Finite Group Databasebeta

← #33119893

Subgroups — #33119893

Order 18688.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988874C211yesmaximal, characteristic, complemented
2C2×C4×#11924015C211yesmaximal, characteristic, complemented
3#22614844C211yesmaximal, characteristic
4#22611124C212yesmaximal, complemented
5#22610932C212yesmaximal, complemented
6#22610893C211yesmaximal, characteristic, complemented
7#22610726C211yesmaximal, characteristic, complemented
8#22610248C212yesmaximal, complemented
9#22610124C212yesmaximal, complemented
10#22609936C211yesmaximal, characteristic, complemented
11C2×#554×C73C211yesmaximal, characteristic, complemented
12C22×#11924015C2211yescharacteristic
13C2×#14556565C2212yes—
14C2×#14556562—21no—
15C2×C4×D292C2211yescharacteristic
16#17988874—24no—
17C4×#11924015—24no—
18C2×#14556522C2212yes—
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2212yes—
23C2×#14556511C2212yes—
24C2×#14556511C2211yescharacteristic
25C2×C22⋊#10596831C2212yes—
26C2×C22⋊#10596831C2212yes—
27C2×C4×#10596831C2211yescharacteristic
28#17988503—22no—
29#17988457C2211yescharacteristic
30#17988447C2212yes—
31#17988444—21no—
32#17988349C2212yes—
33#17988349C2212yes—
34#17988349C2212yes—
35#554×C73—24no—
36#554×C73—24no—
37C2×C4⋊Q8×C73C2211yescharacteristic
38C2×#125×C73—21no—
39C2×C22⋊Q8×C73—22no—
40C2×C22⋊Q8×C73C2212yes—
41C2×C4×Q8×C73C2211yescharacteristic
42C2×C4×D4×C73—21no—
43C2×C4×D4×C73C2211yescharacteristic
44C22×C4⋊C4×C73C2212yes—
45C22×D292C2311yescharacteristic
46C22×C22⋊D73—22no—
47C22×C4×D73—21no—
48C2×#11924015—22no—
49C2×#11924015—22no—
50C2×#11924015—22no—
51C22×#10596831C2312yes—
52C22×#10596831C2311yescharacteristic
53C22×#10596831C2311yescharacteristic
54#14556565—24no—
55#14556565—42no—
56#14556562—44no—
57C4×D292—22no—
58C4×D292—22no—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73—21no—
61C2×C4⋊Dic73C2311yescharacteristic
62C2×C4⋊Dic73C2312yes—
63C2×C22⋊Dic73—22no—
64C2×#11924005—22no—
65C2×#11924005—21no—
66#14556522—24no—
67#14556522—22no—
68C4⋊#10596831—22no—
69#14556515—22no—
70#14556515—42no—
71#14556511—24no—
72#14556511—22no—
73C22⋊#10596831—24no—
74C22⋊#10596831—42no—
75C4×#10596831—22no—
76#14556498—22no—
77C22×Q8×C73—21no—
78C22×Q8×C73C2311yescharacteristic
79C22×D4×C73D411yescharacteristic
80C23×C4×C73—22no—
81C23×C4×C73C2312yes—
82C4⋊Q8×C73—22no—
83C4⋊Q8×C73D414yescomplemented
84#125×C73—28no—
85C22⋊Q8×C73—216no—
86C22⋊Q8×C73—24no—
87C22⋊Q8×C73—24no—
88C4×Q8×C73—22no—
89C4×Q8×C73D414yescomplemented
90C4×D4×C73—28no—
91C4×D4×C73—22no—
92C4×D4×C73—22no—
93C2×C4⋊C4×C73—22no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—24no—
98C2×C4⋊C4×C73—24no—
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2312yes—
102C2×C22⋊C4×C73—22no—
103C2×C22⋊C4×C73D412yes—
104C2×C42×C73—21no—
105C2×C42×C73C2311yescharacteristic
106C23×D73—21no—
107C2×D292—21no—
108C2×D292—21no—
109C2×D292—21no—
110C2×D292—21no—
111C2×D292—21no—
112C2×D292#4112yes—
113C2×C22⋊D73—44no—
114C2×C22⋊D73—44no—
115C2×C22⋊D73—82no—
116C22×Dic73—21no—
