Finite Group Databasebeta

← #32000852

Subgroups — #32000852

Order 18176.

982 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986040C211yesmaximal, characteristic
2C2×#17986040C211yesmaximal, characteristic
3C2×#17985953C211yesmaximal, characteristic
4#22590962C211yesmaximal, characteristic
5#22590955C211yesmaximal, characteristic
6#22589081C211yesmaximal, characteristic, complemented
7#22589080C211yesmaximal, characteristic, complemented
8#22589079C211yesmaximal, characteristic, complemented
9#22589075C211yesmaximal, characteristic, complemented
10#22589072C211yesmaximal, characteristic, complemented
11#22589068C211yesmaximal, characteristic, complemented
12#22589067C211yesmaximal, characteristic, complemented
13#22588055C211yesmaximal, characteristic
14#340×D71C211yesmaximal, characteristic
15C71×#1942C211yesmaximal, characteristic, complemented
16C22×#11765848C2211yescharacteristic
17C22×C8×D71C2211yescharacteristic
18C2×#14556086C2211yescharacteristic
19C2×#14556086C2211yescharacteristic
20C2×#14556085C2211yescharacteristic
21C2×C4⋊C4×D71C2211yescharacteristic
22C2×C4⋊C4×D71C2211yescharacteristic
23#17986040—24no—
24#17986040C418yescomplemented
25C2×#14556061C2211yescharacteristic
26C2×#14556061C2211yescharacteristic
27C2×#14556060C2211yescharacteristic
28C2×#14556060C2211yescharacteristic
29C2×#14556059C2211yescharacteristic
30C2×#14556059C2211yescharacteristic
31#17985953—24no—
32C2×#14556039C2211yescharacteristic
33C2×#14556038C2211yescharacteristic
34C2×C8×Dic71C2211yescharacteristic
35#17985898C2211yescharacteristic
36#17985898C2211yescharacteristic
37#17985897C2211yescharacteristic
38#17985897C2211yescharacteristic
39#17985896—21no—
40#17985895C2211yescharacteristic
41#17985895C2211yescharacteristic
42#17985894—21no—
43#17985873—21no—
44#17985870—21no—
45#17985868—21no—
46#17985863—21no—
47#17985858—21no—
48#17985856—21no—
49#17985762C2211yescharacteristic
50#17985762C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53#17985761C2211yescharacteristic
54#17985761C2211yescharacteristic
55C2×#107×C71C2211yescharacteristic
56C2×#107×C71C2211yescharacteristic
57C2×C4×C8×C71C2211yescharacteristic
58#389×C71C2211yescharacteristic
59#389×C71C2211yescharacteristic
60#340×C71C2211yescharacteristic
61#340×C71C2211yescharacteristic
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C22×C4×D71C2311yescharacteristic
65C2×#11765848C2×C414yes—
66C2×#11765848D414yes—
67C2×#11765848Q814yes—
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C2×C8×D71—22no—
71C2×C8×D71—22no—
72C2×C8×D71Q814yes—
73#14556086—22no—
74#14556086—22no—
75#14556086—22no—
76#14556086C2×C414yes—
77#14556085—22no—
78#14556085—22no—
79C2×C4⋊Dic71—21no—
80C2×C4⋊Dic71—21no—
81C2×C4⋊Dic71C2311yescharacteristic
82C2×C4⋊Dic71C2311yescharacteristic
83C2×C4⋊Dic71C2311yescharacteristic
