Finite Group Databasebeta

← #13176881

Subgroups — #13176881

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×C32×#235C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4#11723381C212yesmaximal, complemented
5#11723379C212yesmaximal, complemented
6#11723378C212yesmaximal, complemented
7#11723377C214yesmaximal, complemented
8#11723367C212yesmaximal, complemented
9#10596293—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×C32×#67—21no—
21C32×#235—21no—
22C32×#235—21no—
23C32×#235C2212yes—
24C32×#235C2212yescomplemented
25C2×C32×D12C2211yescharacteristic, complemented
26C2×C32×C4×D3—21no—
27C2×C32×Dic6C2211yescharacteristic, complemented
28C3×#3748C2212yescomplemented
29C3×#3748C2212yescomplemented
30C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
31#88799C2214yescomplemented
32#88795C2214yescomplemented
33#88794C2214yescomplemented
34#88790C2214yescomplemented
35C2×C32×#41—31no—
36C2×#3828—31no—
37C2×#3828—34no—
38C2×#3828—64no—
39C2×C3×#235D314yescomplemented
40#64160—32no—
41#64159—34no—
42#64158—32no—
43#64151—32no—
44#64141—32no—
45C2×#3748D311yescharacteristic, complemented
46C2×C3×#208—34no—
47#63833—38no—
48#63831—38no—
49#63830—38no—
50#63829—316no—
51#63825—38no—
52C23×C33D411yescharacteristic
53C22×C3⋊(C3⋊D3)—21no—
54C22×C32×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—
70C32×#67—22no—
71C32×#67—22no—
72C2×C32×Dic3—21no—
73C32×D12—21no—
74C32×D12C2312yes—
75C32×C4×D3—22no—
76C32×C4×D3—22no—
77C32×Dic6—21no—
78C32×Dic6C2312yes—
79C3×C3⋊(C3⋊C8)C2314yes—
80#5822—94no—
81#5748—91no—
82C32×#41—32no—
83C32×#41—32no—
84C32×#41—61no—
85C32×#41—61no—
86C2×C32×Q8—31no—
87C2×C32×D4—31no—
88C2×C32×D4—61no—
89C22×C32×C4—24no—
90C22×C32×C4—61no—
91C22×C32×C4C2×D311yescharacteristic
92C22×C32×C4C2×D314yes—
93C2×#638—68no—
94C2×#638—61no—
95C2×#638—64no—
96#3828—32no—
97#3828—32no—
98#3828—38no—
99#3828—38no—
100#3828—68no—
101#3828—68no—
102#3828—68no—
103#3828—68no—
104#3828—61no—
105#3828—61no—
106#3828—64no—
107#3828—64no—
108C2×C4⋊(C3⋊D3)—31no—
109C2×C4⋊(C3⋊D3)—34no—
110C2×C4⋊(C3⋊D3)—64no—
111C2×C4×C3⋊D3—68no—
112C2×C4×C3⋊D3—61no—
113C2×C4×C3⋊D3—64no—
114C2×#634—31no—
115C2×#634—34no—
116C2×#634—64no—
117C2×C3×#67—24no—
118C3×#235—24no—
119C3×#235—24no—
120C3×#235C2×D318yes—
121C3×#235C2×D318yescomplemented
122C2×C3×D12C2×D314yescomplemented
123C2×C3×C4×D3—24no—
124C2×C3×Dic6C2×D314yescomplemented
125#3756—34no—
126#3755—34no—
127#3754—34no—
128#3753—34no—
129#3748C2×D312yescomplemented
130#3748C2×D312yescomplemented
131C2×C3⋊(C3⋊C8)C2×D312yescomplemented
132C3×#208—38no—
133C3×#208—38no—
134C2×C3×C3⋊C8—38no—
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×C32×D3—22no—
146C2×C32×D3—42no—
147C33×C4C2×D411yescharacteristic
148C33×C4C2×D411yescharacteristic
149C33×C4C2×D411yescharacteristic
150C33×C4C2411yescharacteristic
151C3⋊(C3⋊Dic3)—22no—
152C32×Dic3—22no—
153C2×C3×#41—34no—
154C2×#235—94no—
155C2×#235—91no—
156C2×#235—184no—
