Finite Group Databasebeta

← #13176890

Subgroups — #13176890

Order 1728.

2228 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic, complemented
2C2×C3×#3828C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4#11723388C212yesmaximal, complemented
5#11723387C212yesmaximal, complemented
6#11723371C212yesmaximal, complemented
7#11723370C212yesmaximal, complemented
8#11723369C214yesmaximal, complemented
9#10596219—31nomaximal
10#10594795—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×#6312—21no—
13#89076—21no—
14#89076—21no—
15#89076C2212yes—
16#89076C2212yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
18C2×C4×C3⋊(C3⋊D3)—21no—
19C2×#6308C2211yescharacteristic, complemented
20C2×C3×#638—21no—
21C3×#3828—21no—
22C3×#3828—21no—
23C3×#3828C2212yes—
24C3×#3828C2212yescomplemented
25C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
26C2×C3×C4×C3⋊D3—21no—
27C2×C3×#634C2211yescharacteristic, complemented
28C32×#208C2212yescomplemented
29C32×#208C2212yescomplemented
30C2×C32×C3⋊C8C2212yescomplemented
31#88800C2214yescomplemented
32#88797C2214yescomplemented
33#88796C2214yescomplemented
34#88791C2214yescomplemented
35C2×#3828—31no—
36C2×#3828—34no—
37C2×#3828—64no—
38C2×#3828D311yescharacteristic, complemented
39C2×C3×#235—34no—
40C2×C32×M16—31no—
41#64118—32no—
42#64118—32no—
43#64117—32no—
44#64117—32no—
45#64113—34no—
46C2×C3×#208D314yescomplemented
47#63833—38no—
48#63831—38no—
49#63830—38no—
50#63829—316no—
51#63825—38no—
52C23×C33D411yescharacteristic
53C22×C3⋊(C3⋊D3)—21no—
54C22×C3×C3⋊D3—21no—
55C2×C33×C4D411yescharacteristic
56C2×C33×C4D411yescharacteristic
57C2×C33×C4D411yescharacteristic
58C2×C33×C4C2311yescharacteristic
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61#6312—22no—
62#6312—22no—
63C2×C3⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊(C3⋊D3))—21no—
65C4⋊(C3⋊(C3⋊D3))C2312yes—
66C4×C3⋊(C3⋊D3)—22no—
67C4×C3⋊(C3⋊D3)—22no—
68#6308—21no—
69#6308C2312yes—
70C3×#638—22no—
71C3×#638—22no—
72C2×C3×C3⋊Dic3—21no—
73C3×C4⋊(C3⋊D3)—21no—
74C3×C4⋊(C3⋊D3)C2312yes—
75C3×C4×C3⋊D3—22no—
76C3×C4×C3⋊D3—22no—
77C3×#634—21no—
78C3×#634C2312yes—
79C32×C3⋊C8C2314yes—
80#5822—91no—
81#5748—94no—
82C22×C32×C4—24no—
83C22×C32×C4C2×D314yes—
84C22×C32×C4C2×D311yescharacteristic
85C2×#638—21no—
86C2×#638—68no—
87C2×#638—64no—
88C2×#638—61no—
89#3828—21no—
90#3828—21no—
91#3828—32no—
92#3828—32no—
93#3828—38no—
94#3828—38no—
95#3828—68no—
96#3828—68no—
97#3828—68no—
98#3828—68no—
99#3828—64no—
100#3828—64no—
101#3828—61no—
102#3828—61no—
103#3828C2×D312yes—
104#3828C2×D312yescomplemented
105C2×C4⋊(C3⋊D3)—31no—
106C2×C4⋊(C3⋊D3)—34no—
107C2×C4⋊(C3⋊D3)—64no—
108C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
109C2×C4×C3⋊D3—21no—
110C2×C4×C3⋊D3—68no—
111C2×C4×C3⋊D3—64no—
112C2×C4×C3⋊D3—61no—
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—64no—
116C2×#634C2×D311yescharacteristic, complemented
117C2×C3×#67—64no—
118C3×#235—38no—
119C3×#235—38no—
120C3×#235—64no—
121C3×#235—64no—
122C2×C3×D12—34no—
123C2×C3×C4×D3—64no—
124C2×C3×Dic6—34no—
125C32×M16—32no—
126C32×M16—32no—
127C2×C32×C8—32no—
128#3746—34no—
129#3745—34no—
130#3744—34no—
131#3744—34no—
132C3×#208C2×D318yescomplemented
133C3×#208C2×D318yescomplemented
134C2×C3×C3⋊C8C2×D318yescomplemented
135#3719—316no—
136#3717—316no—
137#3716—316no—
