×C71D411yescharacteristic
120C2×C42×C71C2311yescharacteristic
121C2×C42×C71C2311yescharacteristic
122C2×C42×C71C2311yescharacteristic
123#95×C71—21no—
124#95×C71—21no—
125#95×C71—21no—
126C2×Dic142—21no—
127C2×Dic142—21no—
128C2×Dic142—21no—
129C2×Dic142C22×C412yes—
130C2×Dic142#4112yes—
131C2×Dic142#4112yes—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C22×Dic71—21no—
136#11765853—28no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8C22×C412yes—
148C2×C71⋊C8C22×C412yes—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71C22×C412yes—
153C4⋊Dic71#4112yes—
154C4⋊Dic71#4112yes—
155Dic71⋊C4—24no—
156Dic71⋊C4—24no—
157Dic71⋊C4—24no—
158C4×Dic71—24no—
159C4×Dic71—24no—
160C4×Dic71—24no—
161C4×Dic71—42no—
162C4×Dic71—42no—
163C4×Dic71—42no—
164C24×C71—21no—
165C2×D4×C71—21no—
166C2×D4×C71—21no—
167C2×D4×C71—21no—
168C2×D4×C71—21no—
169C2×D4×C71—21no—
170C2×D4×C71C22×C412yes—
171C22×C4×C71—21no—
172C22×C4×C71—24no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—42no—
176C22×C4×C71—42no—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2411yescharacteristic
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—42no—
190C4⋊C4×C71—42no—
191C4⋊C4×C71C22×C412yes—
192C22⋊C4×C71—24no—
193C22⋊C4×C71—42no—
194C22⋊C4×C71—42no—
195C22⋊C4×C71—42no—
196C22⋊C4×C71—42no—
197C42×C71—22no—
198C42×C71—22no—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—41no—
204C42×C71—41no—
205C42×C71—42no—
206C42×C71C22×C412yes—
207C42×C71#4112yes—
208C23×C71—22no—
209C23×C71—41no—
210C23×C71—41no—
211C23×C71—41no—
212C23×C71—41no—
213C23×C71—41no—
214C23×C71—41no—
215C23×C71C22×D411yescharacteristic
216D4×C71—22no—
217D4×C71—22no—
218D4×C71—41no—
219D4×C71—41no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—22no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—48no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—41no—
244C2×C4×C71C4×D412yes—
245C2×C4×C71C4×D412yes—
246C2×C4×C71AGL(1,ℤ/8)12yes—
247C2×C4×C71#13712yes—
248C2×C4×C71C23×C411yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C2×C4×C71C2×#4111yescharacteristic
253C2×C4×C71C2×#4111yescharacteristic
254C2×Dic71—22no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—22no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266Dic142—22no—
267Dic142—22no—
268Dic142—22no—
269Dic142#11714yescomplemented
270C71⋊C8—22no—
271C71⋊C8—22no—
272C71⋊C8—41no—
273C71⋊C8—41no—
274C22×C71—42no—
275C22×C71—42no—
276C22×C71—42no—
277C22×C71—81no—
278C22×C71—81no—
279C22×C71—81no—
280C22×C71—81no—
281C22×C71C2×C4×D411yescharacteristic
282C22×C71#52511yescharacteristic
283C22×C71#52511yescharacteristic
284C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C71C2×#13711yescharacteristic
286C22×C71#57511yescharacteristic
287C22×C71#57511yescharacteristic
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—22no—
291C4×C71—22no—
292C4×C71—41no—
293C4×C71—41no—
294C4×C71—42no—
295C4×C71—81no—
296C4×C71—81no—
297C4×C71C2×#11711yescharacteristic
298C4×C71C4×#4111yescharacteristic
299C4×C71#52911yescharacteristic
300C4×C71#52911yescharacteristic
301Dic71—24no—
302Dic71—42no—
303Dic71—42no—
304Dic71—42no—
305#21124—711nomaximal, Sylow
306C2×C71—82no—
307C2×C71#229011yescharacteristic
308C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
309C2×C71C4×#13711yescharacteristic
310C2×C71#311111yescharacteristic
311C2×C71#311211yescharacteristic
312C2×C71#311511yescharacteristic
313C2×C71#311711yescharacteristic
314C2×#440—711no—
315#2293D7111yescharacteristic
316C42×Q8—711no—
317#1926—711no—
318#1925—711no—
319#1921—711no—
320#1918—711no—
321#1880—711no—
322#1880—711no—
323#1865—711no—
324#1863—711no—
325#1776—711no—
326#1763—711no—
327#1762—711no—
328#1745—711no—
329C71#2112411yescharacteristic, complemented, Sylow
330C22×SD16—711no—
331C2×C4×Q8—711no—
332C2×C4×Q8—711no—
333C2×C4×Q8—711no—
334C2×C4×D4C2×D7111yescharacteristic
335C2×#117—21no—
336#440—718no—
337C2×#107—711no—
338C2×#107—711no—
339C2×#106—711no—
340C2×#103—711no—
341C2×#103—711no—
342C2×#103—711no—
343C2×#102—711no—
344C2×#102—711no—
345C2×#102—711no—
346C2×#97—711no—
347C2×#97—711no—
348#390C2×D7111yescharacteristic
349#390C2×D7111yescharacteristic
350#389C2×D7111yescharacteristic
351#389C2×D7111yescharacteristic
352#388—21no—
353#388—21no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4—711no—
356C4×C4⋊C4—711no—
