Finite Group Databasebeta

← #33118242

Subgroups — #33118242

Order 18688.

1474 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988737C211yesmaximal, characteristic, complemented
2C22×#14556511C211yesmaximal, characteristic, complemented
3#22613287C211yesmaximal, characteristic
4#22611064C212yesmaximal, complemented
5#22610932C212yesmaximal, complemented
6#22610355C211yesmaximal, characteristic, complemented
7#22610286C212yesmaximal, complemented
8#22610248C212yesmaximal, complemented
9#22610224C211yesmaximal, characteristic, complemented
10#22610021C211yesmaximal, characteristic, complemented
11C2×#554×C73C211yesmaximal, characteristic, complemented
12C23×D292C2211yescharacteristic
13C22×#11924015C2212yes—
14C23×#10596831C2211yescharacteristic
15C2×#14556565C2212yes—
16C2×#14556559—21no—
17#17988737—24no—
18C2×#14556515C2212yescomplemented
19C2×#14556511—24no—
20C2×#14556511C2212yes—
21C2×#14556511C2212yes—
22C2×#14556511C2212yes—
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2212yescomplemented
25#17988504—21no—
26#17988496—21no—
27#17988489C2212yes—
28#17988447C2211yescharacteristic
29#17988442—21no—
30#17988349C2212yescomplemented
31#17988349C2212yescomplemented
32#17988349C2212yescomplemented
33#17988342C2212yescomplemented
34#554×C73—24no—
35#554×C73—24no—
36C2×C4⋊Q8×C73C2211yescharacteristic, complemented
37C2×#125×C73—21no—
38C2×C22⋊Q8×C73—21no—
39C2×C22⋊Q8×C73—21no—
40C2×C22⋊Q8×C73C2211yescharacteristic
41C2×C22⋊Q8×C73C2211yescharacteristic, complemented
42C2×C4×Q8×C73C2211yescharacteristic, complemented
43C2×C4×D4×C73—21no—
44C2×C4×D4×C73C2211yescharacteristic
45C22×C4⋊C4×C73C2211yescharacteristic
46C22×C4⋊C4×C73C2211yescharacteristic, complemented
47C24×D73—21no—
48C22×D292—24no—
49C22×D292—22no—
50C22×D292—22no—
51C22×D292—21no—
52C22×D292—42no—
53C22×D292C2312yes—
54C22×C22⋊D73—22no—
55C2×#11924015—24no—
56C2×#11924015—24no—
57C2×#11924015—42no—
58C22×#10596831—22no—
59C22×#10596831—42no—
60C22×#10596831C2312yes—
61C22×#10596831C2312yes—
62#14556565—24no—
63#14556565—42no—
64#14556559—44no—
65C2×C4⋊Dic73C2312yes—
66C2×C22⋊Dic73—21no—
67C2×#11924005—22no—
68C2×#11924005—22no—
69C2×#11924005—22no—
70#14556515—42no—
71#14556511—24no—
72#14556511—24no—
73#14556511—24no—
74C22⋊#10596831—22no—
75C22⋊#10596831—42no—
76#14556498—21no—
77C22×Q8×C73—21no—
78C22×Q8×C73C2311yescharacteristic
79C22×D4×C73D411yescharacteristic
80C23×C4×C73—21no—
81C23×C4×C73—21no—
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2311yescharacteristic
84C4⋊Q8×C73—22no—
85C4⋊Q8×C73—22no—
86#125×C73—28no—
87C22⋊Q8×C73—28no—
88C22⋊Q8×C73—28no—
89C22⋊Q8×C73—22no—
90C22⋊Q8×C73—22no—
91C22⋊Q8×C73—22no—
92C22⋊Q8×C73D414yescomplemented
93C4×Q8×C73—22no—
94C4×Q8×C73—22no—
95C4×D4×C73—28no—
96C4×D4×C73—22no—
97C4×D4×C73—22no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—22no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73D411yescharacteristic
108C2×C4⋊C4×C73D414yescomplemented
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73C2311yescharacteristic
118C23×D73—22no—
119C23×D73—44no—
120C23×D73—42no—
121C23×D73—42no—
122C23×D73—42no—
123C23×D73—82no—
124C2×D292—28no—
125C2×D292—22no—
126C2×D292—22no—
127C2×D292—22no—
128C2×D292—22no—
129C2×D292—22no—
130C2×D292—48no—
131C2×D292—44no—
132C2×D292—44no—
