Finite Group Databasebeta

← #32014117

Subgroups — #32014117

Order 18176.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986709C211yesmaximal, characteristic
2C2×#17986130C211yesmaximal, characteristic, complemented
3#22589312C211yesmaximal, characteristic, complemented
4#22589308C211yesmaximal, characteristic, complemented
5#22589304C211yesmaximal, characteristic, complemented
6#22589297C211yesmaximal, characteristic, complemented
7#22589297C211yesmaximal, characteristic, complemented
8#22589292C211yesmaximal, characteristic, complemented
9#22588347C211yesmaximal, characteristic
10#22588347C211yesmaximal, characteristic
11#22588337C211yesmaximal, characteristic
12#22588168C211yesmaximal, characteristic
13#22588168C211yesmaximal, characteristic
14C4×#14556048C211yesmaximal, characteristic
15C71×#2355C211yesmaximal, characteristic, complemented
16#17986709—24no—
17#17986709C418yescomplemented
18C2×#14556128C2211yescharacteristic
19C2×#14556120C2211yescharacteristic
20C2×#14556120C2211yescharacteristic
21C2×#14556106—21no—
22C2×#14556106—21no—
23C2×#14556104—21no—
24C2×C4×D284C2211yescharacteristic
25C2×C4×D284C2211yescharacteristic
26C2×C4×D284C2211yescharacteristic
27C2×C4⋊C4×D71—21no—
28#17986130—24no—
29C2×C4⋊(C71⋊C8)C2211yescharacteristic
30C2×C4⋊(C71⋊C8)C2211yescharacteristic
31C2×C4⋊(C71⋊C8)C2211yescharacteristic
32C2×#14556048C2211yescharacteristic
33C2×#14556048C2211yescharacteristic
34C2×#14556048C2211yescharacteristic
35C2×#14556048C2211yescharacteristic
36C2×#14556039C2211yescharacteristic
37C2×#14556039C2211yescharacteristic
38C2×C4×C71⋊C8C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51C2×C4×Q8×C71C2211yescharacteristic
52#391×C71—21no—
53#387×C71—21no—
54#387×C71—21no—
55#384×C71C2211yescharacteristic
56#382×C71C2211yescharacteristic
57#382×C71C2211yescharacteristic
58C4×C4⋊C4×C71—21no—
59C4×C4⋊C4×C71C2211yescharacteristic
60C4×C4⋊C4×C71C2211yescharacteristic
61C4×C4⋊C4×C71C2211yescharacteristic
62C22×Dic142—21no—
63C22×D284C2311yescharacteristic
64C22×C4×D71—21no—
65C22×C4×D71—21no—
66C22×C4×D71—21no—
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C22×C71⋊C8C2311yescharacteristic
71#14556128—22no—
72#14556128C2×C414yes—
73#14556120—22no—
74#14556120—22no—
75#14556120C2×C414yes—
76#14556120C2×C414yes—
77#14556106—28no—
78#14556106—28no—
79#14556104—28no—
80C4×D284—22no—
81C4×D284—22no—
82C4×D284—22no—
83C4×D284C2×C414yes—
84C4×D284C2×C414yes—
85C4×D284C2×C414yes—
86C2×C4⋊Dic71—21no—
87C2×C4⋊Dic71C2311yescharacteristic
88C2×C4⋊Dic71C2311yescharacteristic
89C2×C4⋊Dic71C2311yescharacteristic
90C2×Dic71⋊C4—21no—
91C2×Dic71⋊C4—21no—
92C2×Dic71⋊C4—21no—
93C2×Dic71⋊C4—21no—
94C4⋊C4×D71—28no—
95C2×(C2×D71)⋊C4—21no—
96C2×(C2×D71)⋊C4—21no—
97C2×(C2×D71)⋊C4—21no—
98C4⋊(C71⋊C8)—22no—
