Finite Group Databasebeta

← #72246922

Subgroups — #72246922

Order 4032.

780 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72246807C211yesmaximal, characteristic, complemented
2#72246693C211yesmaximal, characteristic, complemented
3C3×#10620516C211yesmaximal, characteristic, complemented
4C3×#10620514C211yesmaximal, characteristic, complemented
5#72246368C211yesmaximal, characteristic, complemented
6#72245887C211yesmaximal, characteristic, complemented
7C7×#63825C211yesmaximal, characteristic, complemented
8#13054800—31nomaximal
9#13053625—31nomaximal
10C2×C4⋊(C3⋊D21)C2211yescharacteristic
11C2×C7×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
12C3×#66728C2211yescharacteristic
13C3×C7×#235C2211yescharacteristic
14#11759626—22no—
15D3×D84—21no—
16D12×D21—21no—
17#11759609—21no—
18#11759605—21no—
19C2×#92678—22no—
20C3×#66627C2211yescharacteristic, complemented
21C3×C7×#208C2211yescharacteristic, complemented
22C32×C4×Dic7C2211yescharacteristic, complemented
23#11759220—21no—
24#11759217—21no—
25C7×#3717—21no—
26C7×#3714—21no—
27#10621662—31no—
28#10621566—31no—
29C7×#949—31no—
30C7×#908—31no—
31#10621013—31no—
32#10621013—31no—
33#10621013—61no—
34#10620866—31no—
35#10620861—31no—
36#10620662—31no—
37C3×#6411—31no—
38C3×#6399—31no—
39#10620516D311yescharacteristic, complemented
40#10620514D311yescharacteristic, complemented
41#10620495—31no—
42#10590397—71nomaximal, Hall
43C22×C3⋊D21D412yes—
44C22×C7×C3⋊D3—21no—
45C2×D3×D21—22no—
46C2×C32×C4×C7C2311yescharacteristic
47C4⋊(C3⋊D21)—21no—
48C4⋊(C3⋊D21)—21no—
49C7×C4⋊(C3⋊D3)—21no—
50C7×C4⋊(C3⋊D3)—21no—
51C3×#4268—21no—
52C3×D84D411yescharacteristic, complemented
53C3×C4×D21—21no—
54C3×Dic42D411yescharacteristic, complemented
55C3×C7×#67—21no—
56C3×C7×D12D411yescharacteristic, complemented
57C3×C4×D3×C7—21no—
58C3×C7×Dic6D411yescharacteristic, complemented
59C2×C32×Dic7—21no—
60#92690—22no—
61#92689—22no—
62#92687—22no—
63#92678—44no—
64C3×C3⋊(C7⋊C8)—21no—
65C3×C7×C3⋊C8—21no—
66#89970—91noHall
67C3×C7×#41—31no—
68#66732—61no—
69D4×D21—61no—
70D4×D21—62no—
71#66728—61no—
72#66728C2×D311yescharacteristic
73C2×D84—31no—
74C2×D84—31no—
75C2×D84—61no—
76C2×D84—61no—
77C7×#235C2×D311yescharacteristic
78C2×C7×D12—31no—
79C2×C7×D12—31no—
80C2×C7×D12—61no—
81C3×#1157—31no—
82#66694—62no—
83C2×#4247—62no—
84C2×#4247—62no—
85C2×#4247—62no—
86C2×#4247—62no—
87D3×D28—61no—
88D7×D12—61no—
89#66677—61no—
90#66673—61no—
91C3×C7×M16—31no—
92#66627C2×D311yescharacteristic, complemented
93C7×#215—61no—
94C7×#213—61no—
95C7×#208C2×D311yescharacteristic, complemented
96C7×D24—61no—
97C7×#204—61no—
98C3×C4×Dic7—21no—
99C3×C4×Dic7C2×D311yescharacteristic, complemented
100C3×C4×Dic7C2×D311yescharacteristic, complemented
101C3×#1136—31no—
102#66567—61no—
103#66564—61no—
104#66562—61no—
105#66560—61no—