138#3714—316no—
139C22×C33—41no—
140C22×C33C2×D411yescharacteristic
141C22×C33C2×D411yescharacteristic
142C22×C33C2×D411yescharacteristic
143C2×C3⋊(C3⋊D3)—22no—
144C2×C3⋊(C3⋊D3)—42no—
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3—42no—
147C33×C4C2×D411yescharacteristic
148C33×C4C2×D411yescharacteristic
149C33×C4C2×D411yescharacteristic
150C33×C4C2411yescharacteristic
151C3⋊(C3⋊Dic3)—22no—
152C3×C3⋊Dic3—22no—
153C2×C3×#41—91no—
154C2×#235—34no—
155C2×#235—94no—
156C2×#235—91no—
157C2×#235—184no—
158C2×C3×M16—34no—
159#951—92no—
160#950—94no—
161#949—92no—
162#942—92no—
163#932—92no—
164C2×#208C3⋊D311yescharacteristic, complemented
165#909—98no—
166#909—98no—
167#908—98no—
168#908—98no—
169#904—916no—
170C23×C32—24no—
171C23×C32D1214yes—
172C23×C32#6711yescharacteristic
173C22×C3⋊D3—21no—
174C22×C3⋊D3—68no—
175C22×C3⋊D3—64no—
176C22×C3⋊D3—61no—
177C22×C3×D3—64no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—24no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4D1214yes—
185C2×C32×C4D1214yes—
186C2×C32×C4D1214yes—
187C2×C32×C4#6711yescharacteristic
188C2×C32×C4#6711yescharacteristic
189C2×C32×C4#6711yescharacteristic
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196#638—22no—
197#638—22no—
198#638—616no—
199#638—616no—
200#638—68no—
201#638—68no—
202#638—62no—
203#638—62no—
204C2×C3⋊Dic3—21no—
205C2×C3⋊Dic3—68no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—61no—
208C4⋊(C3⋊D3)—21no—
209C4⋊(C3⋊D3)—32no—
210C4⋊(C3⋊D3)—38no—
211C4⋊(C3⋊D3)—68no—
212C4⋊(C3⋊D3)—68no—
213C4⋊(C3⋊D3)—64no—
214C4⋊(C3⋊D3)—61no—
215C4⋊(C3⋊D3)C22×D312yes—
216C4×C3⋊D3—22no—
217C4×C3⋊D3—22no—
218C4×C3⋊D3—616no—
219C4×C3⋊D3—616no—
220C4×C3⋊D3—68no—
221C4×C3⋊D3—68no—
222C4×C3⋊D3—62no—
223C4×C3⋊D3—62no—
224#634—21no—
225#634—32no—
226#634—38no—
227#634—68no—
228#634—68no—
229#634—64no—
230#634—61no—
231#634C22×D312yes—
232C3×#67—68no—
233C3×#67—68no—
234C2×C3×Dic3—64no—
235C3×D12—38no—
236C3×D12—64no—
237C3×C4×D3—68no—
238C3×C4×D3—68no—
239C3×Dic6—38no—
240C3×Dic6—64no—
241C32×C8—34no—
242C3×C3⋊C8C22×D3116yes—
243#575—271noSylow
244C2×C33—21no—
245C2×C33—21no—
246C2×C33—21no—
247C2×C33C22×D411yescharacteristic
248C3⋊(C3⋊D3)—42no—
249C3×C3⋊D3—42no—
250C3×#41—92no—
251C3×#41—92no—
252C3×#41—181no—
253C3×#41—181no—
254C2×C3×Q8—91no—
255C2×C3×D4—91no—
256C2×C3×D4—181no—
257C22×C3×C4—24no—
258C22×C3×C4—181no—
259C22×C3×C4D3214yes—
260C22×C3×C4C2×C3⋊D311yescharacteristic
261C2×#67—64no—
262C2×#67—188no—
263C2×#67—184no—
264C2×#67—181no—
265#235—38no—
266#235—38no—
267#235—64no—
268#235—64no—
269#235—98no—
270#235—98no—
271#235—92no—
272#235—92no—
273#235—188no—
274#235—188no—
275#235—188no—
276#235—188no—
277#235—184no—
278#235—184no—
279#235—181no—
280#235—181no—
281C2×D12—34no—
282C2×D12—94no—
283C2×D12—91no—
284C2×D12—184no—
285C2×C4×D3—64no—
286C2×C4×D3—188no—
287C2×C4×D3—184no—
288C2×C4×D3—181no—
289C2×Dic6—34no—
290C2×Dic6—94no—
291C2×Dic6—91no—
292C2×Dic6—184no—
293C3×M16—38no—
294C3×M16—38no—
295C2×C3×C8—38no—
296#216—94no—
297#215—94no—
298#214—94no—
299#213—94no—
300#208C2×C3⋊D312yescomplemented
301#208C2×C3⋊D312yescomplemented
302C2×C3⋊C8C2×C3⋊D312yescomplemented
303Dic12—916no—
304D24—916no—
305#204—916no—
306#204—916no—
307C22×C32—24no—
308C22×C32—24no—
309C22×C32—24no—
310C22×C32—44no—
311C22×C32—41no—
312C22×C32—48no—