357C4×C4⋊C4—1421no—
358C4×C4⋊C4—1421no—
359C4×C4⋊C4—1421no—
360C4×C4⋊C4C2×D7111yescharacteristic
361C4×C22⋊C4—21no—
362C43—1421no—
363C43C2×D7111yescharacteristic
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#336—711no—
371#336—711no—
372#336—711no—
373#336—711no—
374C22×Q8—711no—
375C22×D4C22×D7111yescharacteristic
376C23×C4—21no—
377C2×SD16—714no—
378C2×SD16—1422no—
379C2×SD16—1422no—
380C22×C8—711no—
381C22×C8—711no—
382C22×C8—711no—
383C22×C8—711no—
384C4×Q8—714no—
385C4×Q8—1422no—
386C4×Q8—1422no—
387C4×Q8—1422no—
388C4×Q8—1422no—
389C4×Q8—1422no—
390C4×D4—22no—
391C4×D4C4×D7114yes—
392#117—44no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—1421no—
399C2×C4⋊C4—1421no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4C22×D7111yescharacteristic
402C2×C4⋊C4C22×D7111yescharacteristic
403C2×C4⋊C4C22×D7111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—21no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42—1421no—
413C2×C42—1421no—
414C2×C42—1421no—
415C2×C42—1421no—
416C2×C42C22⋊D7111yescharacteristic
417C2×C42C22⋊D7111yescharacteristic
418C2×C42C22⋊D7111yescharacteristic
419C2×C42C22⋊D7111yescharacteristic
420C2×C42C22×D7111yescharacteristic
421C2×C42C22×D7111yescharacteristic
422C2×C42C22×D7111yescharacteristic
423#107—714no—
424#107—1422no—
425#106—1422no—
426#103—714no—
427#103—1422no—
428#103—1422no—
429#102—714no—
430#102—1422no—
431#102—1422no—
432#97—714no—
433#97—1422no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1421no—
438#95—1421no—
439#95—1421no—
440C24—21no—
441C2×Q8—712no—
442C2×Q8—712no—
443C2×Q8—712no—
444C2×Q8—1421no—
445C2×Q8—1421no—
446C2×Q8—1421no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D7112yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C23×D7111yescharacteristic
470SD16—1428no—
471C2×C8—712no—
472C2×C8—712no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8—1421no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—712no—
491C4⋊C4—712no—
492C4⋊C4—712no—
493C4⋊C4—1424no—
494C4⋊C4—1424no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1424no—
499C4⋊C4C2×C4×D7112yes—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—2842no—
515C42—2842no—
516C42—2842no—
517C42—1424no—
518C42—1424no—
519C42—1424no—
520C42C2×C4×D7112yes—
521C42#1176586512yes—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C22×C22⋊D7111yescharacteristic
530Q8—714no—
531Q8—1422no—
532Q8—1422no—
533Q8—1422no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4C4×C22⋊D7112yes—
575C2×C4C4×C22⋊D7112yes—
576C2×C4#1455615412yes—
577C2×C4#1455615812yes—
578C2×C4C22×C4×D7111yescharacteristic
579C2×C4#41×D7111yescharacteristic
580C2×C4#41×D7111yescharacteristic
581C2×C4#41×D7111yescharacteristic
582C2×C4#41×D7111yescharacteristic
583C2×C4C2×#1176586511yescharacteristic
584C8—2841no—
585C8—2841no—
586C8—1422no—
587C8—1422no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×C4×C22⋊D7111yescharacteristic
596C22#1798650611yescharacteristic
597C22#1798650611yescharacteristic
598C22C2×#1455615411yescharacteristic
599C22C2×#1455615811yescharacteristic
600C22#1798679711yescharacteristic
601C22#1798679711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—2842no—
612C4—2842no—
613C4—2842no—
614C4—1424no—
615C4#117×D7111yescharacteristic
616C4#1798646711yescharacteristic
617C4#1798652711yescharacteristic
618C4#1798652711yescharacteristic
619C2—82no—
620C2#2259077711yescharacteristic
621C2C4×#1455615411yescharacteristic
622C2C4×#1455615811yescharacteristic
623C2#2259844311yescharacteristic
624C2#2259845811yescharacteristic
625C2#2259848511yescharacteristic
626C2#2259850711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2293C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2293D71
hypocenterC71#21124
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C42×Dic142C2semidirect
5. #22588929C2semidirect
6. #22588908C2semidirect
9. #22588607C2semidirect
10. #22588593C2semidirect
11. #22588238C2semidirect
12. #22588173C2semidirect
14. #22588089C2semidirect
20. #17986099C4semidirect
269. Dic142#117semidirect
329. C71#21124semidirect

Maximal subgroups

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

subgroupindex
C2×#179860992
C42×Dic1422
#225889642
#225889572
#225889292
#225889082
#225886982
#225886072
#225885932
#225882382
#225881732
#225881652
#225880892
C71×#22932
#2112471