Finite Group Databasebeta

← #31964584

Subgroups — #31964584

Order 18176.

1116 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986533C211yesmaximal, characteristic
2#22596973C211yesmaximal, characteristic, complemented
3#22586863C214yesmaximal, complemented
4#22586859C212yesmaximal, complemented
5#22586855C212yesmaximal
6#22586824C214yesmaximal, complemented
7C4×#117×C71C211yesmaximal, characteristic, complemented
8#17986533—21no—
9#17986533—21no—
10#17986533—21no—
11#17986533—21no—
12#17986533C412yescomplemented
13#17986533C412yescomplemented
14#17986533C2212yes—
15#17986533C2212yes—
16C2×#14556122—21no—
17C2×#14556120C411yescharacteristic, complemented
18C2×#14556116C411yescharacteristic, complemented
19C2×#14556111—21no—
20C2×#14556107C2211yescharacteristic
21C2×#14556102C2211yescharacteristic
22C2×C4×Dic142C411yescharacteristic, complemented
23C2×C4×D284C411yescharacteristic, complemented
24C2×C4×C22⋊D71—21no—
25C2×#14556085—21no—
26#17986056—21no—
27#17986056C2212yescomplemented
28#17986052—21no—
29#17986052C2212yescomplemented
30#17986023—21no—
31#17985963C2212yes—
32C4×#11765849—21no—
33#17985735C2218yescomplemented
34#17985732C2218yes—
35C2×#117×C71—21no—
36C22×C42×C71C2211yescharacteristic
37C4×C4⋊C4×C71—21no—
38C4×C4⋊C4×C71C2212yescomplemented
39C4×C22⋊C4×C71—21no—
40C4×C22⋊C4×C71C2212yes—
41#376×C71C2212yescomplemented
42C43×C71—21no—
43C22×Dic142C2×C411yescharacteristic
44C22×D284C2×C411yescharacteristic
45C22×C22⋊D71—21no—
46C22×C4×D71—21no—
47C2×#11765849—22no—
48#14556122—22no—
49#14556122—22no—
50#14556122—22no—
51#14556122—22no—
52#14556120—21no—
53#14556120—21no—
54#14556120C2×C412yes—
55#14556120C2×C412yes—
56#14556116—21no—
57#14556116—21no—
58#14556116C2×C412yes—
59#14556116C2×C412yes—
60#14556111—22no—
61#14556111—22no—
62#14556111—22no—
63#14556111—22no—
64#14556107—21no—
65#14556107—21no—
66#14556107C2×C412yes—
67#14556107C2312yes—
68#14556102—21no—
69#14556102—21no—
70#14556102C2×C412yes—
71#14556102C2312yes—
72C4×Dic142—21no—
73C4×Dic142—21no—
74C4×Dic142C2×C412yes—
75C4×Dic142C2×C412yes—
76C4×D284—21no—
77C4×D284—21no—
78C4×D284C2×C412yes—
79C4×D284C2×C412yes—
80C4×C22⋊D71—22no—
81C4×C22⋊D71—22no—
82C4×C22⋊D71—22no—
83C4×C22⋊D71—22no—
84#14556085—22no—
85#14556085—22no—
86#14556085—22no—
87#14556085—22no—
88C2×C4⋊Dic71C2×C411yescharacteristic
89C2×C4⋊Dic71C2×C411yescharacteristic
90C2×Dic71⋊C4—21no—
91C2×Dic71⋊C4—21no—
92C2×Dic71⋊C4—21no—
93C2×Dic71⋊C4—21no—
94C2×C22⋊Dic71—21no—
95C2×(C2×D71)⋊C4—21no—
96C2×(C2×D71)⋊C4—21no—
97C2×(C2×D71)⋊C4—21no—
98C2×(C2×D71)⋊C4—21no—
99C2×C4×Dic71—21no—
100C4⋊(C71⋊C8)—22no—
101C4⋊(C71⋊C8)—22no—
102C22⋊(C71⋊C8)—22no—
103C22⋊(C71⋊C8)C2314yes—
104#14556039—22no—
105C4×C71⋊C8—22no—
106C23×C4×C71D411yescharacteristic
107C23×C4×C71D411yescharacteristic
108C23×C4×C71C2311yescharacteristic
109#117×C71—28no—
110C2×C4⋊C4×C71—22no—
111C2×C22⋊C4×C71—22no—
112C2×C42×C71—22no—