313C22×C32C2×D1214yes—
314C22×C32C2×D1214yes—
315C22×C32C2×D1214yes—
316C22×C32C2×#6711yescharacteristic
317C22×C32C2×#6711yescharacteristic
318C22×C32C2×#6711yescharacteristic
319C2×C3⋊D3—22no—
320C2×C3⋊D3—42no—
321C2×C3⋊D3—616no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—1216no—
325C2×C3⋊D3—128no—
326C2×C3⋊D3—122no—
327C2×C3×D3—68no—
328C2×C3×D3—128no—
329C32×C4—24no—
330C32×C4—24no—
331C32×C4—24no—
332C32×C4—24no—
333C32×C4C2×D1214yes—
334C32×C4C2×D1214yes—
335C32×C4C2×D1214yes—
336C32×C4C2×#6711yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C32×C4C23×D314yes—
340C32×C4C23×D311yescharacteristic
341C3⋊Dic3—22no—
342C3⋊Dic3—616no—
343C3⋊Dic3—68no—
344C3⋊Dic3—62no—
345C3×Dic3—68no—
346C2×#41—91no—
347C2×#41—271no—
348#137—272no—
349#137—272no—
350AGL(1,ℤ/8)—272no—
351AGL(1,ℤ/8)—272no—
352#135—274no—
353C2×M16—31no—
354C33#57511yescharacteristic, complemented, Sylow
355C23×C3—24no—
356C23×C3—181no—
357C23×C3#62614yes—
358C23×C3C4⋊(C3⋊D3)11yescharacteristic
359C22×D3—64no—
360C22×D3—188no—
361C22×D3—184no—
362C22×D3—181no—
363C3×Q8—92no—
364C3×Q8—181no—
365C3×D4—92no—
366C3×D4—182no—
367C3×D4—182no—
368C3×D4—181no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—181no—
376C2×C3×C4—182no—
377C2×C3×C4—182no—
378C2×C3×C4#62614yes—
379C2×C3×C4#62614yes—
380C2×C3×C4#62614yes—
381C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
382C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
383C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
384C2×C3×C4C2×D3214yes—
385C2×C3×C4C2×D3214yes—
386C2×C3×C4C2×D3214yes—
387C2×C3×C4C22×C3⋊D311yescharacteristic
388C2×C3×C4C22×C3⋊D311yescharacteristic
389C2×C3×C4C22×C3⋊D311yescharacteristic
390#67—68no—
391#67—68no—
392#67—1816no—
393#67—1816no—
394#67—188no—
395#67—188no—
396#67—182no—
397#67—182no—
398C2×Dic3—64no—
399C2×Dic3—188no—
400C2×Dic3—184no—
401C2×Dic3—181no—
402D12—38no—
403D12—64no—
404D12—98no—
405D12—92no—
406D12—188no—
407D12—188no—
408D12—184no—
409D12—181no—
410C4×D3—68no—
411C4×D3—68no—
412C4×D3—1816no—
413C4×D3—1816no—
414C4×D3—188no—
415C4×D3—188no—
416C4×D3—182no—
417C4×D3—182no—
418Dic6—38no—
419Dic6—64no—
420Dic6—98no—
421Dic6—92no—
422Dic6—188no—
423Dic6—188no—
424Dic6—184no—
425Dic6—181no—
426C3×C8—316no—
427C3⋊C8C22×C3⋊D314yes—
428C2×C32—28no—
429C2×C32—24no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—24no—
434C2×C32—24no—
435C2×C32—24no—
436C2×C32—44no—
437C2×C32—44no—
438C2×C32C22×D1214yes—
439C2×C32C22×#6711yescharacteristic
440C3⋊D3—42no—
441C3⋊D3—1216no—
442C3⋊D3—128no—
443C3⋊D3—122no—
444C3×D3—128no—
445#41—92no—
446#41—92no—
447#41—181no—
448#41—181no—
449#41—272no—
450#41—272no—
451#41—541no—
452#41—541no—
453C2×Q8—91no—
454C2×Q8—271no—
455C2×D4—91no—
456C2×D4—181no—
457C2×D4—271no—
458C2×D4—541no—
459C22×C4—181no—
460C22×C4—541no—
461C22×C4D3×C3⋊D311yescharacteristic
462Q16—274no—
463SD16—274no—
464SD16—274no—
465D8—274no—
466M16—32no—
467M16—32no—
468C2×C8—32no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—44no—
473C22×C3—41no—
474C22×C3—48no—
475C22×C3—182no—
476C22×C3—362no—
477C22×C3C2×#62614yes—
478C22×C3C2×#62614yes—
479C22×C3C2×#62614yes—
480C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
481C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
482C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
483C2×D3—68no—
