Finite Group Databasebeta

← #32015199

Subgroups — #32015199

Order 18176.

810 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
1#22591233C211yesmaximal, characteristic
2#22591209C211yesmaximal, characteristic
3#22589079C211yesmaximal, characteristic, complemented
4#22589074C211yesmaximal, characteristic, complemented
5#22588934C211yesmaximal, characteristic, complemented
6#22588927C211yesmaximal, characteristic, complemented
7#22588920C211yesmaximal, characteristic, complemented
8#22588913C211yesmaximal, characteristic, complemented
9#22588435C211yesmaximal, characteristic
10#22588430C211yesmaximal, characteristic
11#22588373C211yesmaximal, characteristic
12#22588240C211yesmaximal, characteristic, complemented
13#106×Dic71C211yesmaximal, characteristic, complemented
14#22588121C211yesmaximal, characteristic
15C2×#428×C71C211yesmaximal, characteristic
16C2×#14556087—21no—
17C2×#14556062C2211yescharacteristic
18C2×#14556060C2211yescharacteristic
19C2×C22⋊(C71⋊C8)C2211yescharacteristic
20C2×#14556038C2211yescharacteristic
21C2×C8×Dic71C2211yescharacteristic
22#17985898C2211yescharacteristic
23#17985898C2211yescharacteristic
24#17985898C2211yescharacteristic
25#17985898C2211yescharacteristic
26#17985892—21no—
27#17985892—21no—
28#17985885—21no—
29#17985885—21no—
30#17985867—21no—
31#17985862—21no—
32#17985859C2211yescharacteristic
33#17985859C2211yescharacteristic
34C4⋊C4×Dic71C2211yescharacteristic
35C4⋊C4×Dic71C2211yescharacteristic
36C4⋊C4×Dic71C2211yescharacteristic
37C4⋊C4×Dic71C2211yescharacteristic
38C4×C4⋊Dic71—21no—
39C4×C22⋊Dic71C2211yescharacteristic
40#17985762C2211yescharacteristic
41#17985762C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985753C2211yescharacteristic
51#17985753C2211yescharacteristic
52C22×M16×C71C2211yescharacteristic
53C2×#117×C71C2211yescharacteristic
54C22×C4⋊C4×C71C2211yescharacteristic
55#428×C71C418yescomplemented
56C2×#107×C71C2211yescharacteristic
57C2×#107×C71C2211yescharacteristic
58C2×#106×C71C2211yescharacteristic
59C2×#106×C71C2211yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62#14556087—44no—
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—21no—
65C2×C4⋊Dic71—21no—
66C2×C4⋊Dic71—21no—
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C22⋊Dic71—21no—
72C2×C22⋊Dic71—21no—
73C2×C22⋊Dic71D411yescharacteristic
74C2×C22⋊Dic71D411yescharacteristic
75C2×C4×Dic71—21no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—21no—
79C2×C4×Dic71C2311yescharacteristic
80C2×C4×Dic71C2311yescharacteristic
81#14556062—22no—
82#14556060—22no—
83C22⋊(C71⋊C8)—22no—
84#14556038—22no—
85C8×Dic71—22no—
86C71⋊#95—21no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C71⋊#95D411yescharacteristic
97C71⋊#95D411yescharacteristic
98C23×C4×C71—21no—
99C23×C4×C71C2311yescharacteristic
100C2×M16×C71—22no—
101C2×M16×C71—22no—
102C2×M16×C71C2×C414yes—
103C22×C8×C71C2311yescharacteristic
104C22×C8×C71C2311yescharacteristic
105#117×C71—22no—
106#117×C71C2×C414yes—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—22no—
109C2×C4⋊C4×C71C2×C414yes—
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C22⋊C4×C71—21no—
115C2×C42×C71—21no—
116#107×C71C2×C414yes—
117#107×C71C2×C414yes—
118#106×C71C2×C414yes—
119#106×C71C2×C414yes—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71C2×D411yescharacteristic
125C22×Dic71C2×D411yescharacteristic
126C22×Dic71C2×D411yescharacteristic
127C22×Dic71C2×D411yescharacteristic
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C4⋊Dic71—21no—
135C4⋊Dic71—21no—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—42no—
143C4⋊Dic71—42no—
144C4⋊Dic71—42no—
145C4⋊Dic71—42no—
146C22⋊Dic71—22no—
147C22⋊Dic71—41no—
148C22⋊Dic71—42no—
149C22⋊Dic71—81no—
150C4×Dic71—21no—
151C4×Dic71—21no—
152C4×Dic71—21no—
153C4×Dic71—21no—
154C4×Dic71—42no—
155C4×Dic71—42no—
156C4×Dic71—81no—
157C4×Dic71—81no—
158C24×C71C2×D411yescharacteristic