84C2×C4⋊Dic71C2311yescharacteristic
85C2×Dic71⋊C4—21no—
86C2×Dic71⋊C4—21no—
87C2×Dic71⋊C4—21no—
88C2×Dic71⋊C4—21no—
89C2×Dic71⋊C4—21no—
90C2×Dic71⋊C4—21no—
91C2×Dic71⋊C4D411yescharacteristic
92C2×Dic71⋊C4D411yescharacteristic
93C4⋊C4×D71—22no—
94C4⋊C4×D71—22no—
95C4⋊C4×D71—22no—
96C4⋊C4×D71C2×C414yes—
97C2×(C2×D71)⋊C4—21no—
98C2×(C2×D71)⋊C4—21no—
99C2×(C2×D71)⋊C4D411yescharacteristic
100C2×(C2×D71)⋊C4D411yescharacteristic
101C2×C4×Dic71—21no—
102C2×C4×Dic71—21no—
103C2×C4×Dic71C2311yescharacteristic
104#14556061—22no—
105#14556061C2×C414yescomplemented
106#14556060—22no—
107#14556060C2×C414yescomplemented
108#14556059—22no—
109#14556059C2×C414yescomplemented
110#14556039—22no—
111#14556038—22no—
112C8×Dic71—22no—
113C71⋊#95—21no—
114C71⋊#95—21no—
115C71⋊#95—21no—
116C71⋊#95—21no—
117C71⋊#95—21no—
118C71⋊#95—21no—
119C22×C8×C71C2311yescharacteristic
120C22×C8×C71C2311yescharacteristic
121C2×C4⋊C4×C71—21no—
122C2×C4⋊C4×C71—21no—
123C2×C4⋊C4×C71C2311yescharacteristic
124C2×C4⋊C4×C71C2311yescharacteristic
125C2×C4⋊C4×C71C2311yescharacteristic
126C2×C4⋊C4×C71C2311yescharacteristic
127C2×C42×C71C2311yescharacteristic
128#107×C71—22no—
129#107×C71C2×C414yescomplemented
130C4×C8×C71—22no—
131#95×C71—21no—
132#95×C71—21no—
133C23×D71C2×D411yescharacteristic
134C22×Dic71—21no—
135C22×Dic71—21no—
136C22×Dic71—21no—
137C22×Dic71—21no—
138C22×Dic71C2×D411yescharacteristic
139C22×Dic71C2×D411yescharacteristic
140C22×Dic71C2×D411yescharacteristic
141C2×C4×D71—24no—
142C2×C4×D71—42no—
143C2×C4×D71—81no—
144C2×C4×D71—81no—
145C2×C4×D71—81no—
146C2×C4×D71—81no—
147C2×C4×D71C22×C412yes—
148C2×C4×D71C2×D412yes—
149C2×C4×D71C2×Q812yes—
150C2×C4×D71C2×Q812yes—
151C2×C4×D71#4112yes—
152C2×C4×D71#4112yes—
153#11765848C4⋊C4116yescomplemented
154C2×C71⋊C8—21no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8C22×C412yes—
157C2×C71⋊C8C2×D412yes—
158C2×C71⋊C8C2×Q812yes—
159C2×C71⋊C8C2×Q812yes—
160C8×D71—28no—
161C4⋊Dic71—21no—
162C4⋊Dic71—21no—
163C4⋊Dic71—21no—
164C4⋊Dic71—21no—
165C4⋊Dic71—21no—
166C4⋊Dic71—21no—
167C4⋊Dic71—81no—
168C4⋊Dic71—81no—
169C4⋊Dic71C22×C412yes—
170C4⋊Dic71C22×C412yes—
171Dic71⋊C4—24no—
172Dic71⋊C4—22no—
173Dic71⋊C4—22no—
174Dic71⋊C4—42no—
175Dic71⋊C4—81no—
176Dic71⋊C4—81no—
177Dic71⋊C4—81no—
178Dic71⋊C4—81no—
179(C2×D71)⋊C4—24no—
180(C2×D71)⋊C4—22no—
181(C2×D71)⋊C4—22no—
182(C2×D71)⋊C4—42no—
183C4×Dic71—21no—
184C4×Dic71—21no—
185C4×Dic71—24no—
186C4×Dic71—42no—
187C22×C4×C71—21no—
188C22×C4×C71—21no—
189C22×C4×C71—21no—
190C22×C4×C71—21no—
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2411yescharacteristic
194C2×C8×C71—21no—
195C2×C8×C71—21no—
196C2×C8×C71C22×C412yes—
197C2×C8×C71C2×D412yes—
198C2×C8×C71C2×Q812yes—
199C2×C8×C71C2×Q812yes—
200C4⋊C4×C71—21no—
201C4⋊C4×C71—21no—
202C4⋊C4×C71—21no—
203C4⋊C4×C71—21no—
204C4⋊C4×C71—21no—
205C4⋊C4×C71—21no—