157C2×#235C3⋊D311yescharacteristic, complemented
158C2×C3×M16—91no—
159#951—98no—
160#950—916no—
161#949—98no—
162#942—98no—
163#932—98no—
164C2×#208—34no—
165#909—92no—
166#909—92no—
167#908—92no—
168#908—92no—
169#904—94no—
170C23×C32—24no—
171C23×C32—61no—
172C23×C32D1211yescharacteristic
173C23×C32#6714yes—
174C22×C3⋊D3—68no—
175C22×C3⋊D3—61no—
176C22×C3⋊D3—64no—
177C22×C3×D3—24no—
178C32×Q8—32no—
179C32×Q8—61no—
180C32×D4—32no—
181C32×D4—62no—
182C32×D4—62no—
183C32×D4—61no—
184C2×C32×C4—24no—
185C2×C32×C4—24no—
186C2×C32×C4—24no—
187C2×C32×C4—24no—
188C2×C32×C4—24no—
189C2×C32×C4—24no—
190C2×C32×C4—61no—
191C2×C32×C4—62no—
192C2×C32×C4—62no—
193C2×C32×C4D1211yescharacteristic
194C2×C32×C4D1211yescharacteristic
195C2×C32×C4D1211yescharacteristic
196C2×C32×C4#6714yes—
197C2×C32×C4#6714yes—
198C2×C32×C4#6714yes—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205#638—616no—
206#638—616no—
207#638—62no—
208#638—62no—
209#638—68no—
210#638—68no—
211C2×C3⋊Dic3—68no—
212C2×C3⋊Dic3—61no—
213C2×C3⋊Dic3—64no—
214C4⋊(C3⋊D3)—32no—
215C4⋊(C3⋊D3)—38no—
216C4⋊(C3⋊D3)—68no—
217C4⋊(C3⋊D3)—68no—
218C4⋊(C3⋊D3)—61no—
219C4⋊(C3⋊D3)—64no—
220C4×C3⋊D3—616no—
221C4×C3⋊D3—616no—
222C4×C3⋊D3—62no—
223C4×C3⋊D3—62no—
224C4×C3⋊D3—68no—
225C4×C3⋊D3—68no—
226#634—32no—
227#634—38no—
228#634—68no—
229#634—68no—
230#634—61no—
231#634—64no—
232C3×#67—28no—
233C3×#67—28no—
234C2×C3×Dic3—24no—
235C3×D12—24no—
236C3×D12C22×D318yes—
237C3×C4×D3—28no—
238C3×C4×D3—28no—
239C3×Dic6—24no—
240C3×Dic6C22×D318yes—
241C3⋊(C3⋊C8)C22×D314yes—
242C3×C3⋊C8—316no—
243#575—271noSylow
244C2×C33—21no—
245C2×C33—21no—
246C2×C33—21no—
247C2×C33C22×D411yescharacteristic
248C3⋊(C3⋊D3)—42no—
249C32×D3—42no—
250C3×#41—38no—
251C3×#41—38no—
252C3×#41—64no—
253C3×#41—64no—
254C2×C3×Q8—34no—
255C2×C3×D4—34no—
256C2×C3×D4—64no—
257C22×C3×C4—24no—
258C22×C3×C4—64no—
259C22×C3×C4D3214yes—
260C22×C3×C4C2×C3⋊D311yescharacteristic
261C2×#67—21no—
262C2×#67—188no—
263C2×#67—184no—
264C2×#67—181no—
265#235—21no—
266#235—21no—
267#235—98no—
268#235—98no—
269#235—92no—
270#235—92no—
271#235—188no—
272#235—188no—
273#235—188no—
274#235—188no—
275#235—184no—
276#235—184no—
277#235—181no—
278#235—181no—
279#235C2×C3⋊D312yes—
280#235C2×C3⋊D312yescomplemented
281C2×D12—94no—
282C2×D12—91no—
283C2×D12—184no—
284C2×D12C2×C3⋊D311yescharacteristic, complemented
285C2×C4×D3—21no—
286C2×C4×D3—188no—
287C2×C4×D3—184no—
288C2×C4×D3—181no—
289C2×Dic6—94no—
290C2×Dic6—91no—
291C2×Dic6—184no—
292C2×Dic6C2×C3⋊D311yescharacteristic, complemented
293C3×M16—92no—
294C3×M16—92no—
295C2×C3×C8—92no—
296#216—916no—
297#215—916no—
298#214—916no—
299#213—916no—
300#208—38no—
301#208—38no—
302C2×C3⋊C8—38no—
303Dic12—94no—
304D24—94no—
305#204—94no—
306#204—94no—
307C22×C32—24no—
308C22×C32—24no—
309C22×C32—24no—
310C22×C32—41no—
311C22×C32—44no—
312C22×C32—48no—
313C22×C32—62no—
314C22×C32—122no—
315C22×C32C2×D1211yescharacteristic
