Finite Group Databasebeta

← #32008694

Subgroups — #32008694

Order 18176.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986150C211yesmaximal, characteristic
2C4×#14556107C211yesmaximal, characteristic, complemented
3#22589220C212yesmaximal, complemented
4#22589040C212yesmaximal, complemented
5#22588957C212yesmaximal, complemented
6#22588698C212yesmaximal, complemented
7#22588647C211yesmaximal, characteristic, complemented
8#22588528C211yesmaximal, characteristic, complemented
9#22588271C211yesmaximal, characteristic
10#22588102C211yesmaximal, characteristic, complemented
11C2×#390×C71C211yesmaximal, characteristic
12C22×#11765853C2212yes—
13C2×#14556107C2211yescharacteristic
14C2×C4×Dic142C2211yescharacteristic, complemented
15#17986150C418yescomplemented
16C2×#14556061C2212yescomplemented
17C2×#14556051C2212yescomplemented
18C2×#14556049C2212yes—
19C2×#14556049C2212yes—
20C2×#14556049C2212yes—
21C2×C22⋊(C71⋊C8)C2211yescharacteristic
22C2×C22⋊(C71⋊C8)C2211yescharacteristic
23C2×#14556039C2212yescomplemented
24#17985905C2211yescharacteristic, complemented
25#17985897C2211yescharacteristic, complemented
26#17985883—21no—
27#17985872—21no—
28#17985859C2211yescharacteristic
29C4×Dic71⋊C4—21no—
30C4×C4⋊Dic71C2211yescharacteristic, complemented
31C4×C22⋊Dic71—21no—
32#17985761C2212yescomplemented
33#17985761C2212yescomplemented
34#17985754C2212yescomplemented
35C23×D4×C71C2211yescharacteristic
36C22×C4⋊C4×C71C2211yescharacteristic
37C22×C22⋊C4×C71—21no—
38C22×C22⋊C4×C71—21no—
39C22×C42×C71C2211yescharacteristic
40#390×C71—21no—
41#390×C71—21no—
42#390×C71C2212yes—
43#390×C71C2212yes—
44C22×Dic142C2311yescharacteristic
45C2×#11765853—42no—
46C2×#11765853—42no—
47C2×#11765853C2×C418yescomplemented
48C22×C71⋊C8C2312yes—
49C22×C71⋊C8C2312yes—
50#14556107—22no—
51#14556107C2×C414yes—
52C4×Dic142—42no—
53C2×C4⋊Dic71C2311yescharacteristic
54C2×C4⋊Dic71C2311yescharacteristic
55C2×C4⋊Dic71C2311yescharacteristic
56C2×Dic71⋊C4—21no—
57C2×Dic71⋊C4—21no—
58C2×Dic71⋊C4—21no—
59C2×C22⋊Dic71—21no—
60C2×C22⋊Dic71—21no—
61C2×C4×Dic71—21no—
62C2×C4×Dic71—21no—
63#14556061—42no—
64#14556051—42no—
65#14556049—42no—
66#14556049—42no—
67#14556049C2×C418yescomplemented
68C22⋊(C71⋊C8)—22no—
69C22⋊(C71⋊C8)C2×C414yes—
70#14556039—42no—
71C71⋊#95—21no—
72C71⋊#95—21no—
73C71⋊#95—21no—
74C71⋊#95—21no—
75C71⋊#95—21no—
76C25×C71—21no—
77C22×D4×C71—21no—
78C22×D4×C71—22no—
79C22×D4×C71—42no—
80C22×D4×C71—42no—
81C22×D4×C71C2×C414yes—
82C22×D4×C71C2312yes—
83C23×C4×C71—21no—
84C23×C4×C71D411yescharacteristic
85C23×C4×C71D411yescharacteristic
86C23×C4×C71C2311yescharacteristic
87C2×C4⋊C4×C71—21no—
88C2×C4⋊C4×C71—42no—
89C2×C4⋊C4×C71C2312yes—
90C2×C22⋊C4×C71—22no—
91C2×C22⋊C4×C71—22no—
92C2×C22⋊C4×C71—22no—
93C2×C22⋊C4×C71—22no—
94C2×C22⋊C4×C71—44no—
95C2×C22⋊C4×C71—44no—
96C2×C42×C71—42no—
97C2×C42×C71D411yescharacteristic
98C2×C42×C71D411yescharacteristic
99C2×C42×C71C2311yescharacteristic
100C2×C42×C71C2311yescharacteristic
101C2×Dic142—22no—
102C2×Dic142—22no—
103C2×Dic142—21no—
104C2×Dic142C22×C412yes—
105C22×Dic71—21no—
106C22×Dic71—21no—
107C22×Dic71—21no—
108C22×Dic71—21no—
109#11765853—48no—
110C2×C71⋊C8—22no—
111C2×C71⋊C8—22no—
112C2×C71⋊C8—22no—
113C2×C71⋊C8—22no—
114C2×C71⋊C8—22no—
115C2×C71⋊C8C22×C414yes—
116C4⋊Dic71—21no—
117C4⋊Dic71—22no—
118C4⋊Dic71—22no—
119C4⋊Dic71C22×C412yes—
120Dic71⋊C4—24no—
121Dic71⋊C4—42no—
122Dic71⋊C4—42no—
123C22⋊Dic71—24no—
124C22⋊Dic71—42no—
125C4×Dic71—42no—
126C4×Dic71—42no—
127C24×C71—24no—
128C24×C71—22no—
129C24×C71—42no—
130C24×C71—42no—
131C24×C71—42no—
132C24×C71—42no—
133C24×C71—42no—
134C24×C71—42no—
135C24×C71C2×D411yescharacteristic
136C2×D4×C71—24no—
137C2×D4×C71—22no—
138C2×D4×C71—22no—
139C2×D4×C71—22no—
140C2×D4×C71—22no—
141C2×D4×C71—22no—
142C2×D4×C71—48no—
143C2×D4×C71—42no—
144C2×D4×C71—42no—
145C2×D4×C71—48no—