117C22×Dic73—22no—
118C2×C4×D73—42no—
119C2×C4×D73—42no—
120C2×C4×D73—42no—
121#11924015—28no—
122C2×#10596831—21no—
123C2×#10596831—21no—
124C2×#10596831—21no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—21no—
128C2×#10596831—22no—
129C2×#10596831—21no—
130C2×#10596831#4112yes—
131C4⋊Dic73—21no—
132C4⋊Dic73—21no—
133C4⋊Dic73—22no—
134C4⋊Dic73—22no—
135C4⋊Dic73—42no—
136C4⋊Dic73—81no—
137C22⋊Dic73—44no—
138#11924005—44no—
139#11924005—42no—
140C24×C73C2×D412yes—
141C2×Q8×C73—24no—
142C2×Q8×C73—21no—
143C2×Q8×C73—21no—
144C2×Q8×C73—21no—
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C2×Q8×C73—81no—
148C2×Q8×C73—81no—
149C2×Q8×C73C2×D412yes—
150C2×D4×C73—22no—
151C2×D4×C73—22no—
152C2×D4×C73SD1614yescomplemented
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—28no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—82no—
164C22×C4×C73—82no—
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2411yescharacteristic
169C4⋊C4×C73—28no—
170C4⋊C4×C73—28no—
171C4⋊C4×C73—24no—
172C4⋊C4×C73—24no—
173C4⋊C4×C73—22no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73C2×D412yes—
177C4⋊C4×C73C2×D414yes—
178C22⋊C4×C73—28no—
179C22⋊C4×C73—24no—
180C42×C73—24no—
181C42×C73C2×D412yes—
182C42×C73C2×D412yes—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—22no—
189C23×C73C2×SD1614yes—
190C23×C73C22×D411yescharacteristic
191C22×D73—22no—
192C22×D73—41no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198Q8×C73—22no—
199Q8×C73—22no—
200Q8×C73—41no—
201Q8×C73—41no—
202Q8×C73—84no—
203D4×C73—28no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—42no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—88no—
223C2×C4×C73#12012yes—
224C2×C4×C73#12012yes—
225C2×C4×C73C2×SD1612yes—
226C2×C4×C73#13712yes—
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C22×D411yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73H(5,𝔽2)11yescharacteristic
232C2×C4×C732−1+411yescharacteristic
233C22⋊D73—88no—
234C2×Dic73—41no—
235C2×Dic73—42no—
236C2×Dic73—41no—
237C2×Dic73—42no—
238C2×Dic73—41no—
239C2×Dic73—42no—
240D292—22no—
241D292—22no—
242D292—41no—
243D292—41no—
244C4×D73—84no—
245#10596831—22no—
246#10596831—41no—
247#10596831—42no—
248C22×C73—24no—
249C22×C73—24no—
250C22×C73—24no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—42no—
254C22×C73—42no—
255C22×C73C2×#12011yescharacteristic
256C22×C73#54611yescharacteristic
257C22×C73#54711yescharacteristic
258C22×C73C22×SD1611yescharacteristic
259C22×C73C2×#13711yescharacteristic
260C22×C73#57611yescharacteristic
261C22×C73#57711yescharacteristic
262C2×D73—42no—
263C2×D73—42no—
264C2×D73—42no—
265C2×D73—81no—
266C2×D73—81no—
267C2×D73—81no—
268C2×D73—81no—
269C4×C73—22no—
270C4×C73—42no—
271C4×C73—42no—
272C4×C73—42no—
273C4×C73—44no—
274C4×C73#44814yes—
275C4×C73#52511yescharacteristic
276C4×C73#53711yescharacteristic
277C4×C73#54811yescharacteristic
278C4×C73#55011yescharacteristic
279Dic73—81no—
280Dic73—82no—
281#24118—731nomaximal, Sylow
282C2×C73—44no—
283C2×C73#240711yescharacteristic
