Finite Group Databasebeta

← #33089161

Subgroups — #33089161

Order 18688.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic73C211yesmaximal, characteristic
2#22612576C211yesmaximal, characteristic, complemented
3#22612292C211yesmaximal, characteristic
4#22610568C211yesmaximal, characteristic, complemented
5#22610562C211yesmaximal, characteristic, complemented
6#22610510C211yesmaximal, characteristic, complemented
7#22610508C211yesmaximal, characteristic, complemented
8#22610267C211yesmaximal, characteristic
9#22610192C211yesmaximal, characteristic
10#22610190C211yesmaximal, characteristic
11#22609819C211yesmaximal, characteristic, complemented
12#22609763C211yesmaximal, characteristic, complemented
13#22609758C211yesmaximal, characteristic
14#22609681C211yesmaximal, characteristic, complemented
15C73×#1877C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19SD16×Dic73C418yescomplemented
20C2×#14556523C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988494C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988472—21no—
33#17988470—21no—
34#17988454—21no—
35#17988451—21no—
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C22⋊C4×Dic73—21no—
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73—21no—
43#17988433C2211yescharacteristic
44C2×#14556498—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×SD16×C73C2211yescharacteristic
54C2×C4×D4×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#391×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64Q8×Dic73—22no—
65Q8×Dic73C2×C414yescomplemented
66D4×Dic73—22no—
67D4×Dic73—22no—
68D4×Dic73—22no—
69D4×Dic73C2×C414yes—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86#14556523C2×C414yescomplemented
87#14556522—22no—
88#14556522—22no—
89#14556515—22no—
90#14556515C2×C414yescomplemented
91#14556512—22no—
92#14556512C2×C414yes—
93#14556501—22no—
94C8×Dic73C2×C414yescomplemented
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×Q8×C73C2311yescharacteristic
108C22×D4×C73C2311yescharacteristic
109C23×C4×C73—21no—
110C2×SD16×C73—22no—
111C2×SD16×C73—22no—
112C2×SD16×C73C2×C414yes—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C4×D4×C73—22no—
116C4×D4×C73—22no—
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#103×C73—22no—
122#102×C73—22no—
123#95×C73—21no—
124#95×C73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—24no—
129C22×Dic73—24no—
130C22×Dic73—21no—
131C22×Dic73—81no—
132C22×Dic73—81no—
133C22×Dic73—81no—
134C22×Dic73—81no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831C22×C412yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—24no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73C22×C412yes—
152C4⋊Dic73C22×C412yes—
153C4⋊Dic73#4112yes—
154#11924007—81no—
155#11924007—81no—
156C22⋊Dic73—24no—
157C22⋊Dic73—24no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—24no—
161C4×Dic73—81no—
162C4×Dic73—81no—
163C4×Dic73—81no—
164C4×Dic73—81no—
165C4×Dic73C22×C412yes—
166C4×Dic73#4112yes—
167C24×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73C22×C412yes—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73C22×C412yes—
178C2×D4×C73#4112yes—
179C2×D4×C73#4112yes—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—24no—
183C22×C4×C73—21no—
184C22×C4×C73—81no—
185C22×C4×C73—81no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189SD16×C73—28no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73C22×C412yes—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73#4112yes—
200C22⋊C4×C73—24no—
201C42×C73—21no—
202C42×C73—21no—
203C23×C73—22no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73C22×D411yescharacteristic
211Q8×C73—22no—
212Q8×C73—22no—
213Q8×C73—41no—
214Q8×C73—41no—
215D4×C73—22no—
216D4×C73—22no—
217D4×C73—22no—
218D4×C73#11714yes—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—84no—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—22no—
246C2×Dic73—22no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—22no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—84no—
264C2×Dic73—84no—
265C2×Dic73C4×D412yes—
266C2×Dic73C4×D412yes—
267C8×C73—22no—
268C8×C73—41no—
269#10596831—22no—
270#10596831—41no—
271C22×C73—24no—
272C22×C73—42no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—42no—
278C22×C73C2×C4×D411yescharacteristic
279C22×C73#52511yescharacteristic
280C22×C73#52511yescharacteristic
281C22×C73#57711yescharacteristic
282C22×C73#57711yescharacteristic
283C22×C73#57711yescharacteristic
284C22×C73#57711yescharacteristic
285C4×C73—41no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—81no—
290C4×C73C2×#11711yescharacteristic
291C4×C73C4×#4111yescharacteristic
292C4×C73#52911yescharacteristic
293C4×C73#52911yescharacteristic
294Dic73—22no—
295Dic73—22no—
296Dic73—41no—
297Dic73—42no—
298Dic73—41no—
299Dic73—42no—
300Dic73—42no—
301Dic73—81no—
302#21133—731nomaximal, Sylow
303C2×C73—44no—
304C2×C73#229011yescharacteristic
305C2×C73#294911yescharacteristic
306C2×C73#295211yescharacteristic
307C2×C73#312311yescharacteristic
308C2×C73#312311yescharacteristic
309C2×C73#312511yescharacteristic
310C2×C73#312511yescharacteristic
311C2×C4×SD16—731no—