206C4⋊C4×C71—81no—
207C4⋊C4×C71—81no—
208C4⋊C4×C71C22×C412yes—
209C4⋊C4×C71C22×C412yes—
210C42×C71—21no—
211C42×C71—21no—
212C23×C71C22×D411yescharacteristic
213C22×D71—21no—
214C22×D71—21no—
215C22×D71—21no—
216C22×D71—21no—
217C22×D71C4×D412yes—
218C22×D71#12112yes—
219C22×D71C4⋊Q812yes—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
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—41no—
229C2×C4×C71—41no—
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×C71C23×C411yescharacteristic
239C2×C4×C71C22×D411yescharacteristic
240C2×C4×C71C22×Q811yescharacteristic
241C2×C4×C71C22×Q811yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—21no—
247C2×Dic71—21no—
248C2×Dic71—21no—
249C2×Dic71—21no—
250C2×Dic71—21no—
251C2×Dic71—21no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71C4×D412yes—
263C2×Dic71#12112yes—
264C2×Dic71C4⋊Q812yes—
265C4×D71—84no—
266C4×D71—84no—
267C4×D71C2×C4⋊C414yes—
268C4×D71C4×Q814yes—
269C4×D71C22⋊Q814yes—
270C4×D71#12514yes—
271C8×C71—22no—
272C8×C71C2×C4⋊C414yes—
273C71⋊C8—22no—
274C71⋊C8C2×C4⋊C414yes—
275C22×C71C2×C4×D411yescharacteristic
276C22×C71C2×#12111yescharacteristic
277C22×C71C2×C4⋊Q811yescharacteristic
278C22×C71#57611yescharacteristic
279C22×C71#57711yescharacteristic
280C22×C71#57711yescharacteristic
281C22×C71#57811yescharacteristic
282C2×D71—22no—
283C2×D71—22no—
284C2×D71—22no—
285C2×D71—22no—
286C2×D71—22no—
287C2×D71—22no—
288C2×D71#38914yes—
289C4×C71—42no—
290C4×C71—42no—
291C4×C71—41no—
292C4×C71—41no—
293C4×C71—81no—
294C4×C71—81no—
295C4×C71C22×C4⋊C411yescharacteristic
296C4×C71C2×C4×Q811yescharacteristic
297C4×C71C2×C22⋊Q811yescharacteristic
298C4×C71C2×#12511yescharacteristic
299Dic71—42no—
300Dic71—42no—
301Dic71—41no—
302Dic71—41no—
303Dic71—81no—
304Dic71—81no—
305Dic71#38914yes—
306#21160—711nomaximal, Sylow
307C2×C71C2×#38911yescharacteristic
308C2×C71#294611yescharacteristic
309C2×C71#294711yescharacteristic
310C2×C71#306211yescharacteristic
311C2×C71#306311yescharacteristic
312C2×C71#316311yescharacteristic
313C2×C71#316411yescharacteristic
314D71—24no—
315C2×#428—711no—
316C2×#428—711no—
317C2×#405—711no—
318#2317—711no—
319#2316—711no—
320#1944—711no—
321#1944—711no—
322#1943—711no—
323#1943—711no—
324#1942—711no—
325#1942D7111yescharacteristic, complemented
326#1941—711no—
327#1941—711no—
328#1735—711no—
329C2×#340—711no—
330C71#2116011yescharacteristic, complemented, Sylow
331C22×M16—711no—
332C23×C8—711no—
333C2×#117—711no—
334C2×#117—711no—
335C2×#117—711no—
336C22×C4⋊C4—711no—
337C22×C4⋊C4—711no—
338#428—718no—
339#428—1424no—
340C2×#107—711no—
341C2×#107—711no—
342C2×#107C2×D7111yescharacteristic
343C2×#107C2×D7111yescharacteristic
344C2×#106—711no—
345C2×#106—711no—
346C2×#106—711no—
347C2×#106—711no—
348#405—1424no—
349C2×#97—711no—
350C2×#97—711no—
351C2×C4×C8—711no—
