Finite Group Databasebeta

← #32059979

Subgroups — #32059979

Order 18176.

882 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986281C211yesmaximal, characteristic, complemented
2C22×#14556059C211yesmaximal, characteristic
3#22591343C211yesmaximal, characteristic
4#22589884C211yesmaximal, characteristic, complemented
5#22589841C211yesmaximal, characteristic, complemented
6#22589718C211yesmaximal, characteristic, complemented
7#22589670C211yesmaximal, characteristic, complemented
8#22589101C211yesmaximal, characteristic, complemented
9#22588915C211yesmaximal, characteristic, complemented
10#22588761C211yesmaximal, characteristic
11#22588647C211yesmaximal, characteristic
12#22588607C211yesmaximal, characteristic, complemented
13#22588532C211yesmaximal, characteristic
14#22588520C211yesmaximal, characteristic
15C71×#2440C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556107C2211yescharacteristic
18C2×C4×Dic142C2211yescharacteristic
19C22×C4⋊Dic71C2211yescharacteristic
20#17986281—24no—
21C2×#14556059—24no—
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27C2×#14556059C2211yescharacteristic
28C2×C4⋊(C71⋊C8)C2211yescharacteristic
29C2×C4⋊(C71⋊C8)C2211yescharacteristic
30C2×#14556051C2211yescharacteristic
31C2×#14556051C2211yescharacteristic
32C2×#14556051C2211yescharacteristic
33C2×#14556051C2211yescharacteristic
34C2×C22⋊(C71⋊C8)C2211yescharacteristic
35#17985905C2211yescharacteristic
36#17985897C2211yescharacteristic
37#17985896—21no—
38#17985886—21no—
39#17985871—21no—
40C4×C4⋊Dic71C2211yescharacteristic
41C22⋊C4×Dic71—21no—
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985754C2211yescharacteristic
49#17985754C2211yescharacteristic
50C2×C4⋊Q8×C71C2211yescharacteristic
51C2×C22⋊Q8×C71—21no—
52C22×C4⋊C4×C71C2211yescharacteristic
53#398×C71—21no—
54#395×C71—21no—
55#389×C71C2211yescharacteristic
56#389×C71C2211yescharacteristic
57#385×C71—21no—
58#384×C71C2211yescharacteristic
59#380×C71C2211yescharacteristic
60C4×C22⋊C4×C71C2211yescharacteristic
61C22×Dic142C2311yescharacteristic
62C23×Dic71—21no—
63C22×C71⋊C8—22no—
64C22×C71⋊C8—22no—
65C22×C71⋊C8—22no—
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70#14556107—22no—
71#14556107Q814yescomplemented
72C4×Dic142—22no—
73C4×Dic142—41no—
74C2×C4⋊Dic71—22no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71—41no—
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×Dic71⋊C4—21no—
81C2×Dic71⋊C4—21no—
82C2×Dic71⋊C4—21no—
83C2×C22⋊Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86#14556059—22no—
87#14556059—22no—
88#14556059—22no—
89#14556059—22no—
90#14556059—41no—
91#14556059D414yes—
92C4⋊(C71⋊C8)—22no—
93C4⋊(C71⋊C8)—22no—
94#14556051—22no—
95#14556051—22no—
96#14556051—22no—
97#14556051—41no—
98C22⋊(C71⋊C8)—22no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C71⋊#95—21no—
102C71⋊#95—21no—
103C22×Q8×C71—21no—
104C23×C4×C71—21no—
105C23×C4×C71C2311yescharacteristic
106C4⋊Q8×C71—22no—
107C4⋊Q8×C71—41no—
108C22⋊Q8×C71—44no—
109C2×C4⋊C4×C71—22no—
110C2×C4⋊C4×C71—21no—
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—21no—
113C2×C4⋊C4×C71—21no—
114C2×C4⋊C4×C71—21no—
115C2×C4⋊C4×C71—21no—
116C2×C4⋊C4×C71—41no—
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C4⋊C4×C71C2311yescharacteristic
120C2×C4⋊C4×C71C2311yescharacteristic
121C2×C22⋊C4×C71—21no—
