Finite Group Databasebeta

← #33118142

Subgroups — #33118142

Order 18688.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988725C211yesmaximal, characteristic, complemented
2C22×#14556514C211yesmaximal, characteristic, complemented
3#22613296C211yesmaximal, characteristic, complemented
4#22611069C212yesmaximal, complemented
5#22610934C212yesmaximal, complemented
6#22610360C211yesmaximal, characteristic, complemented
7#22610265C212yesmaximal, complemented
8#22610238C212yesmaximal, complemented
9#22610224C211yesmaximal, characteristic, complemented
10#22610017C211yesmaximal, characteristic, complemented
11C2×#547×C73C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C22×#11924016C2212yes—
14C23×#10596831C2211yescharacteristic
15C2×#14556570C2212yes—
16#17988725—24no—
17C2×#14556514—24no—
18C2×#14556514C2212yes—
19C2×#14556514C2212yes—
20C2×#14556514C2212yes—
21C2×#14556512C2212yes—
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2212yes—
24#17988507—21no—
25#17988499—21no—
26#17988493C2212yes—
27#17988467—21no—
28#17988459—21no—
29#17988447C2211yescharacteristic
30#17988349C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34C22×#41×C73—21no—
35#547×C73—24no—
36#547×C73—24no—
37C2×C4⋊Q8×C73C2211yescharacteristic
38C2×#123×C73—21no—
39C2×#123×C73—21no—
40C2×C22⋊Q8×C73—21no—
41C2×#121×C73C2211yescharacteristic
42C2×#121×C73C2211yescharacteristic
43C2×C4×D4×C73C2211yescharacteristic
44C2×C4×D4×C73C2211yescharacteristic
45C22×C4⋊C4×C73C2211yescharacteristic
46C22×C4⋊C4×C73C2211yescharacteristic
47C22×Dic146—24no—
48C22×Dic146—21no—
49C22×Dic146—22no—
50C22×Dic146—22no—
51C22×Dic146—42no—
52C22×Dic146C2312yes—
53C23×Dic73—21no—
54C2×#11924016—24no—
55C2×#11924016—24no—
56C2×#11924016—42no—
57C22×#10596831—22no—
58C22×#10596831—42no—
59C22×#10596831C2312yes—
60C22×#10596831C2312yes—
61#14556570—24no—
62#14556570—42no—
63C2×C4⋊Dic73C2312yes—
64C2×#11924007—22no—
65C2×C22⋊Dic73—21no—
66C2×C22⋊Dic73—22no—
67#14556514—24no—
68#14556514—24no—
69#14556514—24no—
70#14556512—42no—
71C22⋊#10596831—22no—
72C22⋊#10596831—42no—
73#14556498—21no—
74#14556498—22no—
75#14556498—22no—
76C2×#41×C73—28no—
77C2×#41×C73—44no—
78C2×#41×C73—44no—
79C22×Q8×C73—21no—
80C22×D4×C73—21no—
81C22×D4×C73—21no—
82C22×D4×C73D411yescharacteristic
83C22×D4×C73C2311yescharacteristic
84C23×C4×C73—21no—
85C23×C4×C73—21no—
86C23×C4×C73—21no—
87C23×C4×C73C2311yescharacteristic
88C23×C4×C73C2311yescharacteristic
89C4⋊Q8×C73—22no—
90C4⋊Q8×C73—22no—
91#123×C73—28no—
92#123×C73—28no—
93C22⋊Q8×C73—28no—
94#121×C73—22no—
95#121×C73—22no—
96#121×C73—22no—
97#121×C73D414yes—
98C4×D4×C73—22no—
99C4×D4×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73—22no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—22no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73—22no—
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73D414yes—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×Dic146—28no—
120C2×Dic146—22no—
121C2×Dic146—22no—
122C2×Dic146—22no—
123C2×Dic146—22no—
124C2×Dic146—22no—
125C2×Dic146—48no—
126C2×Dic146—44no—
127C2×Dic146—44no—
128C2×Dic146—42no—
129C2×Dic146#4114yescomplemented
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—44no—
133C22×Dic73—42no—
134#11924016—48no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C2×#10596831—22no—
140C2×#10596831—22no—
141C2×#10596831—44no—
142C4⋊Dic73—22no—
143C4⋊Dic73—22no—
144#11924007—44no—
145C22⋊Dic73—44no—
146C22⋊Dic73—42no—
147C24×C73—21no—
148C24×C73C2×D411yescharacteristic
149C24×C73C2×D411yescharacteristic
150#41×C73—432no—
151C2×Q8×C73—24no—
152C2×Q8×C73—42no—
153C2×Q8×C73—42no—
154C2×D4×C73—22no—
155C2×D4×C73—22no—
156C2×D4×C73—24no—
157C2×D4×C73—24no—
158C2×D4×C73—21no—
159C2×D4×C73—22no—
160C2×D4×C73—22no—
161C2×D4×C73—42no—
162C2×D4×C73—42no—
163C2×D4×C73—42no—
164C2×D4×C73—42no—
165C2×D4×C73SD1614yescomplemented
166C2×D4×C73C2×D412yes—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—24no—
170C22×C4×C73—24no—
171C22×C4×C73—24no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—42no—
180C22×C4×C73—42no—