352C2×C4×C8C2×D7111yescharacteristic
353#389—711no—
354#389—711no—
355#389—711no—
356#389—711no—
357#389—711no—
358#389—711no—
359#389—1421no—
360#389—1421no—
361#389C2×D7111yescharacteristic
362#389C2×D7111yescharacteristic
363#385—1421no—
364#385—1421no—
365#384—1421no—
366#382—1421no—
367#380—1421no—
368#380—1421no—
369#340—711no—
370#340—711no—
371#340—711no—
372#340—711no—
373#340—711no—
374#340—711no—
375#340C2×D7111yescharacteristic
376#340C2×D7111yescharacteristic
377C23×C4—711no—
378C23×C4—1421no—
379C23×C4—1421no—
380C2×M16—714no—
381C2×M16—714no—
382C2×M16—714no—
383C22×C8—711no—
384C22×C8—711no—
385C22×C8—714no—
386C22×C8—1422no—
387C22×C8—1422no—
388C22×C8C22×D7111yescharacteristic
389C22×C8C22×D7111yescharacteristic
390#117—714no—
391#117—1422no—
392#117—1422no—
393#117—1422no—
394#117—1422no—
395#117—1422no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4—711no—
400C2×C4⋊C4—714no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—711no—
403C2×C4⋊C4—711no—
404C2×C4⋊C4—711no—
405C2×C4⋊C4—1422no—
406C2×C4⋊C4—1422no—
407C2×C4⋊C4—1422no—
408C2×C4⋊C4—1421no—
409C2×C4⋊C4—1421no—
410C2×C4⋊C4—1421no—
411C2×C4⋊C4—1421no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1421no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4C22×D7111yescharacteristic
417C2×C4⋊C4C22×D7111yescharacteristic
418C2×C4⋊C4C22×D7111yescharacteristic
419C2×C4⋊C4C22×D7111yescharacteristic
420C2×C22⋊C4—711no—
421C2×C22⋊C4—711no—
422C2×C22⋊C4—1421no—
423C2×C22⋊C4—1421no—
424C2×C42—711no—
425C2×C42—1421no—
426C2×C42—1421no—
427C2×C42C22×D7111yescharacteristic
428#107—22no—
429#107—714no—
430#107—1422no—
431#107C4×D7114yescomplemented
432#106—714no—
433#106—714no—
434#106—1422no—
435#106—1422no—
436#97—1422no—
437#97—1422no—
438C4×C8—22no—
439C4×C8—1422no—
440#95—21no—
441#95—21no—
442#95—1421no—
443#95—1421no—
444#95—1421no—
445#95—1421no—
446#95—1421no—
447#95—1421no—
448C24—711no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—2842no—
454C22×C4—5681no—
455C22×C4—5681no—
456C22×C4—5681no—
457C22×C4—5681no—
458C22×C4—711no—
459C22×C4—712no—
460C22×C4—712no—
461C22×C4—712no—
462C22×C4—712no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—711no—
466C22×C4—711no—
467C22×C4—1424no—
468C22×C4—1421no—
469C22×C4—1421no—
470C22×C4—1421no—
471C22×C4—1421no—
472C22×C4D4×D7111yescharacteristic
473C22×C4D4×D7111yescharacteristic
474C22×C4C23×D7111yescharacteristic
475M16—7116no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—712no—
479C2×C8—712no—
480C2×C8—712no—
481C2×C8—712no—
482C2×C8—1428no—
483C2×C8—1421no—
484C2×C8—1421no—
485C2×C8C2×C4×D7112yes—
486C2×C8D4×D7112yes—
487C2×C8Q8×D7112yes—
488C2×C8Q8×D7112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—2842no—
498C4⋊C4—5681no—
499C4⋊C4—5681no—
500C4⋊C4—5681no—
501C4⋊C4—5681no—
502C4⋊C4—5681no—
503C4⋊C4—5681no—
504C4⋊C4—712no—