122C2×C22⋊C4×C71D411yescharacteristic
123C2×C22⋊C4×C71D411yescharacteristic
124C2×C42×C71C2311yescharacteristic
125C2×C42×C71C2311yescharacteristic
126#95×C71—21no—
127#95×C71—21no—
128#95×C71—21no—
129#95×C71D411yescharacteristic
130#95×C71D411yescharacteristic
131C2×Dic142—21no—
132C2×Dic142—21no—
133C2×Dic142—21no—
134C2×Dic142—21no—
135C2×Dic142—21no—
136C2×Dic142C2×Q812yes—
137C22×Dic71—21no—
138C22×Dic71—21no—
139C22×Dic71—21no—
140C22×Dic71—21no—
141C22×Dic71—42no—
142C22×Dic71—42no—
143C22×Dic71—81no—
144C2×C71⋊C8—28no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8C2×D412yes—
156C2×C71⋊C8#4112yes—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—42no—
162C4⋊Dic71C2×D412yes—
163C4⋊Dic71C2×Q812yes—
164Dic71⋊C4—24no—
165Dic71⋊C4—42no—
166Dic71⋊C4—81no—
167C22⋊Dic71—24no—
168C4×Dic71—42no—
169C4×Dic71—81no—
170C24×C71C2×D411yescharacteristic
171C2×Q8×C71—42no—
172C2×Q8×C71—42no—
173C2×Q8×C71—81no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71—21no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—21no—
181C22×C4×C71—21no—
182C22×C4×C71—42no—
183C22×C4×C71—42no—
184C22×C4×C71—81no—
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×Q812yes—
191C22×C4×C71#4112yes—
192C22×C4×C71C2411yescharacteristic
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71—42no—
201C4⋊C4×C71—42no—
202C4⋊C4×C71—42no—
203C4⋊C4×C71—42no—
204C4⋊C4×C71—42no—
205C4⋊C4×C71—81no—
206C4⋊C4×C71C2×D412yes—
207C22⋊C4×C71—22no—
208C22⋊C4×C71—42no—
209C22⋊C4×C71SD1614yes—
210C42×C71—21no—
211C42×C71—21no—
212C42×C71—21no—
213C42×C71—21no—
214C23×C71—21no—
215C23×C71—21no—
216C23×C71—21no—
217C23×C71—21no—
218C23×C71C22⋊Q812yes—
219C23×C71C2×SD1612yes—
220C23×C71#13712yes—
221C23×C71C22×D411yescharacteristic
222Q8×C71—84no—
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—22no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—21no—
234C2×C4×C71—22no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
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—84no—
252C2×C4×C71C22⋊Q812yes—
253C2×C4×C71C2×SD1612yes—
254C2×C4×C71C2×SD1612yes—
255C2×C4×C71C22×D411yescharacteristic
256C2×C4×C71C22×Q811yescharacteristic
257C2×C4×C71C2×#4111yescharacteristic
258C2×C4×C71C2×#4111yescharacteristic
259C2×C4×C71H(5,𝔽2)11yescharacteristic
260C2×C4×C712−1+411yescharacteristic
261C2×Dic71—22no—
262C2×Dic71—22no—
263C2×Dic71—22no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—41no—
268C2×Dic71—41no—
269C2×Dic71—41no—
270C2×Dic71—41no—
271C2×Dic71—41no—
272C2×Dic71—41no—
273C2×Dic71—84no—
274Dic142—22no—
275Dic142—22no—
276Dic142—41no—
277Dic142—41no—
278C71⋊C8—22no—
279C71⋊C8—22no—
280C71⋊C8—41no—
281C71⋊C8—41no—
282C22×C71—22no—
283C22×C71—22no—
284C22×C71—22no—
285C22×C71—22no—
286C22×C71—22no—
287C22×C71—22no—
288C22×C71#47514yescomplemented
289C22×C71C2×C22⋊Q811yescharacteristic
290C22×C71D4211yescharacteristic
291C22×C71#54711yescharacteristic
292C22×C71C22×SD1611yescharacteristic
293C22×C71C2×#13711yescharacteristic
294C22×C71#57711yescharacteristic
295C22×C71#57711yescharacteristic
296C4×C71—22no—
297C4×C71—22no—
298C4×C71—22no—
299C4×C71—41no—
300C4×C71—41no—
301C4×C71—42no—
302C4×C71—81no—
303C4×C71—81no—
304C4×C71—81no—
305C4×C71#52511yescharacteristic
306C4×C71#53911yescharacteristic
307C4×C71D4×Q811yescharacteristic
308C4×C71#55411yescharacteristic
