Finite Group Databasebeta

← #33120335

Subgroups — #33120335

Order 18688.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989137C211yesmaximal, characteristic
2C2×#17988872C211yesmaximal, characteristic, complemented
3C2×#17988787C211yesmaximal, characteristic, complemented
4#22611120C212yesmaximal, complemented
5#22610932C212yesmaximal, complemented
6#22610883C211yesmaximal, characteristic, complemented
7#22610858C211yesmaximal, characteristic, complemented
8#22610268C212yesmaximal, complemented
9#22610191C212yesmaximal, complemented
10#22609911C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C73C211yesmaximal, characteristic, complemented
12C22×#11924028—21no—
13C22×#11924015C2212yes—
14#17989137—24no—
15#17989137—24no—
16C2×#14556591C2211yescharacteristic
17C2×#14556588C2211yescharacteristic
18C2×#14556579—21no—
19C2×#14556565—21no—
20C2×#14556565C2212yes—
21C2×C4×D292—21no—
22C2×C4×D292C2212yes—
23#17988872—24no—
24#17988787—24no—
25C2×#14556522C2212yes—
26C2×C4⋊#10596831C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556515C2212yes—
29C2×#14556511C2212yes—
30C2×#14556511C2212yes—
31C2×#14556511C2212yes—
32C2×C22⋊#10596831C2212yes—
33#17988349C2212yes—
34#17988349C2212yes—
35#17988342C2212yes—
36C2×C22⋊Q8×C73C2211yescharacteristic
37C2×C4×Q8×C73C2211yescharacteristic
38C22×C42×C73C2211yescharacteristic
39#391×C73C2211yescharacteristic
40#389×C73C2211yescharacteristic
41#387×C73—21no—
42#385×C73—21no—
43#380×C73C2211yescharacteristic
44C4×C4⋊C4×C73—21no—
45C4×C4⋊C4×C73C2211yescharacteristic
46C4×C22⋊C4×C73—21no—
47C2×#11924028—28no—
48C2×#11924028—44no—
49C2×#11924028—44no—
50C22×Dic146—21no—
51C22×D292—21no—
52C22×D292—21no—
53C22×D292C2312yes—
54C22×C22⋊D73—22no—
55C22×C4×D73—22no—
56C2×#11924015—24no—
57C2×#11924015—24no—
58C2×#11924015—42no—
59C22×#10596831C2312yes—
60C22×#10596831C2312yes—
61#14556591—22no—
62#14556591D414yes—
63#14556588—22no—
64#14556588D414yes—
65#14556579—28no—
66#14556565—28no—
67#14556565—24no—
68#14556565—24no—
69C4×D292—28no—
70C4×D292—24no—
71C4×D292—24no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2312yes—
74C2×#11924005—22no—
75C2×#11924005—22no—
76#14556522—24no—
77C4⋊#10596831—22no—
78C4⋊#10596831—22no—
79#14556515—24no—
80#14556511—24no—
81#14556511—24no—
82#14556511—42no—
83C22⋊#10596831—42no—
84C22×Q8×C73C2311yescharacteristic
85C23×C4×C73D411yescharacteristic
86C23×C4×C73D411yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C22⋊Q8×C73—22no—
89C22⋊Q8×C73—22no—
90C4×Q8×C73—22no—
91C4×Q8×C73—22no—
92C2×C4⋊C4×C73—21no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73C2311yescharacteristic
96C2×C4⋊C4×C73C2311yescharacteristic
97C2×C4⋊C4×C73C2311yescharacteristic
98C2×C22⋊C4×C73—21no—
99C2×C22⋊C4×C73—21no—
100C2×C42×C73—22no—
101C2×C42×C73—22no—
102C2×C42×C73—21no—
103C2×C42×C73—21no—
104C2×C42×C73D411yescharacteristic
105C2×C42×C73D411yescharacteristic
106C2×C42×C73C2311yescharacteristic
107C2×C42×C73C2311yescharacteristic
108#95×C73—21no—
109#95×C73—21no—
110#95×C73—21no—
111#95×C73—21no—
112#95×C73—21no—
113C23×D73—22no—
114#11924028—432no—
115C2×Dic146—24no—