106#64308—71no—
107#64083—71no—
108#63825—71no—
109#63825D711yescharacteristic, complemented
110C3×#805—71no—
111C3×#805—71no—
112#63559—71no—
113C22×C32×C7C2×D411yescharacteristic
114C2×C3⋊D21—22no—
115C2×C3⋊D21—42no—
116C2×C7×C3⋊D3—42no—
117C2×C7×C3⋊D3—41no—
118C2×C3×D21—21no—
119C2×C3×D3×C7—21no—
120D3×D21—82no—
121C32×C4×C7C2×D411yescharacteristic
122C32×C4×C7C2×D411yescharacteristic
123C3×Dic21—21no—
124C3×C7×Dic3—21no—
125C32×Dic7—22no—
126#6553—91no—
127C7×AGL(1,ℤ/8)—91no—
128#6503—91no—
129#6495—91no—
130#6411—31no—
131#6399—31no—
132#6397—91no—
133#4822—211no—
134#4822—211no—
135C22×D21—32no—
136C22×D21—32no—
137C22×D21—62no—
138C22×D21—62no—
139C22×D3×C7—61no—
140C22×D3×C7—61no—
141C22×D3×C7—61no—
142C22×D3×C7—61no—
143C2×D3×D7—62no—
144C3×C7×Q8—31no—
145C3×C7×D4—31no—
146C3×C7×D4—61no—
147C2×C3×C4×C7—21no—
148C2×C3×C4×C7—61no—
149C2×C3×C4×C7C22×D311yescharacteristic
150C2×C3×C4×C7C22×D311yescharacteristic
151#4268—21no—
152#4268—62no—
153D84—62no—
154D84—61no—
155D84—61no—
156D84—61no—
157D84—61no—
158D84—121no—
159D84—121no—
160D84D1211yescharacteristic, complemented
161C4×D21—21no—
162C4×D21—62no—
163Dic42D1211yescharacteristic, complemented
164C7×#67—21no—
165C7×D12—61no—
166C7×D12—61no—
167C7×D12—61no—
168C7×D12—61no—
169C7×D12—121no—
170C7×D12—121no—
171C7×D12D1211yescharacteristic, complemented
172C4×D3×C7—21no—
173C7×Dic6D1211yescharacteristic, complemented
174C3×C22⋊D7—61no—
175C2×C3×Dic7—21no—
176C2×C3×Dic7—21no—
177C2×C3×Dic7—21no—
178C2×C3×Dic7—21no—
179C3×D28—31no—
180C3×C4×D7—61no—
181C3×Dic14—31no—
182#4247—62no—
183#4247—124no—
184#4247—124no—
185#4247—124no—
186#4247—124no—
187#4246—62no—
188D21⋊C4—62no—
189C3×C7×C8—61no—
190C3⋊(C7⋊C8)—21no—
191C3×C7⋊C8—61no—
192C7×C3⋊C8—21no—
193C2×C4⋊(C3⋊D3)—71no—
194C2×C4⋊(C3⋊D3)—142no—
195C2×C4⋊(C3⋊D3)C2×D711yescharacteristic, complemented
196C3×#235—71no—
197C3×#235C2×D711yescharacteristic
198#3811—142no—
199D3×D12—141no—
200D3×D12—141no—
201#3799—141no—
202#3799—141no—
203C32×C42—71no—
204C3×#208—71no—
205C3×#208C2×D711yescharacteristic, complemented
206#3717—21no—
207#3717—141no—
208#3714—21no—
209#3714—141no—
210C2×C32×C7—21no—
211C2×C32×C7#12011yescharacteristic
212C3⋊D21—42no—
213C7×C3⋊D3—81no—
214C3×D21—41no—
215C3×D3×C7—41no—
216C7×#41—31no—
217C2×C7×D4—91no—
218C2×C22⋊D7—182no—
219#1161—181no—
220D4×D7—181no—
221D4×D7—182no—
222#1157—31no—
223#1157—181no—
224C2×D28—91no—
225C2×D28—181no—
226C7×SD16—181no—
227C7×D8—181no—
228C7×M16—31no—
229#1142—181no—
230#1141—181no—
231C4×Dic7D3211yescharacteristic, complemented
232#1136—31no—
233#1009—211no—
234#1009—211no—
235#949—31no—
236#949—211no—
237#908—31no—
238#908—211no—
239#874—211no—
240#874—211no—
241#874—421no—
242C3×#104—211no—
243C3×#104—211no—
244#823—211no—
245#823—211no—
246#805—71no—
247#805—71no—
