Finite Group Databasebeta

← #33114677

Subgroups — #33114677

Order 18688.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988629C211yesmaximal, characteristic, complemented
2C2×#17988610C211yesmaximal, characteristic, complemented
3C2×#17988588C211yesmaximal, characteristic
4#22612422C211yesmaximal, characteristic
5C2×#17988483C211yesmaximal, characteristic, complemented
6#22611324C212yesmaximal, complemented
7#22611317C212yesmaximal, complemented
8#22611310C212yesmaximal, complemented
9#22611303C212yesmaximal, complemented
10#22609650C211yesmaximal, characteristic, complemented
11C2×#417×C73C211yesmaximal, characteristic
12C22×#11924012C2211yescharacteristic
13C2×#14556550C2211yescharacteristic
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924007—21no—
16C22×C4×Dic73C2211yescharacteristic
17#17988629—24no—
18C2×#14556525C2212yes—
19C2×#14556523C2212yes—
20#17988610—24no—
21C2×#14556520C2212yes—
22C2×#14556520C2212yes—
23#17988588—24no—
24C2×#14556515C2212yes—
25C2×#14556512C2212yes—
26#17988494C2211yescharacteristic
27#17988494C2211yescharacteristic
28#17988490C2211yescharacteristic
29#17988490C2211yescharacteristic
30#17988483—21no—
31#17988483—21no—
32#17988483C2212yes—
33#17988483C2212yes—
34#17988467—21no—
35#17988467—21no—
36C4×C22⋊Dic73—21no—
37C4×C22⋊Dic73—21no—
38C2×#14556498—21no—
39#17988349C2212yes—
40#17988349C2212yes—
41C22×#41×C73C2211yescharacteristic
42C22×M16×C73C2211yescharacteristic
43C22×C4⋊C4×C73C2211yescharacteristic
44#417×C73—24no—
45C2×#103×C73C2212yes—
46C2×#102×C73C2212yes—
47C23×Dic73—21no—
48C23×Dic73D411yescharacteristic
49C23×Dic73D411yescharacteristic
50C2×#11924012—22no—
51C2×#11924012—22no—
52C2×#11924012—22no—
53C22×#10596831C2312yes—
54#14556550—22no—
55#14556550—22no—
56C2×C4⋊Dic73—21no—
57C2×C4⋊Dic73—22no—
58C2×C4⋊Dic73D414yes—
59C2×C4⋊Dic73C2311yescharacteristic
60C2×C4⋊Dic73C2311yescharacteristic
61C2×C4⋊Dic73C2312yes—
62C2×#11924007—22no—
63C2×#11924007—22no—
64C2×#11924007—44no—
65C2×C22⋊Dic73—21no—
66C2×C22⋊Dic73—21no—
67C2×C22⋊Dic73—21no—
68C2×C22⋊Dic73—21no—
69C2×C22⋊Dic73—21no—
70C2×C4×Dic73—22no—
71C2×C4×Dic73—21no—
72C2×C4×Dic73—22no—
73C2×C4×Dic73D411yescharacteristic
74C2×C4×Dic73D411yescharacteristic
75C2×C4×Dic73C2311yescharacteristic
76C2×C4×Dic73C2311yescharacteristic
77#14556525—24no—
78#14556523—24no—
79#14556520—24no—
80#14556520—24no—
81#14556515—24no—
82#14556512—24no—
83#14556498—22no—
84#14556498—22no—
85#14556498—21no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89C2×#41×C73—22no—
90C2×#41×C73—22no—
91C2×#41×C73—22no—
92C2×#41×C73—22no—
93C2×#41×C73—22no—
94C2×#41×C73Q814yes—
95C22×Q8×C73C2311yescharacteristic
96C22×D4×C73—21no—
97C22×D4×C73C2311yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73—21no—
100C23×C4×C73C2311yescharacteristic
101C2×M16×C73—22no—
102C2×M16×C73—22no—
103C2×M16×C73D414yes—
104C22×C8×C73C2312yes—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2312yes—
109#103×C73—24no—
110#102×C73—24no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—24no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—21no—
118C22×Dic73—21no—
119C22×Dic73—41no—
120C22×Dic73—41no—
121C22×Dic73—42no—
122C22×Dic73—42no—
123C22×Dic73C2×D411yescharacteristic
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127#11924012—44no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831—22no—
131C4⋊Dic73—24no—
132C4⋊Dic73—22no—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73C2×D414yes—
138#11924007—44no—
139#11924007—44no—
140C22⋊Dic73—42no—
141C22⋊Dic73—42no—
142C22⋊Dic73—42no—
143C22⋊Dic73—42no—
144C22⋊Dic73—42no—
145C4×Dic73—22no—
146C4×Dic73—22no—
147C4×Dic73—21no—
148C4×Dic73—21no—
149C4×Dic73—21no—
150C4×Dic73—21no—
151C4×Dic73—42no—
152C24×C73—21no—
153C24×C73C2×D411yescharacteristic
154#41×C73—28no—
155#41×C73—28no—
156#41×C73—44no—
157#41×C73—44no—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73C2×Q812yes—
164C2×D4×C73—24no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—42no—
171C2×D4×C73—42no—
172C2×D4×C73C2×Q812yes—
