Finite Group Databasebeta

← #33085587

Subgroups — #33085587

Order 18688.

1438 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556599C211yesmaximal, characteristic
2C2×#17988629C211yesmaximal, characteristic, complemented
3#22612524C211yesmaximal, characteristic
4#22612519C211yesmaximal, characteristic
5#22609965C212yesmaximal, complemented
6#22609961C212yesmaximal, complemented
7#22609957C212yesmaximal, complemented
8#22609953C212yesmaximal, complemented
9#22609647C211yesmaximal, characteristic, complemented
10C2×#17988349C211yesmaximal, characteristic, complemented
11C2×#428×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C2×#14556599—24no—
14C2×#14556599—24no—
15C2×#14556599C418yescomplemented
16C22×#11924011C2212yes—
17C23×#10596831C2211yescharacteristic
18C22×C8×D73C2212yes—
19C2×#14556550C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21#17988629—24no—
22C2×#14556525C2212yes—
23C2×#14556523C2212yes—
24#17988493—21no—
25#17988492—21no—
26#17988489—21no—
27#17988488—21no—
28#17988466—22no—
29#17988464—22no—
30#17988444C2212yes—
31#17988444C2212yes—
32#17988444C2212yes—
33#17988444C2212yes—
34#17988349C2211yescharacteristic
35#17988349C2212yes—
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2212yes—
39#17988349C2211yescharacteristic
40C22×M16×C73C2211yescharacteristic
41C2×#117×C73C2211yescharacteristic
42C22×C4⋊C4×C73C2211yescharacteristic
43#428×C73—24no—
44C2×#107×C73C2212yes—
45C2×#106×C73C2212yes—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028—22no—
50C2×#11924028C2×C414yes—
51C2×#11924028Q814yes—
52C22×Dic146D411yescharacteristic
53C22×D292D411yescharacteristic
54C22×C22⋊D73D412yes—
55C23×Dic73—21no—
56C22×C4×D73C2312yes—
57#14556599—232no—
58C2×#11924011—24no—
59C2×#11924011—24no—
60C2×#11924011C2×C418yescomplemented
61C22×#10596831—22no—
62C22×#10596831—22no—
63C22×#10596831C2×C414yes—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C2×C8×D73—24no—
67C2×C8×D73—24no—
68C2×C8×D73C2×C418yescomplemented
69#14556550—22no—
70#14556550—22no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2312yes—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C22⋊Dic73—21no—
78C2×#11924005—22no—
79C2×#11924005—22no—
80C2×#11924005—22no—
81C2×#11924005—22no—
82C2×C4×Dic73—21no—
83#14556525—24no—
84#14556523—24no—
85#14556498—22no—
86#14556498—22no—
87#14556498—22no—
88#14556498—22no—
89C23×C4×C73—21no—
90C23×C4×C73C2311yescharacteristic
91C2×M16×C73C2×C414yes—
92C2×M16×C73D414yes—
93C2×M16×C73D414yes—
94C22×C8×C73C2312yes—
95#117×C73—22no—
96#117×C73—22no—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2312yes—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C42×C73—21no—
105#107×C73—24no—
106#106×C73—24no—
107C23×D73C2×D412yes—
108#11924028—28no—
109#11924028—28no—
110#11924028—28no—
111#11924028—28no—
112C2×Dic146—22no—
113C2×Dic146—22no—
114C2×Dic146C4⋊C414yes—
115C2×D292—22no—
116C2×D292—22no—
117C2×D292C4⋊C414yes—
118C2×C22⋊D73—24no—
119C2×C22⋊D73—24no—
120C2×C22⋊D73C4⋊C418yes—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—22no—
124C22×Dic73—42no—
125C22×Dic73—81no—
126C22×Dic73—81no—
127C22×Dic73C2×D412yes—
128C2×C4×D73—22no—