248C22×C3×C7—21no—
249C22×C3×C7—61no—
250C22×C3×C7D3×D411yescharacteristic
251C22×C3×C7D3×D411yescharacteristic
252C2×D21—21no—
253C2×D21—62no—
254C2×D21—62no—
255C2×D21—122no—
256C2×D21—122no—
257C2×D21—122no—
258C2×D21—122no—
259C2×D21—122no—
260C2×D21—242no—
261C2×D3×C7—21no—
262C2×D3×C7—122no—
263C2×D3×C7—122no—
264C2×D3×C7—122no—
265C2×D3×C7—122no—
266C2×D3×C7—121no—
267C2×D3×C7—121no—
268C2×D3×C7—121no—
269C2×D3×C7—121no—
270C2×C3×D7—61no—
271D3×D7—242no—
272C3×C4×C7—21no—
273C3×C4×C7—21no—
274C3×C4×C7—61no—
275C3×C4×C7C2×D1211yescharacteristic
276C3×C4×C7C2×D1211yescharacteristic
277C3×C4×C7D3×D411yescharacteristic
278C3×C4×C7D3×D411yescharacteristic
279Dic21—21no—
280C3×Dic7—22no—
281C3×Dic7—22no—
282C3×Dic7—22no—
283C3×Dic7—22no—
284C3×Dic7—61no—
285C7×Dic3—21no—
286C22×C3⋊D3—21no—
287C22×C3⋊D3—72no—
288C2×D32—142no—
289C2×C32×C4—141no—
290C2×C32×C4C22×D711yescharacteristic
291C4⋊(C3⋊D3)—21no—
292C4⋊(C3⋊D3)—21no—
293C4⋊(C3⋊D3)—141no—
294C4⋊(C3⋊D3)—141no—
295C4⋊(C3⋊D3)—284no—
296C3×#67—21no—
297C3×#67—141no—
298C3×D12—71no—
299C3×D12C22⋊D711yescharacteristic, complemented
300C3×C4×D3—21no—
301C3×C4×D3—141no—
302C3×Dic6—71no—
303C3×Dic6C22⋊D711yescharacteristic, complemented
304#626—142no—
305#626—142no—
306#624—142no—
307C3×C3⋊C8—21no—
308C3×C3⋊C8—141no—
309#453—631noSylow
310C32×C7#45311yescharacteristic, complemented, Hall
311C23×C7—181no—
312C22×D7—92no—
313C22×D7—182no—
314C7×Q8—31no—
315C7×D4—31no—
316C7×D4—61no—
317C7×D4—181no—
318C7×D4—181no—
319C2×C4×C7—61no—
320C2×C4×C7C2×D3211yescharacteristic
321C22⋊D7—61no—
322C22⋊D7—182no—
323C22⋊D7—364no—
324C2×Dic7—21no—
325D28—31no—
326D28—182no—
327D28—181no—
328D28—181no—
329C4×D7—61no—
330C4×D7—182no—
331Dic14—31no—
332C7×C8—61no—
333C7⋊C8—61no—
334C3×#41—31no—
335C3×#41—211no—
336#239—421no—
337#239—421no—
338D3×D4—421no—
339D3×D4—421no—
340D3×D4—422no—
341D3×D4—422no—
342#235—71no—
343#235—421no—
344#235—421no—
345#235D3×D711yescharacteristic
346C2×D12—31no—
347C2×D12—31no—
348C2×D12—61no—
349C2×D12—211no—
350C2×D12—211no—
351C2×D12—421no—
352C2×D12—421no—
353C2×D12—421no—
354C2×D12—422no—
355C2×D12—422no—
356C2×D12—422no—
357C2×D12—422no—
358C3×M16—31no—
359C3×M16—211no—
360C3×C42—71no—
361C3×C42—71no—
362C3×C42—141no—
363#215—61no—
364#215—421no—
365#213—61no—
366#213—421no—
367#208—71no—
368#208D3×D711yescharacteristic, complemented
369D24—61no—
370D24—421no—
371#204—61no—
372#204—421no—
373C2×C3×C7—21no—
374C2×C3×C7—21no—
375C2×C3×C7—21no—
376C2×C3×C7—21no—
377C2×C3×C7—21no—
378C2×C3×C7—121no—
379C2×C3×C7#88211yescharacteristic
380C2×C3×C7#88211yescharacteristic
381D21—41no—
382D21—122no—
383D21—122no—
384D21—242no—
385C3×D7—121no—
386D3×C7—41no—
387D3×C7—241no—
388D3×C7—241no—
389D3×C7—241no—
390D3×C7—241no—
391C22×C32D4×D711yescharacteristic
392C2×C3⋊D3—42no—
393C2×C3⋊D3—41no—
