Finite Group Databasebeta

← #33087222

Subgroups — #33087222

Order 18688.

1934 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556567C211yesmaximal, characteristic
2C2×#117×D73C211yesmaximal, characteristic
3C2×#17988596C211yesmaximal, characteristic, complemented
4C2×#17988550C211yesmaximal, characteristic
5#22610240C214yesmaximal, complemented
6#22610235C214yesmaximal, complemented
7#22609652C211yesmaximal, characteristic, complemented
8#22609651C211yesmaximal, characteristic, complemented
9C2×#419×C73C211yesmaximal, characteristic, complemented
10C23×C4×D73C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12C2×#14556567—24no—
13C2×#14556567—21no—
14C2×#14556567—21no—
15C2×#14556567C418yescomplemented
16C2×#14556567C2212yes—
17C2×#14556567C2212yes—
18C2×#14556550C2211yescharacteristic
19C2×#14556549C2212yes—
20C2×#14556548—21no—
21C2×#14556547—21no—
22#117×D73—24no—
23#117×D73—24no—
24C22×C4⋊Dic73C2211yescharacteristic
25C22×#11924007—21no—
26C2×C4⋊C4×D73C2212yes—
27C22×#11924005—21no—
28C2×C22⋊C4×D73—21no—
29C2×C42×D73—21no—
30#17988596—24no—
31C2×#14556515C2214yescomplemented
32#17988550—24no—
33C2×#14556503C2212yes—
34C2×#14556503C2212yes—
35#17988350C2214yescomplemented
36#17988349C2214yescomplemented
37C23×Q8×C73C2211yescharacteristic
38C22×M16×C73C2211yescharacteristic
39C2×#117×C73C2211yescharacteristic
40#419×C73—24no—
41C2×#103×C73C2214yescomplemented
42C24×D73D411yescharacteristic
43C23×Dic73—21no—
44C23×Dic73D411yescharacteristic
45C22×C4×D73—22no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—22no—
49C22×C4×D73—22no—
50C22×C4×D73C2×C414yes—
51C22×C4×D73D411yescharacteristic
52C22×C4×D73D411yescharacteristic
53C22×C4×D73D414yes—
54C22×C4×D73C2311yescharacteristic
55C22×C4×D73C2311yescharacteristic
56C2×#11924012—22no—
57C2×#11924012—22no—
58C2×#11924012—22no—
59C22×#10596831C2312yes—
60#14556567—28no—
61#14556567—28no—
62#14556567—24no—
63#14556567—24no—
64#14556567C2×C418yes—
65#14556567C2×C418yescomplemented
66#14556550—22no—
67#14556550D414yes—
68#14556549—24no—
69#14556549—24no—
70#14556548—28no—
71#14556547—28no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73C2×C414yescomplemented
75C2×C4⋊Dic73C2312yes—
76C2×C4⋊Dic73C2312yes—
77C2×#11924007—28no—
78C2×#11924007—22no—
79C2×#11924007—22no—
80C2×#11924007—22no—
81C4⋊C4×D73—24no—
82C4⋊C4×D73—24no—
83C2×C22⋊Dic73—21no—
84C2×#11924005—28no—
85C2×#11924005—22no—
86C2×#11924005—22no—
87C2×#11924005—22no—
88C22⋊C4×D73—28no—
89C2×C4×Dic73—22no—
90C2×C4×Dic73—21no—
91C42×D73—28no—
92#14556515—28no—
93#14556503—24no—
94#14556503—24no—
95C22×Q8×C73—22no—
96C22×Q8×C73—21no—
97C22×Q8×C73—22no—
98C22×Q8×C73—22no—
99C22×Q8×C73—22no—
100C22×Q8×C73—22no—
101C22×Q8×C73C2×C414yescomplemented
102C22×Q8×C73C2312yes—
103C23×C4×C73—21no—
104C23×C4×C73C2311yescharacteristic