99C4⋊(C71⋊C8)—22no—
100C4⋊(C71⋊C8)—22no—
101#14556048—22no—
102#14556048—22no—
103#14556048—22no—
104#14556048Q814yescomplemented
105#14556039—22no—
106#14556039—22no—
107C4×C71⋊C8—22no—
108C22×Q8×C71D411yescharacteristic
109C4×Q8×C71C2×C414yes—
110C4×Q8×C71D414yes—
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—21no—
113C2×C4⋊C4×C71D411yescharacteristic
114C2×C4⋊C4×C71D411yescharacteristic
115C2×C4⋊C4×C71D411yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C4⋊C4×C71C2311yescharacteristic
120C2×C42×C71—21no—
121C2×C42×C71—21no—
122C2×C42×C71—21no—
123C2×C42×C71C2311yescharacteristic
124C2×C42×C71C2311yescharacteristic
125C2×C42×C71C2311yescharacteristic
126#95×C71—21no—
127#95×C71—21no—
128#95×C71—21no—
129#95×C71—21no—
130#95×C71—21no—
131#95×C71—21no—
132#95×C71—21no—
133C23×D71—21no—
134C2×Dic142—24no—
135C2×Dic142—81no—
136C2×Dic142—81no—
137C2×D284—21no—
138C2×D284—21no—
139C2×D284—21no—
140C2×D284C22×C412yes—
141C2×D284C2×Q812yes—
142C2×D284#4112yes—
143C22×Dic71—21no—
144C22×Dic71—21no—
145C22×Dic71—21no—
146C2×C4×D71—24no—
147C2×C4×D71—24no—
148C2×C4×D71—24no—
149C2×C4×D71—42no—
150C2×C4×D71—42no—
151C2×C4×D71—81no—
152C2×C4×D71—81no—
153C2×C4×D71—81no—
154C2×C4×D71—81no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8—21no—
160C2×C71⋊C8—21no—
161C2×C71⋊C8—21no—
162C2×C71⋊C8—21no—
163C2×C71⋊C8—21no—
164C2×C71⋊C8—21no—
165C2×C71⋊C8—21no—
166C2×C71⋊C8C2×Q812yes—
167C4⋊Dic71—24no—
168C4⋊Dic71—21no—
169C4⋊Dic71—21no—
170C4⋊Dic71—21no—
171C4⋊Dic71C22×C412yes—
172C4⋊Dic71C22×C412yes—
173C4⋊Dic71C22×C412yes—
174Dic71⋊C4—24no—
175Dic71⋊C4—24no—
176Dic71⋊C4—24no—
177Dic71⋊C4—24no—
178(C2×D71)⋊C4—24no—
179(C2×D71)⋊C4—24no—
180(C2×D71)⋊C4—24no—
181C2×Q8×C71—41no—
182C2×Q8×C71—41no—
183C2×Q8×C71C22⋊C414yes—
184C22×C4×C71—21no—
185C22×C4×C71—21no—
186C22×C4×C71—21no—
187C22×C4×C71—21no—
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71C2411yescharacteristic
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71—21no—
201C4⋊C4×C71—21no—
202C4⋊C4×C71—42no—
203C4⋊C4×C71—81no—
204C4⋊C4×C71C22⋊C414yes—
205C4⋊C4×C71C22⋊C414yes—
206C4⋊C4×C71C22⋊C414yes—
207C4⋊C4×C71C2×Q812yes—
208C42×C71—42no—
209C42×C71—81no—
210C42×C71—81no—
211C42×C71C22×C412yes—
212C42×C71C22×C412yes—
213C42×C71C22×C412yes—
214C42×C71C2×D412yes—
215C42×C71C2×D412yes—
216C42×C71C2×D412yes—
217C23×C71C22×D411yescharacteristic
218C22×D71—22no—
219C22×D71—22no—
220C22×D71—22no—
221C22×D71—41no—
222C22×D71—41no—
223C22×D71—41no—
224C22×D71—41no—
225Q8×C71—44no—
226C2×C4×C71—21no—
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—21no—
237C2×C4×C71—21no—
238C2×C4×C71—22no—
239C2×C4×C71—41no—
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×C71C2×C22⋊C412yes—
251C2×C4×C71C2×C22⋊C412yes—
252C2×C4×C71C2×C22⋊C412yes—
