Finite Group Databasebeta

← #32016794

Subgroups — #32016794

Order 18176.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986097C211yesmaximal, characteristic
2#22591311C211yesmaximal, characteristic
3#22589036C211yesmaximal, characteristic
4#22589022C211yesmaximal, characteristic
5#22588929C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588695C211yesmaximal, characteristic
8#22588695C211yesmaximal, characteristic
9#22588475C211yesmaximal, characteristic, complemented
10#22588355C211yesmaximal, characteristic, complemented
11#22588337C211yesmaximal, characteristic
12#22588215C211yesmaximal, characteristic, complemented
13#22588170C211yesmaximal, characteristic, complemented
14C4×#14556052C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C71C211yesmaximal, characteristic, complemented
16C22×#11765852C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18#17986097C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×#14556052C2211yescharacteristic
22C2×#14556052C2211yescharacteristic
23C2×#14556052C2211yescharacteristic
24C2×#14556048C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556039C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×C4×C71⋊C8C2211yescharacteristic
30#17985901C2211yescharacteristic
31#17985901C2211yescharacteristic
32#17985889—21no—
33#17985889—21no—
34#17985865C2211yescharacteristic
35#17985861—21no—
36C4×C4⋊Dic71C2211yescharacteristic
37C4×C4⋊Dic71C2211yescharacteristic
38#17985846—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×Q8×C71C2211yescharacteristic
50C2×C4×Q8×C71C2211yescharacteristic
51C2×C4×Q8×C71C2211yescharacteristic
52#389×C71—21no—
53#389×C71—21no—
54#389×C71C2211yescharacteristic
55#389×C71C2211yescharacteristic
56#384×C71—21no—
57#384×C71C2211yescharacteristic
58C4×C4⋊C4×C71—21no—
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C2×#11765852—22no—
63C2×#11765852—22no—
64C2×#11765852C2×C414yes—
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C4×D284—22no—
70C4×D284C2×C414yes—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×(C2×D71)⋊C4—21no—
75C2×(C2×D71)⋊C4—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81#14556061C2×C414yes—
82C4⋊(C71⋊C8)—22no—
83#14556052—22no—
84#14556052—22no—
85#14556052C2×C414yes—
86#14556048—41no—
87#14556048—41no—
88#14556048C2×C414yes—
89#14556039—41no—
90#14556039C2×C414yes—
91C4×C71⋊C8—41no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C22×Q8×C71C2311yescharacteristic
97C4×Q8×C71—22no—
98C4×Q8×C71—22no—
99C4×Q8×C71—22no—
100C4×Q8×C71—41no—
101C4×Q8×C71—41no—
102C4×Q8×C71C2×C414yes—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71D411yescharacteristic
111C2×C4⋊C4×C71D411yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C42×C71—21no—
116C2×C42×C71—21no—
117C2×C42×C71—21no—
118C2×C42×C71C2311yescharacteristic
119C2×C42×C71C2311yescharacteristic
120C2×C42×C71C2311yescharacteristic
121#95×C71—21no—
122#95×C71—21no—
123#95×C71D411yescharacteristic
124#95×C71D411yescharacteristic
125C23×D71—21no—
126C2×D284—21no—
127C2×D284—21no—
128C2×D284—21no—
129C2×D284—21no—
130C2×D284—21no—
131C2×D284C22×C412yes—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C2×C4×D71—24no—
136C2×C4×D71—81no—
137C2×C4×D71—81no—
138#11765852—28no—
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⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8C22×C412yes—
150C2×C71⋊C8C22×C412yes—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71C22×C412yes—
157(C2×D71)⋊C4—24no—
158(C2×D71)⋊C4—42no—
159(C2×D71)⋊C4—42no—
160(C2×D71)⋊C4—81no—
161(C2×D71)⋊C4—81no—
162C4×Dic71—81no—
163C4×Dic71—81no—
164C2×Q8×C71—21no—
165C2×Q8×C71—21no—
166C2×Q8×C71—21no—
167C2×Q8×C71C22×C412yes—
168C2×Q8×C71#4112yes—
169C2×Q8×C71#4112yes—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2411yescharacteristic
