Finite Group Databasebeta

← #11865409

Subgroups — #11865409

Order 1152.

1567 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2#10596296C211yesmaximal, characteristic, complemented
3#10594856C212yesmaximal, complemented
4C3×#5240C211yesmaximal, characteristic, complemented
5#10591534C218yesmaximal, complemented
6#10591231C212yesmaximal, complemented
7#84124—31nomaximal
8#78692—31nomaximal
9C2×C32×#41C2211yescharacteristic
10C22×#634C2211yescharacteristic, complemented
11C3×#1010—22no—
12C3×#1009—22no—
13C3×D3×#41—24no—
14C3×#1007—21no—
15C3×#1004—21no—
16C3×#1004C2214yes—
17C2×C3×#235—22no—
18C2×C3×#235C2212yes—
19#64153—21no—
20#64145—21no—
21#64144C2214yescomplemented
22#64135C2211yescharacteristic, complemented
23#64133—22no—
24C3×#926C2212yescomplemented
25#63962C2218yescomplemented
26#63866—22no—
27#63860—22no—
28#63857—24no—
29C3×#834C2214yescomplemented
30#63580C2218yescomplemented
31C3×#585—31no—
32#5964D311yescharacteristic
33#5825—31no—
34#5825—31no—
35#5825—32no—
36#5825—61no—
37#5526—32no—
38C3×#410—31no—
39#5243—38no—
40#5240—32no—
41#5240D311yescharacteristic, complemented
42#5006—32no—
43#4742—38no—
44C32×#41—22no—
45C32×#41—22no—
46C2×C32×Q8D411yescharacteristic
47C2×C32×D4D411yescharacteristic
48C2×C32×D4C2312yes—
49C22×C32×C4D412yes—
50C22×C32×C4C2311yescharacteristic
51C22×C3⋊Dic3C2312yes—
52C2×#634—21no—
53C2×#634—21no—
54C2×#634—22no—
55C2×#634—22no—
56C2×C3×#67—24no—
57C2×C3×#67C2314yes—
58C3×#239—24no—
59C3×D3×Q8—24no—
60C3×#237—24no—
61C3×#237—28no—
62C3×D3×D4—24no—
63C3×D3×D4—28no—
64C3×#235—24no—
65C3×#235—24no—
66C3×#235—22no—
67C3×#235—24no—
68C3×#235—22no—
69C3×#235—22no—
70C3×#235—22no—
71C2×C3×D12—22no—
72C2×C3×D12D412yes—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3D414yes—
75C2×C3×Dic6—22no—
76C2×C3×Dic6D412yes—
77#3757—24no—
78#3757C2312yes—
79C4⋊(C3⋊Dic3)—21no—
80#3750—24no—
81C4×C3⋊Dic3—21no—
82C3×#217C2314yes—
83C3×C4⋊Dic3—22no—
84C3×C4×Dic3—22no—
85Dic3⋊Dic3—28no—
86(C2×D3)⋊Dic3—28no—
87#3028—91noSylow
88C3×2−1+4—62no—
89C3×2−1+4—61no—
90C3×H(5,𝔽2)—34no—
91C3×H(5,𝔽2)—62no—
92C3×H(5,𝔽2)—61no—
93C2×C3×#41—21no—
94C2×C3×#41—32no—
95C2×C3×#41—64no—
96C2×C3×#41—62no—
97C2×C3×#41C2×D311yescharacteristic
98C2×C3×#41C2×D311yescharacteristic
99#1010—22no—
100#1009—22no—
101D3×#41—24no—
102#1007—21no—
103#1004—21no—
104#1004C2×D314yes—
105C2×#235—22no—
106C2×#235C2×D312yes—
107C22×Dic6—31no—
108C22×Dic6—31no—
109C22×Dic6—61no—
110C3×#117—32no—
111#944—62no—
112#944—62no—
113#944—61no—
114#944—61no—
115#936—62no—
116#936—62no—
117#936—61no—
118#936—61no—
119#935—34no—
120#935—34no—
121#935—38no—
122#935—64no—
123#926—31no—
124#926—31no—
125#926—61no—
126#926C2×D312yescomplemented
127#924—64no—
128#924—64no—
129#924—62no—
130#924—62no—
131#897—64no—
132#886—38no—
133#875—62no—
134#869—62no—
135C3×#99—34no—