116C2×Dic146—42no—
117C2×Dic146—42no—
118C2×D292—22no—
119C2×D292—22no—
120C2×D292—22no—
121C2×D292—22no—
122C2×D292—24no—
123C2×D292—24no—
124C2×D292—22no—
125C2×D292—42no—
126C2×D292—42no—
127C2×D292—42no—
128C2×D292—42no—
129C2×D292C2×D414yes—
130C2×C22⋊D73—28no—
131C2×C22⋊D73—44no—
132C2×C22⋊D73—44no—
133C22×Dic73—22no—
134C2×C4×D73—28no—
135C2×C4×D73—44no—
136C2×C4×D73—44no—
137#11924015—48no—
138C2×#10596831—22no—
139C2×#10596831—22no—
140C2×#10596831—22no—
141C2×#10596831—22no—
142C2×#10596831—22no—
143C2×#10596831—22no—
144C4⋊Dic73—24no—
145C4⋊Dic73—22no—
146C4⋊Dic73C2×D414yes—
147#11924005—28no—
148#11924005—28no—
149C24×C73C2×D411yescharacteristic
150C2×Q8×C73—21no—
151C2×Q8×C73—21no—
152C2×Q8×C73—21no—
153C2×Q8×C73—21no—
154C2×Q8×C73—21no—
155C2×Q8×C73#4112yescomplemented
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—41no—
170C22×C4×C73—41no—
171C22×C4×C73SD1614yescomplemented
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—24no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—42no—
185C4⋊C4×C73—42no—
186C4⋊C4×C73#4112yescomplemented
187C22⋊C4×C73—24no—
188C22⋊C4×C73—42no—
189C42×C73—22no—
190C42×C73—22no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—42no—
194C42×C73—42no—
195C42×C73C2×D412yes—
196C42×C73C2×D412yes—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73C2×SD1612yes—
204C23×C73C22×D411yescharacteristic
205C22×D73—24no—
206C22×D73—42no—
207C22×D73—42no—
208C22×D73—42no—
209C22×D73—42no—
210C22×D73—42no—
211C22×D73—42no—
212Q8×C73—22no—
213Q8×C73—22no—
214Q8×C73—41no—
215Q8×C73—41no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—28no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—44no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73#12012yes—
245C2×C4×C73#12012yes—
246C2×C4×C73C2×SD1612yes—
247C2×C4×C73C2×SD1612yes—
248C2×C4×C73AGL(1,ℤ/8)12yes—
249C2×C4×C73C22×D411yescharacteristic
250C2×C4×C73C22×D411yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73H(5,𝔽2)11yescharacteristic
254C2×C4×C732−1+411yescharacteristic
255C22⋊D73—416no—
256C2×Dic73—24no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259D292—24no—
260D292—24no—
261D292—48no—
262D292—42no—
263D292—42no—
264D292—48no—
265C4×D73—416no—
266Dic146—48no—
267#10596831—42no—
268#10596831—42no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73C2×#12011yescharacteristic
277C22×C73#52511yescharacteristic
278C22×C73#53711yescharacteristic
279C22×C73C22×SD1611yescharacteristic
280C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C73#57511yescharacteristic
282C22×C73#57511yescharacteristic
283C2×D73—44no—
284C2×D73—44no—
285C2×D73—44no—
286C2×D73—82no—
287C2×D73—82no—
288C2×D73—82no—
289C2×D73—82no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—42no—
293C4×C73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73—81no—
297C4×C73—81no—
298C4×C73#45014yes—
299C4×C73#54611yescharacteristic
300C4×C73#54711yescharacteristic
301C4×C73#54811yescharacteristic
302C4×C73#55011yescharacteristic
303Dic73—44no—
304#24154—731nomaximal, Sylow
305C2×C73—42no—
306C2×C73#240711yescharacteristic
307C2×C73C2×#45011yescharacteristic
