Finite Group Databasebeta

← #31889495

Subgroups — #31889495

Order 18176.

760 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588360C211yesmaximal, characteristic
2C4×#14556052C211yesmaximal, characteristic, complemented
3#22586839C211yesmaximal, characteristic, complemented
4#22586795C211yesmaximal, characteristic, complemented
5#22586769C211yesmaximal, characteristic, complemented
6#22586722C211yesmaximal, characteristic
7Q8×#41×C71C211yesmaximal, characteristic
8#17986052C2211yescharacteristic
9#17986023C2211yescharacteristic
10C2×#14556055—21no—
11C2×#14556052—21no—
12C2×C4×C71⋊C8C2211yescharacteristic
13C4×C4⋊Dic71C2211yescharacteristic
14#17985761—21no—
15#17985758—21no—
16#17985736—22no—
17Q8⋊(C71⋊C8)—21no—
18Q8⋊(C71⋊C8)—21no—
19Q8⋊(C71⋊C8)—21no—
20Q8⋊(C71⋊C8)—22no—
21D4⋊(C71⋊C8)—21no—
22Q82×C71C411yescharacteristic, complemented
23#557×C71—21no—
24#557×C71C411yescharacteristic, complemented
25#554×C71—22no—
26#554×C71—21no—
27#554×C71—21no—
28#554×C71C411yescharacteristic, complemented
29D4×Q8×C71—21no—
30D4×Q8×C71C411yescharacteristic, complemented
31#533×C71—22no—
32#533×C71—21no—
33#533×C71—21no—
34#533×C71C2211yescharacteristic
35C4×#41×C71—21no—
36C4×#41×C71C2211yescharacteristic
37C2×C4×Q8×C71—21no—
38C2×C4×Q8×C71C2211yescharacteristic
39C2×#11765849—21no—
40C22×C71⋊C8—21no—
41C2×C4⋊Dic71—21no—
42C2×C4×Dic71—21no—
43C4⋊(C71⋊C8)—21no—
44C4⋊(C71⋊C8)—21no—
45C4⋊(C71⋊C8)—21no—
46#14556055—42no—
47#14556052—24no—
48C22⋊(C71⋊C8)—21no—
49C4×C71⋊C8—21no—
50C4×C71⋊C8—21no—
51C4×C71⋊C8—21no—
52C71⋊#95—21no—
53C2×#41×C71D411yescharacteristic
54C22×Q8×C71—21no—
55C22×Q8×C71D411yescharacteristic
56C4⋊Q8×C71—22no—
57C4⋊Q8×C71—21no—
58C4⋊Q8×C71—22no—
59C4⋊Q8×C71—21no—
60C4⋊Q8×C71—21no—
61C4⋊Q8×C71—21no—
62C4⋊Q8×C71C2×C411yescharacteristic
63C4⋊Q8×C71C2×C411yescharacteristic
64#125×C71—22no—
65#125×C71—22no—
66#125×C71—22no—
67#125×C71—21no—
68#125×C71—21no—
69#125×C71—21no—
70C22⋊Q8×C71—22no—
71C22⋊Q8×C71—22no—
72C22⋊Q8×C71—22no—
73C22⋊Q8×C71—22no—
74C22⋊Q8×C71—22no—
75C22⋊Q8×C71—21no—
76C22⋊Q8×C71—22no—
77C22⋊Q8×C71—21no—
78C22⋊Q8×C71—21no—
79C22⋊Q8×C71—22no—
80C22⋊Q8×C71D412yes—
81C4×Q8×C71—22no—
82C4×Q8×C71—22no—
83C4×Q8×C71—22no—
84C4×Q8×C71—21no—
85C4×Q8×C71—21no—
86C4×Q8×C71—21no—
87C4×Q8×C71—21no—
88C4×Q8×C71—21no—
89C4×Q8×C71—21no—
90C4×Q8×C71—21no—
91C4×Q8×C71C2×C411yescharacteristic
92C4×Q8×C71C2×C411yescharacteristic
93C4×Q8×C71C2×C411yescharacteristic
94C4×Q8×C71D411yescharacteristic
95C4×Q8×C71D411yescharacteristic
96C4×Q8×C71D411yescharacteristic
97C4×Q8×C71D412yes—
98C4×D4×C71—22no—
99C4×D4×C71—22no—
100C4×D4×C71—21no—
101C4×D4×C71—21no—
102C4×D4×C71—21no—
103C4×D4×C71—21no—
104C4×D4×C71C2×C411yescharacteristic
105C4×D4×C71D411yescharacteristic
106#117×C71—22no—
107#117×C71—21no—
108#117×C71—21no—
109#117×C71D411yescharacteristic
110C2×C4⋊C4×C71—22no—
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—21no—
113C2×C4⋊C4×C71D411yescharacteristic
114C2×C42×C71—22no—
115C2×C42×C71—21no—
116C2×C42×C71—21no—
117C2×C42×C71C2311yescharacteristic
118C22×Dic71—41no—