181C4⋊C4×C71—24no—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—41no—
189C4⋊C4×C71—42no—
190C4⋊C4×C71—42no—
191C4⋊C4×C71—42no—
192C4⋊C4×C71—81no—
193C4⋊C4×C71—81no—
194C4⋊C4×C71—81no—
195C4⋊C4×C71C22×C412yes—
196C42×C71—24no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—42no—
202C42×C71—42no—
203C42×C71C22×C412yes—
204C42×C71#4112yes—
205C23×C71C22×D411yescharacteristic
206C22×D71—22no—
207C22×D71—41no—
208C22×D71—41no—
209C22×D71—41no—
210C22×D71—41no—
211C22×D71—41no—
212C22×D71—41no—
213Q8×C71—22no—
214Q8×C71—22no—
215Q8×C71—22no—
216Q8×C71#11714yes—
217C2×C4×C71—22no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—22no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
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—21no—
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×C71#13712yes—
247C2×C4×C71C23×C411yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71H(5,𝔽2)11yescharacteristic
252C2×C4×C712−1+411yescharacteristic
253C2×Dic71—22no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262D284—22no—
263D284—22no—
264D284—41no—
265D284—41no—
266C4×D71—84no—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C71⋊C8—41no—
270C71⋊C8—41no—
271C22×C71C2×C4×D411yescharacteristic
272C22×C71#54611yescharacteristic
273C22×C71#54811yescharacteristic
274C22×C71C2×#13711yescharacteristic
275C22×C71#57511yescharacteristic
276C22×C71#57611yescharacteristic
277C22×C71#57711yescharacteristic
278C2×D71—42no—
279C2×D71—42no—
280C2×D71—42no—
281C2×D71—81no—
282C2×D71—81no—
283C2×D71—81no—
284C2×D71—81no—
285C4×C71—24no—
286C4×C71—22no—
287C4×C71—22no—
288C4×C71—22no—
289C4×C71—42no—
290C4×C71—41no—
291C4×C71—41no—
292C4×C71—41no—
293C4×C71—81no—
294C4×C71—81no—
295C4×C71C2×#11711yescharacteristic
296C4×C71#52011yescharacteristic
297C4×C71#55211yescharacteristic
298C4×C71#55311yescharacteristic
299Dic71—42no—
300Dic71—81no—
301Dic71—81no—
302#22775—711nomaximal, Sylow
303C2×C71#235211yescharacteristic
304C2×C71C4×#13711yescharacteristic
305C2×C71#294611yescharacteristic
306C2×C71#330911yescharacteristic
307C2×C71#332011yescharacteristic
308C2×C71#337111yescharacteristic
309C2×C71#337511yescharacteristic
310D71—82no—
311C2×#440—711no—
312#2357—711no—
313Q8×C4⋊C4D7111yescharacteristic, complemented
314#1936—711no—
315#1934—711no—
316#1921—711no—
317#1918—711no—
318#1880—711no—
319#1880—711no—
320#1836—711no—
321#1805—711no—
322#1802—711no—
323#1771—711no—
324#1763—711no—
325C4×#103—711no—
326C71#2277511yescharacteristic, complemented, Sylow
327C22×SD16—711no—
328C2×C4×Q8C2×D7111yescharacteristic
329C2×C4×Q8C2×D7111yescharacteristic
330C2×C4×Q8C2×D7111yescharacteristic
331C2×C4×D4—711no—
332#440—718no—
333C2×#107—711no—
334C2×C4⋊C8—711no—
335C2×#103—711no—
336C2×#103—711no—
337C2×#103—711no—
338C2×#102—711no—
339C2×#102—711no—
340C2×#102—711no—
341C2×#97—711no—
342C2×#97—711no—
343C2×C4×C8—711no—
344#390—711no—
345#390—711no—
346#389—21no—
347#389—21no—
348#389C2×D7111yescharacteristic
349#389C2×D7111yescharacteristic
350#388—1421no—
351#388—1421no—
352#384—21no—
353#384C2×D7111yescharacteristic
354#382—711no—
355#381—1421no—
356C4×C4⋊C4—21no—
357C4×C4⋊C4—711no—
358C4×C4⋊C4—711no—
359C4×C4⋊C4C2×D7111yescharacteristic
360C4×C22⋊C4—1421no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#336—711no—
370#336—711no—
371C22×Q8C22×D7111yescharacteristic
372C22×D4—711no—
373C23×C4—1421no—
374C2×SD16—714no—
375C2×SD16—1422no—
376C2×SD16—1422no—
377C22×C8—711no—
378C22×C8—711no—
379C22×C8—711no—
380C22×C8—711no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—41no—
385C4×Q8—41no—
386C4×Q8C4×D7114yes—
387C4×D4—714no—
388C4×D4—1422no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4C22⋊D7111yescharacteristic
400C2×C4⋊C4C22⋊D7111yescharacteristic
401C2×C4⋊C4C22×D7111yescharacteristic
402C2×C4⋊C4C22×D7111yescharacteristic