133C2×D292—42no—
134C2×D292#4114yes—
135C2×C22⋊D73—44no—
136C2×C22⋊D73—44no—
137C2×C22⋊D73—82no—
138C22×Dic73—22no—
139#11924015—48no—
140C2×#10596831—22no—
141C2×#10596831—22no—
142C2×#10596831—22no—
143C2×#10596831—22no—
144C2×#10596831—22no—
145C2×#10596831—22no—
146C2×#10596831—44no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C22⋊Dic73—42no—
150#11924005—44no—
151#11924005—44no—
152#11924005—82no—
153C24×C73C2×D411yescharacteristic
154C24×C73C2×D411yescharacteristic
155C2×Q8×C73—24no—
156C2×Q8×C73—21no—
157C2×Q8×C73—22no—
158C2×Q8×C73—22no—
159C2×Q8×C73—82no—
160C2×Q8×C73C2×D412yes—
161C2×D4×C73—22no—
162C2×D4×C73—22no—
163C2×D4×C73SD1614yescomplemented
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—24no—
171C22×C4×C73—24no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—82no—
177C22×C4×C73—82no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D412yes—
182C22×C4×C73C2×D412yes—
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C4⋊C4×C73—24no—
186C4⋊C4×C73—24no—
187C4⋊C4×C73—24no—
188C4⋊C4×C73—24no—
189C4⋊C4×C73—24no—
190C4⋊C4×C73—24no—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73C2×D412yes—
196C4⋊C4×C73C2×D412yes—
197C4⋊C4×C73C2×D412yes—
198C22⋊C4×C73—24no—
199C22⋊C4×C73—24no—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—22no—
202C42×C73—24no—
203C42×C73—21no—
204C42×C73—21no—
205C23×C73—22no—
206C23×C73—22no—
207C23×C73—21no—
208C23×C73—22no—
209C23×C73—21no—
210C23×C73—22no—
211C23×C73#12012yes—
212C23×C73C2×SD1612yes—
213C23×C73C2×SD1612yes—
214C23×C73#13712yes—
215C23×C73C22×D411yescharacteristic
216C22×D73—24no—
217C22×D73—48no—
218C22×D73—42no—
219C22×D73—42no—
220C22×D73—42no—
221C22×D73—42no—
222C22×D73—42no—
223C22×D73—42no—
224C22×D73—88no—
225C22×D73—88no—
226C22×D73—84no—
227C22×D73—84no—
228Q8×C73—24no—
229Q8×C73—41no—
230Q8×C73—41no—
231Q8×C73—84no—
232D4×C73—28no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—24no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—21no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—42no—
254C2×C4×C73—84no—
255C2×C4×C73—84no—
256C2×C4×C73#12012yes—
257C2×C4×C73C2×SD1612yes—
258C2×C4×C73C22×D411yescharacteristic
259C2×C4×C73C22×D411yescharacteristic
260C2×C4×C73C2×#4112yes—
261C2×C4×C73H(5,𝔽2)12yes—
262C22⋊D73—88no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266D292—24no—
267D292—24no—
268D292—48no—
269D292—42no—
270D292—42no—
271#10596831—42no—
272#10596831—42no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—24no—
276C22×C73—24no—
277C22×C73—22no—
278C22×C73—24no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—42no—
282C22×C73#44814yes—
283C22×C73C2×#12011yescharacteristic
284C22×C73#54612yes—
285C22×C73C22×SD1611yescharacteristic
286C22×C73C2×#13711yescharacteristic
287C22×C73#57612yes—
288C2×D73—44no—
289C2×D73—44no—
290C2×D73—44no—
291C2×D73—816no—
292C2×D73—82no—
293C2×D73—82no—
294C2×D73—82no—
295C2×D73—82no—
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—41no—
299C4×C73—42no—
300C4×C73—41no—
301C4×C73—42no—
302C4×C73—42no—
303C4×C73—42no—
304C4×C73#52511yescharacteristic
305C4×C73#53511yescharacteristic
306C4×C73#54612yes—
307Dic73—82no—
308#23969—731nomaximal, Sylow
309C2×C73—24no—
310C2×C73—42no—
311C2×C73#240411yescharacteristic
312C2×C73C2×#44811yescharacteristic
