Finite Group Databasebeta

← #32019455

Subgroups — #32019455

Order 18176.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986296C211yesmaximal, characteristic
2C4⋊C4×D284C211yesmaximal, characteristic
3#22589310C211yesmaximal, characteristic, complemented
4#22589103C211yesmaximal, characteristic, complemented
5#22589075C211yesmaximal, characteristic, complemented
6#22589065C211yesmaximal, characteristic, complemented
7#22588925C211yesmaximal, characteristic, complemented
8#22588911C211yesmaximal, characteristic, complemented
9#22588600C211yesmaximal, characteristic
10#22588600C211yesmaximal, characteristic
11#22588215C211yesmaximal, characteristic, complemented
12#22588168C211yesmaximal, characteristic
13#22588168C211yesmaximal, characteristic
14C4×#14556048C211yesmaximal, characteristic
15C71×#2370C211yesmaximal, characteristic, complemented
16C2×C4×D284C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18C2×C4×D284C2211yescharacteristic
19C2×C4⋊C4×D71—21no—
20#17986296C418yescomplemented
21C2×#14556061C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×C4⋊(C71⋊C8)C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556048C2211yescharacteristic
29C2×#14556048C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×#14556039C2211yescharacteristic
32C2×C4×C71⋊C8C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985897C2211yescharacteristic
35#17985873—21no—
36#17985873—21no—
37#17985869C2211yescharacteristic
38#17985856—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×#125×C71C2211yescharacteristic
50#389×C71—21no—
51#389×C71C2211yescharacteristic
52#389×C71C2211yescharacteristic
53#387×C71—21no—
54#387×C71—21no—
55C4×C4⋊C4×C71—21no—
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C4×D284—22no—
69C4×D284—22no—
70C4×D284—22no—
71C4×D284—41no—
72C4×D284—41no—
73C4×D284C2×C414yes—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71C2311yescharacteristic
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×Dic71⋊C4—21no—
80C2×Dic71⋊C4—21no—
81C4⋊C4×D71—44no—
82C2×(C2×D71)⋊C4—21no—
83C2×(C2×D71)⋊C4—21no—
84C2×(C2×D71)⋊C4—21no—
85#14556061C2×C414yescomplemented
86#14556059—41no—
87#14556059—41no—
88#14556059C2×C414yescomplemented
89C4⋊(C71⋊C8)C2×C414yescomplemented
90#14556048—22no—
91#14556048C2×C414yes—
92#14556048C2×C414yes—
93#14556048Q814yes—
94#14556039—22no—
95#14556039—41no—
96C4×C71⋊C8—41no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99#125×C71—22no—
100#125×C71C2×C414yescomplemented
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71D411yescharacteristic
106C2×C4⋊C4×C71D411yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C42×C71—21no—
112C2×C42×C71—21no—
113C2×C42×C71—21no—
114C2×C42×C71C2311yescharacteristic
115C2×C42×C71C2311yescharacteristic
116C2×C42×C71C2311yescharacteristic
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120#95×C71—21no—
121#95×C71—21no—
122#95×C71D411yescharacteristic
123#95×C71D411yescharacteristic
124C23×D71—21no—
125C2×D284—21no—
126C2×D284—21no—
127C2×D284—21no—
128C2×D284C22×C412yes—
129C2×D284C2×Q812yes—
130C2×D284#4112yes—
131C22×Dic71—21no—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C2×C4×D71—24no—
135C2×C4×D71—42no—
136C2×C4×D71—42no—
137C2×C4×D71—42no—
138C2×C4×D71—42no—
139C2×C4×D71—42no—
140C2×C4×D71—42no—
141C2×C4×D71—81no—
142C2×C4×D71—81no—
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⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8C22×C412yes—
153C2×C71⋊C8C22×C412yes—
154C2×C71⋊C8C2×Q812yes—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—42no—
161C4⋊Dic71—81no—
162C4⋊Dic71C22×C412yes—
163Dic71⋊C4—42no—
164Dic71⋊C4—42no—
165(C2×D71)⋊C4—24no—
166(C2×D71)⋊C4—42no—
167(C2×D71)⋊C4—42no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2411yescharacteristic
179C4⋊C4×C71—24no—
180C4⋊C4×C71—24no—
181C4⋊C4×C71—22no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—41no—