309Dic71—42no—
310Dic71—42no—
311Dic71—42no—
312Dic71—81no—
313#26340—711nomaximal, Sylow
314C2×C71—24no—
315C2×C71#244911yescharacteristic
316C2×C71C2×#47511yescharacteristic
317C2×C71#308411yescharacteristic
318C2×C71#326611yescharacteristic
319C2×C71D4×SD1611yescharacteristic
320C2×C71#333511yescharacteristic
321C2×C71#334411yescharacteristic
322C2×#475—711no—
323C22×#106—711no—
324#2440D7111yescharacteristic
325#2360—711no—
326#2096—711no—
327#2088—711no—
328#2066—711no—
329#2058—711no—
330#1948—711no—
331#1919—711no—
332#1892—711no—
333#1872—711no—
334#1865—711no—
335#1853—711no—
336#1850—711no—
337C71#2634011yescharacteristic, complemented, Sylow
338C23×C8—711no—
339C2×C4⋊Q8C2×D7111yescharacteristic
340C2×C22⋊Q8—21no—
341C2×C22⋊Q8—711no—
342C2×C4×Q8—711no—
343C22×C4⋊C4—711no—
344C22×C4⋊C4C2×D7111yescharacteristic
345#475—1424no—
346C2×#106—711no—
347C2×#106—711no—
348C2×#106—711no—
349C2×#106—711no—
350C2×#106—711no—
351C2×#106—711no—
352C2×#106—1424no—
353C2×C4⋊C8—711no—
354C2×C4⋊C8—711no—
355C2×#103—711no—
356C2×#103—711no—
357C2×#103—711no—
358C2×#103—711no—
359C2×C22⋊C8—711no—
360#398—21no—
361#395—21no—
362#391—711no—
363#389—711no—
364#389—1421no—
365#389C2×D7111yescharacteristic
366#389C2×D7111yescharacteristic
367#387—1421no—
368#385—21no—
369#385—1421no—
370#384C2×D7111yescharacteristic
371#380C2×D7111yescharacteristic
372C4×C4⋊C4—711no—
373C4×C22⋊C4—1421no—
374C4×C22⋊C4C2×D7111yescharacteristic
375#340—711no—
376#340—711no—
377#340—711no—
378#340—711no—
379#340—711no—
380#340—711no—
381#336—711no—
382#336—711no—
383C22×Q8—21no—
384C22×Q8—711no—
385C23×C4—21no—
386C23×C4—1421no—
387C23×C4C22×D7111yescharacteristic
388C22×C8—711no—
389C22×C8—711no—
390C22×C8—711no—
391C22×C8—711no—
392C22×C8—1422no—
393C22×C8—1422no—
394C22×C8—1422no—
395C4⋊Q8—22no—
396C4⋊Q8—41no—
397C22⋊Q8—44no—
398C22⋊Q8—714no—
399C22⋊Q8—1422no—
400C4×Q8—2841no—
401C4×Q8—1422no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—2841no—
411C2×C4⋊C4—711no—
412C2×C4⋊C4—711no—
413C2×C4⋊C4—711no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1422no—
417C2×C4⋊C4—1421no—
418C2×C4⋊C4—1421no—
419C2×C4⋊C4C22×D7111yescharacteristic
420C2×C4⋊C4C22×D7111yescharacteristic
421C2×C4⋊C4C22×D7111yescharacteristic
422C2×C4⋊C4C22×D7111yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1421no—
425C2×C22⋊C4C22⋊D7111yescharacteristic
426C2×C22⋊C4C22⋊D7111yescharacteristic
427C2×C42—1421no—
428C2×C42—1421no—
429C2×C42C22×D7111yescharacteristic
430C2×C42C22×D7111yescharacteristic
431#106—2841no—
432#106—714no—
433#106—1422no—
434#106—1422no—
435#106—1422no—
436#106—1422no—
437C4⋊C8—1422no—
438C4⋊C8—1422no—
439#103—2841no—
440#103—1422no—
441#103—1422no—
442#103—1422no—
443C22⋊C8—1422no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1421no—
448#95—1421no—
449#95—1421no—
450#95—1421no—
451#95C22⋊D7111yescharacteristic
452#95C22⋊D7111yescharacteristic
453C24C2×C22⋊D7111yescharacteristic
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8—712no—
458C2×Q8—1421no—
459C2×Q8—1421no—
460C2×Q8—1421no—
461C2×Q8—1421no—
462C2×Q8—1421no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—2842no—
475C22×C4—2842no—
476C22×C4—5681no—
477C22×C4—1421no—
478C22×C4—1421no—
479C22×C4—1421no—
480C22×C4—1421no—
481C22×C4C2×C22⋊D7111yescharacteristic
482C22×C4C2×C22⋊D7111yescharacteristic