394C2×C3⋊D3—142no—
395C2×C3⋊D3—282no—
396C2×C3×D3—21no—
397C2×C3×D3—141no—
398D32—562no—
399C32×C4—142no—
400C32×C4D4×D711yescharacteristic
401C32×C4C2×C22⋊D711yescharacteristic
402C3×Dic3—21no—
403C3×Dic3—141no—
404H(5,𝔽2)—631no—
405AGL(1,ℤ/8)—91no—
406AGL(1,ℤ/8)—631no—
407#127—631no—
408#104—211no—
409#104—211no—
410#100—631no—
411C22×C7—61no—
412C22×C7—362no—
413C22×C7—361no—
414C22×C7#381111yescharacteristic
415C2×D7—61no—
416C2×D7—182no—
417C2×D7—362no—
418C2×D7—722no—
419C4×C7—61no—
420C4×C7D3×D1211yescharacteristic
421C4×C7D3×D1211yescharacteristic
422Dic7—22no—
423Dic7—61no—
424C22×D3—61no—
425C22×D3—61no—
426C22×D3—61no—
427C22×D3—61no—
428C22×D3—212no—
429C22×D3—212no—
430C22×D3—422no—
431C22×D3—422no—
432C22×D3—422no—
433C3×Q8—31no—
434C3×Q8—211no—
435C3×D4—31no—
436C3×D4—61no—
437C3×D4—211no—
438C3×D4—421no—
439C2×C3×C4—21no—
440C2×C3×C4—61no—
441C2×C3×C4—141no—
442C2×C3×C4—141no—
443C2×C3×C4—141no—
444C2×C3×C4—141no—
445C2×C3×C4—421no—
446C2×C3×C4C2×D3×D711yescharacteristic
447C2×C3×C4C2×D3×D711yescharacteristic
448#67—21no—
449#67—141no—
450#67—422no—
451#67—422no—
452D12—61no—
453D12—61no—
454D12—61no—
455D12—61no—
456D12—71no—
457D12—121no—
458D12—121no—
459D12—422no—
460D12—422no—
461D12—421no—
462D12—421no—
463D12—421no—
464D12—421no—
465D12—841no—
466D12—841no—
467D12—844no—
468D12—844no—
469D12—844no—
470D12—844no—
471D12#424711yescharacteristic, complemented
472C4×D3—21no—
473C4×D3—141no—
474C4×D3—422no—
475C4×D3—422no—
476Dic6—71no—
477Dic6#424711yescharacteristic, complemented
478C3×C8—61no—
479C3×C8—421no—
480C3⋊C8—21no—
481C3⋊C8—141no—
482C3×C7—21no—
483C3×C7#482211yescharacteristic, complemented
484C3×C7#482211yescharacteristic, complemented
485C2×C32—21no—
486C2×C32#650011yescharacteristic
487C3⋊D3—81no—
488C3⋊D3—282no—
489C3×D3—41no—
490C3×D3—281no—
491#41—31no—
492#41—211no—
493#41—1261no—
494#41—1261no—
495C2×D4—91no—
496C2×D4—631no—
497C2×D4—1261no—
498C2×D4—1262no—
499C2×D4—1261no—
500C2×D4—1262no—
501SD16—181no—
502SD16—1261no—
503D8—181no—
504D8—1261no—
505M16—31no—
506M16—211no—
507C42—71no—
508C2×C7—21no—
509C2×C7—121no—
510C2×C7—721no—
511C2×C7#6392311yescharacteristic
512D7—121no—
513D7—362no—
514C22×C3—21no—
515C22×C3—61no—
516C22×C3—421no—
517C22×C3#6668111yescharacteristic
518C22×C3#6669411yescharacteristic
519C2×D3—21no—
520C2×D3—121no—
521C2×D3—122no—
522C2×D3—122no—
523C2×D3—122no—
524C2×D3—122no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—121no—
528C2×D3—141no—
529C2×D3—422no—
530C2×D3—422no—
531C2×D3—842no—
532C2×D3—842no—
533C2×D3—842no—
534C2×D3—842no—
535C2×D3—842no—
536C2×D3—1682no—
537C2×D3—1682no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—61no—
541C3×C4—142no—
542C3×C4—142no—
543C3×C4—142no—
544C3×C4—142no—
545C3×C4—421no—
546C3×C4D7×D1211yescharacteristic