173C22×C4×C73—21no—
174C22×C4×C73—24no—
175C22×C4×C73—22no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73—42no—
179C22×C4×C73—42no—
180C22×C4×C73—42no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D412yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×Q812yes—
185C22×C4×C73#4112yes—
186C22×C4×C73#4112yes—
187C22×C4×C73#4112yes—
188C22×C4×C73C2411yescharacteristic
189M16×C73—28no—
190C2×C8×C73—22no—
191C2×C8×C73—22no—
192C2×C8×C73C2×D414yes—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—42no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—22no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73#12112yes—
208C23×C73#12112yes—
209C23×C73C22⋊Q812yes—
210C23×C73C22×D411yescharacteristic
211Q8×C73—22no—
212Q8×C73—22no—
213Q8×C73—41no—
214Q8×C73—41no—
215D4×C73—22no—
216D4×C73—22no—
217D4×C73—48no—
218D4×C73—41no—
219D4×C73—41no—
220C2×C4×C73—22no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—48no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—84no—
228C2×C4×C73#12112yes—
229C2×C4×C73#12114yes—
230C2×C4×C73#12112yes—
231C2×C4×C73C22⋊Q814yes—
232C2×C4×C73C22⋊Q812yes—
233C2×C4×C73C22⋊Q814yes—
234C2×C4×C73#12614yes—
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C22×Q811yescharacteristic
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—21no—
246C2×Dic73—21no—
247C2×Dic73—24no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—44no—
253C2×Dic73—44no—
254C2×Dic73—48no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73AGL(1,ℤ/8)12yes—
264C2×Dic73#13712yes—
265C8×C73—24no—
266#10596831—42no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—81no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73#39614yes—
281C22×C73C2×#12111yescharacteristic
282C22×C73C2×#12111yescharacteristic
283C22×C73C2×C22⋊Q811yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C73C2×#13711yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57511yescharacteristic
288C4×C73—42no—
289C4×C73—82no—
290C4×C73#39614yes—
291C4×C73C2×#12111yescharacteristic
292C4×C73C2×C22⋊Q811yescharacteristic
293C4×C73C2×C22⋊Q811yescharacteristic
294C4×C73C2×#12611yescharacteristic
295Dic73—22no—
296Dic73—22no—
297Dic73—44no—
298Dic73—41no—
299Dic73—41no—
300Dic73—81no—
301Dic73—81no—
302#23617—731nomaximal, Sylow
303C2×C73—24no—
304C2×C73—82no—
305C2×C73C2×#39611yescharacteristic
306C2×C73#303611yescharacteristic
307C2×C73#303711yescharacteristic
308C2×C73#305511yescharacteristic
309C2×C73#305611yescharacteristic
310C2×C73#307911yescharacteristic
311C2×C73#308011yescharacteristic
312C2×#428—731no—
313C2×#423—731no—
314C2×#417—731no—
315C2×#417D7311yescharacteristic
316#2297—731no—
317C2×#389—731no—
318#2068—732no—
319#2067—732no—
320#2066—732no—
321#2065—732no—
322#1736—731no—
323C73#2361711yescharacteristic, complemented, Sylow
324C22×#41C2×D7311yescharacteristic
325C22×M16—731no—
326C22×M16C2×D7311yescharacteristic
327C2×#117—731no—
328C22×C4⋊C4—731no—
329C22×C4⋊C4—1461no—
330C22×C4⋊C4C2×D7311yescharacteristic
331C22×C42—731no—
332#428—1464no—
333C2×#107—732no—
334C2×#106—732no—
335#423—1464no—
336C2×C4⋊C8—732no—
337C2×C4⋊C8—732no—
338#417—24no—
339#417—1464no—
340C2×#103—732no—
341C2×#103C2×D7312yes—
342C2×#102—732no—
343C2×#102C2×D7312yes—
344#391—731no—
345#391—731no—
346#390—731no—
347#390—731no—
348#389—732no—
349#389—732no—
350#389—1461no—
351#389—1461no—
352#386—1461no—
353#386—1461no—
354C4×C22⋊C4—1461no—
355C4×C22⋊C4—1461no—
356C2×#95—1461no—
357#340—732no—
358#340—732no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41Dic14614yes—
365C22×Q8C22×D7311yescharacteristic
366C22×D4—21no—
367C22×D4C22×D7311yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—731no—
371C23×C4—731no—
372C23×C4—1461no—
373C23×C4C22×D7311yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—1462no—
377C2×M16—1462no—
378C2×M16—1462no—
379C2×M16C22⋊D7314yes—
380C22×C8—732no—
381C22×C8C22×D7312yes—
382#117—1462no—
383#117—1462no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—2924no—
388C2×C4⋊C4—734no—