119#11765849—41no—
120#11765849—41no—
121C2×C71⋊C8—22no—
122C2×C71⋊C8—41no—
123C2×C71⋊C8—41no—
124C2×C71⋊C8—41no—
125C2×C71⋊C8—41no—
126C4⋊Dic71—22no—
127C4⋊Dic71—41no—
128C4×Dic71—41no—
129C4×Dic71—41no—
130#41×C71—21no—
131#41×C71—21no—
132#41×C71SD1612yes—
133#41×C71Q1612yes—
134C2×Q8×C71—22no—
135C2×Q8×C71—22no—
136C2×Q8×C71—21no—
137C2×Q8×C71—21no—
138C2×Q8×C71—22no—
139C2×Q8×C71—41no—
140C2×Q8×C71—42no—
141C2×Q8×C71—42no—
142C2×Q8×C71C22⋊C411yescharacteristic
143C2×Q8×C71C22⋊C411yescharacteristic
144C2×Q8×C71C22⋊C411yescharacteristic
145C2×Q8×C71C22⋊C412yes—
146C2×D4×C71—21no—
147C2×D4×C71C22⋊C411yescharacteristic
148C22×C4×C71—21no—
149C22×C4×C71—22no—
150C22×C4×C71—21no—
151C22×C4×C71—22no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71C2×D411yescharacteristic
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C4⋊C4×C71—22no—
159C4⋊C4×C71—22no—
160C4⋊C4×C71—22no—
161C4⋊C4×C71—22no—
162C4⋊C4×C71—22no—
163C4⋊C4×C71—22no—
164C4⋊C4×C71—22no—
165C4⋊C4×C71—22no—
166C4⋊C4×C71—21no—
167C4⋊C4×C71—21no—
168C4⋊C4×C71—21no—
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—22no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—22no—
178C4⋊C4×C71C22⋊C411yescharacteristic
179C4⋊C4×C71C22⋊C411yescharacteristic
180C4⋊C4×C71C22⋊C411yescharacteristic
181C4⋊C4×C71C22⋊C411yescharacteristic
182C4⋊C4×C71C22⋊C412yes—
183C22⋊C4×C71—22no—
184C22⋊C4×C71—22no—
185C22⋊C4×C71—22no—
186C22⋊C4×C71—21no—
187C22⋊C4×C71—21no—
188C22⋊C4×C71—21no—
189C42×C71—22no—
190C42×C71—22no—
191C42×C71—21no—
192C42×C71—21no—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—22no—
196C42×C71—21no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71C22×C411yescharacteristic
200C42×C71C2×D411yescharacteristic
201C42×C71C2×D411yescharacteristic
202C42×C71C2×D411yescharacteristic
203C23×C71—21no—
204C23×C71#12011yescharacteristic
205Q8×C71—21no—
206Q8×C71—21no—
207Q8×C71—21no—
208Q8×C71—21no—
209Q8×C71—42no—
210Q8×C71—42no—
211Q8×C71—44no—
212Q8×C71—41no—
213Q8×C71—41no—
214Q8×C71—82no—
215Q8×C71#10312yescomplemented
216Q8×C71#10412yescomplemented
217D4×C71—22no—
218D4×C71—22no—
219D4×C71—21no—
220D4×C71#10312yescomplemented
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—21no—
228C2×C4×C71—22no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—22no—
233C2×C4×C71—21no—
234C2×C4×C71—41no—
235C2×C4×C71—42no—
236C2×C4×C71—44no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—81no—
240C2×C4×C71—81no—
241C2×C4×C71—82no—
242C2×C4×C71C2×C22⋊C411yescharacteristic
243C2×C4×C71C2×C22⋊C411yescharacteristic
244C2×C4×C71C2×C22⋊C411yescharacteristic
245C2×C4×C71#12011yescharacteristic
246C2×C4×C71#12011yescharacteristic
247C2×C4×C71#12011yescharacteristic
248C2×C4×C71C2×SD1611yescharacteristic
249C2×C4×C71C2×Q1611yescharacteristic
250C2×C4×C71#13711yescharacteristic
251C2×C4×C71#13711yescharacteristic
252C2×Dic71—42no—
253C2×Dic71—81no—
254C2×Dic71—81no—
255C71⋊C8—42no—
256C71⋊C8—81no—
257C71⋊C8—81no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—22no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71#37911yescharacteristic