505C4⋊C4—712no—
506C4⋊C4—1424no—
507C4⋊C4—1422no—
508C4⋊C4—1422no—
509C4⋊C4—1421no—
510C4⋊C4—1421no—
511C4⋊C4—1421no—
512C4⋊C4—1421no—
513C4⋊C4—1421no—
514C4⋊C4—1421no—
515C4⋊C4C2×C4×D7112yes—
516C4⋊C4C2×C4×D7112yes—
517C22⋊C4—2842no—
518C22⋊C4—1422no—
519C22⋊C4—1422no—
520C22⋊C4—1424no—
521C42—21no—
522C42—21no—
523C42—2842no—
524C42—1424no—
525C42—1421no—
526C42—1421no—
527C23—712no—
528C23—712no—
529C23—712no—
530C23—1421no—
531C23—1421no—
532C23—1421no—
533C23—1421no—
534C23C2×D4×D7111yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—5684no—
564C2×C4—5684no—
565C2×C4—712no—
566C2×C4—712no—
567C2×C4—712no—
568C2×C4—714no—
569C2×C4—714no—
570C2×C4—714no—
571C2×C4—714no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1421no—
575C2×C4—1421no—
576C2×C4—1421no—
577C2×C4—1421no—
578C2×C4—1421no—
579C2×C4—1421no—
580C2×C4C22×C4×D7111yescharacteristic
581C2×C4C2×D4×D7111yescharacteristic
582C2×C4C2×Q8×D7111yescharacteristic
583C2×C4C2×Q8×D7111yescharacteristic
584C2×C4#41×D7111yescharacteristic
585C2×C4#41×D7111yescharacteristic
586C8—22no—
587C8—714no—
588C8—1422no—
589C8C4⋊C4×D7114yes—
590C22—714no—
591C22—1422no—
592C22—1422no—
593C22—1422no—
594C22—1422no—
595C22—1422no—
596C22—1422no—
597C22C4×D4×D7111yescharacteristic
598C22#121×D7111yescharacteristic
599C22C4⋊Q8×D7111yescharacteristic
600C22#1798680011yescharacteristic
601C22#1798680611yescharacteristic
602C22#1798680611yescharacteristic
603C22#1798681211yescharacteristic
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—2842no—
611C4—2842no—
612C4—2841no—
613C4—2841no—
614C4—5681no—
615C4—5681no—
616C4—714no—
617C4C2×C4⋊C4×D7111yescharacteristic
618C4C4×Q8×D7111yescharacteristic
619C4C22⋊Q8×D7111yescharacteristic
620C4#125×D7111yescharacteristic
621C2—1424no—
622C2#389×D7111yescharacteristic
623C2#2259716511yescharacteristic
624C2#2259717811yescharacteristic
625C2#2259801911yescharacteristic
626C2#2259803011yescharacteristic
627C2#2259886911yescharacteristic
628C2#2259887311yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1942C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1942D71
hypocenterC71#21160
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
6. #22589081C2semidirect
7. #22589080C2semidirect
8. #22589079C2semidirect
9. #22589075C2semidirect
10. #22589072C2semidirect
11. #22589068C2semidirect
12. #22589067C2semidirect
15. C71×#1942C2semidirect
24. #17986040C4semidirect
105. #14556061C2×C4semidirect
107. #14556060C2×C4semidirect
109. #14556059C2×C4semidirect
129. #107×C71C2×C4semidirect
153. #11765848C4⋊C4semidirect
325. #1942D71semidirect
330. C71#21160semidirect
431. #107C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860402
C2×#179859532
#225909622
#225909552
#225890812
#225890802
#225890792
#225890752
#225890722
#225890682
#225890672
#225880552
#340×D712
C71×#19422
#2116071