188C4⋊C4×C71—42no—
189C4⋊C4×C71—81no—
190C4⋊C4×C71C22×C412yes—
191C4⋊C4×C71C22×C412yes—
192C4⋊C4×C71C2×Q812yes—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—21no—
196C42×C71—21no—
197C42×C71—42no—
198C42×C71—81no—
199C42×C71—81no—
200C42×C71C22×C412yes—
201C42×C71#4112yes—
202C23×C71C22×D411yescharacteristic
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—41no—
207C22×D71—41no—
208C22×D71—41no—
209C22×D71—41no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
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×C71C4×D412yes—
238C2×C4×C71C4×D412yes—
239C2×C4×C71AGL(1,ℤ/8)12yes—
240C2×C4×C71C23×C411yescharacteristic
241C2×C4×C71C22×Q811yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71H(5,𝔽2)11yescharacteristic
245C2×C4×C712−1+411yescharacteristic
246C2×Dic71—22no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255D284—22no—
256D284—22no—
257D284—22no—
258D284C4×Q814yes—
259C4×D71—48no—
260C4×D71—84no—
261C4×D71—84no—
262C71⋊C8—22no—
263C71⋊C8—22no—
264C71⋊C8—22no—
265C71⋊C8—41no—
266C22×C71C2×C4×D411yescharacteristic
267C22×C71#54611yescharacteristic
268C22×C71D4×Q811yescharacteristic
269C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C71#57511yescharacteristic
271C22×C71#57711yescharacteristic
272C22×C71#57711yescharacteristic
273C2×D71—24no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C2×D71—42no—
279C2×D71—42no—
280C4×C71—22no—
281C4×C71—22no—
282C4×C71—22no—
283C4×C71—42no—
284C4×C71—42no—
285C4×C71—42no—
286C4×C71—41no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—81no—
290C4×C71C2×C4×Q811yescharacteristic
291C4×C71#52011yescharacteristic
292C4×C71#55211yescharacteristic
293C4×C71#55411yescharacteristic
294Dic71—42no—
295Dic71—81no—
296Dic71—81no—
297#22978—711nomaximal, Sylow
298C2×C71#236011yescharacteristic
299C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
300C2×C71#295211yescharacteristic
301C2×C71#330711yescharacteristic
302C2×C71#331011yescharacteristic
303C2×C71#338411yescharacteristic
304C2×C71#338811yescharacteristic
305D71—44no—
306C2×#478—711no—
307#2370D7111yescharacteristic, complemented
308D4×C4⋊C4—711no—
309#1992—711no—
310#1948—711no—
311#1943—711no—
312#1941—711no—
313#1920—711no—
314#1918—711no—
315#1864—711no—
316#1864—711no—
317#1771—711no—
318#1762—711no—
319#1762—711no—
320C4×#102—711no—
321C71#2297811yescharacteristic, complemented, Sylow
322C2×#125C2×D7111yescharacteristic
323C2×C4×D4—711no—
324C2×C4×D4—711no—
325C2×C4×D4—711no—
326C22×C4⋊C4—1421no—
327#478—718no—
328C2×#107—711no—
329C2×#106—711no—
330C2×#106—711no—
331C2×#106—711no—
332C2×C4⋊C8—711no—
333C2×#102—711no—
334C2×#102—711no—
335C2×#102—711no—
336C2×#102—711no—
337C2×#97—711no—
338C2×#97—711no—
339C2×C4×C8—711no—
340#389—21no—
341#389—711no—
342#389—711no—
343#389C2×D7111yescharacteristic
344#389C2×D7111yescharacteristic
345#387—21no—
346#387—21no—
347#385—1421no—
348#385—1421no—
349#384—711no—
350#380—1421no—
351C4×C4⋊C4—21no—
352C4×C4⋊C4C2×D7111yescharacteristic
353C4×C4⋊C4C2×D7111yescharacteristic
354C4×C4⋊C4C2×D7111yescharacteristic
355C4×C4⋊C4C2×D7111yescharacteristic
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#336—711no—
365#336—711no—
366C22×D4—711no—
367C23×C4—1421no—
368C23×C4—1421no—
369C23×C4—1421no—
370C22×C8—711no—
371C22×C8—711no—
372C22×C8—711no—
373C22×C8—711no—
374#125—22no—
375#125C4×D7114yescomplemented
376C4×D4—2841no—
377C4×D4—2841no—
378C4×D4—714no—
379C4×D4—1422no—
380C4×D4—1422no—
381C4×D4—1422no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—2844no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4C22⋊D7111yescharacteristic
395C2×C4⋊C4C22⋊D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C4⋊C4C22×D7111yescharacteristic
399C2×C4⋊C4C22×D7111yescharacteristic
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C22⋊C4—1421no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C22×D7111yescharacteristic