264C22×C71#44811yescharacteristic
265C22×C71#45111yescharacteristic
266C4×C71—21no—
267C4×C71—21no—
268C4×C71—22no—
269C4×C71—21no—
270C4×C71—22no—
271C4×C71—41no—
272C4×C71—42no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—42no—
277C4×C71—41no—
278C4×C71C2×#10311yescharacteristic
279C4×C71#41911yescharacteristic
280C4×C71C2×#10411yescharacteristic
281C4×C71#42111yescharacteristic
282C4×C71#44811yescharacteristic
283C4×C71#45111yescharacteristic
284Dic71—82no—
285#8439—711nomaximal, Sylow
286C2×C71—21no—
287C2×C71—21no—
288C2×C71—42no—
289C2×C71#178711yescharacteristic
290C2×C71#178911yescharacteristic
291C2×C71#179411yescharacteristic
292Q8×#41D7111yescharacteristic
293#1806—711no—
294C4×#103—711no—
295#1503—711no—
296#1493—711no—
297#1487—711no—
298#1476—711no—
299C71#843911yescharacteristic, complemented, Sylow
300Q82Dic7111yescharacteristic, complemented
301#557—21no—
302#557Dic7111yescharacteristic, complemented
303#554—22no—
304#554—21no—
305#554—21no—
306#554Dic7111yescharacteristic, complemented
307D4×Q8—21no—
308D4×Q8Dic7111yescharacteristic, complemented
309#533—22no—
310#533—21no—
311#533—21no—
312#533C2×D7111yescharacteristic
313C4×#41—21no—
314C4×#41C2×D7111yescharacteristic
315C2×C4×Q8—21no—
316C2×C4×Q8C2×D7111yescharacteristic
317#431—711no—
318#423—711no—
319C2×#104—1421no—
320C2×#103—1421no—
321C2×C4×C8—711no—
322C4×C4⋊C4—711no—
323#340—1421no—
324#339—1421no—
325#328—1422no—
326Q8⋊C8—1421no—
327Q8⋊C8—1421no—
328Q8⋊C8—1421no—
329Q8⋊C8—1422no—
330D4⋊C8—1421no—
331C2×#41C22⋊D7111yescharacteristic
332C22×Q8—21no—
333C22×Q8C22⋊D7111yescharacteristic
334C2×M16—1421no—
335C22×C8—1421no—
336C4⋊Q8—22no—
337C4⋊Q8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8C2×Dic7111yescharacteristic
343C4⋊Q8C2×Dic7111yescharacteristic
344#125—22no—
345#125—22no—
346#125—22no—
347#125—21no—
348#125—21no—
349#125—21no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—22no—
360C22⋊Q8C22⋊D7112yes—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8C2×Dic7111yescharacteristic
372C4×Q8C2×Dic7111yescharacteristic
373C4×Q8C2×Dic7111yescharacteristic
374C4×Q8C22⋊D7111yescharacteristic
375C4×Q8C22⋊D7111yescharacteristic
376C4×Q8C22⋊D7111yescharacteristic
377C4×Q8C22⋊D7112yes—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4C2×Dic7111yescharacteristic
385C4×D4C22⋊D7111yescharacteristic
386#117—22no—
387#117—21no—
388#117—21no—
389#117C22⋊D7111yescharacteristic
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4C22⋊D7111yescharacteristic
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—1421no—
399C2×C42C22×D7111yescharacteristic
400C4⋊C8—1421no—
401C4⋊C8—1421no—
402C4⋊C8—1421no—
403#104—2842no—
404#103—1424no—
405C22⋊C8—1421no—
406C4×C8—1421no—
407C4×C8—1421no—
408C4×C8—1421no—
409#95—1421no—
410#41—21no—
411#41—21no—
412#41#1176585212yes—
413#41#1176585512yes—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8C22⋊Dic7111yescharacteristic
423C2×Q8C22⋊Dic7111yescharacteristic
424C2×Q8C22⋊Dic7111yescharacteristic
