Finite Group Databasebeta

← #32019297

Subgroups — #32019297

Order 18176.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986299C211yesmaximal, characteristic
2#22591434C211yesmaximal, characteristic, complemented
3#22589304C211yesmaximal, characteristic, complemented
4#22589106C211yesmaximal, characteristic, complemented
5#22589070C211yesmaximal, characteristic, complemented
6#22589065C211yesmaximal, characteristic, complemented
7#22588929C211yesmaximal, characteristic, complemented
8#22588922C211yesmaximal, characteristic, complemented
9#22588601C211yesmaximal, characteristic
10#22588587C211yesmaximal, characteristic
11C4⋊C4×C71⋊C8C211yesmaximal, characteristic, complemented
12C4×#14556049C211yesmaximal, characteristic
13C4×#14556049C211yesmaximal, characteristic
14C4×#14556049C211yesmaximal, characteristic
15C71×#2349C211yesmaximal, characteristic
16C2×#14556120C2211yescharacteristic
17#17986299C418yescomplemented
18C2×#14556061C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×#14556049C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×C4×C71⋊C8C2211yescharacteristic
28C2×C4×C71⋊C8C2211yescharacteristic
29C2×C4×C71⋊C8C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985897C2211yescharacteristic
32#17985894—21no—
33#17985881—21no—
34#17985880—21no—
35C4×C4⋊Dic71C2211yescharacteristic
36C4×C4⋊Dic71C2211yescharacteristic
37C4×C4⋊Dic71C2211yescharacteristic
38#17985850—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×D4×C71C2211yescharacteristic
50C2×C4×D4×C71C2211yescharacteristic
51C2×C4×D4×C71C2211yescharacteristic
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#385×C71—21no—
55#385×C71—21no—
56#382×C71C2211yescharacteristic
57#380×C71—21no—
58C4×C4⋊C4×C71C2211yescharacteristic
59C2×#95×C71—21no—
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556120—22no—
65#14556120C2×C414yescomplemented
66C2×C4⋊Dic71C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—21no—
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76C2×C4×Dic71—21no—
77#14556061—41no—
78#14556061C2×C414yescomplemented
79#14556059—41no—
80#14556059C2×C414yescomplemented
81C4⋊(C71⋊C8)C2×C414yescomplemented
82#14556049—22no—
83#14556049C2×C414yes—
84#14556049C2×C414yes—
85#14556049Q814yes—
86C4×C71⋊C8—22no—
87C4×C71⋊C8—41no—
88C4×C71⋊C8—41no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C22×D4×C71C2311yescharacteristic
95C23×C4×C71—21no—
96C23×C4×C71—21no—
97C23×C4×C71—21no—
98C4×D4×C71—22no—
99C4×D4×C71—22no—
100C4×D4×C71—22no—
101C4×D4×C71—41no—
102C4×D4×C71—41no—
103C4×D4×C71C2×C414yes—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71D411yescharacteristic
107C2×C4⋊C4×C71D411yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×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×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8C22×C412yes—
137C2×C71⋊C8C22×C412yes—
138C2×C71⋊C8C2×Q812yes—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71C22×C412yes—
145C4⋊Dic71C22×C412yes—
146C4⋊Dic71C2×Q812yes—
147Dic71⋊C4—24no—
148Dic71⋊C4—24no—
149Dic71⋊C4—42no—
150Dic71⋊C4—81no—
151C4×Dic71—42no—
152C4×Dic71—81no—
153C4×Dic71—81no—
154C24×C71—21no—
155C2×D4×C71—21no—
156C2×D4×C71—21no—
157C2×D4×C71—21no—
158C2×D4×C71C22×C412yes—
159C2×D4×C71C2×Q812yes—
160C2×D4×C71#4112yes—
161C22×C4×C71—21no—
162C22×C4×C71—24no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—42no—
166C22×C4×C71—42no—
167C22×C4×C71—42no—
168C22×C4×C71—42no—
169C22×C4×C71—81no—
170C22×C4×C71—81no—
171C22×C4×C71—81no—
172C22×C4×C71—81no—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2411yescharacteristic
180C4⋊C4×C71—22no—
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—81no—
188C4⋊C4×C71—81no—
189C4⋊C4×C71C22×C412yes—