316C22×C32C2×D1211yescharacteristic
317C22×C32C2×D1211yescharacteristic
318C22×C32C2×#6714yes—
319C22×C32C2×#6714yes—
320C22×C32C2×#6714yes—
321C2×C3⋊D3—616no—
322C2×C3⋊D3—62no—
323C2×C3⋊D3—68no—
324C2×C3⋊D3—1216no—
325C2×C3⋊D3—122no—
326C2×C3⋊D3—128no—
327C2×C3×D3—28no—
328C2×C3×D3—48no—
329C32×C4—24no—
330C32×C4—24no—
331C32×C4—24no—
332C32×C4—24no—
333C32×C4—62no—
334C32×C4C2×D1211yescharacteristic
335C32×C4C2×D1211yescharacteristic
336C32×C4C2×D1211yescharacteristic
337C32×C4C2×#6714yes—
338C32×C4C2×#6714yes—
339C32×C4C2×#6714yes—
340C32×C4C23×D314yes—
341C32×C4C23×D311yescharacteristic
342C3⋊Dic3—616no—
343C3⋊Dic3—62no—
344C3⋊Dic3—68no—
345C3×Dic3—28no—
346C2×#41—31no—
347C2×#41—271no—
348#137—272no—
349#137—272no—
350AGL(1,ℤ/8)—272no—
351AGL(1,ℤ/8)—272no—
352#135—274no—
353C2×M16—91no—
354C33#57511yescharacteristic, complemented, Sylow
355C23×C3—24no—
356C23×C3—64no—
357C23×C3#62614yes—
358C23×C3#63811yescharacteristic
359C22×D3—21no—
360C22×D3—188no—
361C22×D3—184no—
362C22×D3—181no—
363C3×Q8—38no—
364C3×Q8—64no—
365C3×D4—38no—
366C3×D4—68no—
367C3×D4—68no—
368C3×D4—64no—
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—64no—
376C2×C3×C4—68no—
377C2×C3×C4—68no—
378C2×C3×C4#62614yes—
379C2×C3×C4#62614yes—
380C2×C3×C4#62614yes—
381C2×C3×C4#63811yescharacteristic
382C2×C3×C4#63811yescharacteristic
383C2×C3×C4#63811yescharacteristic
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—22no—
391#67—22no—
392#67—1816no—
393#67—1816no—
394#67—188no—
395#67—188no—
396#67—182no—
397#67—182no—
398C2×Dic3—21no—
399C2×Dic3—188no—
400C2×Dic3—184no—
401C2×Dic3—181no—
402D12—21no—
403D12—98no—
404D12—92no—
405D12—188no—
406D12—188no—
407D12—184no—
408D12—181no—
409D12C22×C3⋊D312yes—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—1816no—
413C4×D3—1816no—
414C4×D3—188no—
415C4×D3—188no—
416C4×D3—182no—
417C4×D3—182no—
418Dic6—21no—
419Dic6—98no—
420Dic6—92no—
421Dic6—188no—
422Dic6—188no—
423Dic6—184no—
424Dic6—181no—
425Dic6C22×C3⋊D312yes—
426C3×C8—94no—
427C3⋊C8—316no—
428C2×C32—28no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—24no—
435C2×C32—24no—
436C2×C32—44no—
437C2×C32—44no—
438C2×C32—122no—
439C2×C32C22×D1211yescharacteristic
440C2×C32C22×#6714yes—
441C3⋊D3—1216no—
442C3⋊D3—122no—
443C3⋊D3—128no—
444C3×D3—48no—
445#41—32no—
446#41—32no—
447#41—61no—
448#41—61no—
449#41—272no—
450#41—272no—
451#41—541no—
452#41—541no—
453C2×Q8—31no—
454C2×Q8—271no—
455C2×D4—31no—
456C2×D4—61no—
457C2×D4—271no—
458C2×D4—541no—
459C22×C4—61no—
460C22×C4—541no—
461C22×C4D3×C3⋊D311yescharacteristic
462Q16—274no—
463SD16—274no—
464SD16—274no—
465D8—274no—
466M16—92no—
467M16—92no—
468C2×C8—92no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—44no—
473C22×C3—41no—
474C22×C3—48no—
475C22×C3—68no—
476C22×C3—128no—
477C22×C3C2×#62614yes—
478C22×C3C2×#62614yes—
479C22×C3C2×#62614yes—
480C22×C3C2×#63811yescharacteristic
481C22×C3C2×#63811yescharacteristic
482C22×C3C2×#63811yescharacteristic