389C2×C4⋊C4—732no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—1462no—
393C2×C4⋊C4—1462no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1462no—
396C2×C4⋊C4C22×D7312yes—
397C2×C22⋊C4—1461no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—1461no—
407C2×C42—1462no—
408C2×C42—1462no—
409#107—1464no—
410#106—1464no—
411C4⋊C8—1464no—
412C4⋊C8—1464no—
413#103—24no—
414#103—1464no—
415#102—24no—
416#102—1464no—
417#95—1462no—
418#95—1462no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423C24—21no—
424C24D4×D7311yescharacteristic
425#41—28no—
426#41—28no—
427#41—44no—
428#41—44no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C2×Dic14612yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C2×Dic14612yes—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—2922no—
453C22×C4—2922no—
454C22×C4—2921no—
455C22×C4—2921no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1464no—
467C22×C4—1462no—
468C22×C4D4×D7311yescharacteristic
469C22×C4D4×D7312yes—
470C22×C4C2×C22⋊D7312yes—
471C22×C4C2×Dic14612yes—
472C22×C4#1192402712yes—
473C22×C4#1192402712yes—
474C22×C4#1192402812yes—
475C22×C4C23×D7311yescharacteristic
476M16—28no—
477M16—2924no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—1462no—
481C2×C8—1462no—
482C2×C8—1462no—
483C2×C8C2×C22⋊D7314yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—2924no—
488C4⋊C4—2924no—
489C4⋊C4—734no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1464no—
495C4⋊C4—1462no—
496C22⋊C4—2922no—
497C22⋊C4—2922no—
498C22⋊C4—2922no—
499C22⋊C4—2922no—
500C22⋊C4—2922no—
501C42—2922no—
502C42—1462no—
503C42—1462no—
504C42—1461no—
505C42—1461no—
506C42—1461no—
507C42—1461no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#1455656212yes—
520C23#1455656412yes—
521C23#1455656812yes—
522C23C2×D4×D7311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—2928no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2924no—
544C2×C4—2924no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—732no—
552C2×C4—732no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1464no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4#1455656212yes—
565C2×C4#1455656314yes—
566C2×C4#1455656412yes—
567C2×C4#1455656812yes—
568C2×C4#1455656814yes—
569C2×C4#1455657014yes—
570C2×C4#1455658414yes—
571C2×C4C2×D4×D7311yescharacteristic
572C2×C4C22×C22⋊D7311yescharacteristic
573C2×C4C22×Dic14611yescharacteristic
574C2×C4C2×#1192402711yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C2×C4C2×#1192402811yescharacteristic
577C8—24no—
578C8—2922no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22#1798851014yes—
593C22C2×#1455656211yescharacteristic
594C22C2×#1455656411yescharacteristic
595C22C2×#1455656811yescharacteristic
596C22AGL(1,ℤ/8)×D7311yescharacteristic
597C22#137×D7311yescharacteristic
598C22#1798938311yescharacteristic
599C22#1798938611yescharacteristic
600C4—42no—
601C4—82no—
602C4—2924no—
603C4—2921no—
604C4—2921no—
605C4—5841no—
606C4—5841no—
607C4—1462no—
608C4—1462no—
609C4#1798851014yes—
610C4C2×#1455656311yescharacteristic
611C4C2×#1455656811yescharacteristic
612C4C2×#1455657011yescharacteristic
613C4C2×#1455658411yescharacteristic
614C2—24no—
615C2—82no—
616C2C2×#1798851011yescharacteristic
617C2#2261939411yescharacteristic
618C2#2261940211yescharacteristic
619C2#2261954411yescharacteristic
620C2#2261955511yescharacteristic
621C2#2261974211yescharacteristic
622C2#2261975111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#417×C73C2
socleC23×C73C22×D4
prime radicalC22×#41×C73C22
hypercenterC2×#417D73
hypocenterC73#23617
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C2×#17988629C2semidirect
2. C2×#17988610C2semidirect
5. C2×#17988483C2semidirect
6. #22611324C2semidirect
7. #22611317C2semidirect
8. #22611310C2semidirect
9. #22611303C2semidirect
10. #22609650C2semidirect
323. C73#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#179886292
C2×#179886102
C2×#179885882
#226124222
C2×#179884832
#226113242
#226113172
#226113102
#226113032
#226096502
C2×#417×C732
#2361773