129C2×C4×D73—22no—
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73C22×C414yes—
133C2×C4×D73C2×Q814yes—
134#11924011—216no—
135C2×#10596831—28no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831C22×C412yes—
141C2×#10596831C22×C412yes—
142C8×D73—216no—
143C4⋊Dic73—21no—
144C4⋊Dic73—22no—
145C4⋊Dic73—21no—
146C4⋊Dic73—22no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—42no—
150C22⋊Dic73—42no—
151#11924005—82no—
152#11924005—82no—
153#11924005—82no—
154#11924005—82no—
155C4×Dic73—42no—
156C24×C73C2×D411yescharacteristic
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—22no—
160C22×C4×C73—42no—
161C22×C4×C73—81no—
162C22×C4×C73—81no—
163C22×C4×C73C22×C412yes—
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D412yes—
167C22×C4×C73C2×Q812yes—
168C22×C4×C73#4112yes—
169C22×C4×C73#4112yes—
170C22×C4×C73C2411yescharacteristic
171M16×C73C22⋊C4116yescomplemented
172C2×C8×C73C22×C414yes—
173C2×C8×C73C2×D414yes—
174C2×C8×C73C2×D414yes—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—42no—
182C22⋊C4×C73—42no—
183C42×C73—42no—
184C23×C73—21no—
185C23×C73—21no—
186C23×C73—21no—
187C23×C73—21no—
188C23×C73C2×C4⋊C412yes—
189C23×C73#12112yes—
190C23×C73#12112yes—
191C23×C73C22×D411yescharacteristic
192C22×D73—22no—
193C22×D73—22no—
194C22×D73—22no—
195C22×D73—22no—
196C22×D73—22no—
197C22×D73—22no—
198C22×D73C2×C4⋊C414yes—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—42no—
202C2×C4×C73—42no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—42no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—84no—
209C2×C4×C73C2×C22⋊C414yes—
210C2×C4×C73C2×C4⋊C412yes—
211C2×C4×C73#11714yes—
212C2×C4×C73#12112yes—
213C2×C4×C73#12112yes—
214C2×C4×C73C22⋊Q814yes—
215C2×C4×C73C22⋊Q814yes—
216C2×C4×C73C23×C411yescharacteristic
217C2×C4×C73C22×D411yescharacteristic
218C2×C4×C73C22×D411yescharacteristic
219C2×C4×C73C22×Q811yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C22⋊D73—216no—
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—41no—
226C2×Dic73—41no—
227C2×Dic73—42no—
228C2×Dic73—41no—
229C2×Dic73—42no—
230C2×Dic73—41no—
231C2×Dic73—42no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—84no—
235C2×Dic73C2×C4⋊C414yes—
236D292—28no—
237C4×D73—24no—
238C4×D73—24no—
239C4×D73—24no—
240C4×D73—24no—
241Dic146—28no—
242C8×C73C2×C22⋊C418yes—
243#10596831—22no—
244#10596831—22no—
245C22×C73—22no—
246C22×C73—22no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73—22no—
250C22×C73—22no—
251C22×C73#38014yes—
252C22×C73C22×C4⋊C411yescharacteristic
253C22×C73C2×#12111yescharacteristic
254C22×C73C2×#12111yescharacteristic
255C22×C73#57611yescharacteristic
256C22×C73#57711yescharacteristic
257C22×C73#57711yescharacteristic
258C22×C73#57811yescharacteristic
259C2×D73—24no—
260C2×D73—24no—
261C2×D73—24no—
262C2×D73—42no—
263C2×D73—42no—
264C2×D73—42no—
265C2×D73—42no—
266C4×C73—81no—
267C4×C73—82no—
268C4×C73—81no—
269C4×C73#38014yes—
270C4×C73C22×C22⋊C411yescharacteristic
271C4×C73C2×#11711yescharacteristic
272C4×C73C2×C22⋊Q811yescharacteristic
273C4×C73C2×C22⋊Q811yescharacteristic
274Dic73—24no—
275Dic73—81no—
276Dic73—82no—
277Dic73—81no—
278#20664—731nomaximal, Sylow