105C2×M16×C73—22no—
106C2×M16×C73—22no—
107C2×M16×C73—22no—
108C22×C8×C73C2312yes—
109#117×C73—22no—
110#117×C73D414yes—
111C2×C4⋊C4×C73C2312yes—
112C2×C22⋊C4×C73—21no—
113C2×C42×C73—21no—
114#103×C73—28no—
115C23×D73—22no—
116C23×D73—22no—
117C23×D73—24no—
118C23×D73—42no—
119C23×D73C22⋊C414yes—
120C23×D73C2×D411yescharacteristic
121C23×D73C2×D411yescharacteristic
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—24no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—44no—
128C22×Dic73C22⋊C414yes—
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C2×C4×D73—28no—
132C2×C4×D73—28no—
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—28no—
136C2×C4×D73—28no—
137C2×C4×D73—22no—
138C2×C4×D73—22no—
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—21no—
144C2×C4×D73—21no—
145C2×C4×D73—21no—
146C2×C4×D73—21no—
147C2×C4×D73—28no—
148C2×C4×D73—44no—
149C2×C4×D73—44no—
150C2×C4×D73C22⋊C414yes—
151C2×C4×D73C22⋊C414yes—
152C2×C4×D73C22×C412yes—
153C2×C4×D73C22×C412yes—
154C2×C4×D73C2×D412yes—
155C2×C4×D73C2×D412yes—
156#11924012—44no—
157C2×#10596831—22no—
158C2×#10596831—22no—
159C2×#10596831—22no—
160C4⋊Dic73—24no—
161C4⋊Dic73—22no—
162C4⋊Dic73—22no—
163C4⋊Dic73C22×C414yes—
164C4⋊Dic73C2×D414yes—
165#11924007—28no—
166#11924007—28no—
167#11924007—28no—
168C22⋊Dic73—24no—
169#11924005—28no—
170#11924005—28no—
171#11924005—28no—
172C4×Dic73—28no—
173C4×Dic73—24no—
174C24×C73C2×D411yescharacteristic
175C2×Q8×C73—24no—
176C2×Q8×C73—22no—
177C2×Q8×C73—28no—
178C2×Q8×C73—28no—
179C2×Q8×C73—22no—
180C2×Q8×C73—22no—
181C2×Q8×C73—22no—
182C2×Q8×C73—22no—
183C2×Q8×C73—42no—
184C2×Q8×C73—44no—
185C2×Q8×C73—44no—
186C2×Q8×C73—42no—
187C2×Q8×C73C22×C414yes—
188C22×C4×C73—24no—
189C22×C4×C73—22no—
190C22×C4×C73—22no—
191C22×C4×C73—42no—
192C22×C4×C73—42no—
193C22×C4×C73C22×C412yes—
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D412yes—
196C22×C4×C73C2×D412yes—
197C22×C4×C73C2×D412yes—
198C22×C4×C73#4112yes—
199C22×C4×C73#4112yes—
200C22×C4×C73C2411yescharacteristic
201M16×C73—44no—
202C2×C8×C73—22no—
203C2×C8×C73—22no—
204C2×C8×C73—22no—
205C4⋊C4×C73—22no—
206C4⋊C4×C73C2×D414yes—
207C22⋊C4×C73—24no—
208C42×C73—24no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73C2×C22⋊C412yes—
212C23×C73#12012yes—
213C23×C73#12312yes—
214C23×C73C22×D411yescharacteristic
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—22no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—22no—
221C22×D73—22no—
222C22×D73—28no—
223C22×D73—28no—
224C22×D73—44no—
225C22×D73—48no—
226C22×D73—48no—
227C22×D73C2×C22⋊C412yes—
228C22×D73C2×C22⋊C412yes—
229C22×D73#13714yes—
230Q8×C73—24no—
231Q8×C73—24no—
232Q8×C73—42no—
233Q8×C73—42no—
234Q8×C73—48no—
235C2×C4×C73—24no—
236C2×C4×C73—24no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—48no—