113C2×C42×C71—22no—
114C2×C42×C71—22no—
115C2×C42×C71C2×C411yescharacteristic
116C2×C42×C71C2×C411yescharacteristic
117C2×C42×C71C2×C411yescharacteristic
118C2×C42×C71C2×C411yescharacteristic
119C2×C42×C71D411yescharacteristic
120C2×C42×C71D411yescharacteristic
121C2×C42×C71D411yescharacteristic
122C2×C42×C71D411yescharacteristic
123C2×C42×C71D411yescharacteristic
124C2×C42×C71D411yescharacteristic
125C2×C42×C71C2311yescharacteristic
126C2×C42×C71C2311yescharacteristic
127#95×C71—22no—
128#95×C71C2314yes—
129C23×D71—21no—
130C2×Dic142—21no—
131C2×Dic142—42no—
132C2×Dic142C22×C412yes—
133C2×D284—21no—
134C2×D284—42no—
135C2×D284C22×C412yes—
136C2×C22⋊D71—22no—
137C2×C22⋊D71—22no—
138C2×C22⋊D71—44no—
139C22×Dic71—21no—
140C22×Dic71—21no—
141C22×Dic71—21no—
142C2×C4×D71—22no—
143C2×C4×D71—22no—
144C2×C4×D71—44no—
145#11765849—44no—
146C2×C71⋊C8—24no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71C22×C412yes—
150C4⋊Dic71C22×C412yes—
151Dic71⋊C4—22no—
152Dic71⋊C4—22no—
153Dic71⋊C4—22no—
154Dic71⋊C4—22no—
155Dic71⋊C4—22no—
156Dic71⋊C4—22no—
157Dic71⋊C4—22no—
158Dic71⋊C4—22no—
159C22⋊Dic71—22no—
160C22⋊Dic71—22no—
161(C2×D71)⋊C4—22no—
162(C2×D71)⋊C4—22no—
163(C2×D71)⋊C4—22no—
164(C2×D71)⋊C4—22no—
165(C2×D71)⋊C4—22no—
166(C2×D71)⋊C4—22no—
167(C2×D71)⋊C4—22no—
168(C2×D71)⋊C4—22no—
169C4×Dic71—22no—
170C4×Dic71—22no—
171C24×C71C2×D411yescharacteristic
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—42no—
175C22×C4×C71—42no—
176C22×C4×C71C22⋊C411yescharacteristic
177C22×C4×C71C22⋊C411yescharacteristic
178C22×C4×C71C22⋊C411yescharacteristic
179C22×C4×C71C22⋊C411yescharacteristic
180C22×C4×C71C22⋊C411yescharacteristic
181C22×C4×C71C22⋊C411yescharacteristic
182C22×C4×C71C22⋊C411yescharacteristic
183C22×C4×C71C22⋊C411yescharacteristic
184C22×C4×C71C22×C411yescharacteristic
185C22×C4×C71C22×C411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2411yescharacteristic
194C4⋊C4×C71—28no—
195C22⋊C4×C71—28no—
196C42×C71—28no—
197C42×C71—44no—
198C42×C71—44no—
199C42×C71C22⋊C411yescharacteristic
200C42×C71C22⋊C411yescharacteristic
201C42×C71C22⋊C411yescharacteristic
202C42×C71C22⋊C411yescharacteristic
203C42×C71C22⋊C411yescharacteristic
204C42×C71C22⋊C411yescharacteristic
205C42×C71C22⋊C411yescharacteristic
206C42×C71C22⋊C411yescharacteristic
207C42×C71C22×C411yescharacteristic
208C42×C71C22×C411yescharacteristic
209C42×C71C22×C411yescharacteristic
210C42×C71C22×C411yescharacteristic
211C42×C71C2×D411yescharacteristic
212C42×C71C2×D411yescharacteristic
213C42×C71C2×D411yescharacteristic
214C42×C71C2×D411yescharacteristic
215C23×C71—22no—
216C23×C71—22no—
217C23×C71—41no—
218C23×C71C2×C22⋊C411yescharacteristic
219C23×C71C2×C22⋊C411yescharacteristic
220C23×C71C22×D411yescharacteristic
221C22×D71—22no—
222C22×D71—42no—
223C22×D71—42no—
224C22×D71—42no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—28no—
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—44no—