483C22×C4C2×C22⋊D7111yescharacteristic
484C22×C4C2×C22⋊D7111yescharacteristic
485C22×C4C2×C22⋊D7111yescharacteristic
486C22×C4Q8×D7112yes—
487C22×C4#1176586612yes—
488C22×C4C23×D7111yescharacteristic
489C2×C8—712no—
490C2×C8—712no—
491C2×C8—1421no—
492C2×C8—1421no—
493C2×C8—1421no—
494C2×C8—1421no—
495C2×C8—1421no—
496C2×C8—1421no—
497C2×C8—1428no—
498C2×C8—1421no—
499C2×C8—1421no—
500C2×C8—1421no—
501C2×C8—1421no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—2842no—
516C4⋊C4—2842no—
517C4⋊C4—5681no—
518C4⋊C4—712no—
519C4⋊C4—712no—
520C4⋊C4—1421no—
521C4⋊C4—1421no—
522C4⋊C4—1421no—
523C4⋊C4—1421no—
524C4⋊C4—1424no—
525C4⋊C4D4×D7112yes—
526C22⋊C4—22no—
527C22⋊C4—42no—
528C22⋊C4—1424no—
529C22⋊C4#1176585314yes—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—2842no—
535C42—5681no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1455610412yes—
541C23C2×#1176585312yes—
542C23#1455615912yes—
543C23C22×C22⋊D7111yescharacteristic
544Q8—84no—
545Q8—2841no—
546Q8—2841no—
547Q8—1422no—
548Q8—1422no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
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—84no—
578C2×C4—2841no—
579C2×C4—2841no—
580C2×C4—2841no—
581C2×C4—2841no—
582C2×C4—2841no—
583C2×C4—2841no—
584C2×C4—2841no—
585C2×C4—2841no—
586C2×C4—2841no—
587C2×C4—5684no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4—1422no—
591C2×C4#1455610412yes—
592C2×C4C2×#1176585312yes—
593C2×C4C2×#1176585312yes—
594C2×C4C2×D4×D7111yescharacteristic
595C2×C4C2×Q8×D7111yescharacteristic
596C2×C4#41×D7111yescharacteristic
597C2×C4C2×#1176586611yescharacteristic
598C2×C4#1455617311yescharacteristic
599C2×C4#1455617511yescharacteristic
600C8—2841no—
601C8—2841no—
602C8—1422no—
603C8—1422no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#1798628114yescomplemented
611C22C2×#1455610411yescharacteristic
612C22D4×C22⋊D7111yescharacteristic
613C22#1798664811yescharacteristic
614C22C22×#1176585311yescharacteristic
615C22C2×#1455615911yescharacteristic
616C22#1798680711yescharacteristic
617C22#1798681011yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—2842no—
628C4—2842no—
629C4—2842no—
630C4—5681no—
631C4#1798651211yescharacteristic
632C4#1798659811yescharacteristic
633C4#1798666211yescharacteristic
634C4#1798670311yescharacteristic
635C2—24no—
636C2#2259214511yescharacteristic
637C2C2×#1798628111yescharacteristic
638C2#2259819811yescharacteristic
639C2#2259985111yescharacteristic
640C2D4×#1176585311yescharacteristic
641C2#2260069711yescharacteristic
642C2#2260078911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2440C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2440D71
hypocenterC71#26340
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#17986281C2semidirect
4. #22589884C2semidirect
5. #22589841C2semidirect
6. #22589718C2semidirect
7. #22589670C2semidirect
8. #22589101C2semidirect
9. #22588915C2semidirect
12. #22588607C2semidirect
71. #14556107Q8semidirect
288. C22×C71#475semidirect
337. C71#26340semidirect
610. C22#17986281semidirect

Maximal subgroups

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

subgroupindex
C2×#179862812
C22×#145560592
#225913432
#225898842
#225898412
#225897182
#225896702
#225891012
#225889152
#225887612
#225886472
#225886072
#225885322
#225885202
C71×#24402
#2634071