Finite Group Databasebeta

← #33101636

Subgroups — #33101636

Order 18688.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988665C211yesmaximal, characteristic, complemented
2C2×#17988588C212yesmaximal, complemented
3D4×C4⋊Dic73C211yesmaximal, characteristic, complemented
4#22610855C211yesmaximal, characteristic, complemented
5#22610734C211yesmaximal, characteristic, complemented
6#22610365C212yesmaximal, complemented
7#22610239C212yesmaximal, complemented
8#22609931C211yesmaximal, characteristic, complemented
9#22609871C212yesmaximal, complemented
10C4×#14556512C211yesmaximal, characteristic, complemented
11C2×#547×C73C211yesmaximal, characteristic
12C22×#11924012C2212yes—
13C2×D4×Dic73—21no—
14C22×C4⋊Dic73C2212yes—
15#17988665—24no—
16C2×C4⋊#10596831C2211yescharacteristic
17#17988588—28no—
18C2×#14556515C2212yes—
19C2×#14556515C2212yes—
20C2×#14556515C2212yes—
21C2×#14556515C2212yes—
22C2×#14556512C2212yes—
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×C22⋊#10596831C2212yes—
26C2×C4×#10596831C2211yescharacteristic
27#17988486—21no—
28#17988462—22no—
29#17988457C2211yescharacteristic
30#17988447C2212yes—
31C4×C4⋊Dic73C2211yescharacteristic
32#17988349C2212yes—
33#17988349C2211yescharacteristic
34#17988349C2211yescharacteristic
35C22×#41×C73C2212yes—
36#547×C73—24no—
37#547×C73C418yescomplemented
38C2×C4⋊Q8×C73C2211yescharacteristic
39C2×#123×C73—22no—
40C2×C22⋊Q8×C73C2212yes—
41C2×#121×C73—21no—
42C2×C4×D4×C73C2211yescharacteristic
43C2×C4×D4×C73C2211yescharacteristic
44C22×C4⋊C4×C73—21no—
45C23×Dic73—22no—
46C2×#11924012—24no—
47C2×#11924012—24no—
48C2×#11924012—42no—
49C22×#10596831C2312yes—
50C22×#10596831C2311yescharacteristic
51C22×#10596831C2311yescharacteristic
52D4×Dic73—44no—
53C2×C4⋊Dic73—24no—
54C2×C4⋊Dic73—21no—
55C2×C4⋊Dic73—22no—
56C2×C4⋊Dic73—21no—
57C2×C4⋊Dic73—42no—
58C2×C4⋊Dic73C2311yescharacteristic
59C2×C4⋊Dic73C2312yes—
60C2×C4⋊Dic73C2311yescharacteristic
61C2×C22⋊Dic73—22no—
62C2×C4×Dic73—21no—
63C4⋊#10596831—22no—
64#14556515—24no—
65#14556515—24no—
66#14556515—42no—
67#14556515—42no—
68#14556512—24no—
69#14556512—22no—
70#14556512D414yescomplemented
71C22⋊#10596831—24no—
72C4×#10596831—22no—
73#14556498—21no—
74#14556498—22no—
75C2×#41×C73—24no—
76C2×#41×C73—24no—
77C2×#41×C73—24no—
78C2×#41×C73—42no—
79C2×#41×C73—42no—
80C2×#41×C73C2×C418yescomplemented
81C22×Q8×C73C2312yes—
82C22×D4×C73—22no—
83C22×D4×C73D411yescharacteristic
84C22×D4×C73C2311yescharacteristic
85C23×C4×C73—21no—
86C23×C4×C73—22no—
87C23×C4×C73C2312yes—
88C4⋊Q8×C73—22no—
89C4⋊Q8×C73C2×C414yescomplemented
90#123×C73—216no—
91C22⋊Q8×C73—24no—
92C22⋊Q8×C73C2×C418yescomplemented
93#121×C73—28no—
94C4×D4×C73—22no—
95C4×D4×C73C2×C414yes—
96C4×D4×C73C2×C414yescomplemented
97C4×D4×C73D414yes—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—22no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—28no—
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C22⋊C4×C73—22no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73D412yes—
107C2×C42×C73C2311yescharacteristic
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C22×Dic73—44no—
112C22×Dic73—44no—
113C22×Dic73—82no—
114#11924012—48no—
115C2×#10596831—21no—
116C2×#10596831—21no—