253C2×C4×C71C2×C22⋊C412yes—
254C2×C4×C71C2×C22⋊C412yes—
255C2×C4×C71C2×C22⋊C412yes—
256C2×C4×C71C23×C411yescharacteristic
257C2×C4×C71C22×D411yescharacteristic
258C2×C4×C71C22×Q811yescharacteristic
259C2×C4×C71C2×#4111yescharacteristic
260C2×C4×C712−1+411yescharacteristic
261C2×C4×C712−1+411yescharacteristic
262C2×Dic71—22no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—41no—
268C2×Dic71—41no—
269C2×Dic71—41no—
270C2×Dic71—41no—
271D284—22no—
272D284—22no—
273D284—22no—
274D284C4×Q814yes—
275C4×D71—48no—
276C4×D71—84no—
277C4×D71—84no—
278Dic142—84no—
279C71⋊C8—22no—
280C71⋊C8—41no—
281C71⋊C8—41no—
282C71⋊C8—41no—
283C22×C71C22×C22⋊C411yescharacteristic
284C22×C71#54811yescharacteristic
285C22×C71D4×Q811yescharacteristic
286C22×C71#57511yescharacteristic
287C22×C71#57511yescharacteristic
288C22×C71#57611yescharacteristic
289C22×C71#57711yescharacteristic
290C2×D71—24no—
291C2×D71—42no—
292C2×D71—42no—
293C2×D71—42no—
294C2×D71—42no—
295C2×D71—42no—
296C2×D71—42no—
297C4×C71—22no—
298C4×C71—22no—
299C4×C71—22no—
300C4×C71—22no—
301C4×C71—22no—
302C4×C71—22no—
303C4×C71—42no—
304C4×C71—81no—
305C4×C71—81no—
306C4×C71—81no—
307C4×C71C2×C4×Q811yescharacteristic
308C4×C71#51911yescharacteristic
309C4×C71#54811yescharacteristic
310C4×C71D4×Q811yescharacteristic
311Dic71—42no—
312Dic71—42no—
313Dic71—42no—
314#22526—711nomaximal, Sylow
315C2×C71Q8×C22⋊C411yescharacteristic
316C2×C71#289611yescharacteristic
317C2×C71#289711yescharacteristic
318C2×C71#337011yescharacteristic
319C2×C71#337611yescharacteristic
320C2×C71#338811yescharacteristic
321C2×C71#338911yescharacteristic
322D71—44no—
323C2×#554—711no—
324C2×#446—711no—
325#2355D7111yescharacteristic, complemented
326#1993—711no—
327#1992—711no—
328#1991—711no—
329#1989—711no—
330#1989—711no—
331#1988—711no—
332#1804—711no—
333#1804—711no—
334#1802—711no—
335#1762—711no—
336#1762—711no—
337C4×#102—711no—
338C71#2252611yescharacteristic, complemented, Sylow
339#554—718no—
340#554—1424no—
341C2×C4⋊Q8—711no—
342C2×#125—711no—
343C2×#125—711no—
344C2×C22⋊Q8—1421no—
345C2×C22⋊Q8—1421no—
346C2×C22⋊Q8—1421no—
347C2×C4×Q8C2×D7111yescharacteristic
348C2×C4×D4—711no—
349C2×C4×D4—711no—
350C2×C4×D4—711no—
351C22×C4⋊C4—1421no—
352#446—1424no—
353C2×C4⋊C8—711no—
354C2×C4⋊C8—711no—
355C2×C4⋊C8—711no—
356C2×#102—711no—
357C2×#102—711no—
358C2×#102—711no—
359C2×#102—711no—
360C2×#97—711no—
361C2×#97—711no—
362C2×C4×C8—711no—
363#391—21no—
364#387—21no—
365#387—21no—
366#384C2×D7111yescharacteristic
367#382C2×D7111yescharacteristic
368#382C2×D7111yescharacteristic
369C4×C4⋊C4—21no—
370C4×C4⋊C4C2×D7111yescharacteristic
371C4×C4⋊C4C2×D7111yescharacteristic
372C4×C4⋊C4C2×D7111yescharacteristic
373#340—711no—
374#340—711no—
375#340—711no—
376#340—711no—