244C2×C4×C71—44no—
245C2×C4×C71#10414yes—
246C2×C4×C71C2×C22⋊C411yescharacteristic
247C2×C4×C71C2×C22⋊C411yescharacteristic
248C2×C4×C71C2×C22⋊C411yescharacteristic
249C2×C4×C71C2×C22⋊C411yescharacteristic
250C2×C4×C71C2×C22⋊C411yescharacteristic
251C2×C4×C71C2×C22⋊C411yescharacteristic
252C2×C4×C71C2×C22⋊C411yescharacteristic
253C2×C4×C71C2×C22⋊C411yescharacteristic
254C2×C4×C71C2×C22⋊C411yescharacteristic
255C2×C4×C71C2×C22⋊C411yescharacteristic
256C2×C4×C71C23×C411yescharacteristic
257C2×C4×C71C22×D411yescharacteristic
258C22⋊D71—44no—
259C22⋊D71—44no—
260C2×Dic71—22no—
261C2×Dic71—22no—
262C2×Dic71—22no—
263C2×Dic71—42no—
264C2×Dic71—42no—
265C2×Dic71—42no—
266D284—42no—
267D284—42no—
268C4×D71—44no—
269C4×D71—44no—
270Dic142—42no—
271Dic142—42no—
272C71⋊C8—44no—
273C22×C71—21no—
274C22×C71—21no—
275C22×C71—21no—
276C22×C71—21no—
277C22×C71—21no—
278C22×C71—21no—
279C22×C71—21no—
280C22×C71—44no—
281C22×C71C2×#10412yes—
282C22×C71C22×C22⋊C411yescharacteristic
283C22×C71#57512yes—
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—42no—
287C2×D71—84no—
288C4×C71—21no—
289C4×C71—21no—
290C4×C71—21no—
291C4×C71—21no—
292C4×C71—21no—
293C4×C71—21no—
294C4×C71—21no—
295C4×C71—21no—
296C4×C71—21no—
297C4×C71—48no—
298C4×C71#41012yes—
299C4×C71C2×#10412yes—
300C4×C71C2×#10412yes—
301Dic71—42no—
302Dic71—42no—
303Dic71—42no—
304#17537—711nomaximal, Sylow
305C2×C71—22no—
306C2×C71—22no—
307C2×C71—41no—
308C2×C71C2×#41011yescharacteristic
309C2×C71#289611yescharacteristic
310C2×C71C22×#10411yescharacteristic
311D71—82no—
312C2×#529—711no—
313#2920—711no—
314C4×#117D7111yescharacteristic, complemented
315#1509—714no—
316#1508—712no—
317#1507—712no—
318#1500—714no—
319C71#1753711yescharacteristic, complemented, Sylow
320#529—712no—
321#529—712no—
322#529—712no—
323#529—712no—
324#529—1421no—
325#529—1421no—
326#529—1421no—
327#529—1421no—
328C2×#126—1421no—
329C2×#125—711no—
330C2×#124—711no—
331C2×#123—1421no—
332C2×C22⋊Q8—711no—
333C2×#121—711no—
334C2×C4×Q8—711no—
335C2×C4×D4—711no—
336C2×C4×D4—1421no—
337C2×#117—21no—
338C2×#117—1421no—
339C22×C42C2×D7111yescharacteristic
340#432—712no—
341#432—1421no—
342#431—712no—
343#431—1421no—
344#423—1421no—
345#407—712no—
346C4×M16—1421no—
347C4×C4⋊C4—21no—
348C4×C4⋊C4C2×D7112yescomplemented
349C4×C22⋊C4—21no—
350C4×C22⋊C4C2×D7112yes—
351#376C2×D7112yescomplemented
352C43—21no—
353#328—718no—
354#327—718no—
355C22×Q8—711no—
356C22×D4—711no—
357C22×D4—1421no—
358C23×C4—1421no—
359C23×C4C22⋊D7111yescharacteristic
360C23×C4C22⋊D7111yescharacteristic
361C23×C4C22×D7111yescharacteristic
362C2×M16—1422no—
363#126—1422no—
364#126—1422no—
365#126—1422no—
366#126—1422no—
367#125—712no—
368#125—712no—
369#125—1421no—
370#125—1421no—
371#124—712no—
372#124—712no—
373#124—1421no—
374#124—1421no—
375#123—1422no—
376#123—1422no—
377#123—1422no—
378#123—1422no—
379C22⋊Q8—712no—