279C2×C73—24no—
280C2×C73C2×#38011yescharacteristic
281C2×C73#291211yescharacteristic
282C2×C73#291311yescharacteristic
283C2×C73#306211yescharacteristic
284C2×C73#306211yescharacteristic
285C2×C73#306311yescharacteristic
286C2×C73#306311yescharacteristic
287D73—44no—
288C22×#131—731no—
289C2×#428—731no—
290C2×#428D7311yescharacteristic, complemented
291#2311—731no—
292#2311—731no—
293#1809—732no—
294#1809—732no—
295#1808—732no—
296#1807—732no—
297#1735—731no—
298C2×#340—731no—
299C73#2066411yescharacteristic, complemented, Sylow
300C22×#41—731no—
301C2×#131—738no—
302C2×#131—1464no—
303C2×#131—1464no—
304C22×M16—732no—
305C22×M16C2×D7311yescharacteristic
306C23×C8—731no—
307C23×C8—732no—
308C2×#117—731no—
309C2×#117C2×D7311yescharacteristic
310C22×C4⋊C4—731no—
311C22×C4⋊C4C2×D7311yescharacteristic
312#428—24no—
313#428—1464no—
314C2×#107—732no—
315C2×#107C2×D7312yes—
316C2×#106—732no—
317C2×#106C2×D7312yes—
318#391—1461no—
319#391—1461no—
320#390—1461no—
321#390—1461no—
322#386—1462no—
323#386—1462no—
324#380—732no—
325#380—732no—
326#380—732no—
327#380—732no—
328#340—731no—
329#340—732no—
330#340—731no—
331#340—732no—
332#340—731no—
333#340—731no—
334C2×#41—734no—
335C2×#41—734no—
336C2×#41—1462no—
337C2×#41—1462no—
338C2×#41—1462no—
339C2×#41—1462no—
340C22×Q8—731no—
341C22×D4—732no—
342C22×D4—731no—
343C23×C4—21no—
344C23×C4—732no—
345C23×C4—1461no—
346C23×C4C22×D7311yescharacteristic
347#131—14632no—
348C2×M16—738no—
349C2×M16—1464no—
350C2×M16—1464no—
351C2×M16C4×D7314yes—
352C2×M16D29214yes—
353C2×M16C22⋊D7314yes—
354C22×C8—734no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8—738no—
358C22×C8—1462no—
359C22×C8—1462no—
360C22×C8—1464no—
361C22×C8—1464no—
362C22×C8C22×D7312yes—
363#117—22no—
364#117—22no—
365#117—1462no—
366#117—1462no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—732no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4C22×D7312yes—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1462no—
382C2×C22⋊C4—1462no—
383C2×C22⋊C4—1462no—
384C2×C22⋊C4—1462no—
385C2×C42—21no—
386C2×C42—1461no—
387#107—24no—
388#107—1464no—
389#106—24no—
390#106—1464no—
391#95—1462no—
392#95—1462no—
393#95—1462no—
394#95—1462no—
395C24—732no—
396C24D4×D7311yescharacteristic
397#41—1468no—
398#41—1468no—
399#41—1468no—
400#41—1468no—
401C2×Q8—734no—
402C2×Q8—1462no—
403C2×Q8—1462no—
404C2×D4—738no—
405C2×D4—734no—
406C2×D4—1464no—
407C2×D4—1464no—
408C2×D4—1462no—
409C2×D4—1462no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—2922no—
417C22×C4—5841no—
418C22×C4—5841no—
419C22×C4—732no—
420C22×C4—734no—
421C22×C4—734no—
422C22×C4—1461no—
423C22×C4—1461no—
424C22×C4—1462no—
425C22×C4—1462no—
426C22×C4—1462no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4C2×C4×D7312yes—
430C22×C4D4×D7311yescharacteristic
431C22×C4C2×C22⋊D7312yes—
432C22×C4C2×D29212yes—
433C22×C4Q8×D7312yes—
434C22×C4#1192402712yes—
435C22×C4#1192402912yes—
436C22×C4C23×D7311yescharacteristic
437M16—14616no—
438M16#11924005116yescomplemented
439C2×C8—732no—
440C2×C8—732no—
441C2×C8—1468no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—14616no—
447C2×C8C2×C4×D7314yes—