181C22×C4×C73—42no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D412yes—
189C22×C4×C73C2×D412yes—
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192C4⋊C4×C73—24no—
193C4⋊C4×C73—24no—
194C4⋊C4×C73—24no—
195C4⋊C4×C73—22no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73C2×D412yes—
200C4⋊C4×C73C2×D412yes—
201C4⋊C4×C73C2×D412yes—
202C22⋊C4×C73—24no—
203C22⋊C4×C73—24no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—24no—
206C22⋊C4×C73—22no—
207C42×C73—21no—
208C42×C73—21no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73—21no—
212C23×C73—22no—
213C23×C73—21no—
214C23×C73—22no—
215C23×C73—22no—
216C23×C73—42no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—42no—
220C23×C73#12012yes—
221C23×C73C2×SD1612yes—
222C23×C73C2×SD1612yes—
223C23×C73AGL(1,ℤ/8)12yes—
224C23×C73C22×D411yescharacteristic
225Q8×C73—48no—
226D4×C73—28no—
227D4×C73—24no—
228D4×C73—48no—
229D4×C73—48no—
230D4×C73—41no—
231D4×C73—41no—
232C2×C4×C73—21no—
233C2×C4×C73—22no—
234C2×C4×C73—21no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—24no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—48no—
246C2×C4×C73—48no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—48no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—42no—
257C2×C4×C73#12012yes—
258C2×C4×C73C2×SD1612yes—
259C2×C4×C73C22×D411yescharacteristic
260C2×C4×C73C22×D411yescharacteristic
261C2×C4×C73C2×#4112yes—
262C2×C4×C73H(5,𝔽2)12yes—
263C2×Dic73—24no—
264C2×Dic73—48no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270Dic146—24no—
271Dic146—24no—
272Dic146—48no—
273Dic146—42no—
274Dic146—42no—
275#10596831—42no—
276#10596831—42no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—24no—
280C22×C73—24no—
281C22×C73—22no—
282C22×C73—24no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—42no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—82no—
290C22×C73—82no—
291C22×C73#45014yes—
292C22×C73C2×#12011yescharacteristic
293C22×C73#54612yes—
294C22×C73C22×SD1611yescharacteristic
295C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C73#57812yes—
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—41no—
300C4×C73—42no—
301C4×C73—42no—
302C4×C73—42no—
303C4×C73—41no—
304C4×C73#52511yescharacteristic
305C4×C73#53511yescharacteristic
306C4×C73#54612yes—
307Dic73—44no—
308Dic73—82no—
309#23961—731nomaximal, Sylow
310C2×C73—24no—
311C2×C73—42no—
312C2×C73—82no—
313C2×C73#240411yescharacteristic
314C2×C73C2×#45011yescharacteristic
315C2×C73#301111yescharacteristic
316C2×C73#329812yes—
317C2×C73#330812yes—
318C2×#547D7311yescharacteristic
319C2×#448—731no—
320C22×#103—731no—
321#2409—731no—
322#2024—732no—
323#2000—732no—
324#1893—731no—
325#1877—732no—
326#1872—732no—
327#1869—731no—
328#1823—731no—
329C73#2396111yescharacteristic, complemented, Sylow
330C22×#41—21no—
331C23×Q8—731no—
332C22×SD16—732no—
333C23×C8—731no—
334#547—24no—
335#547—24no—
336C2×C4⋊Q8C2×D7311yescharacteristic
337C2×#123—21no—
338C2×#123—21no—
339C2×C22⋊Q8—21no—
340C2×C22⋊Q8—732no—
341C2×#121C2×D7311yescharacteristic
342C2×#121C2×D7311yescharacteristic
343C2×C4×D4C2×D7311yescharacteristic
344C2×C4×D4C2×D7311yescharacteristic
345C22×C4⋊C4C2×D7311yescharacteristic
346C22×C4⋊C4C2×D7311yescharacteristic
347#448—1464no—
348C2×#103—732no—
349C2×#103—732no—
350C2×#103—732no—
351C2×#103—1464no—
352C2×#102—732no—
353C2×C22⋊C8—732no—
354C2×C22⋊C8—731no—
355#395—1461no—
356#393—1461no—
357#391—732no—
358#386—1461no—
359#385—1461no—
360#380—731no—
361#340—732no—
362#340—732no—
363#340—732no—
364#336—732no—
365C2×#41—28no—
366C2×#41—44no—
367C2×#41—44no—
368C22×Q8—21no—
369C22×Q8—2922no—
370C22×Q8—732no—
371C22×Q8—1464no—
372C22×Q8—1462no—
373C22×Q8—1462no—
374C22×Q8—1461no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4C22⋊D7311yescharacteristic
378C22×D4C22×D7311yescharacteristic
379C23×C4—21no—
380C23×C4—21no—
381C23×C4—21no—
382C23×C4—1461no—
383C23×C4C22×D7311yescharacteristic
384C23×C4C22×D7311yescharacteristic
385C2×SD16—2922no—
386C2×SD16—1464no—
387C2×SD16—1464no—
388C22×C8—2922no—