136#834—38no—
137#834C2×D314yescomplemented
138#806—38no—
139C23×C32C2×D412yes—
140C23×C32C2411yescharacteristic
141C22×C3×D3—24no—
142C22×C3×D3C2×D414yes—
143C32×Q8—22no—
144C32×D4—24no—
145C32×D4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D412yes—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—24no—
156#634—41no—
157#634—42no—
158#634—41no—
159C3×#67—28no—
160C3×#67—24no—
161C3×#67—24no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3C2×D414yes—
165C3×D12—24no—
166C3×D12—22no—
167C3×D12—22no—
168C3×C4×D3—28no—
169C3×C4×D3—24no—
170C3×C4×D3—24no—
171C3×Dic6—24no—
172C3×Dic6—22no—
173C3×Dic6—22no—
174#585—31no—
175#536—92no—
176#536—91no—
177#458—98no—
178#410—31no—
179#410—92no—
180C3×#41—22no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41—42no—
185C3×#41—42no—
186C3×#41—64no—
187C3×#41—64no—
188C3×#41—68no—
189C3×#41—64no—
190C3×#41—64no—
191C3×#41—62no—
192C3×#41—62no—
193C3×#41—64no—
194C3×#41—62no—
195C3×#41—62no—
196C2×C3×Q8—21no—
197C2×C3×Q8—32no—
198C2×C3×Q8—62no—
199C2×C3×Q8—64no—
200C2×C3×Q8D1211yescharacteristic
201C2×C3×Q8#6711yescharacteristic
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—34no—
205C2×C3×D4—32no—
206C2×C3×D4—62no—
207C2×C3×D4—68no—
208C2×C3×D4—64no—
209C2×C3×D4—64no—
210C2×C3×D4D1211yescharacteristic
211C2×C3×D4#6711yescharacteristic
212C2×C3×D4C22×D312yes—
213C2×C3×D4C22×D312yes—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—34no—
217C22×C3×C4—64no—
218C22×C3×C4D1212yes—
219C22×C3×C4#6712yes—
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C22×D311yescharacteristic
222C2×#67—24no—
223C2×#67C22×D314yes—
224C22×Dic3—32no—
225C22×Dic3—32no—
226C22×Dic3—62no—
227#239—24no—
228D3×Q8—24no—
229#237—24no—
230#237—28no—
231D3×D4—24no—
232D3×D4—28no—
233#235—24no—
234#235—24no—
235#235—22no—
236#235—24no—
237#235—22no—
238#235—22no—
239#235—22no—
240C2×D12—22no—
241C2×D12#6712yes—
242C2×C4×D3—24no—
243C2×C4×D3#6714yes—
244C2×Dic6—22no—
245C2×Dic6—62no—
246C2×Dic6—62no—
247C2×Dic6—61no—
248C2×Dic6—61no—
249C2×Dic6—62no—
250C2×Dic6—62no—
251C2×Dic6—62no—
252C2×Dic6—62no—
253C2×Dic6—61no—
254C2×Dic6—61no—
255C2×Dic6—122no—
256C2×Dic6—122no—
257C2×Dic6#6712yes—
258C3×C4⋊C4—62no—
259C3×C22⋊C4—34no—
260C3×C42—62no—
261#217—32no—
262#217—32no—
263#217—62no—
264#217—68no—
265#217—68no—
266#217—64no—
267#217—64no—
268#217C22×D314yes—
269(C2×D3)⋊C4—68no—
270C4⋊Dic3—22no—
271C4⋊Dic3—62no—
272C4⋊Dic3—61no—
273C4⋊Dic3—61no—
274C4⋊Dic3—68no—
275Dic3⋊C4—68no—
276Dic3⋊C4—68no—
277Dic3⋊C4—64no—
278Dic3⋊C4—64no—
279C4×Dic3—22no—
280C4×Dic3—62no—
281C4×Dic3—61no—
282C4×Dic3—61no—
283C22×C32—22no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32C22×D411yescharacteristic
287C22×C32C22×D412yes—
288C2×C3×D3—24no—
289C2×C3×D3—44no—
290C2×C3×D3—84no—
291C32×C4—22no—
292C32×C4#12011yescharacteristic