242C2×C4×C73C2×C22⋊C412yes—
243C2×C4×C73C4×D414yes—
244C2×C4×C73C4×D414yes—
245C2×C4×C73#12012yes—
246C2×C4×C73#12114yes—
247C2×C4×C73#12114yes—
248C2×C4×C73#12312yes—
249C2×C4×C73C23×C411yescharacteristic
250C2×C4×C73C22×D411yescharacteristic
251C2×C4×C73C22×D411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—28no—
260C2×Dic73—24no—
261C2×Dic73—24no—
262C2×Dic73—48no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—44no—
266C2×Dic73C2×C22⋊C412yes—
267C2×Dic73C2×C22⋊C412yes—
268C4×D73—22no—
269C4×D73—22no—
270C4×D73—22no—
271C4×D73—22no—
272C4×D73—22no—
273C4×D73—22no—
274C4×D73—22no—
275C4×D73—22no—
276C4×D73—28no—
277C4×D73—28no—
278C4×D73—416no—
279C8×C73—42no—
280#10596831—42no—
281C22×C73—24no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—24no—
285C22×C73#38614yes—
286C22×C73C22×C22⋊C411yescharacteristic
287C22×C73C2×#12011yescharacteristic
288C22×C73C2×#12311yescharacteristic
289C22×C73C2×#13712yes—
290C22×C73#57512yes—
291C2×D73—24no—
292C2×D73—22no—
293C2×D73—24no—
294C2×D73—22no—
295C2×D73—22no—
296C2×D73—22no—
297C2×D73—22no—
298C2×D73—416no—
299C2×D73—42no—
300C2×D73—42no—
301C2×D73#41914yescomplemented
302C4×C73—44no—
303C4×C73—44no—
304C4×C73#38614yes—
305C4×C73C2×C4×D411yescharacteristic
306C4×C73C2×C4×D411yescharacteristic
307C4×C73C2×#12111yescharacteristic
308C4×C73C2×#12111yescharacteristic
309Dic73—22no—
310Dic73—22no—
311Dic73—44no—
312Dic73—44no—
313#20911—731nomaximal, Sylow
314C2×C73—24no—
315C2×C73C2×#38611yescharacteristic
316C2×C73C2×#41911yescharacteristic
317C2×C73#289611yescharacteristic
318C2×C73#300712yes—
319C2×C73#309512yes—
320D73—24no—
321D73—42no—
322C22×C22⋊Q8—731no—
323C22×#117—731no—
324C2×#419—731no—
325C2×#419D7311yescharacteristic, complemented
326C2×#407—731no—
327#1873—734no—
328#1872—734no—
329#1736—731no—
330#1736—731no—
331C73#2091111yescharacteristic, complemented, Sylow
332C23×Q8C2×D7311yescharacteristic
333C24×C4—731no—
334C22×M16—731no—
335C22×M16C2×D7311yescharacteristic
336C2×C22⋊Q8—738no—
337C2×C22⋊Q8—732no—
338C2×C22⋊Q8—732no—
339C2×C22⋊Q8—1461no—
340C2×C22⋊Q8—1464no—
341C2×C22⋊Q8—1461no—
342C2×#117—731no—
343C2×#117—732no—
344C2×#117—1461no—
345C2×#117—1461no—
346C2×#117—1464no—
347C2×#117—1464no—
348C2×#117C2×D7311yescharacteristic
349C22×C4⋊C4—731no—
350C22×C4⋊C4—732no—
351C22×C4⋊C4—1461no—
352C22×C22⋊C4—1461no—
353C22×C22⋊C4—1461no—
354C22×C42—1461no—
355#419—24no—
356#419—1464no—
357C2×#103—734no—
358C2×#103C2×D7314yescomplemented
359#407—1464no—
360C2×C22⋊C8—732no—
361C2×C22⋊C8—732no—
362#340—734no—
363#340—734no—
364C25—731no—
365C22×Q8—22no—
366C22×Q8—21no—
367C22×Q8—22no—
368C22×Q8—22no—
369C22×Q8—22no—
370C22×Q8—22no—
371C22×Q8C4×D7314yescomplemented
372C22×Q8C22×D7312yes—
373C23×C4—21no—
374C23×C4—731no—
375C23×C4—734no—
376C23×C4—731no—
377C23×C4—734no—