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—24no—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—42no—
169C22×C4×C71—42no—
170C22×C4×C71C22×C412yes—
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71#4112yes—
173C22×C4×C71C2411yescharacteristic
174M16×C71—28no—
175C2×C8×C71—21no—
176C2×C8×C71—21no—
177C2×C8×C71C22×C412yes—
178C2×C8×C71C22×C412yes—
179C2×C8×C71#4112yes—
180C2×C8×C71#4112yes—
181C4⋊C4×C71—24no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71C22×C412yes—
187C4⋊C4×C71C22×C412yes—
188C4⋊C4×C71C22×C412yes—
189C4⋊C4×C71C22×C412yes—
190C22⋊C4×C71—24no—
191C42×C71—24no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71C4×D412yes—
199C23×C71C22×D411yescharacteristic
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—48no—
211C2×C4×C71—41no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71C4×D412yes—
220C2×C4×C71C23×C411yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C2×C4×C71C2×#4111yescharacteristic
223C2×C4×C71C2×#4111yescharacteristic
224C2×C4×C712−1+411yescharacteristic
225C2×C4×C712−1+411yescharacteristic
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71AGL(1,ℤ/8)12yes—
250C8×C71—22no—
251C8×C71#11714yescomplemented
252C71⋊C8—41no—
253C71⋊C8—41no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71—41no—
261C22×C71C2×C4×D411yescharacteristic
262C22×C71#54711yescharacteristic
263C22×C71#54711yescharacteristic
264C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C71#57511yescharacteristic
266C22×C71#57711yescharacteristic
267C22×C71#57811yescharacteristic
268C4×C71—22no—
269C4×C71—42no—
270C4×C71—42no—
271C4×C71—42no—
272C4×C71—42no—
273C4×C71C2×#11711yescharacteristic
274C4×C71#51911yescharacteristic
275C4×C71#55211yescharacteristic
276C4×C71#55211yescharacteristic
277Dic71—22no—
278Dic71—41no—
279Dic71—41no—
280Dic71—41no—
281Dic71—81no—
282Dic71—81no—
283Dic71—81no—
284Dic71—81no—
285#22622—711nomaximal, Sylow
286C2×C71—42no—
287C2×C71#234411yescharacteristic
288C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C71#294711yescharacteristic
290C2×C71#334511yescharacteristic
291C2×C71#334911yescharacteristic
292C2×C71#335111yescharacteristic
293C2×C71#335311yescharacteristic
294C2×#428D7111yescharacteristic
295#2347—711no—
296#2344—711no—
297#1943—711no—
298#1942—711no—
299#1921—711no—
300#1920—711no—
301#1919—711no—
302#1918—711no—
303#1825—711no—
304#1824—711no—
305#1809—711no—
306#1776—711no—
307C4×#106—711no—
308#1753—711no—
309C71#2262211yescharacteristic, complemented, Sylow
310C22×M16C2×D7111yescharacteristic
311C2×#117—1421no—
312C2×#117C2×D7111yescharacteristic
313C22×C4⋊C4C2×D7111yescharacteristic
314#428Dic7118yescomplemented
315C2×#107—711no—
316C2×#107C2×D7111yescharacteristic
317C2×#107C2×D7111yescharacteristic
318C2×#106—711no—
319C2×#106C2×D7111yescharacteristic
320C2×#106C2×D7111yescharacteristic
321C2×C22⋊C8—711no—
322C2×#97—711no—
323C2×C4×C8—711no—
324#389—711no—
325#389—711no—
326#389—711no—
327#389—711no—
328#388—1421no—
329#388—1421no—
330#387—1421no—
331#387—1421no—
332#383—1421no—
333#381—1421no—
334#380—711no—
335#380—711no—
336C4×C4⋊C4—711no—
337C4×C4⋊C4—711no—
338C4×C4⋊C4—711no—
339C4×C4⋊C4—711no—
340C4×C4⋊C4—1421no—
341C4×C22⋊C4—711no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#336—711no—
353#336—711no—
354C23×C4—21no—
355C23×C4C22×D7111yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C2×Dic7114yes—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8C22×D7111yescharacteristic
362C22×C8C22×D7111yescharacteristic
363#117—22no—
364#117—2844no—
365#117C2×Dic7114yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—711no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4C2×Dic7114yes—
377C2×C4⋊C4C22×D7111yescharacteristic
378C2×C4⋊C4C22×D7111yescharacteristic
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—711no—
383C2×C22⋊C4—711no—
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C42—21no—
387C2×C42—711no—