312#2320—731no—
313#2275—731no—
314#1927—731no—
315#1927—731no—
316#1919—731no—
317#1919—731no—
318#1877—731no—
319#1877D7311yescharacteristic
320#1864—731no—
321#1864—731no—
322C4×#106—731no—
323#1763—731no—
324#1762—731no—
325#1746—731no—
326C73#2113311yescharacteristic, complemented, Sylow
327C22×SD16C2×D7311yescharacteristic
328C2×C4×Q8—731no—
329C2×C4×D4—731no—
330C2×C4×D4—731no—
331C2×C4×D4C2×D7311yescharacteristic
332C4×SD16—738no—
333C2×#106—731no—
334C2×#106—731no—
335C2×#106—731no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#103C2×D7311yescharacteristic
339C2×#102—731no—
340C2×#102—731no—
341C2×#102C2×D7311yescharacteristic
342C2×#97—731no—
343C2×C4×C8—731no—
344#391—731no—
345#391C2×D7311yescharacteristic
346#389—731no—
347#389—731no—
348#387—1461no—
349#387—1461no—
350#383—1461no—
351#382—1461no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C22⋊C4—1461no—
357C4×C22⋊C4—1461no—
358C4×C22⋊C4—1461no—
359#376—731no—
360C2×#95—1461no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340C2×D7311yescharacteristic
367#336—731no—
368#336—731no—
369#336—731no—
370#336C2×D7311yescharacteristic
371C22×Q8C22×D7311yescharacteristic
372C22×D4C22×D7311yescharacteristic
373C23×C4—21no—
374C23×C4—1461no—
375C23×C4—1461no—
376C2×SD16—22no—
377C2×SD16—22no—
378C2×SD16C4×D7314yes—
379C22×C8—731no—
380C22×C8—731no—
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C4×Q8—734no—
384C4×Q8—1462no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—734no—
388C4×D4—1462no—
389C4×D4—1462no—
390C4×D4—1462no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—1461no—
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42C22×D7311yescharacteristic
411#106—734no—
412#106—1462no—
413#106—1462no—
414#103—22no—
415#103—734no—
416#103—1462no—
417#102—22no—
418#102—734no—
419#102—1462no—
420#97—1462no—
421C4×C8—734no—
422#95—21no—
423#95—21no—
424#95—731no—
425#95—731no—
426#95—731no—
427#95—731no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D7312yes—
447C2×D4#1192402712yes—
448C2×D4#1192402712yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—5841no—
456C22×C4—5841no—
457C22×C4—5841no—
458C22×C4—5841no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1464no—
468C22×C4—1464no—
469C22×C4D4×D7311yescharacteristic
470C22×C4D4×D7311yescharacteristic
471C22×C4C23×D7311yescharacteristic
472SD16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—732no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8C2×C4×D7312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—5841no—
489C4⋊C4—5841no—
490C4⋊C4—732no—
491C4⋊C4—732no—
492C4⋊C4—732no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4#1192402712yes—
500C22⋊C4—24no—
501C22⋊C4—1464no—
502C22⋊C4—1464no—
503C42—21no—
504C42—21no—
505C42—5841no—
506C42—5841no—
507C42—5841no—
508C42—5841no—
509C42—732no—
510C42—732no—
511C42—1464no—
512C42—1461no—
513C42—1461no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D7311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1455654714yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—5844no—
556C2×C4—5844no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1462no—
564C2×C4—1462no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1462no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4C22×C4×D7311yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4#41×D7311yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C2×C4C2×#1192402711yescharacteristic
577C2×C4C2×#1192402711yescharacteristic
578C8—22no—
579C8—41no—
580C8—2921no—
581C8—1462no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C4×D4×D7311yescharacteristic
590C22#1798909311yescharacteristic
591C22#1798909711yescharacteristic
592C22#1798939311yescharacteristic
593C22#1798939411yescharacteristic
594C22#1798939611yescharacteristic
595C22#1798939711yescharacteristic
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—2922no—
602C4—2921no—
603C4—2921no—
604C4—2922no—
605C4—2922no—
606C4—5841no—
607C4—1462no—
608C4—1462no—
609C4C2×#1455654711yescharacteristic
610C4C4×#1192402711yescharacteristic
611C4#1798911411yescharacteristic
612C4#1798911811yescharacteristic
613C2—44no—
614C2#2261237111yescharacteristic
615C2#2261878711yescharacteristic
616C2#2261880911yescharacteristic
617C2#2262017711yescharacteristic
618C2#2262018111yescharacteristic
619C2#2262020111yescharacteristic
620C2#2262020511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1877C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1877D73
hypocenterC73#21133
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #22612576C2semidirect
4. #22610568C2semidirect
5. #22610562C2semidirect
6. #22610510C2semidirect
7. #22610508C2semidirect
11. #22609819C2semidirect
12. #22609763C2semidirect
14. #22609681C2semidirect
19. SD16×Dic73C4semidirect
65. Q8×Dic73C2×C4semidirect
86. #14556523C2×C4semidirect
90. #14556515C2×C4semidirect
94. C8×Dic73C2×C4semidirect
326. C73#21133semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic732
#226125762
#226122922
#226105682
#226105622
#226105102
#226105082
#226102672
#226101922
#226101902
#226098192
#226097632
#226097582
#226096812
C73×#18772
#2113373