117C2×#10596831—21no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—21no—
121C2×#10596831—21no—
122C2×#10596831—22no—
123C2×#10596831C2×D412yes—
124C4⋊Dic73—21no—
125C4⋊Dic73—22no—
126C4⋊Dic73—21no—
127C4⋊Dic73—21no—
128C4⋊Dic73—22no—
129C4⋊Dic73—44no—
130C4⋊Dic73—42no—
131C4⋊Dic73—81no—
132C4⋊Dic73C2×D412yes—
133C22⋊Dic73—44no—
134C4×Dic73—42no—
135C24×C73—21no—
136C24×C73C2×D412yes—
137#41×C73—216no—
138#41×C73—48no—
139#41×C73—48no—
140#41×C73—48no—
141C2×Q8×C73—22no—
142C2×Q8×C73—22no—
143C2×Q8×C73—22no—
144C2×Q8×C73—22no—
145C2×Q8×C73—22no—
146C2×Q8×C73C22×C414yes—
147C2×D4×C73—28no—
148C2×D4×C73—22no—
149C2×D4×C73—22no—
150C2×D4×C73—21no—
151C2×D4×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—44no—
154C2×D4×C73—44no—
155C2×D4×C73C22⋊C414yes—
156C2×D4×C73C22×C412yes—
157C2×D4×C73C2×D412yes—
158C2×D4×C73#4112yes—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—21no—
164C22×C4×C73—22no—
165C22×C4×C73—24no—
166C22×C4×C73—28no—
167C22×C4×C73—44no—
168C22×C4×C73—44no—
169C22×C4×C73—42no—
170C22×C4×C73—42no—
171C22×C4×C73C22×C414yes—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D412yes—
174C22×C4×C73C2×D414yes—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2411yescharacteristic
177C4⋊C4×C73—24no—
178C4⋊C4×C73—28no—
179C4⋊C4×C73—24no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73C22⋊C414yes—
182C4⋊C4×C73C22×C412yes—
183C22⋊C4×C73—28no—
184C22⋊C4×C73—24no—
185C22⋊C4×C73C22⋊C418yes—
186C42×C73C22×C412yes—
187C42×C73C2×D412yes—
188C23×C73—22no—
189C23×C73—22no—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73—22no—
193C23×C73—22no—
194C23×C73—22no—
195C23×C73—22no—
196C23×C73—22no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73C2×C22⋊C414yes—
202C23×C73C22×D411yescharacteristic
203Q8×C73—24no—
204Q8×C73—24no—
205Q8×C73—42no—
206Q8×C73—42no—
207D4×C73—22no—
208D4×C73—22no—
209D4×C73—22no—
210D4×C73—28no—
211D4×C73—416no—
212D4×C73C4×D414yescomplemented
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—24no—
223C2×C4×C73—416no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—48no—
229C2×C4×C73C2×C22⋊C412yes—
230C2×C4×C73C2×C22⋊C414yes—
231C2×C4×C73C2×C22⋊C412yes—
232C2×C4×C73AGL(1,ℤ/8)12yes—
233C2×C4×C73#13712yes—
234C2×C4×C73C23×C411yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C73H(5,𝔽2)11yescharacteristic
240C2×Dic73—22no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—42no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—41no—
249C2×Dic73—88no—
250#10596831—22no—
251#10596831—42no—
252#10596831—41no—
253C22×C73—24no—
254C22×C73—24no—
255C22×C73—24no—
256C22×C73—24no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73C22×C22⋊C411yescharacteristic
268C22×C73D4211yescharacteristic
269C22×C73#54611yescharacteristic
270C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C73C2×#13711yescharacteristic
272C22×C73#57711yescharacteristic
273C22×C73#57811yescharacteristic
274C4×C73—24no—
275C4×C73—22no—
276C4×C73—44no—
277C4×C73—42no—
278C4×C73#41714yescomplemented
279C4×C73C2×C4×D411yescharacteristic
280C4×C73#51811yescharacteristic
281C4×C73D4211yescharacteristic
282C4×C73#54611yescharacteristic
283Dic73—42no—
284Dic73—81no—
285Dic73—82no—
286#22431—731nomaximal, Sylow
287C2×C73—44no—
288C2×C73—44no—