484C2×D3—128no—
485C2×D3—1816no—
486C2×D3—188no—
487C2×D3—182no—
488C2×D3—3616no—
489C2×D3—362no—
490C2×D3—368no—
491C3×C4—24no—
492C3×C4—24no—
493C3×C4—24no—
494C3×C4—24no—
495C3×C4—182no—
496C3×C4C2×#62614yes—
497C3×C4C2×#62614yes—
498C3×C4C2×#62614yes—
499C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
500C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
501C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
502C3×C4C22×D3214yes—
503C3×C4C23×C3⋊D311yescharacteristic
504Dic3—68no—
505Dic3—1816no—
506Dic3—188no—
507Dic3—182no—
508C32—24no—
509C32#574814yescomplemented
510C32#582211yescharacteristic, complemented
511C23—181no—
512C23—541no—
513C23#629211yescharacteristic
514Q8—92no—
515Q8—181no—
516Q8—272no—
517Q8—541no—
518D4—92no—
519D4—182no—
520D4—182no—
521D4—181no—
522D4—272no—
523D4—542no—
524D4—542no—
525D4—541no—
526C2×C4—181no—
527C2×C4—182no—
528C2×C4—182no—
529C2×C4—541no—
530C2×C4—542no—
531C2×C4—542no—
532C2×C4#629211yescharacteristic
533C2×C4#629211yescharacteristic
534C2×C4#629211yescharacteristic
535C2×C4C2×D3×C3⋊D311yescharacteristic
536C2×C4C2×D3×C3⋊D311yescharacteristic
537C2×C4C2×D3×C3⋊D311yescharacteristic
538C8—34no—
539C2×C3—28no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—24no—
544C2×C3—24no—
545C2×C3—24no—
546C2×C3—24no—
547C2×C3—44no—
548C2×C3—44no—
549C2×C3—362no—
550C2×C3C22×#62614yes—
551C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
552D3—128no—
553D3—3616no—
554D3—362no—
555D3—368no—
556C22—41no—
557C22—182no—
558C22—362no—
559C22—542no—
560C22—1082no—
561C22C2×#629211yescharacteristic
562C22C2×#629211yescharacteristic
563C22C2×#629211yescharacteristic
564C4—182no—
565C4—542no—
566C4C2×#629211yescharacteristic
567C4C2×#629211yescharacteristic
568C4C2×#629211yescharacteristic
569C4C22×D3×C3⋊D311yescharacteristic
570C3—24no—
571C3#1059479514yescomplemented
572C3#1059621911yescharacteristic, complemented
573C2—21no—
574C2—21no—
575C2—21no—
576C2—362no—
577C2—1082no—
578C2C22×#629211yescharacteristic
subgroupG/H
centerC4C2×#6292
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC2×C33C22×D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C2×C3×#3828C2semidirect
3. C2×C32×#208C2semidirect
4. #11723388C2semidirect
5. #11723387C2semidirect
6. #11723371C2semidirect
7. #11723370C2semidirect
8. #11723369C2semidirect
16. #89076C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×#6308C22semidirect
24. C3×#3828C22semidirect
25. C2×C3×C4⋊(C3⋊D3)C22semidirect
27. C2×C3×#634C22semidirect
28. C32×#208C22semidirect
29. C32×#208C22semidirect
30. C2×C32×C3⋊C8C22semidirect
31. #88800C22semidirect
32. #88797C22semidirect
33. #88796C22semidirect
34. #88791C22semidirect
38. C2×#3828D3semidirect
46. C2×C3×#208D3semidirect
104. #3828C2×D3semidirect
108. C2×C4⋊(C3⋊D3)C2×D3semidirect
116. C2×#634C2×D3semidirect
132. C3×#208C2×D3semidirect
133. C3×#208C2×D3semidirect
134. C2×C3×C3⋊C8C2×D3semidirect
164. C2×#208C3⋊D3semidirect
300. #208C2×C3⋊D3semidirect
301. #208C2×C3⋊D3semidirect
302. C2×C3⋊C8C2×C3⋊D3semidirect
354. C33#575semidirect
509. C32#5748semidirect
510. C32#5822semidirect
571. C3#10594795semidirect
572. C3#10596219semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C3×#38282
C2×C32×#2082
#117233882
#117233872
#117233712
#117233702
#117233692
#105962193
#105947953