403C2×C4⋊C4C22×D7111yescharacteristic
404C2×C22⋊C4—1421no—
405C2×C22⋊C4—1421no—
406C2×C22⋊C4—1421no—
407C2×C22⋊C4—1421no—
408C2×C22⋊C4—1421no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—1421no—
413C2×C42—1421no—
414C2×C42C22×D7111yescharacteristic
415C2×C42C22×D7111yescharacteristic
416C2×C42C22×D7111yescharacteristic
417#107—714no—
418C4⋊C8—1422no—
419#103—714no—
420#103—1422no—
421#103—1422no—
422#102—2841no—
423#102—2841no—
424#102—714no—
425#97—2841no—
426#97—714no—
427C4×C8—2841no—
428#95—21no—
429#95—21no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95C22⋊D7111yescharacteristic
435#95C22⋊D7111yescharacteristic
436C24—1421no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D7112yes—
441C2×Q8#1176586412yes—
442C2×Q8#1176586612yes—
443C2×D4—712no—
444C2×D4—1421no—
445C2×D4—1421no—
446C2×D4—1421no—
447C2×D4—1421no—
448C2×D4—1421no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—5681no—
454C22×C4—5681no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1424no—
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C23×D7111yescharacteristic
466SD16—1428no—
467C2×C8—712no—
468C2×C8—712no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—712no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4C2×C4×D7112yes—
500C22⋊C4—2842no—
501C22⋊C4—2842no—
502C22⋊C4—5681no—
503C22⋊C4—5681no—
504C22⋊C4—1424no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—24no—
510C42—42no—
511C42—42no—
512C42—5681no—
513C42—5681no—
514C42C2×C4×D7112yes—
515C42#1176586512yes—
516C23—2841no—
517C23—2841no—
518C23—2841no—
519C23—2841no—
520C23—2841no—
521C23—2841no—
522C23—1422no—
523C23C22×C22⋊D7111yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8#1455608614yes—
528D4—2841no—
529D4—2841no—
530D4—1422no—
531D4—1422no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—5684no—
568C2×C4—1422no—
569C2×C4C4×C22⋊D7112yes—
570C2×C4C4×C22⋊D7112yes—
571C2×C4#1455615912yes—
572C2×C4C22×C4×D7111yescharacteristic
573C2×C4C2×#1176586411yescharacteristic
574C2×C4C2×#1176586511yescharacteristic
575C2×C4C2×#1176586611yescharacteristic
576C2×C4#1455617311yescharacteristic
577C2×C4#1455617611yescharacteristic
578C8—2841no—
579C8—2841no—
580C8—1422no—
581C8—1422no—
582C22—2842no—
583C22—2842no—
584C22—2842no—
585C22—5681no—
586C22—5681no—
587C22—5681no—
588C22—5681no—
589C22C2×C4×C22⋊D7111yescharacteristic
590C22#1798664311yescharacteristic
591C22#1798666011yescharacteristic
592C22C2×#1455615911yescharacteristic
593C22#1798679711yescharacteristic
594C22#1798680211yescharacteristic
595C22#1798680711yescharacteristic
596C4—24no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—2842no—
607C4—5681no—
608C4—5681no—
609C4C2×#1455608611yescharacteristic
610C4#1798648011yescharacteristic
611C4#1798668411yescharacteristic
612C4#1798669711yescharacteristic
613C2—5682no—
614C2#2259126511yescharacteristic
615C2C4×#1455615911yescharacteristic
616C2#2259716811yescharacteristic
617C2#2260034611yescharacteristic
618C2#2260051111yescharacteristic
619C2#2260110811yescharacteristic
620C2#2260115411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingQ8×C4⋊C4×C71C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenterQ8×C4⋊C4D71
hypocenterC71#22775
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #22588929C2semidirect
6. #22588908C2semidirect
9. #22588475C2semidirect
10. #22588355C2semidirect
12. #22588215C2semidirect
13. #22588170C2semidirect
14. C4×#14556052C2semidirect
15. Q8×C4⋊C4×C71C2semidirect
18. #17986097C4semidirect
313. Q8×C4⋊C4D71semidirect
326. C71#22775semidirect

Maximal subgroups

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

subgroupindex
C2×#179860972
#225913112
#225890362
#225890222
#225889292
#225889082
#225886952
#225884752
#225883552
#225883372
#225882152
#225881702
C4×#145560522
Q8×C4⋊C4×C712
#2277571