483C2×D3—22no—
484C2×D3—42no—
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—68no—
496C3×C4C2×#62614yes—
497C3×C4C2×#62614yes—
498C3×C4C2×#62614yes—
499C3×C4C2×#63811yescharacteristic
500C3×C4C2×#63811yescharacteristic
501C3×C4C2×#63811yescharacteristic
502C3×C4C22×D3214yes—
503C3×C4C23×C3⋊D311yescharacteristic
504Dic3—22no—
505Dic3—1816no—
506Dic3—188no—
507Dic3—182no—
508C32—24no—
509C32#574811yescharacteristic, complemented
510C32#582214yescomplemented
511C23—61no—
512C23—541no—
513C23#629111yescharacteristic
514Q8—32no—
515Q8—61no—
516Q8—272no—
517Q8—541no—
518D4—32no—
519D4—62no—
520D4—62no—
521D4—61no—
522D4—272no—
523D4—542no—
524D4—542no—
525D4—541no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—541no—
530C2×C4—542no—
531C2×C4—542no—
532C2×C4#629111yescharacteristic
533C2×C4#629111yescharacteristic
534C2×C4#629111yescharacteristic
535C2×C4C2×D3×C3⋊D311yescharacteristic
536C2×C4C2×D3×C3⋊D311yescharacteristic
537C2×C4C2×D3×C3⋊D311yescharacteristic
538C8—94no—
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—128no—
550C2×C3C22×#62614yes—
551C2×C3C22×#63811yescharacteristic
552D3—42no—
553D3—3616no—
554D3—362no—
555D3—368no—
556C22—41no—
557C22—62no—
558C22—122no—
559C22—542no—
560C22—1082no—
561C22C2×#629111yescharacteristic
562C22C2×#629111yescharacteristic
563C22C2×#629111yescharacteristic
564C4—62no—
565C4—542no—
566C4C2×#629111yescharacteristic
567C4C2×#629111yescharacteristic
568C4C2×#629111yescharacteristic
569C4C22×D3×C3⋊D311yescharacteristic
570C3—24no—
571C3#1059479514yescomplemented
572C3#1059629311yescharacteristic, complemented
573C2—21no—
574C2—21no—
575C2—21no—
576C2—122no—
577C2—1082no—
578C2C22×#629111yescharacteristic
subgroupG/H
centerC4C2×#6291
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×C32×#235C2semidirect
3. C2×C3×#3748C2semidirect
4. #11723381C2semidirect
5. #11723379C2semidirect
6. #11723378C2semidirect
7. #11723377C2semidirect
8. #11723367C2semidirect
16. #89076C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×#6308C22semidirect
24. C32×#235C22semidirect
25. C2×C32×D12C22semidirect
27. C2×C32×Dic6C22semidirect
28. C3×#3748C22semidirect
29. C3×#3748C22semidirect
30. C2×C3×C3⋊(C3⋊C8)C22semidirect
31. #88799C22semidirect
32. #88795C22semidirect
33. #88794C22semidirect
34. #88790C22semidirect
39. C2×C3×#235D3semidirect
45. C2×#3748D3semidirect
121. C3×#235C2×D3semidirect
122. C2×C3×D12C2×D3semidirect
124. C2×C3×Dic6C2×D3semidirect
129. #3748C2×D3semidirect
130. #3748C2×D3semidirect
131. C2×C3⋊(C3⋊C8)C2×D3semidirect
157. C2×#235C3⋊D3semidirect
280. #235C2×C3⋊D3semidirect
284. C2×D12C2×C3⋊D3semidirect
292. C2×Dic6C2×C3⋊D3semidirect
354. C33#575semidirect
509. C32#5748semidirect
510. C32#5822semidirect
571. C3#10594795semidirect
572. C3#10596293semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C32×#2352
C2×C3×#37482
#117233812
#117233792
#117233782
#117233772
#117233672
#105962933
#105947953