146C2×D4×C71C22×C414yes—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—41no—
154C22×C4×C71—41no—
155C22×C4×C71—41no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71C22×C412yes—
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71#4112yes—
166C22×C4×C71C2411yescharacteristic
167C4⋊C4×C71—22no—
168C4⋊C4×C71—22no—
169C4⋊C4×C71—42no—
170C22⋊C4×C71—44no—
171C22⋊C4×C71—44no—
172C22⋊C4×C71—44no—
173C22⋊C4×C71—44no—
174C42×C71—22no—
175C42×C71—22no—
176C42×C71—41no—
177C42×C71—41no—
178C23×C71—21no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—24no—
183C23×C71—48no—
184C23×C71—48no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—42no—
190C23×C71—42no—
191C23×C71—48no—
192C23×C71—84no—
193C23×C71—84no—
194C23×C71—84no—
195C23×C71—84no—
196C23×C71C4×D412yes—
197C23×C71AGL(1,ℤ/8)12yes—
198C23×C71#13712yes—
199C23×C71C22×D411yescharacteristic
200D4×C71—24no—
201D4×C71—24no—
202D4×C71—48no—
203D4×C71—42no—
204D4×C71—42no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—24no—
210C2×C4×C71—24no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—44no—
221C2×C4×C71—44no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—84no—
226C2×C4×C71C4×D412yes—
227C2×C4×C71AGL(1,ℤ/8)12yes—
228C2×C4×C71C23×C411yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C2×C4×C71H(5,𝔽2)12yes—
231C2×C4×C71H(5,𝔽2)12yes—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—41no—
235C2×Dic71—41no—
236C2×Dic71—41no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244Dic142—24no—
245Dic142—42no—
246C71⋊C8—24no—
247C71⋊C8—42no—
248C22×C71—22no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71—22no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—44no—
255C22×C71—44no—
256C22×C71—44no—
257C22×C71—816no—
258C22×C71—82no—
259C22×C71—82no—
260C22×C71—82no—
261C22×C71—82no—
262C22×C71#44014yes—
263C22×C71C2×C4×D411yescharacteristic
264C22×C71#53411yescharacteristic
265C22×C71#53511yescharacteristic
266C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C71C2×#13711yescharacteristic
269C22×C71#57511yescharacteristic
270C4×C71—22no—
271C4×C71—22no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—82no—
276C4×C71#51812yes—
277C4×C71#53812yes—
278Dic71—42no—
279Dic71—42no—
280Dic71—81no—
281Dic71—81no—
282#22049—711nomaximal, Sylow
283C2×C71—24no—
284C2×C71—84no—
285C2×C71#232111yescharacteristic
286C2×C71C2×#44011yescharacteristic
287C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
288C2×C71#319311yescharacteristic
289C2×C71#319411yescharacteristic
290C2×C71#321511yescharacteristic
291C2×C71#321711yescharacteristic
292C2×#450—711no—
293C4×C22⋊Q8—711no—
294C2×#390D7111yescharacteristic
295#1973—712no—
296#1936—712no—
297#1925—712no—
298#1880—712no—
299#1872—711no—
300#1852—711no—
301#1786—711no—
302#1749—711no—
303C71#2204911yescharacteristic, complemented, Sylow
304C23×D4C2×D7111yescharacteristic
305C22×SD16—712no—
306C2×C22⋊Q8—711no—
307C2×C4×Q8—711no—
308C22×C4⋊C4C2×D7111yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4—21no—
311C22×C42C2×D7111yescharacteristic
312#450—718no—
313C2×#107—712no—
314C2×#103—712no—
315C2×#102—712no—
316C2×#102—712no—
317C2×#102—712no—
318C2×C22⋊C8—711no—
319C2×C22⋊C8—711no—
320C2×#97—712no—
321#391—711no—
322#390—21no—
323#390—21no—
324#390C2×D7112yes—
325#390C2×D7112yes—
326#389—711no—
327#387—1421no—
328#385—1421no—
329#380—711no—
330C4×C4⋊C4—711no—
331C4×C4⋊C4—1421no—
332C4×C22⋊C4—1421no—
333#340—712no—
334#340—712no—
335#336—712no—
336C25—21no—
337C22×Q8—711no—
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C4×D7114yes—
343C22×D4C22×D7112yes—
344C23×C4—21no—
345C23×C4C22⋊D7111yescharacteristic
346C23×C4C22⋊D7111yescharacteristic
347C23×C4C22×D7111yescharacteristic
348C2×SD16—2842no—
349C2×SD16—2842no—
350C2×SD16—718no—
351C22×C8—712no—
352C22×C8—712no—
353C22⋊Q8—714no—
354C22⋊Q8—1422no—