308C2×C73#300511yescharacteristic
309C2×C73#310311yescharacteristic
310C2×C73#311511yescharacteristic
311C2×C73#324611yescharacteristic
312C2×C73#325111yescharacteristic
313D73—84no—
314C2×#532—731no—
315C2×#475—731no—
316C2×#460—731no—
317C4×C22⋊Q8D7311yescharacteristic, complemented
318#2034—732no—
319#2000—732no—
320#1988—731no—
321#1982—731no—
322#1877—732no—
323#1864—732no—
324#1797—731no—
325C73#2415411yescharacteristic, complemented, Sylow
326C22×#41—1461no—
327C22×SD16—732no—
328#532—1464no—
329#532—1464no—
330C2×C4⋊Q8—731no—
331C2×#127—731no—
332C2×#124—1461no—
333C2×C22⋊Q8C2×D7311yescharacteristic
334C2×#121—732no—
335C2×#121—1461no—
336C2×C4×Q8C2×D7311yescharacteristic
337C2×C4×D4—732no—
338C2×C4×D4—1461no—
339C22×C42C2×D7311yescharacteristic
340#475—1464no—
341#460—1464no—
342C2×#106—732no—
343C2×C4⋊C8—731no—
344C2×C4⋊C8—731no—
345C2×#103—732no—
346C2×#102—732no—
347C2×#102—732no—
348C2×#102—732no—
349C2×C22⋊C8—732no—
350#391C2×D7311yescharacteristic
351#389C2×D7311yescharacteristic
352#387—21no—
353#385—21no—
354#380C2×D7311yescharacteristic
355C4×C4⋊C4—21no—
356C4×C4⋊C4C2×D7311yescharacteristic
357C4×C22⋊C4—21no—
358#340—732no—
359#340—732no—
360#336—732no—
361C2×#41—2924no—
362C2×#41—2924no—
363C2×#41—1468no—
364C22×Q8—1461no—
365C22×Q8C22×D7311yescharacteristic
366C22×D4—732no—
367C22×D4—1462no—
368C22×D4—1461no—
369C22×D4—1461no—
370C23×C4—1462no—
371C23×C4C22⋊D7311yescharacteristic
372C23×C4C22⋊D7311yescharacteristic
373C23×C4C22×D7311yescharacteristic
374C2×SD16—2922no—
375C2×SD16—1464no—
376C2×SD16—1464no—
377C22×C8—732no—
378C22×C8—732no—
379C4⋊Q8—734no—
380C4⋊Q8—1462no—
381#127—734no—
382#127—1462no—
383#124—1468no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386#121—1464no—
387#121—1464no—
388#121—1468no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×D4—1464no—
392C4×D4—1464no—
393C4×D4—1468no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—732no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C4⋊C4C22×D7311yescharacteristic
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1462no—
405C2×C22⋊C4—1462no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C22⋊D7311yescharacteristic
411C2×C42C22⋊D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413C2×C42C22×D7311yescharacteristic
414#106—1464no—
415C4⋊C8—1462no—
416C4⋊C8—1462no—
417#103—1464no—
418#102—2922no—
419#102—1464no—
420#102—1464no—
421C22⋊C8—2922no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427C24—1462no—
428C24C2×C22⋊D7311yescharacteristic
429#41—29232no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—2922no—
436C2×Q8—2922no—
437C2×Q8—1464no—
438C2×Q8#1192402812yescomplemented
439C2×D4—2922no—
440C2×D4—2922no—
441C2×D4—2924no—
442C2×D4—2924no—
443C2×D4—2922no—
444C2×D4—2922no—
445C2×D4—734no—
446C2×D4—1468no—
447C2×D4—1464no—
448C2×D4—1464no—
449C2×D4—1462no—
450C2×D4—1462no—
451C2×D4—1462no—
452C2×D4—1462no—
453C2×D4—1462no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—2924no—
470C22×C4—2924no—
471C22×C4—1468no—
472C22×C4—1462no—
473C22×C4#1192401514yescomplemented