378C23×C4—731no—
379C23×C4—731no—
380C23×C4—731no—
381C23×C4—1462no—
382C23×C4—1461no—
383C23×C4—1462no—
384C23×C4—1462no—
385C23×C4—1462no—
386C23×C4—1462no—
387C23×C4C22×D7311yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—1462no—
392C2×M16—1462no—
393C2×M16—1462no—
394C22×C8—732no—
395C22×C8C22×D7312yes—
396C22⋊Q8—738no—
397C22⋊Q8—738no—
398C22⋊Q8—1468no—
399C22⋊Q8—1468no—
400C22⋊Q8—1464no—
401C22⋊Q8—1464no—
402#117—22no—
403#117—734no—
404#117—1468no—
405#117—1468no—
406#117—1464no—
407#117—1462no—
408#117—1464no—
409#117C22⋊D7314yes—
410C2×C4⋊C4—734no—
411C2×C4⋊C4—732no—
412C2×C4⋊C4—732no—
413C2×C4⋊C4—1464no—
414C2×C4⋊C4—1462no—
415C2×C4⋊C4—1468no—
416C2×C4⋊C4—1462no—
417C2×C4⋊C4—1462no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1462no—
420C2×C4⋊C4—1464no—
421C2×C4⋊C4C22×D7312yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1468no—
424C2×C22⋊C4—1462no—
425C2×C22⋊C4—1468no—
426C2×C22⋊C4—1462no—
427C2×C22⋊C4—1462no—
428C2×C22⋊C4—1461no—
429C2×C42—21no—
430C2×C42—1468no—
431C2×C42—1462no—
432C2×C42—1461no—
433#103—28no—
434#103—1468no—
435C22⋊C8—1464no—
436C22⋊C8—1464no—
437C24—2922no—
438C24—734no—
439C24—731no—
440C24—731no—
441C24—1464no—
442C24—1462no—
443C24—1462no—
444C24D4×D7311yescharacteristic
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—28no—
448C2×Q8—28no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—42no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8C2×C4×D7314yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—2924no—
464C22×C4—2924no—
465C22×C4—2924no—
466C22×C4—734no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—734no—
470C22×C4—734no—
471C22×C4—732no—
472C22×C4—732no—
473C22×C4—732no—
474C22×C4—732no—
475C22×C4—1468no—
476C22×C4—1464no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1468no—
482C22×C4—1468no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1468no—
486C22×C4—1468no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4C2×C4×D7312yes—
498C22×C4D4×D7311yescharacteristic
499C22×C4D4×D7312yes—
500C22×C4D4×D7312yes—
501C22×C4C2×C22⋊D7312yes—
502C22×C4#1192402712yes—
503C22×C4#1192402812yes—
504C22×C4C23×D7311yescharacteristic
505M16—44no—
506M16—2924no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—1462no—
511C2×C8—1462no—
512C2×C8—1462no—
513C4⋊C4—22no—
514C4⋊C4—734no—
515C4⋊C4—734no—
516C4⋊C4—1468no—
517C4⋊C4—1468no—
518C4⋊C4—1468no—
519C4⋊C4—1464no—
520C4⋊C4—1462no—
521C4⋊C4—1462no—
522C4⋊C4C2×C22⋊D7314yes—
523C22⋊C4—24no—
524C22⋊C4—1468no—
525C22⋊C4—1468no—
526C22⋊C4—1468no—
527C22⋊C4—1464no—
528C42—24no—
529C42—1468no—
530C42—1464no—
531C23—22no—
532C23—22no—
533C23—2928no—
534C23—2928no—
535C23—2924no—
536C23—734no—
537C23—732no—
538C23—732no—
539C23—1468no—
540C23—1462no—
541C23—1462no—
542C23—1468no—
543C23—1461no—
544C23—1461no—