190C22⋊C4×C71—24no—
191C22⋊C4×C71—42no—
192C22⋊C4×C71—42no—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—21no—
196C42×C71—21no—
197C42×C71C22×C412yes—
198C42×C71#4112yes—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—41no—
203C23×C71—41no—
204C23×C71—41no—
205C23×C71—41no—
206C23×C71C22×D411yescharacteristic
207D4×C71—22no—
208D4×C71—22no—
209D4×C71—22no—
210D4×C71C4×Q814yes—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
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—84no—
235C2×C4×C71—84no—
236C2×C4×C71—84no—
237C2×C4×C71C4×D412yes—
238C2×C4×C71C4×D412yes—
239C2×C4×C71#13512yes—
240C2×C4×C71#13512yes—
241C2×C4×C71C23×C411yescharacteristic
242C2×C4×C71C22×Q811yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71H(5,𝔽2)11yescharacteristic
246C2×C4×C712−1+411yescharacteristic
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—22no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C71⋊C8—22no—
260C71⋊C8—22no—
261C71⋊C8—22no—
262C71⋊C8—41no—
263C22×C71—24no—
264C22×C71—42no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71C2×C4×D411yescharacteristic
271C22×C71#54611yescharacteristic
272C22×C71D4×Q811yescharacteristic
273C22×C71C2×#13511yescharacteristic
274C22×C71C2×#13511yescharacteristic
275C22×C71#57611yescharacteristic
276C22×C71#57711yescharacteristic
277C4×C71—22no—
278C4×C71—22no—
279C4×C71—22no—
280C4×C71—22no—
281C4×C71—42no—
282C4×C71—41no—
283C4×C71—41no—
284C4×C71—81no—
285C4×C71—81no—
286C4×C71C2×C4×Q811yescharacteristic
287C4×C71#52011yescharacteristic
288C4×C71#55211yescharacteristic
289C4×C71#55411yescharacteristic
290Dic71—42no—
291Dic71—42no—
292Dic71—42no—
293Dic71—81no—
294#22966—711nomaximal, Sylow
295C2×C71—44no—
296C2×C71#236011yescharacteristic
297C2×C71C4×#13511yescharacteristic
298C2×C71#294611yescharacteristic
299C2×C71#329611yescharacteristic
300C2×C71#329711yescharacteristic
301C2×C71#338011yescharacteristic
302C2×C71#338111yescharacteristic
303C2×#478—711no—
304#2370—711no—
305#2349D7111yescharacteristic
306#1991—711no—
307#1949—711no—
308#1942—711no—
309#1941—711no—
310#1921—711no—
311#1920—711no—
312#1864—711no—
313#1862—711no—
314C8×C4⋊C4—711no—
315C4×#102—711no—
316C4×#102—711no—
317C4×#102—711no—
318C71#2296611yescharacteristic, complemented, Sylow
319C2×#125—711no—
320C2×C4×D4C2×D7111yescharacteristic
321C2×C4×D4C2×D7111yescharacteristic
322C2×C4×D4C2×D7111yescharacteristic
323#478—718no—
324C2×#107—711no—
325C2×#107—711no—
326C2×#106—711no—
327C2×#106—711no—
328C2×C4⋊C8—711no—
329C2×#102—711no—
330C2×#102—711no—
331C2×#102—711no—
332C2×#102—711no—
333C2×C4×C8—711no—
334C2×C4×C8—711no—
335C2×C4×C8—711no—
336#389—711no—
337#389—711no—
338#389—1421no—
339#389C2×D7111yescharacteristic
340#389C2×D7111yescharacteristic
341#387—1421no—
342#387—1421no—
343#385—21no—
344#385—21no—
345#382C2×D7111yescharacteristic
346#380—21no—
347C4×C4⋊C4—711no—
348C4×C4⋊C4—711no—
349C4×C4⋊C4—711no—
350C4×C4⋊C4—1421no—
351C4×C4⋊C4C2×D7111yescharacteristic
352C2×#95—21no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#336—711no—
362#336—711no—
363C22×D4C22×D7111yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—21no—
367C22×C8—711no—
368C22×C8—711no—
369C22×C8—711no—
370C22×C8—711no—
371#125—714no—
372#125—1422no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—41no—
377C4×D4—41no—
378C4×D4C4×D7114yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—711no—
382C2×C4⋊C4—711no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—1421no—
386C2×C4⋊C4—1421no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4C22⋊D7111yescharacteristic
390C2×C4⋊C4C22⋊D7111yescharacteristic
391C2×C4⋊C4C22×D7111yescharacteristic
392C2×C4⋊C4C22×D7111yescharacteristic