388C2×C42—711no—
389C2×C42—1421no—
390C2×C42—1421no—
391C2×C42—1421no—
392C2×C42—1421no—
393#107—1422no—
394#107C2×Dic7114yes—
395#107C2×Dic7114yes—
396#106—1422no—
397#106C2×Dic7114yes—
398#106C2×Dic7114yes—
399C22⋊C8—1422no—
400#97—1422no—
401C4×C8—1422no—
402#95—711no—
403#95—711no—
404#95—1421no—
405#95—1421no—
406#95—1421no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95—1421no—
411#95—1421no—
412#95—1421no—
413#95—1421no—
414C24D4×D7111yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—711no—
427C22×C4—711no—
428C22×C4—711no—
429C22×C4—711no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4—1421no—
434C22×C4C22×Dic7112yes—
435C22×C4D4×D7111yescharacteristic
436C22×C4#1176586412yes—
437C22×C4C23×D7111yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—1421no—
442C2×C8—1421no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8C22×Dic7112yes—
448C2×C8C22×Dic7112yes—
449C2×C8#1176586512yes—
450C2×C8#1176586512yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—2842no—
457C4⋊C4—2842no—
458C4⋊C4—2842no—
459C4⋊C4—2842no—
460C4⋊C4—1421no—
461C4⋊C4—1421no—
462C4⋊C4—1421no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4C22×Dic7112yes—
469C4⋊C4C22×Dic7112yes—
470C4⋊C4C22×Dic7112yes—
471C4⋊C4C22×Dic7112yes—
472C22⋊C4—24no—
473C22⋊C4—2842no—
474C22⋊C4—2841no—
475C22⋊C4—5681no—
476C22⋊C4—1422no—
477C42—24no—
478C42—2842no—
479C42—2842no—
480C42—5681no—
481C42—5681no—
482C42—1421no—
483C42—1421no—
484C42—1421no—
485C42—1421no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23D4×Dic7112yes—
493C23C2×D4×D7111yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—2841no—
514C2×C4—2841no—
515C2×C4—2841no—
516C2×C4—2841no—
517C2×C4—2841no—
518C2×C4—2841no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—712no—
526C2×C4—1421no—
527C2×C4—1421no—
528C2×C4—1421no—
529C2×C4—1421no—
530C2×C4—1421no—
531C2×C4—1421no—
532C2×C4—1421no—
533C2×C4—1421no—
534C2×C4—1421no—
535C2×C4—1421no—
536C2×C4—1421no—
537C2×C4D4×Dic7112yes—
538C2×C4C23×Dic7111yescharacteristic
539C2×C4C2×#1176586411yescharacteristic
540C2×C4C2×#1176586511yescharacteristic
541C2×C4C2×#1176586511yescharacteristic
542C2×C4#1455617611yescharacteristic
543C2×C4#1455617611yescharacteristic
544C8—22no—
545C8—2841no—
546C8—2841no—
547C8#1455608714yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×D4×Dic7111yescharacteristic
556C22#1798665011yescharacteristic
557C22#1798665011yescharacteristic
558C22AGL(1,ℤ/8)×D7111yescharacteristic
559C22#1798679511yescharacteristic
560C22#1798680811yescharacteristic
561C22#1798681511yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—2841no—
568C4—2841no—
569C4—2841no—
570C4—5681no—
571C4—5681no—
572C4—5681no—
573C4—5681no—
574C4—1422no—
575C4C2×#1455608711yescharacteristic
576C4#1798647711yescharacteristic
577C4#1798668411yescharacteristic
578C4#1798668411yescharacteristic
579C2—42no—
580C2#2259120911yescharacteristic
581C2AGL(1,ℤ/8)×Dic7111yescharacteristic
582C2#2259718411yescharacteristic
583C2#2260081111yescharacteristic
584C2#2260087111yescharacteristic
585C2#2260089611yescharacteristic
586C2#2260092011yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#428×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenterC2×#428D71
hypocenterC71#22622
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #22589079C2semidirect
4. #22589074C2semidirect
5. #22588934C2semidirect
6. #22588927C2semidirect
7. #22588920C2semidirect
8. #22588913C2semidirect
12. #22588240C2semidirect
13. #106×Dic71C2semidirect
55. #428×C71C4semidirect
251. C8×C71#117semidirect
309. C71#22622semidirect
314. #428Dic71semidirect
547. C8#14556087semidirect

Maximal subgroups

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

subgroupindex
#225912332
#225912092
#225890792
#225890742
#225889342
#225889272
#225889202
#225889132
#225884352
#225884302
#225883732
#225882402
#106×Dic712
#225881212
C2×#428×C712
#2262271