545C23—1462no—
546C23—1461no—
547C23—1461no—
548C23C22⋊C4×D7312yes—
549C23#1455655912yes—
550C23#1455657312yes—
551C23C2×D4×D7311yescharacteristic
552Q8—24no—
553Q8—24no—
554Q8—48no—
555Q8—42no—
556Q8—42no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—29216no—
565C2×C4—2928no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2924no—
569C2×C4—732no—
570C2×C4—732no—
571C2×C4—1464no—
572C2×C4—1464no—
573C2×C4—1468no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1468no—
579C2×C4—1468no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4—1462no—
583C2×C4—1462no—
584C2×C4—1462no—
585C2×C4—1462no—
586C2×C4—1462no—
587C2×C4—1462no—
588C2×C4C22⋊C4×D7312yes—
589C2×C4#1455655114yes—
590C2×C4C4×C22⋊D7314yes—
591C2×C4#1455655912yes—
592C2×C4#1455656214yes—
593C2×C4#1455656414yes—
594C2×C4#1455657312yes—
595C2×C4C22×C4×D7311yescharacteristic
596C2×C4C2×D4×D7311yescharacteristic
597C2×C4C2×D4×D7311yescharacteristic
598C2×C4C22×C22⋊D7311yescharacteristic
599C2×C4C2×#1192402711yescharacteristic
600C2×C4C2×#1192402811yescharacteristic
601C8—42no—
602C8—2922no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—29216no—
608C22—2922no—
609C22—2922no—
610C22—734no—
611C22—1462no—
612C22—1462no—
613C22—1462no—
614C22—1464no—
615C22—1462no—
616C22—1462no—
617C22—1464no—
618C22#1798846314yes—
619C22C2×C22⋊C4×D7311yescharacteristic
620C22C2×#1455655911yescharacteristic
621C22C2×#1455657311yescharacteristic
622C22#137×D7312yes—
623C22#1798938612yes—
624C4—44no—
625C4—44no—
626C4—2924no—
627C4—2924no—
628C4—1462no—
629C4—1462no—
630C4#1798846314yes—
631C4C2×#1455655111yescharacteristic
632C4C2×C4×C22⋊D7311yescharacteristic
633C4C2×#1455656211yescharacteristic
634C4C2×#1455656411yescharacteristic
635C2—24no—
636C2—2922no—
637C2—1464no—
638C2C2×#1798846311yescharacteristic
639C2#419×D7311yescharacteristic
640C2#2261844911yescharacteristic
641C2#2261918712yes—
642C2#2261987712yes—
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#419×C73C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenterC2×#419D73
hypocenterC73#20911
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×#17988596C2semidirect
5. #22610240C2semidirect
6. #22610235C2semidirect
7. #22609652C2semidirect
8. #22609651C2semidirect
9. C2×#419×C73C2semidirect
15. C2×#14556567C4semidirect
31. C2×#14556515C22semidirect
35. #17988350C22semidirect
36. #17988349C22semidirect
41. C2×#103×C73C22semidirect
65. #14556567C2×C4semidirect
74. C2×C4⋊Dic73C2×C4semidirect
101. C22×Q8×C73C2×C4semidirect
301. C2×D73#419semidirect
325. C2×#419D73semidirect
331. C73#20911semidirect
358. C2×#103C2×D73semidirect
371. C22×Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C22×#145565672
C2×#117×D732
C2×#179885962
C2×#179885502
#226102402
#226102352
#226096522
#226096512
C2×#419×C732
#2091173