377#340—711no—
378#340—711no—
379#340—711no—
380#340—711no—
381#340—711no—
382#340—711no—
383#340—711no—
384#340—711no—
385C22×Q8—1421no—
386C22×Q8C22⋊D7111yescharacteristic
387C22×D4—711no—
388C23×C4—1421no—
389C23×C4—1421no—
390C23×C4—1421no—
391C22×C8—711no—
392C22×C8—711no—
393C22×C8—711no—
394C22×C8—711no—
395C4⋊Q8—714no—
396C4⋊Q8—1422no—
397#125—714no—
398#125—714no—
399#125—1422no—
400#125—1422no—
401C22⋊Q8—1428no—
402C22⋊Q8—1428no—
403C22⋊Q8—1428no—
404C4×Q8C4×D7114yes—
405C4×Q8D28414yes—
406C4×D4—714no—
407C4×D4—714no—
408C4×D4—714no—
409C4×D4—1422no—
410C4×D4—1422no—
411C4×D4—1422no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—711no—
415C2×C4⋊C4—711no—
416C2×C4⋊C4—711no—
417C2×C4⋊C4—1421no—
418C2×C4⋊C4—1421no—
419C2×C4⋊C4—1421no—
420C2×C4⋊C4—1421no—
421C2×C4⋊C4—1421no—
422C2×C4⋊C4—1428no—
423C2×C4⋊C4C22⋊D7111yescharacteristic
424C2×C4⋊C4C22⋊D7111yescharacteristic
425C2×C4⋊C4C22⋊D7111yescharacteristic
426C2×C4⋊C4C22×D7111yescharacteristic
427C2×C4⋊C4C22×D7111yescharacteristic
428C2×C4⋊C4C22×D7111yescharacteristic
429C2×C4⋊C4C22×D7111yescharacteristic
430C2×C22⋊C4—1421no—
431C2×C22⋊C4—1421no—
432C2×C22⋊C4—1421no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D7111yescharacteristic
437C2×C42C22×D7111yescharacteristic
438C2×C42C22×D7111yescharacteristic
439C4⋊C8—1422no—
440C4⋊C8—1422no—
441C4⋊C8—1422no—
442#102—714no—
443#102—1422no—
444#102—1422no—
445#102—1422no—
446#97—1422no—
447#97—1422no—
448C4×C8—1422no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456C24—1421no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—5681no—
460C2×Q8—5681no—
461C2×Q8—1424no—
462C2×Q8(C2×D71)⋊C414yes—
463C2×D4—712no—
464C2×D4—712no—
465C2×D4—712no—
466C2×D4—1421no—
467C2×D4—1421no—
468C2×D4—1421no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—2842no—
474C22×C4—2842no—
475C22×C4—5681no—
476C22×C4—5681no—
477C22×C4—5681no—
478C22×C4—5681no—
479C22×C4—1421no—
480C22×C4—1421no—
481C22×C4—1421no—
482C22×C4—1424no—
483C22×C4—1424no—
484C22×C4—1424no—
485C22×C4C2×C22⋊D7111yescharacteristic
486C22×C4C2×C22⋊D7111yescharacteristic
487C22×C4C2×C22⋊D7111yescharacteristic
488C22×C4C2×C22⋊D7111yescharacteristic
489C22×C4C2×C22⋊D7111yescharacteristic
490C22×C4C2×C22⋊D7111yescharacteristic
491C22×C4C23×D7111yescharacteristic
492C2×C8—712no—
493C2×C8—1421no—
494C2×C8—1421no—
495C2×C8—1421no—
496C2×C8—1421no—
497C2×C8—1421no—
498C2×C8—1421no—
499C2×C8—1421no—
500C2×C8—1421no—
501C2×C8—1421no—
502C2×C8—1421no—
503C2×C8—1421no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—712no—
514C4⋊C4—712no—
515C4⋊C4—712no—
516C4⋊C4—1421no—
517C4⋊C4—1421no—
518C4⋊C4—1421no—
519C4⋊C4—1424no—
520C4⋊C4—1424no—
521C4⋊C4—1424no—
522C4⋊C4—1424no—
523C4⋊C4—1424no—
524C4⋊C4(C2×D71)⋊C414yes—
525C4⋊C4(C2×D71)⋊C414yes—