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C23×D7311yescharacteristic
480SD16—2928no—
481C2×C8—1462no—
482C2×C8—1462no—
483C2×C8—1462no—
484C2×C8—1462no—
485C2×C8—1462no—
486C2×C8—1462no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—734no—
496C4⋊C4—1464no—
497C4⋊C4—1462no—
498C4⋊C4#1192402812yescomplemented
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1468no—
502C22⋊C4—1468no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42D4×D7312yes—
510C42D4×D7312yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—2922no—
518C23—2922no—
519C23—2922no—
520C23—2922no—
521C23—2922no—
522C23—2922no—
523C23—1464no—
524C23C2×#1192401512yes—
525C23C22×C22⋊D7311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2928no—
531D4—2928no—
532D4—29216no—
533D4—2922no—
534D4—2928no—
535D4—2922no—
536D4—1464no—
537D4—1464no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—29216no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—1464no—
570C2×C4#1455655912yes—
571C2×C4#1455655912yes—
572C2×C4C2×#1192401512yes—
573C2×C4C2×#1192401512yes—
574C2×C4#1455661612yes—
575C2×C4C2×D4×D7311yescharacteristic
576C2×C4C2×D4×D7311yescharacteristic
577C2×C4C2×#1192402811yescharacteristic
578C2×C4C2×#1192402811yescharacteristic
579C2×C4#1455663711yescharacteristic
580C2×C4#1455663811yescharacteristic
581C8—2922no—
582C8—2922no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—2924no—
591C22—2924no—
592C22—2924no—
593C22—5842no—
594C22—5842no—
595C22—5842no—
596C22—5842no—
597C22C2×#1455655911yescharacteristic
598C22#1798909811yescharacteristic
599C22#1798916511yescharacteristic
600C22C22×#1192401511yescharacteristic
601C22C2×#1455661611yescharacteristic
602C22#1798938511yescharacteristic
603C22#1798938511yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—2924no—
613C4#1798873714yes—
614C4#1798923311yescharacteristic
615C4#1798924211yescharacteristic
616C4#1798924611yescharacteristic
617C4#1798925811yescharacteristic
618C2—42no—
619C2—5844no—
620C2#2261327611yescharacteristic
621C2C2×#1798873711yescharacteristic
622C2#2261917111yescharacteristic
623C2#2261996211yescharacteristic
624C2#2262008211yescharacteristic
625C2#2262124311yescharacteristic
626C2#2262129311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×C22⋊Q8×C73C2
socleC23×C73C22×D4
prime radicalC2×#17989137C2
hypercenterC4×C22⋊Q8D73
hypocenterC73#24154
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#17988872C2semidirect
3. C2×#17988787C2semidirect
4. #22611120C2semidirect
5. #22610932C2semidirect
6. #22610883C2semidirect
7. #22610858C2semidirect
8. #22610268C2semidirect
9. #22610191C2semidirect
10. #22609911C2semidirect
11. C4×C22⋊Q8×C73C2semidirect
155. C2×Q8×C73#41semidirect
171. C22×C4×C73SD16semidirect
186. C4⋊C4×C73#41semidirect
317. C4×C22⋊Q8D73semidirect
325. C73#24154semidirect
438. C2×Q8#11924028semidirect
473. C22×C4#11924015semidirect
498. C4⋊C4#11924028semidirect

Maximal subgroups

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

subgroupindex
C2×#179891372
C2×#179888722
C2×#179887872
#226111202
#226109322
#226108832
#226108582
#226102682
#226101912
#226099112
C4×C22⋊Q8×C732
#2415473