407C2×C42C22×D7111yescharacteristic
408C2×C42C22×D7111yescharacteristic
409#107—714no—
410#106—2841no—
411#106—2841no—
412#106—714no—
413C4⋊C8—714no—
414#102—714no—
415#102—714no—
416#102—714no—
417#102—1422no—
418#97—2841no—
419#97—1422no—
420C4×C8—2841no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—1421no—
427#95—1421no—
428#95C22⋊D7111yescharacteristic
429#95C22⋊D7111yescharacteristic
430C24—1421no—
431C2×D4—712no—
432C2×D4—712no—
433C2×D4—712no—
434C2×D4—1421no—
435C2×D4—1421no—
436C2×D4—1421no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2842no—
442C22×C4—2842no—
443C22×C4—2842no—
444C22×C4—2842no—
445C22×C4—2842no—
446C22×C4—2842no—
447C22×C4—5681no—
448C22×C4—5681no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1421no—
452C22×C4—1424no—
453C22×C4C2×C22⋊D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C2×C22⋊D7111yescharacteristic
456C22×C4C2×C22⋊D7111yescharacteristic
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C23×D7111yescharacteristic
460C2×C8—712no—
461C2×C8—712no—
462C2×C8—712no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—2842no—
484C4⋊C4—2842no—
485C4⋊C4—2842no—
486C4⋊C4—5681no—
487C4⋊C4—712no—
488C4⋊C4—1421no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4C2×C4×D7112yes—
494C4⋊C4C2×C4×D7112yes—
495C4⋊C4Q8×D7112yes—
496C22⋊C4—2842no—
497C22⋊C4—2842no—
498C22⋊C4—1424no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—81no—
505C42—81no—
506C42C2×C4×D7112yes—
507C42#1176586512yes—
508C23—2841no—
509C23—2841no—
510C23—2841no—
511C23—2841no—
512C23—1422no—
513C23—1422no—
514C23—1422no—
515C23C22×C22⋊D7111yescharacteristic
516D4—714no—
517D4—1422no—
518D4—1422no—
519D4—1422no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—2848no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—5684no—
557C2×C4—5684no—
558C2×C4—1422no—
559C2×C4C4×C22⋊D7112yes—
560C2×C4C4×C22⋊D7112yes—
561C2×C4#1455615312yes—
562C2×C4C22×C4×D7111yescharacteristic
563C2×C4C2×Q8×D7111yescharacteristic
564C2×C4#41×D7111yescharacteristic
565C2×C4C2×#1176586511yescharacteristic
566C2×C4#1455617411yescharacteristic
567C2×C4#1455617611yescharacteristic
568C8—2841no—
569C8—1422no—
570C8—1422no—
571C8—1422no—
572C22—2842no—
573C22—2842no—
574C22—2842no—
575C22—2842no—
576C22—2842no—
577C22—2842no—
578C22—1424no—
579C22C2×C4×C22⋊D7111yescharacteristic
580C22#1798663711yescharacteristic
581C22Q8×C22⋊D7111yescharacteristic
582C22C2×#1455615311yescharacteristic
583C22#1798679711yescharacteristic
584C22#1798680711yescharacteristic
585C22#1798681011yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—2842no—
597C4—5681no—
598C4—5681no—
599C4C4×Q8×D7111yescharacteristic
600C4#1798648111yescharacteristic
601C4#1798669011yescharacteristic
602C4#1798670911yescharacteristic
603C2—2844no—
604C2#2259134111yescharacteristic
605C2C4×#1455615311yescharacteristic
606C2#2259722111yescharacteristic
607C2#2260031011yescharacteristic
608C2#2260035511yescharacteristic
609C2#2260123911yescharacteristic
610C2#2260127911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2370C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2370D71
hypocenterC71#22978
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
3. #22589310C2semidirect
4. #22589103C2semidirect
5. #22589075C2semidirect
6. #22589065C2semidirect
7. #22588925C2semidirect
8. #22588911C2semidirect
11. #22588215C2semidirect
15. C71×#2370C2semidirect
20. #17986296C4semidirect
85. #14556061C2×C4semidirect
88. #14556059C2×C4semidirect
89. C4⋊(C71⋊C8)C2×C4semidirect
100. #125×C71C2×C4semidirect
307. #2370D71semidirect
321. C71#22978semidirect
375. #125C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862962
C4⋊C4×D2842
#225893102
#225891032
#225890752
#225890652
#225889252
#225889112
#225886002
#225882152
#225881682
C4×#145560482
C71×#23702
#2297871