289C2×C73D4×C22⋊C411yescharacteristic
290C2×C73C2×#41711yescharacteristic
291C2×C73#289811yescharacteristic
292C2×C73#327211yescharacteristic
293C2×C73#327611yescharacteristic
294C2×C73#333011yescharacteristic
295C2×C73#333111yescharacteristic
296C2×#547D7311yescharacteristic
297C2×#436—731no—
298C2×#417—732no—
299D4×C4⋊C4—731no—
300#1981—731no—
301#1956—731no—
302#1894—732no—
303#1872—732no—
304#1802—731no—
305#1788—732no—
306C4×#102—731no—
307C73#2243111yescharacteristic, complemented, Sylow
308C22×#41C2×D7312yes—
309C22×M16—732no—
310#547—24no—
311#547Dic7318yescomplemented
312C2×C4⋊Q8C2×D7311yescharacteristic
313C2×#123—22no—
314C2×C22⋊Q8C2×D7312yes—
315C2×#121—21no—
316C2×C4×D4—1461no—
317C2×C4×D4C2×D7311yescharacteristic
318C2×C4×D4C2×D7311yescharacteristic
319C22×C4⋊C4—21no—
320C22×C4⋊C4—732no—
321#436—1464no—
322C2×C4⋊C8—731no—
323#417—1468no—
324C2×#103—732no—
325C2×#103—732no—
326C2×#103—732no—
327C2×#103—732no—
328C2×#102—731no—
329C2×#102—732no—
330C2×#102—731no—
331C2×C22⋊C8—732no—
332C2×C4×C8—731no—
333#389—1461no—
334#385—1462no—
335#384—731no—
336#380—732no—
337C4×C4⋊C4—731no—
338#340—732no—
339#340—731no—
340#340—731no—
341C2×#41—24no—
342C2×#41—24no—
343C2×#41—24no—
344C2×#41—42no—
345C2×#41—42no—
346C2×#41C2×Dic7318yescomplemented
347C22×Q8C22×D7312yes—
348C22×D4—22no—
349C22×D4C22⋊D7311yescharacteristic
350C22×D4C22×D7311yescharacteristic
351C23×C4—21no—
352C23×C4—22no—
353C23×C4—1462no—
354C23×C4C22×D7312yes—
355C2×M16—2922no—
356C2×M16—1464no—
357C2×M16—1464no—
358C22×C8—731no—
359C22×C8—732no—
360C22×C8—731no—
361C4⋊Q8—22no—
362C4⋊Q8C2×Dic7314yescomplemented
363#123—216no—
364C22⋊Q8—24no—
365C22⋊Q8C2×Dic7318yescomplemented
366#121—28no—
367C4×D4—22no—
368C4×D4—2924no—
369C4×D4C2×Dic7314yes—
370C4×D4C2×Dic7314yescomplemented
371C4×D4C22⋊D7314yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—2922no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—732no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—1462no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1464no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4C22⋊D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1462no—
389C2×C22⋊C4C22⋊D7312yes—
390C2×C42—1461no—
391C2×C42C22×D7311yescharacteristic
392C4⋊C8—1462no—
393#103—2922no—
394#103—2922no—
395#103—1464no—
396#103—1464no—
397#102—734no—
398#102—1464no—
399#102—1462no—
400C22⋊C8—1464no—
401C4×C8—1462no—
402#95—1461no—
403#95—1462no—
404C24—21no—
405C24C2×C22⋊D7312yes—
406#41—216no—
407#41—48no—
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C22×Dic7314yes—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4C22⋊Dic7314yes—
425C2×D4C22×Dic7312yes—
426C2×D4D4×D7312yes—
427C2×D4#1192402712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—2924no—
441C22×C4—2924no—
442C22×C4—5842no—
443C22×C4—1461no—
444C22×C4—1462no—
445C22×C4—1461no—
446C22×C4C22×Dic7314yes—
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7312yes—
449C22×C4C2×C22⋊D7314yes—
450C22×C4C2×C22⋊D7311yescharacteristic
451C22×C4C23×D7311yescharacteristic
452M16—2928no—
453C2×C8—732no—
454C2×C8—1462no—
455C2×C8—1461no—
456C2×C8—1462no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1462no—
460C2×C8—1461no—