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C42—1421no—
398C2×C42—1421no—
399C2×C42—1421no—
400C2×C42C22×D7111yescharacteristic
401C2×C42C22×D7111yescharacteristic
402C2×C42C22×D7111yescharacteristic
403#107—2841no—
404#107—714no—
405#106—2841no—
406#106—714no—
407C4⋊C8—714no—
408#102—714no—
409#102—714no—
410#102—714no—
411#102—1422no—
412C4×C8—2841no—
413C4×C8—2841no—
414C4×C8—1422no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1421no—
420#95—1421no—
421#95—1421no—
422#95—1421no—
423#95—1421no—
424#95C22⋊D7111yescharacteristic
425#95C22⋊D7111yescharacteristic
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D7112yes—
431C2×D4Q8×D7112yes—
432C2×D4#1176586412yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—1421no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4C2×C22⋊D7111yescharacteristic
450C22×C4C2×C22⋊D7111yescharacteristic
451C22×C4C2×C22⋊D7111yescharacteristic
452C22×C4C2×C22⋊D7111yescharacteristic
453C22×C4C2×C22⋊D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C23×D7111yescharacteristic
456C2×C8—712no—
457C2×C8—712no—
458C2×C8—712no—
459C2×C8—1421no—
460C2×C8—1421no—
461C2×C8—1421no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—2842no—
478C4⋊C4—5681no—
479C4⋊C4—712no—
480C4⋊C4—712no—
481C4⋊C4—712no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4—1421no—
487C4⋊C4—1424no—
488C4⋊C4—1424no—
489C4⋊C4C2×C4×D7112yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—2842no—
498C42—5681no—
499C42—5681no—
500C42C2×C4×D7112yes—
501C42#1176586512yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C22⋊D7111yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4#1455609314yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—2841no—
541C2×C4—2841no—
542C2×C4—2841no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4C4×C22⋊D7112yes—
553C2×C4C4×C22⋊D7112yes—
554C2×C4#1455614812yes—
555C2×C4#1455614812yes—
556C2×C4C22×C4×D7111yescharacteristic
557C2×C4C2×Q8×D7111yescharacteristic
558C2×C4C2×#1176586411yescharacteristic
559C2×C4C2×#1176586511yescharacteristic
560C2×C4#1455617311yescharacteristic
561C2×C4#1455617611yescharacteristic
562C8—2841no—
563C8—1422no—
564C8—1422no—
565C8—1422no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×C4×C22⋊D7111yescharacteristic
574C22#1798664311yescharacteristic
575C22Q8×C22⋊D7111yescharacteristic
576C22C2×#1455614811yescharacteristic
577C22C2×#1455614811yescharacteristic
578C22#1798680211yescharacteristic
579C22#1798681011yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—2842no—
590C4—2842no—
591C4—2842no—
592C4—5681no—
593C4C2×#1455609311yescharacteristic
594C4#1798648011yescharacteristic
595C4#1798668411yescharacteristic
596C4#1798670411yescharacteristic
597C2—44no—
598C2#2259134511yescharacteristic
599C2C4×#1455614811yescharacteristic
600C2#2259717211yescharacteristic
601C2#2260017511yescharacteristic
602C2#2260019011yescharacteristic
603C2#2260120311yescharacteristic
604C2#2260121511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2349C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2349D71
hypocenterC71#22966
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #22591434C2semidirect
3. #22589304C2semidirect
4. #22589106C2semidirect
5. #22589070C2semidirect
6. #22589065C2semidirect
7. #22588929C2semidirect
8. #22588922C2semidirect
11. C4⋊C4×C71⋊C8C2semidirect
17. #17986299C4semidirect
65. #14556120C2×C4semidirect
78. #14556061C2×C4semidirect
80. #14556059C2×C4semidirect
81. C4⋊(C71⋊C8)C2×C4semidirect
318. C71#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#179862992
#225914342
#225893042
#225891062
#225890702
#225890652
#225889292
#225889222
#225886012
#225885872
C4⋊C4×C71⋊C82
C4×#145560492
C71×#23492
#2296671