293C32×C4#12012yes—
294C32×C4#12011yescharacteristic
295C3⋊Dic3—42no—
296C3⋊Dic3—42no—
297C3×Dic3—24no—
298C3×Dic3—44no—
2992−1+4—62no—
3002−1+4—61no—
301H(5,𝔽2)—34no—
302H(5,𝔽2)—62no—
303H(5,𝔽2)—61no—
304C2×#41—32no—
305C2×#41—64no—
306C2×#41—62no—
307C2×#41D3211yescharacteristic
308C22×Q8—91no—
309C4⋊Q8—182no—
310C4⋊Q8—181no—
311#124—182no—
312#124—181no—
313#123—98no—
314#123—94no—
315C22⋊Q8—184no—
316C22⋊Q8—182no—
317#117—32no—
318#117—91no—
319#99—34no—
320#99—98no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—34no—
324C23×C3—64no—
325C23×C3C2×D1212yes—
326C23×C3C2×#6712yes—
327C23×C3C23×D311yescharacteristic
328C23×C3C23×D311yescharacteristic
329C22×D3—24no—
330C22×D3C2×#6714yes—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—42no—
334C3×Q8—64no—
335C3×Q8—62no—
336C3×Q8—62no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—42no—
342C3×D4—44no—
343C3×D4—64no—
344C3×D4—68no—
345C3×D4—64no—
346C3×D4—64no—
347C3×D4—62no—
348C3×D4—62no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—34no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—68no—
362C2×C3×C4—64no—
363C2×C3×C4—64no—
364C2×C3×C4—64no—
365C2×C3×C4—64no—
366C2×C3×C4C2×D1212yes—
367C2×C3×C4C2×D1211yescharacteristic
368C2×C3×C4C2×D1211yescharacteristic
369C2×C3×C4D3×D412yes—
370C2×C3×C4D3×D412yes—
371C2×C3×C4C2×#6712yes—
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C2×#6711yescharacteristic
374C2×C3×C4C2×#6712yes—
375C2×C3×C4C2×#6712yes—
376#67—28no—
377#67—24no—
378#67—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—64no—
382C2×Dic3—64no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—64no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—64no—
389C2×Dic3—124no—
390C2×Dic3C2×#6714yes—
391D12—24no—
392D12—22no—
393D12—22no—
394C4×D3—28no—
395C4×D3—24no—
396C4×D3—24no—
397Dic6—24no—
398Dic6—22no—
399Dic6—22no—
400Dic6—122no—
401Dic6—122no—
402Dic6—124no—
403Dic6—121no—
404Dic6—121no—
405Dic6—122no—
406Dic6—121no—
407Dic6—122no—
408Dic6—121no—
409C2×C32—21no—
410C2×C32—22no—
411C2×C32—42no—
412C2×C32C2×#12011yescharacteristic
413C3×D3—44no—
414#41—22no—
415#41—22no—
416#41—64no—
417#41—64no—
418#41—68no—
419#41—64no—
420#41—64no—
421#41—62no—
422#41—62no—
423#41—64no—
424#41—62no—
425#41—62no—
426C2×Q8—32no—
427C2×Q8—62no—
428C2×Q8—64no—
429C2×Q8—181no—
430C2×Q8—181no—
431C2×Q8—182no—
432C2×Q8—182no—
433C2×Q8#62611yescharacteristic
434C2×D4—34no—
435C2×D4—32no—
436C2×D4—62no—
437C2×D4—68no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4#62611yescharacteristic
441C2×D4C2×D3212yes—
442C22×C4—34no—
443C22×C4—64no—
444C22×C4—92no—
445C22×C4#62612yes—
446C22×C4C2×D3211yescharacteristic
447C4⋊C4—62no—
448C4⋊C4—188no—
449C4⋊C4—181no—
450C4⋊C4—184no—
451C22⋊C4—34no—
452C22⋊C4—92no—
453C22⋊C4—188no—
454C22⋊C4—184no—
455C42—62no—