284C2×C73C2×#44811yescharacteristic
285C2×C73#301211yescharacteristic
286C2×C73#329511yescharacteristic
287C2×C73#333111yescharacteristic
288C2×C73#333511yescharacteristic
289C2×C73#335711yescharacteristic
290D73—82no—
291C2×#554D7311yescharacteristic, complemented
292C2×#476—731no—
293C2×C4×SD16—731no—
294#2537—731no—
295#2035—732no—
296#2000—732no—
297#1990—731no—
298#1954—731no—
299#1874—732no—
300#1853—732no—
301#1803—731no—
302C73#2411811yescharacteristic, complemented, Sylow
303C22×SD16—731no—
304#554—24no—
305#554—24no—
306C2×C4⋊Q8C2×D7311yescharacteristic
307C2×#125—21no—
308C2×C22⋊Q8—22no—
309C2×C22⋊Q8C2×D7312yes—
310C2×#121—732no—
311C2×#121—1461no—
312C2×C4×Q8C2×D7311yescharacteristic
313C2×C4×D4—21no—
314C2×C4×D4—731no—
315C2×C4×D4C2×D7311yescharacteristic
316C22×C4⋊C4C2×D7312yes—
317#476—1464no—
318C4×SD16—1464no—
319C2×#106—731no—
320C2×#106—732no—
321C2×C4⋊C8—731no—
322C2×#103—732no—
323C2×#103—731no—
324C2×#102—731no—
325C2×#102—732no—
326C2×C22⋊C8—732no—
327C2×C22⋊C8—732no—
328C2×C4×C8—731no—
329#394—1462no—
330#384—731no—
331#380—732no—
332#380—1461no—
333#340—732no—
334#340—732no—
335#340—732no—
336C22×Q8—21no—
337C22×Q8C22×D7311yescharacteristic
338C22×D4—731no—
339C22×D4—1462no—
340C22×D4C22⋊D7311yescharacteristic
341C23×C4—22no—
342C23×C4—1461no—
343C23×C4C22×D7312yes—
344C2×SD16—1462no—
345C2×SD16—1462no—
346C2×SD16—1462no—
347C22×C8—731no—
348C22×C8—731no—
349C22×C8—732no—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D7314yescomplemented
352#125—28no—
353C22⋊Q8—216no—
354C22⋊Q8—24no—
355C22⋊Q8—24no—
356#121—2924no—
357#121—2922no—
358#121—1464no—
359C4×Q8—22no—
360C4×Q8C22⋊D7314yescomplemented
361C4×D4—28no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—1462no—
365C4×D4—1462no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—732no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4C22⋊D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7312yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—1462no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1462no—
383C2×C22⋊C4C22⋊D7312yes—
384C2×C42—21no—
385C2×C42C22×D7311yescharacteristic
386#106—1462no—
387#106—1464no—
388C4⋊C8—1462no—
389#103—2922no—
390#103—1462no—
391#102—1462no—
392#102—1464no—
393C22⋊C8—2922no—
394C22⋊C8—1464no—
395C4×C8—1462no—
396#95—1462no—
397C24—1461no—
398C24C2×C22⋊D7312yes—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×C22⋊D7312yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—2924no—
411C2×D4—2924no—
412C2×D4—5842no—
413C2×D4—732no—
414C2×D4—1461no—
415C2×D4—1461no—
416C2×D4—1461no—
417C2×D4—1461no—
418C2×D4—1461no—
419C2×D4#1192401514yescomplemented
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—2922no—
433C22×C4—2922no—
434C22×C4—2922no—
435C22×C4—1461no—
436C22×C4—1462no—
437C22×C4C2×C22⋊D7312yes—
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4C2×C22⋊D7311yescharacteristic
440C22×C4C23×D7311yescharacteristic
441SD16—1468no—
442C2×C8—732no—
443C2×C8—1461no—
444C2×C8—1462no—
445C2×C8—1462no—