526C4⋊C4(C2×D71)⋊C414yes—
527C4⋊C4Q8×D7112yes—
528C22⋊C4—1424no—
529C22⋊C4—1424no—
530C22⋊C4—1424no—
531C42—42no—
532C42—81no—
533C42—81no—
534C42C2×C4×D7112yes—
535C42C2×C4×D7112yes—
536C42C2×C4×D7112yes—
537C42C2×D28412yes—
538C42C2×D28412yes—
539C42C2×D28412yes—
540C23—2841no—
541C23—2841no—
542C23—2841no—
543C23—2841no—
544C23—1422no—
545C23—1422no—
546C23—1422no—
547C23C22×C22⋊D7111yescharacteristic
548Q8—44no—
549Q8—5684no—
550D4—714no—
551D4—1422no—
552D4—1422no—
553D4—1422no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
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—41no—
578C2×C4—2848no—
579C2×C4—2841no—
580C2×C4—2841no—
581C2×C4—2841no—
582C2×C4—2841no—
583C2×C4—2841no—
584C2×C4—2841no—
585C2×C4—5684no—
586C2×C4—5684no—
587C2×C4—1422no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4C2×(C2×D71)⋊C412yes—
591C2×C4C2×(C2×D71)⋊C412yes—
592C2×C4C2×(C2×D71)⋊C412yes—
593C2×C4C2×(C2×D71)⋊C412yes—
594C2×C4C2×(C2×D71)⋊C412yes—
595C2×C4C2×(C2×D71)⋊C412yes—
596C2×C4C22×C4×D7111yescharacteristic
597C2×C4C22×D28411yescharacteristic
598C2×C4C2×Q8×D7111yescharacteristic
599C2×C4#41×D7111yescharacteristic
600C2×C4#1455617511yescharacteristic
601C2×C4#1455617611yescharacteristic
602C8—2841no—
603C8—2841no—
604C8—2841no—
605C8—1422no—
606C22—2842no—
607C22—2842no—
608C22—2842no—
609C22—2842no—
610C22—2842no—
611C22—2842no—
612C22—1424no—
613C22C22×(C2×D71)⋊C411yescharacteristic
614C22#1798665611yescharacteristic
615C22Q8×C22⋊D7111yescharacteristic
616C22#1798679711yescharacteristic
617C22#1798679711yescharacteristic
618C22#1798680211yescharacteristic
619C22#1798680711yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—2842no—
631C4—2842no—
632C4—2842no—
633C4C4×Q8×D7111yescharacteristic
634C4#1798647511yescharacteristic
635C4#1798666111yescharacteristic
636C4Q8×D28411yescharacteristic
637C2—2844no—
638C2Q8×(C2×D71)⋊C411yescharacteristic
639C2#2259685711yescharacteristic
640C2#2259686311yescharacteristic
641C2#2260108711yescharacteristic
642C2#2260116111yescharacteristic
643C2#2260127911yescharacteristic
644C2#2260129411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2355C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2355D71
hypocenterC71#22526
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C2×#17986130C2semidirect
3. #22589312C2semidirect
4. #22589308C2semidirect
5. #22589304C2semidirect
6. #22589297C2semidirect
7. #22589297C2semidirect
8. #22589292C2semidirect
15. C71×#2355C2semidirect
17. #17986709C4semidirect
104. #14556048Q8semidirect
325. #2355D71semidirect
338. C71#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#179867092
C2×#179861302
#225893122
#225893082
#225893042
#225892972
#225892922
#225883472
#225883372
#225881682
C4×#145560482
C71×#23552
#2252671