425C2×Q8C22⋊Dic7112yes—
426C2×D4—21no—
427C2×D4C22⋊Dic7111yescharacteristic
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—2841no—
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439M16—2841no—
440M16—2841no—
441C2×C8—2841no—
442C2×C8—2841no—
443C2×C8—2841no—
444C2×C8—2841no—
445C2×C8—1422no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—2841no—
467C4⋊C4—1422no—
468C4⋊C4C22⋊Dic7111yescharacteristic
469C4⋊C4C22⋊Dic7111yescharacteristic
470C4⋊C4C22⋊Dic7111yescharacteristic
471C4⋊C4C22⋊Dic7111yescharacteristic
472C4⋊C4C22⋊Dic7112yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—2841no—
490C42—2841no—
491C42C22×Dic7111yescharacteristic
492C42C2×C22⋊D7111yescharacteristic
493C42C2×C22⋊D7111yescharacteristic
494C42C2×C22⋊D7111yescharacteristic
495C23—21no—
496C23#1455609711yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—41no—
505Q8—41no—
506Q8—82no—
507Q8#1455605212yescomplemented
508Q8#1455605512yescomplemented
509D4—22no—
510D4—22no—
511D4—21no—
512D4#1455605212yescomplemented
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—2842no—
535C2×C4—5681no—
536C2×C4—5681no—
537C2×C4C2×C22⋊Dic7111yescharacteristic
538C2×C4C2×C22⋊Dic7111yescharacteristic
539C2×C4C2×C22⋊Dic7111yescharacteristic
540C2×C4#1455609711yescharacteristic
541C2×C4#1455609711yescharacteristic
542C2×C4#1455609711yescharacteristic
543C2×C4C2×#1176585211yescharacteristic
544C2×C4C2×#1176585511yescharacteristic
545C2×C4#1455615911yescharacteristic
546C2×C4#1455615911yescharacteristic
547C8—2842no—
548C8—5681no—
549C8—5681no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22#1798585511yescharacteristic
556C22#1798613811yescharacteristic
557C22#1798615511yescharacteristic
558C4—22no—
559C4—21no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—5682no—
571C4C2×#1455605211yescharacteristic
572C4#1798600811yescharacteristic
573C4C2×#1455605511yescharacteristic
574C4#1798601611yescharacteristic
575C4#1798613811yescharacteristic
576C4#1798615511yescharacteristic
577C2—21no—
578C2—21no—
579C2—42no—
580C2#2258827511yescharacteristic
581C2#2258828411yescharacteristic
582C2#2258830411yescharacteristic
subgroupG/H
centerC2×C4#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingQ8×#41×C71C2
socleC22×C71#379
prime radicalC2×#41×C71D4
cosocleC2×C42×C71C23
hypercenterQ8×#41D71
hypocenterC71#8439
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
2. C4×#14556052C2semidirect
3. #22586839C2semidirect
4. #22586795C2semidirect
5. #22586769C2semidirect
22. Q82×C71C4semidirect
24. #557×C71C4semidirect
28. #554×C71C4semidirect
30. D4×Q8×C71C4semidirect
215. Q8×C71#103semidirect
216. Q8×C71#104semidirect
220. D4×C71#103semidirect
299. C71#8439semidirect
300. Q82Dic71semidirect
302. #557Dic71semidirect
306. #554Dic71semidirect
308. D4×Q8Dic71semidirect
507. Q8#14556052semidirect
508. Q8#14556055semidirect
512. D4#14556052semidirect

Maximal subgroups

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

subgroupindex
#225883602
C4×#145560522
#225868392
#225867952
#225867692
#225867222
Q8×#41×C712
#843971