461C2×C8—1461no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—2924no—
467C4⋊C4—2922no—
468C4⋊C4—5841no—
469C4⋊C4—732no—
470C4⋊C4—1462no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1462no—
474C4⋊C4—1461no—
475C4⋊C4C22⋊Dic7314yes—
476C4⋊C4C22×Dic7312yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—2924no—
480C22⋊C4C22⋊Dic7318yes—
481C42—2922no—
482C42C22×Dic7312yes—
483C42C2×C22⋊D7312yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×C22⋊Dic7314yes—
498C23C22×C22⋊D7311yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—28no—
507D4—416no—
508D4D4×Dic7314yescomplemented
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—416no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—48no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2922no—
528C2×C4—2921no—
529C2×C4—2922no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2922no—
533C2×C4—5848no—
534C2×C4—1462no—
535C2×C4C2×C22⋊Dic7312yes—
536C2×C4C2×C22⋊Dic7314yes—
537C2×C4C2×C22⋊Dic7312yes—
538C2×C4#1455661612yes—
539C2×C4#1455662112yes—
540C2×C4C23×Dic7311yescharacteristic
541C2×C4C2×D4×D7311yescharacteristic
542C2×C4C22×C22⋊D7311yescharacteristic
543C2×C4C2×#1192402711yescharacteristic
544C2×C4#1455663611yescharacteristic
545C2×C4#1455663611yescharacteristic
546C8—2922no—
547C8—2921no—
548C8—1462no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22C22×C22⋊Dic7311yescharacteristic
564C22D4×C22⋊D7311yescharacteristic
565C22#1798923111yescharacteristic
566C22C2×#1455661611yescharacteristic
567C22C2×#1455662111yescharacteristic
568C22#1798939511yescharacteristic
569C22#1798940211yescharacteristic
570C4—24no—
571C4—22no—
572C4—44no—
573C4—42no—
574C4—2922no—
575C4—5841no—
576C4—5842no—
577C4#1798858814yescomplemented
578C4C2×D4×Dic7311yescharacteristic
579C4#1798906211yescharacteristic
580C4D4×C22⋊D7311yescharacteristic
581C4#1798923111yescharacteristic
582C2—44no—
583C2—44no—
584C2D4×C22⋊Dic7311yescharacteristic
585C2C2×#1798858811yescharacteristic
586C2#2261846411yescharacteristic
587C2#2262153111yescharacteristic
588C2#2262157311yescharacteristic
589C2#2262223611yescharacteristic
590C2#2262224711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#547×C73C2
socleC23×C73C22×D4
cosocleC22×C4×C73C24
hypercenterC2×#547D73
hypocenterC73#22431
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#17988665C2semidirect
2. C2×#17988588C2semidirect
3. D4×C4⋊Dic73C2semidirect
4. #22610855C2semidirect
5. #22610734C2semidirect
6. #22610365C2semidirect
7. #22610239C2semidirect
8. #22609931C2semidirect
9. #22609871C2semidirect
10. C4×#14556512C2semidirect
37. #547×C73C4semidirect
70. #14556512D4semidirect
80. C2×#41×C73C2×C4semidirect
89. C4⋊Q8×C73C2×C4semidirect
92. C22⋊Q8×C73C2×C4semidirect
96. C4×D4×C73C2×C4semidirect
212. D4×C73C4×D4semidirect
278. C4×C73#417semidirect
307. C73#22431semidirect
311. #547Dic73semidirect
346. C2×#41C2×Dic73semidirect
362. C4⋊Q8C2×Dic73semidirect
365. C22⋊Q8C2×Dic73semidirect
370. C4×D4C2×Dic73semidirect
508. D4D4×Dic73semidirect
577. C4#17988588semidirect

Maximal subgroups

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

subgroupindex
C2×#179886652
C2×#179885882
D4×C4⋊Dic732
#226108552
#226107342
#226103652
#226102392
#226099312
#226098712
C4×#145565122
C2×#547×C732
#2243173