448C2×C8C2×C22⋊D7314yes—
449C2×C8C2×D29214yes—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—2922no—
458C4⋊C4—1461no—
459C4⋊C4—1462no—
460C4⋊C4—1461no—
461C4⋊C4—1462no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C22⋊C4—42no—
465C22⋊C4—2922no—
466C22⋊C4—5842no—
467C22⋊C4—5842no—
468C22⋊C4—5842no—
469C22⋊C4—5842no—
470C42—42no—
471C42—2922no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—734no—
477C23—1462no—
478C23—1462no—
479C23—1462no—
480C23—1462no—
481C23—1462no—
482C23—1462no—
483C23C4⋊C4×D7312yes—
484C23#1455656212yes—
485C23#1455656612yes—
486C23C2×D4×D7311yescharacteristic
487Q8—1468no—
488D4—14616no—
489D4—1468no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—84no—
500C2×C4—2921no—
501C2×C4—2922no—
502C2×C4—2922no—
503C2×C4—2921no—
504C2×C4—2921no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2922no—
509C2×C4—5844no—
510C2×C4—734no—
511C2×C4—1462no—
512C2×C4—1462no—
513C2×C4—1464no—
514C2×C4—1464no—
515C2×C4—1464no—
516C2×C4—1464no—
517C2×C4C2×#1192400514yes—
518C2×C4C4⋊C4×D7312yes—
519C2×C4#1455654914yes—
520C2×C4#1455656212yes—
521C2×C4#1455656612yes—
522C2×C4#1455656714yes—
523C2×C4#1455657114yes—
524C2×C4C22×C4×D7311yescharacteristic
525C2×C4C22×C22⋊D7311yescharacteristic
526C2×C4C22×D29211yescharacteristic
527C2×C4C2×Q8×D7311yescharacteristic
528C2×C4C2×#1192402711yescharacteristic
529C2×C4C2×#1192402911yescharacteristic
530C8—1462no—
531C8—1462no—
532C8C2×#1192400518yes—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—2922no—
540C22—2922no—
541C22—2922no—
542C22—2922no—
543C22—1464no—
544C22—1464no—
545C22—1464no—
546C22#1798844414yes—
547C22C2×C4⋊C4×D7311yescharacteristic
548C22C2×#1455656211yescharacteristic
549C22C2×#1455656611yescharacteristic
550C22#1798939111yescharacteristic
551C22#1798939611yescharacteristic
552C22#1798939711yescharacteristic
553C22#1798940311yescharacteristic
554C4—82no—
555C4—81no—
556C4—81no—
557C4—5842no—
558C4—5841no—
559C4—5841no—
560C4—1464no—
561C4#1798844414yes—
562C4C22×#1192400511yescharacteristic
563C4C2×#1455654911yescharacteristic
564C4C2×#1455656711yescharacteristic
565C4C2×#1455657111yescharacteristic
566C2—24no—
567C2—2924no—
568C2C2×#1798844411yescharacteristic
569C2#2261852511yescharacteristic
570C2#2261853011yescharacteristic
571C2#2261960511yescharacteristic
572C2#2261961211yescharacteristic
573C2#2261961611yescharacteristic
574C2#2261962311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#428D73
hypocenterC73#20664
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#17988629C2semidirect
5. #22609965C2semidirect
6. #22609961C2semidirect
7. #22609957C2semidirect
8. #22609953C2semidirect
9. #22609647C2semidirect
10. C2×#17988349C2semidirect
11. C2×#428×C73C2semidirect
15. C2×#14556599C4semidirect
60. C2×#11924011C2×C4semidirect
68. C2×C8×D73C2×C4semidirect
171. M16×C73C22⋊C4semidirect
290. C2×#428D73semidirect
299. C73#20664semidirect
438. M16#11924005semidirect

Maximal subgroups

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

subgroupindex
C22×#145565992
C2×#179886292
#226125242
#226125192
#226099652
#226099612
#226099572
#226099532
#226096472
C2×#179883492
C2×#428×C732
#2066473