355C4×Q8—2842no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—711no—
359C2×C4⋊C4—711no—
360C2×C4⋊C4—711no—
361C2×C4⋊C4—1421no—
362C2×C4⋊C4—1421no—
363C2×C4⋊C4—1421no—
364C2×C4⋊C4C22×D7112yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—1421no—
372C2×C22⋊C4—1421no—
373C2×C42—42no—
374C2×C42—1421no—
375C2×C42—1421no—
376C2×C42C22⋊D7111yescharacteristic
377C2×C42C22⋊D7111yescharacteristic
378C2×C42C22×D7111yescharacteristic
379C2×C42C22×D7111yescharacteristic
380#107—2842no—
381#103—2842no—
382#102—2842no—
383#102—2842no—
384#102—718no—
385C22⋊C8—714no—
386C22⋊C8—1422no—
387#97—2842no—
388#95—1421no—
389#95—1421no—
390#95—1421no—
391#95—1421no—
392#95—1421no—
393C24—22no—
394C24—24no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C2×C22⋊D7111yescharacteristic
402C2×Q8—712no—
403C2×Q8—1422no—
404C2×Q8—1422no—
405C2×Q8—1421no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C2×C4×D7114yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—1421no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4C2×C4×D7112yes—
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4#1176586512yes—
440C22×C4C23×D7111yescharacteristic
441SD16—2848no—
442C2×C8—714no—
443C2×C8—1422no—
444C2×C8—1422no—
445C2×C8—1422no—
446C2×C8—1422no—
447C2×C8—1422no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—2842no—
452C4⋊C4—2842no—
453C4⋊C4—712no—
454C4⋊C4—1422no—
455C4⋊C4—1422no—
456C4⋊C4—1421no—
457C4⋊C4—1424no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—2842no—
463C22⋊C4—1424no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—2842no—
469C42—2842no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—48no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23C4×C22⋊D7112yes—
489C23#1455615412yes—
490C23#1455615812yes—
491C23C22×C22⋊D7111yescharacteristic
492Q8—2842no—
493Q8—1424no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—1422no—
531C2×C4—1422no—
532C2×C4C4×C22⋊D7112yes—
533C2×C4#1455615412yes—
534C2×C4C22×C4×D7111yescharacteristic
535C2×C4C2×#1176586511yescharacteristic
536C2×C4#1455617312yes—
537C2×C4#1455617312yes—
538C8—2842no—
539C8—1424no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#1798609914yes—
555C22C2×C4×C22⋊D7111yescharacteristic
556C22#1798655811yescharacteristic
557C22#1798656611yescharacteristic
558C22C2×#1455615411yescharacteristic
559C22C2×#1455615411yescharacteristic
560C22C2×#1455615811yescharacteristic
561C22#1798679711yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—82no—
568C4—2842no—
569C4—2842no—
570C4—5681no—
571C4—5681no—
572C4#1798647212yes—
573C4#1798658712yes—
574C2—24no—
575C2—84no—
576C2#2259099811yescharacteristic
577C2C2×#1798609911yescharacteristic
578C2C4×#1455615411yescharacteristic
579C2#2259914611yescharacteristic
580C2#2259915411yescharacteristic
581C2#2259934311yescharacteristic
582C2#2259935811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#390×C71C2
socleC23×C71C22×D4
prime radicalC23×D4×C71C22
cosocleC22×C4×C71C24
hypercenterC2×#390D71
hypocenterC71#22049
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#14556107C2semidirect
3. #22589220C2semidirect
4. #22589040C2semidirect
5. #22588957C2semidirect
6. #22588698C2semidirect
7. #22588647C2semidirect
8. #22588528C2semidirect
10. #22588102C2semidirect
14. C2×C4×Dic142C22semidirect
15. #17986150C4semidirect
16. C2×#14556061C22semidirect
17. C2×#14556051C22semidirect
23. C2×#14556039C22semidirect
24. #17985905C22semidirect
25. #17985897C22semidirect
30. C4×C4⋊Dic71C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
34. #17985754C22semidirect
47. C2×#11765853C2×C4semidirect
67. #14556049C2×C4semidirect
303. C71#22049semidirect

Maximal subgroups

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

subgroupindex
C2×#179861502
C4×#145561072
#225892202
#225890402
#225889572
#225886982
#225886472
#225885282
#225882712
#225881022
C2×#390×C712
#2204971