313C2×C73#300811yescharacteristic
314C2×C73#329512yes—
315C2×C73#331412yes—
316D73—84no—
317C2×#554D7311yescharacteristic, complemented
318C2×#450—731no—
319C22×#102—731no—
320#2408—731no—
321#2023—732no—
322#2000—732no—
323#1892—731no—
324#1880—732no—
325#1874—732no—
326#1869—731no—
327#1824—731no—
328C73#2396911yescharacteristic, complemented, Sylow
329C23×D4—731no—
330C22×SD16—732no—
331C23×C8—731no—
332#554—24no—
333#554—24no—
334C2×C4⋊Q8C2×D7311yescharacteristic, complemented
335C2×#125—21no—
336C2×C22⋊Q8—21no—
337C2×C22⋊Q8—21no—
338C2×C22⋊Q8C2×D7311yescharacteristic
339C2×C22⋊Q8C2×D7311yescharacteristic, complemented
340C2×#121—732no—
341C2×#120—1461no—
342C2×C4×Q8C2×D7311yescharacteristic, complemented
343C2×C4×D4—21no—
344C2×C4×D4C2×D7311yescharacteristic
345C22×C4⋊C4C2×D7311yescharacteristic
346C22×C4⋊C4C2×D7311yescharacteristic, complemented
347#450—1464no—
348C2×#103—732no—
349C2×#102—732no—
350C2×#102—732no—
351C2×#102—732no—
352C2×#102—1464no—
353C2×C22⋊C8—732no—
354C2×C22⋊C8—731no—
355#394—1461no—
356#392—1461no—
357#390—732no—
358#380—731no—
359#379—1461no—
360#340—732no—
361#340—732no—
362#340—732no—
363#336—732no—
364C25—1461no—
365C22×Q8—21no—
366C22×Q8C22×D7311yescharacteristic
367C22×D4—2922no—
368C22×D4—732no—
369C22×D4—1464no—
370C22×D4—1462no—
371C22×D4—1461no—
372C22×D4—1462no—
373C22×D4—1462no—
374C22×D4C22⋊D7311yescharacteristic
375C23×C4—21no—
376C23×C4—21no—
377C23×C4C22×D7311yescharacteristic
378C23×C4C22×D7311yescharacteristic
379C2×SD16—2922no—
380C2×SD16—1464no—
381C2×SD16—1464no—
382C22×C8—2922no—
383C22×C8—732no—
384C22×C8—732no—
385C22×C8—1462no—
386C4⋊Q8—22no—
387C4⋊Q8—22no—
388#125—28no—
389C22⋊Q8—28no—
390C22⋊Q8—28no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8C22⋊D7314yescomplemented
395#121—2922no—
396#121—1464no—
397#120—2924no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×D4—28no—
401C4×D4—22no—
402C4×D4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—732no—
413C2×C4⋊C4C22⋊D7311yescharacteristic
414C2×C4⋊C4C22⋊D7314yescomplemented
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1461no—
421C2×C22⋊C4—1462no—
422C2×C22⋊C4—1462no—
423C2×C22⋊C4—1462no—
424C2×C22⋊C4C22⋊D7311yescharacteristic
425C2×C22⋊C4C22⋊D7311yescharacteristic
426C2×C42—21no—
427C2×C42C22×D7311yescharacteristic
428#103—2922no—
429#102—1464no—
430#102—1464no—
431#102—1464no—
432C22⋊C8—2922no—
433C22⋊C8—1462no—
434#95—1461no—
435C24—2924no—
436C24—2922no—
437C24—2922no—
438C24—2922no—
439C24—5842no—
440C24—1462no—
441C24C2×C22⋊D7311yescharacteristic
442C24C2×C22⋊D7311yescharacteristic
443C2×Q8—24no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—82no—
448C2×Q8C2×C22⋊D7312yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—2924no—
452C2×D4—2924no—
453C2×D4—2928no—
454C2×D4—2924no—
455C2×D4—2924no—
456C2×D4—2922no—
457C2×D4—5842no—
458C2×D4—734no—
459C2×D4—1468no—
460C2×D4—1462no—
461C2×D4—1462no—
462C2×D4—1462no—
463C2×D4—1462no—
464C2×D4—1462no—
465C2×D4#1192401514yescomplemented
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—1462no—
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C2×C22⋊D7312yes—
485C22×C4C2×C22⋊D7312yes—
486C22×C4C2×C22⋊D7311yescharacteristic
487C22×C4C23×D7311yescharacteristic
488SD16—2928no—
489C2×C8—2924no—
490C2×C8—1462no—