547C3×C4C2×#424711yescharacteristic
548C3×C4D3×C22⋊D711yescharacteristic
549C3×C4#6669411yescharacteristic
550Dic3—21no—
551Dic3—141no—
552C32#8997011yescharacteristic, complemented, Sylow
553C23—181no—
554C23—632no—
555C23—1262no—
556Q8—31no—
557Q8—211no—
558D4—31no—
559D4—61no—
560D4—181no—
561D4—181no—
562D4—211no—
563D4—421no—
564D4—1262no—
565D4—1262no—
566D4—1261no—
567D4—1261no—
568D4—2524no—
569C2×C4—61no—
570C2×C4—141no—
571C2×C4—421no—
572C2×C4—1262no—
573C2×C4D32×D711yescharacteristic
574C8—61no—
575C8—421no—
576C7#1059039711yescharacteristic, complemented, Sylow
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—121no—
583C2×C3—841no—
584C2×C3#1062103711yescharacteristic
585C2×C3#1062113511yescharacteristic
586D3—41no—
587D3—241no—
588D3—241no—
589D3—241no—
590D3—241no—
591D3—281no—
592D3—842no—
593D3—842no—
594D3—1682no—
595C22—61no—
596C22—362no—
597C22—361no—
598C22—421no—
599C22—1262no—
600C22—5042no—
601C22—2522no—
602C22#1175942711yescharacteristic
603C4—61no—
604C4—142no—
605C4—421no—
606C4D3×#424711yescharacteristic
607C4#1175943011yescharacteristic
608C3—21no—
609C3#1305362511yescharacteristic, complemented
610C3#1305480011yescharacteristic, complemented
611C2—21no—
612C2—121no—
613C2—721no—
614C2—841no—
615C2—2522no—
616C2#7224691911yescharacteristic
subgroupG/H
centerC2#72246919
commutator, Fitting, cosocleC2×C32×C4×C7C23
Frattini, hypercenterC2×C4D32×D7
socleC2×C32×C7#120
hypocenterC32×C7#453
LCSG ▷ C2×C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72246807C2semidirect
2. #72246693C2semidirect
3. C3×#10620516C2semidirect
4. C3×#10620514C2semidirect
5. #72246368C2semidirect
6. #72245887C2semidirect
7. C7×#63825C2semidirect
11. C2×C7×C4⋊(C3⋊D3)C22semidirect
20. C3×#66627C22semidirect
21. C3×C7×#208C22semidirect
22. C32×C4×Dic7C22semidirect
39. #10620516D3semidirect
40. #10620514D3semidirect
52. C3×D84D4semidirect
54. C3×Dic42D4semidirect
56. C3×C7×D12D4semidirect
58. C3×C7×Dic6D4semidirect
92. #66627C2×D3semidirect
95. C7×#208C2×D3semidirect
99. C3×C4×Dic7C2×D3semidirect
100. C3×C4×Dic7C2×D3semidirect
109. #63825D7semidirect
160. D84D12semidirect
163. Dic42D12semidirect
171. C7×D12D12semidirect
173. C7×Dic6D12semidirect
195. C2×C4⋊(C3⋊D3)C2×D7semidirect
205. C3×#208C2×D7semidirect
231. C4×Dic7D32semidirect
299. C3×D12C22⋊D7semidirect
303. C3×Dic6C22⋊D7semidirect
310. C32×C7#453semidirect
368. #208D3×D7semidirect
471. D12#4247semidirect
477. Dic6#4247semidirect
483. C3×C7#4822semidirect
484. C3×C7#4822semidirect
552. C32#89970semidirect
576. C7#10590397semidirect
609. C3#13053625semidirect
610. C3#13054800semidirect

Maximal subgroups

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

subgroupindex
#722468072
#722466932
C3×#106205162
C3×#106205142
#722463682
#722458872
C7×#638252
#130548003
#130536253
#105903977