389C22×C8—732no—
390C22×C8—732no—
391C22×C8—1462no—
392C4⋊Q8—22no—
393C4⋊Q8—22no—
394#123—28no—
395#123—28no—
396C22⋊Q8—28no—
397C22⋊Q8—2922no—
398C22⋊Q8—1464no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121C22⋊D7314yes—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—732no—
414C2×C4⋊C4—1462no—
415C2×C4⋊C4C22⋊D7311yescharacteristic
416C2×C4⋊C4C22⋊D7314yes—
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C4⋊C4C22×D7311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1462no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4C22⋊D7311yescharacteristic
426C2×C22⋊C4C22⋊D7311yescharacteristic
427C2×C42C22×D7311yescharacteristic
428#103—1464no—
429#103—1464no—
430#103—1464no—
431#102—2922no—
432C22⋊C8—2922no—
433C22⋊C8—1462no—
434#95—1461no—
435#95—1462no—
436#95—1462no—
437C24—21no—
438C24C2×C22⋊D7311yescharacteristic
439C24C2×C22⋊D7311yescharacteristic
440#41—432no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—2924no—
445C2×Q8—2928no—
446C2×Q8—2924no—
447C2×Q8—2922no—
448C2×Q8—734no—
449C2×Q8—1468no—
450C2×Q8—1462no—
451C2×Q8—1462no—
452C2×Q8—1462no—
453C2×Q8—1462no—
454C2×Q8—1462no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4#1192401614yescomplemented
467C2×D4C2×C22⋊D7312yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—2922no—
487C22×C4—2924no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4C2×C22⋊D7311yescharacteristic
491C22×C4C2×C22⋊D7311yescharacteristic
492C22×C4C2×C22⋊D7311yescharacteristic
493C22×C4C2×C22⋊D7312yes—
494C22×C4C2×C22⋊D7312yes—
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C23×D7311yescharacteristic
497SD16—2928no—
498C2×C8—2924no—
499C2×C8—1462no—
500C2×C8—1462no—
501C2×C8—1462no—
502C2×C8—1462no—
503C2×C8—1462no—
504C2×C8—1462no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—2924no—
513C4⋊C4—1462no—
514C4⋊C4—1462no—
515C4⋊C4C2×C22⋊D7312yes—
516C4⋊C4C2×C22⋊D7312yes—
517C4⋊C4C2×C22⋊D7312yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—2922no—
524C22⋊C4—2924no—
525C42—21no—
526C42—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23#1455656012yes—
539C23C2×#1192401612yes—
540C23C2×#1192401612yes—
541C23#1455661712yes—
542C23C22×C22⋊D7311yescharacteristic
543Q8—48no—
544Q8—2928no—
545Q8—2922no—
546Q8—2922no—
547Q8—1464no—
548Q8—1464no—
549D4—28no—
550D4—24no—
551D4—48no—
552D4—48no—
553D4—41no—
554D4—41no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—2928no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—1464no—
587C2×C4#1455656012yes—
588C2×C4C2×#1192401612yes—
589C2×C4C22×C22⋊D7311yescharacteristic
590C2×C4C22×C22⋊D7311yescharacteristic
591C2×C4#41×D7312yes—
592C2×C4#1455663712yes—
593C8—2922no—
594C8—2922no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—22no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22#1798873814yes—
610C22C2×#1455656011yescharacteristic
611C22#1798922512yes—
612C22C22×#1192401611yescharacteristic
613C22C2×#1455661711yescharacteristic
614C22#1798940212yes—
615C4—22no—
616C4—22no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—2924no—
623C4—5842no—
624C4#1798909411yescharacteristic
625C4#1798915111yescharacteristic
626C4#1798922512yes—
627C2—24no—
628C2—42no—
629C2—82no—
630C2#2261324611yescharacteristic
631C2C2×#1798873811yescharacteristic
632C2#2261922611yescharacteristic
633C2#2262179012yes—
634C2#2262191612yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#547×C73C2
socleC23×C73C22×D4
hypercenterC2×#547D73
hypocenterC73#23961
LCSG ▷ C22×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×#17988725C2semidirect
2. C22×#14556514C2semidirect
3. #22613296C2semidirect
4. #22611069C2semidirect
5. #22610934C2semidirect
6. #22610360C2semidirect
7. #22610265C2semidirect
8. #22610238C2semidirect
9. #22610224C2semidirect
10. #22610017C2semidirect
129. C2×Dic146#41semidirect
165. C2×D4×C73SD16semidirect
329. C73#23961semidirect
466. C2×D4#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179887252
C22×#145565142
#226132962
#226110692
#226109342
#226103602
#226102652
#226102382
#226102242
#226100172
C2×#547×C732
#2396173