380C22⋊Q8—712no—
381C22⋊Q8—1421no—
382C22⋊Q8—1421no—
383#121—712no—
384#121—712no—
385#121—1421no—
386#121—1421no—
387C4×Q8—712no—
388C4×Q8—712no—
389C4×Q8—1421no—
390C4×Q8—1421no—
391C4×D4—712no—
392C4×D4—712no—
393C4×D4—1422no—
394C4×D4—1422no—
395C4×D4—1422no—
396C4×D4—1422no—
397C4×D4—1421no—
398C4×D4—1421no—
399#117—28no—
400#117—1422no—
401#117—1422no—
402#117—1422no—
403#117—1422no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—711no—
406C2×C4⋊C4—711no—
407C2×C4⋊C4—1421no—
408C2×C4⋊C4—1421no—
409C2×C4⋊C4—1421no—
410C2×C4⋊C4—1421no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—1421no—
413C2×C22⋊C4—1421no—
414C2×C22⋊C4—1421no—
415C2×C22⋊C4—1421no—
416C2×C22⋊C4—1421no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—1421no—
421C2×C42C4×D7111yescharacteristic
422C2×C42C4×D7111yescharacteristic
423C2×C42C4×D7111yescharacteristic
424C2×C42C4×D7111yescharacteristic
425C2×C42D28411yescharacteristic
426C2×C42D28411yescharacteristic
427C2×C42D28411yescharacteristic
428C2×C42D28411yescharacteristic
429C2×C42C22⋊D7111yescharacteristic
430C2×C42C22⋊D7111yescharacteristic
431C2×C42C22×D7111yescharacteristic
432C2×C42C22×D7111yescharacteristic
433C4⋊C8—1422no—
434C4⋊C8—1422no—
435C22⋊C8—714no—
436C22⋊C8—1422no—
437#97—1422no—
438C4×C8—1422no—
439#95—22no—
440#95C22×D7114yes—
441C24—1421no—
442C24C2×C22⋊D7111yescharacteristic
443C2×Q8—2842no—
444C2×Q8—712no—
445C2×Q8—1421no—
446C2×D4—2842no—
447C2×D4—2844no—
448C2×D4—712no—
449C2×D4—1422no—
450C2×D4—1422no—
451C2×D4—1421no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—2844no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4(C2×D71)⋊C411yescharacteristic
463C22×C4(C2×D71)⋊C411yescharacteristic
464C22×C4(C2×D71)⋊C411yescharacteristic
465C22×C4(C2×D71)⋊C411yescharacteristic
466C22×C4(C2×D71)⋊C411yescharacteristic
467C22×C4(C2×D71)⋊C411yescharacteristic
468C22×C4(C2×D71)⋊C411yescharacteristic
469C22×C4(C2×D71)⋊C411yescharacteristic
470C22×C4C2×C4×D7111yescharacteristic
471C22×C4C2×C4×D7111yescharacteristic
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4C2×C22⋊D7111yescharacteristic
474C22×C4C2×C22⋊D7111yescharacteristic
475C22×C4C2×C22⋊D7111yescharacteristic
476C22×C4C2×C22⋊D7111yescharacteristic
477C22×C4C2×D28411yescharacteristic
478C22×C4C2×D28411yescharacteristic
479C22×C4C23×D7111yescharacteristic
480M16—2844no—
481C2×C8—1424no—
482C4⋊C4—28no—
483C4⋊C4—712no—
484C4⋊C4—712no—
485C4⋊C4—1422no—
486C4⋊C4—1422no—
487C4⋊C4—1422no—
488C4⋊C4—1422no—
489C4⋊C4—1422no—
490C4⋊C4—1422no—
491C4⋊C4—1422no—
492C4⋊C4—1422no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C22⋊C4—28no—
496C22⋊C4—1422no—
497C22⋊C4—1422no—
498C22⋊C4—1422no—
499C22⋊C4—1422no—
500C22⋊C4—1422no—
501C22⋊C4—1422no—
502C22⋊C4—1422no—
503C22⋊C4—1422no—
504C22⋊C4—1422no—
505C22⋊C4—1422no—
506C42—28no—
507C42—44no—
508C42—44no—
509C42—1422no—
510C42—1422no—
511C42(C2×D71)⋊C411yescharacteristic