456C42—181no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—21no—
460C22×C3—22no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—44no—
468C22×C3—64no—
469C22×C3—124no—
470C22×C3—244no—
471C22×C3C22×D1211yescharacteristic
472C22×C3C2×D3×D412yes—
473C22×C3C22×#6711yescharacteristic
474C22×C3C22×#6712yes—
475C2×D3—24no—
476C2×D3—44no—
477C2×D3—84no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—22no—
483C3×C4—42no—
484C3×C4—64no—
485C3×C4—124no—
486C3×C4#88211yescharacteristic
487C3×C4#88211yescharacteristic
488C3×C4#88212yes—
489C3×C4#95311yescharacteristic
490C3×C4#95312yes—
491C3×C4#95311yescharacteristic
492Dic3—24no—
493Dic3—44no—
494Dic3—124no—
495Dic3—124no—
496Dic3—122no—
497Dic3—122no—
498Dic3—122no—
499Dic3—122no—
500C32#302811yescharacteristic, complemented, Sylow
501C23—34no—
502C23—64no—
503C23C2×#62612yes—
504C23C22×D3211yescharacteristic
505Q8—22no—
506Q8—64no—
507Q8—62no—
508Q8—62no—
509Q8—361no—
510Q8—361no—
511Q8—362no—
512D4—24no—
513D4—22no—
514D4—64no—
515D4—68no—
516D4—64no—
517D4—64no—
518D4—62no—
519D4—62no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—34no—
523C2×C4—68no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—182no—
529C2×C4—182no—
530C2×C4—184no—
531C2×C4C2×#62612yes—
532C2×C4C2×#62611yescharacteristic
533C2×C4C2×#62611yescharacteristic
534C2×C4D3×#6712yes—
535C2×C4D3×#6712yes—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—42no—
543C2×C3—42no—
544C2×C3—42no—
545C2×C3—82no—
546C2×C3—124no—
547C2×C3C2×#88211yescharacteristic
548C2×C3C2×#95311yescharacteristic
549D3—44no—
550C22—22no—
551C22—41no—
552C22—42no—
553C22—64no—
554C22—124no—
555C22—244no—
556C22C22×#62611yescharacteristic
557C22C2×D3×#6712yes—
558C4—22no—
559C4—64no—
560C4—124no—
561C4—362no—
562C4—362no—
563C4#6397711yescharacteristic
564C4#6397711yescharacteristic
565C4#6397712yes—
566C3—21no—
567C3#7869211yescharacteristic, complemented
568C3#8412411yescharacteristic, complemented
569C2—22no—
570C2—21no—
571C2—42no—
572C2—124no—
573C2C2×#6397711yescharacteristic
subgroupG/H
centerC2C2×#63977
commutator, cosocleC23×C32C24
FrattiniC23C22×D32
FittingC2×C32×#41C22
socleC2×C32C2×#120
prime radicalC3×#5964C2
hypercenterC2×#41D32
hypocenterC32#3028
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #10596296C2semidirect
3. #10594856C2semidirect
4. C3×#5240C2semidirect
5. #10591534C2semidirect
6. #10591231C2semidirect
10. C22×#634C22semidirect
21. #64144C22semidirect
22. #64135C22semidirect
24. C3×#926C22semidirect
25. #63962C22semidirect
29. C3×#834C22semidirect
30. #63580C22semidirect
41. #5240D3semidirect
126. #926C2×D3semidirect
137. #834C2×D3semidirect
500. C32#3028semidirect
567. C3#78692semidirect
568. C3#84124semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
#105962962
#105948562
C3×#52402
#105915342
#105912312
#841243
#786923