446C2×C8—1461no—
447C2×C8—1462no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—2922no—
459C4⋊C4—5841no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1462no—
463C4⋊C4—1462no—
464C4⋊C4C2×C22⋊D7312yes—
465C4⋊C4C2×C22⋊D7314yes—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—2924no—
469C22⋊C4—2924no—
470C22⋊C4—2922no—
471C42—24no—
472C42C2×C22⋊D7312yes—
473C42C2×C22⋊D7312yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—2921no—
481C23—2921no—
482C23—2921no—
483C23—2921no—
484C23—2921no—
485C23—2921no—
486C23—1462no—
487C23C2×#1192401514yes—
488C23C22×C22⋊D7311yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8—84no—
494D4—28no—
495D4—2921no—
496D4—2921no—
497D4—5848no—
498D4—1462no—
499D4—1462no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—2921no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2922no—
525C2×C4—5844no—
526C2×C4#1455656012yes—
527C2×C4#1455656012yes—
528C2×C4C2×#1192401512yes—
529C2×C4#1455662212yes—
530C2×C4C22×C22⋊D7311yescharacteristic
531C2×C4C22×C22⋊D7311yescharacteristic
532C2×C4#41×D7311yescharacteristic
533C2×C4#41×D7311yescharacteristic
534C2×C4#1455663711yescharacteristic
535C2×C4#1455663811yescharacteristic
536C8—2921no—
537C8—2922no—
538C8—1462no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—2922no—
547C22—2922no—
548C22—2922no—
549C22—5841no—
550C22—5841no—
551C22—5841no—
552C22—5841no—
553C22C2×#1455656011yescharacteristic
554C22#1798922511yescharacteristic
555C22#1798923611yescharacteristic
556C22C22×#1192401511yescharacteristic
557C22C2×#1455662211yescharacteristic
558C22#1798939011yescharacteristic
559C22#1798939811yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—5842no—
566C4—5841no—
567C4#1798872614yes—
568C4#1798909411yescharacteristic
569C4#1798916511yescharacteristic
570C4#1798924411yescharacteristic
571C4#1798925611yescharacteristic
572C2—44no—
573C2—5842no—
574C2#2261327011yescharacteristic
575C2C2×#1798872611yescharacteristic
576C2#2261923511yescharacteristic
577C2#2262174911yescharacteristic
578C2#2262224111yescharacteristic
579C2#2262228211yescharacteristic
580C2#2262255711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#554×C73C2
socleC23×C73C22×D4
prime radical#22614844C2
hypercenterC2×#554D73
hypocenterC73#24118
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×#17988874C2semidirect
2. C2×C4×#11924015C2semidirect
4. #22611124C2semidirect
5. #22610932C2semidirect
6. #22610893C2semidirect
7. #22610726C2semidirect
8. #22610248C2semidirect
9. #22610124C2semidirect
10. #22609936C2semidirect
11. C2×#554×C73C2semidirect
83. C4⋊Q8×C73D4semidirect
89. C4×Q8×C73D4semidirect
152. C2×D4×C73SD16semidirect
291. C2×#554D73semidirect
302. C73#24118semidirect
351. C4⋊Q8C22⋊D73semidirect
360. C4×Q8C22⋊D73semidirect
419. C2×D4#11924015semidirect

Maximal subgroups

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

subgroupindex
C2×#179888742
C2×C4×#119240152
#226148442
#226111242
#226109322
#226108932
#226107262
#226102482
#226101242
#226099362
C2×#554×C732
#2411873