512C42(C2×D71)⋊C411yescharacteristic
513C42(C2×D71)⋊C411yescharacteristic
514C42(C2×D71)⋊C411yescharacteristic
515C42(C2×D71)⋊C411yescharacteristic
516C42(C2×D71)⋊C411yescharacteristic
517C42(C2×D71)⋊C411yescharacteristic
518C42(C2×D71)⋊C411yescharacteristic
519C42C2×C4×D7111yescharacteristic
520C42C2×C4×D7111yescharacteristic
521C42C2×C4×D7111yescharacteristic
522C42C2×C4×D7111yescharacteristic
523C42C2×D28411yescharacteristic
524C42C2×D28411yescharacteristic
525C42C2×D28411yescharacteristic
526C42C2×D28411yescharacteristic
527C23—22no—
528C23—22no—
529C23—41no—
530C23—2842no—
531C23—2842no—
532C23—2842no—
533C23—1422no—
534C23C2×(C2×D71)⋊C411yescharacteristic
535C23C2×(C2×D71)⋊C411yescharacteristic
536C23C22×C22⋊D7111yescharacteristic
537Q8—2842no—
538Q8—2842no—
539D4—2842no—
540D4—2844no—
541D4—2844no—
542D4—2842no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2844no—
567C2×C4—2844no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4#1455605614yes—
572C2×C4C2×(C2×D71)⋊C411yescharacteristic
573C2×C4C2×(C2×D71)⋊C411yescharacteristic
574C2×C4C2×(C2×D71)⋊C411yescharacteristic
575C2×C4C2×(C2×D71)⋊C411yescharacteristic
576C2×C4C2×(C2×D71)⋊C411yescharacteristic
577C2×C4C2×(C2×D71)⋊C411yescharacteristic
578C2×C4C2×(C2×D71)⋊C411yescharacteristic
579C2×C4C2×(C2×D71)⋊C411yescharacteristic
580C2×C4C2×(C2×D71)⋊C411yescharacteristic
581C2×C4C2×(C2×D71)⋊C411yescharacteristic
582C2×C4C22×C4×D7111yescharacteristic
583C2×C4C22×D28411yescharacteristic
584C8—2844no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—44no—
593C22—2842no—
594C22—2842no—
595C22—2842no—
596C22—5684no—
597C22C2×#1455605612yes—
598C22C22×(C2×D71)⋊C411yescharacteristic
599C22#1798679712yes—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—48no—
610C4—2842no—
611C4—2842no—
612C4—2842no—
613C4#1798597212yes—
614C4C2×#1455605612yes—
615C4C2×#1455605612yes—
616C2—22no—
617C2—22no—
618C2—41no—
619C2—5682no—
620C2C2×#1798597211yescharacteristic
621C2#2259685711yescharacteristic
622C2C22×#1455605611yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×#117×C71C2
socleC22×C71C22×C22⋊C4
prime radicalC2×#14556102C22
cosocleC22×C4×C71C24
hypercenterC4×#117D71
hypocenterC71#17537
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #22596973C2semidirect
3. #22586863C2semidirect
4. #22586859C2semidirect
6. #22586824C2semidirect
7. C4×#117×C71C2semidirect
12. #17986533C4semidirect
13. #17986533C4semidirect
17. C2×#14556120C4semidirect
18. C2×#14556116C4semidirect
22. C2×C4×Dic142C4semidirect
23. C2×C4×D284C4semidirect
27. #17986056C22semidirect
29. #17986052C22semidirect
33. #17985735C22semidirect
38. C4×C4⋊C4×C71C22semidirect
41. #376×C71C22semidirect
314. C4×#117D71semidirect
319. C71#17537semidirect
348. C4×C4⋊C4C2×D71semidirect
351. #376C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179865332
#225969732
#225868632
#225868592
#225868552
#225868242
C4×#117×C712
#1753771