491C2×C8—1462no—
492C2×C8—1462no—
493C2×C8—1462no—
494C2×C8—1462no—
495C2×C8—1462no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—1462no—
507C4⋊C4—1462no—
508C4⋊C4C2×C22⋊D7312yes—
509C4⋊C4C2×C22⋊D7312yes—
510C4⋊C4C2×C22⋊D7312yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—2924no—
516C22⋊C4—2922no—
517C22⋊C4—2924no—
518C22⋊C4—5842no—
519C42—24no—
520C42—21no—
521C42—21no—
522C23—22no—
523C23—22no—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—2928no—
529C23—2922no—
530C23—2922no—
531C23—2922no—
532C23—2922no—
533C23—2922no—
534C23—2922no—
535C23—5848no—
536C23—5848no—
537C23—5844no—
538C23—5844no—
539C23—1464no—
540C23#1455656012yes—
541C23C2×#1192401512yes—
542C23C2×#1192401512yes—
543C23#1455662212yes—
544C23C22×C22⋊D7311yescharacteristic
545Q8—24no—
546Q8—41no—
547Q8—41no—
548Q8—84no—
549D4—28no—
550D4—2928no—
551D4—2922no—
552D4—2922no—
553D4—5848no—
554D4—1464no—
555D4—1464no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4#1455656012yes—
583C2×C4C2×#1192401512yes—
584C2×C4C22×C22⋊D7311yescharacteristic
585C2×C4C22×C22⋊D7311yescharacteristic
586C2×C4#41×D7312yes—
587C2×C4#1455663712yes—
588C8—2922no—
589C8—2922no—
590C22—24no—
591C22—24no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—22no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—2924no—
600C22—2924no—
601C22—2924no—
602C22—58416no—
603C22—5842no—
604C22—5842no—
605C22—5842no—
606C22—5842no—
607C22#1798872614yes—
608C22C2×#1455656011yescharacteristic
609C22#1798922512yes—
610C22C22×#1192401511yescharacteristic
611C22C2×#1455662211yescharacteristic
612C22#1798939012yes—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—5842no—
622C4#1798909411yescharacteristic
623C4#1798915111yescharacteristic
624C4#1798922512yes—
625C2—24no—
626C2—42no—
627C2—5844no—
628C2#2261324611yescharacteristic
629C2C2×#1798872611yescharacteristic
630C2#2261919911yescharacteristic
631C2#2262174912yes—
632C2#2262200612yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#554×C73C2
socleC23×C73C22×D4
prime radical#22613287C2
hypercenterC2×#554D73
hypocenterC73#23969
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#17988737C2semidirect
2. C22×#14556511C2semidirect
4. #22611064C2semidirect
5. #22610932C2semidirect
6. #22610355C2semidirect
7. #22610286C2semidirect
8. #22610248C2semidirect
9. #22610224C2semidirect
10. #22610021C2semidirect
11. C2×#554×C73C2semidirect
18. C2×#14556515C22semidirect
24. C2×C22⋊#10596831C22semidirect
30. #17988349C22semidirect
31. #17988349C22semidirect
32. #17988349C22semidirect
33. #17988342C22semidirect
36. C2×C4⋊Q8×C73C22semidirect
41. C2×C22⋊Q8×C73C22semidirect
42. C2×C4×Q8×C73C22semidirect
46. C22×C4⋊C4×C73C22semidirect
92. C22⋊Q8×C73D4semidirect
108. C2×C4⋊C4×C73D4semidirect
163. C2×D4×C73SD16semidirect
317. C2×#554D73semidirect
328. C73#23969semidirect
334. C2×C4⋊Q8C2×D73semidirect
339. C2×C22⋊Q8C2×D73semidirect
342. C2×C4×Q8C2×D73semidirect
346. C22×C4⋊C4C2×D73semidirect
394. C22⋊Q8C22⋊D73semidirect
414. C2×C4⋊C4C22⋊D73semidirect
465. C2×D4#11924015semidirect

Maximal subgroups

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

subgroupindex
C2×#179887372
C22×#145565112
#226132872
#226110642
#226109322
#226103552
#226102862
#226102482
#226102242
#226100212
C2×#554×C732
#2396973