Finite Group Databasebeta

← #11865060

Subgroups — #11865060

Order 1152.

1617 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1004C211yesmaximal, characteristic, complemented
2C3×#5805C211yesmaximal, characteristic, complemented
3#10594786C212yesmaximal, complemented
4C2×#63660C211yesmaximal, characteristic, complemented
5#10591512C214yesmaximal, complemented
6#10591509C214yesmaximal, complemented
7#10591111C212yesmaximal, complemented
8#84104—31nomaximal
9#83394—31nomaximal
10C22×C32×D4C2211yescharacteristic
11C22×C3×#67—22no—
12C3×#1004—22no—
13C3×#1004—24no—
14C3×#1004C2214yes—
15C2×C3×#237—22no—
16C2×C3×D3×D4—22no—
17C2×C3×#235C2212yes—
18C2×C3×#235C2211yescharacteristic, complemented
19C2×#3748C2212yescomplemented
20C3×#953—22no—
21C3×#940C2212yescomplemented
22C3×#938—22no—
23C3×#936C2212yescomplemented
24C3×#926C2211yescharacteristic, complemented
25#63827—22no—
26#63827C2214yescomplemented
27#63823—22no—
28#63823C2214yescomplemented
29#63822—24no—
30#63660C2214yescomplemented
31#63569C2218yescomplemented
32C2×C3×H(5,𝔽2)—31no—
33C2×#1004D311yescharacteristic, complemented
34C3×#535—31no—
35#5822—32no—
36#5822—32no—
37#5805D311yescharacteristic, complemented
38#5242—34no—
39#5241—34no—
40#5236—32no—
41C2×#833—31no—
42C2×#833—31no—
43C2×#833—61no—
44#5077—34no—
45#5061—34no—
46#5005—32no—
47C24×C32D412yes—
48C23×C3×D3—22no—
49C2×C32×D4—22no—
50C2×C32×D4—22no—
51C2×C32×D4D412yes—
52C2×C32×D4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C2×C3×#67—24no—
55C2×C3×#67—28no—
56C2×C3×#67—22no—
57C2×C3×#67—28no—
58C2×C3×#67—21no—
59C2×C3×#67—24no—
60C22×C3×Dic3—22no—
61C3×#237—28no—
62C3×#237—28no—
63C3×D3×D4—28no—
64C3×D3×D4—28no—
65C3×#235—22no—
66C3×#235—22no—
67C3×#235—22no—
68C3×#235—22no—
69C3×#235D414yescomplemented
70C3×#235C2314yes—
71C2×C3×D12D412yescomplemented
72C2×C3×D12C2311yescharacteristic, complemented
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3—21no—
75C2×C3×Dic6D412yescomplemented
76C2×C3×Dic6C2311yescharacteristic, complemented
77#3748—22no—
78#3748C2314yescomplemented
79C2×C3⋊(C3⋊C8)—22no—
80C3×#217—21no—
81C3×#217—24no—
82C3×C4⋊Dic3—21no—
83C3×C4×Dic3C2312yescomplemented
84#3718—24no—
85#3715—24no—
86#3715—24no—
87#3713—24no—
88#3019—91noSylow
89C3×H(5,𝔽2)—34no—
90C3×H(5,𝔽2)—64no—
91C3×H(5,𝔽2)—62no—
92C2×C3×#41—32no—
93C2×C3×#41—31no—
94C2×C3×#41—62no—
95C22×C3×D4—21no—
96C22×C3×D4—62no—
97C22×C3×D4—62no—
98C22×C3×D4C2×D311yescharacteristic
99C22×C3×D4C2×D311yescharacteristic
100C22×#67—22no—
101#1004—22no—
102#1004—24no—
103#1004C2×D314yes—
104C2×#237—22no—
105C2×D3×D4—22no—
106C2×#235C2×D312yes—
107C2×#235C2×D311yescharacteristic, complemented
108C3×#127—32no—
109C3×#124—32no—
110C3×#121—62no—
111C3×#120—62no—
112C3×#117—31no—
113#953—22no—
114#951—34no—
115#951—62no—
116#950—64no—
117#950—64no—
118#949—34no—
119#949—62no—
120#942—34no—
121#942—62no—
122#940C2×D312yescomplemented
123#938—22no—
124#936C2×D312yescomplemented
125#932—34no—
126#932—62no—
127#926C2×D311yescharacteristic, complemented
128C2×#208—32no—
129C2×#208—32no—
130C2×#208—62no—
131#837—38no—
132#833—34no—
133#833—34no—
134#833—64no—
135#805—38no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—82no—
139C23×C32C2×D414yes—
140C23×C32C2×D411yescharacteristic
141C22×C3×D3—22no—
142C22×C3×D3—28no—
143C22×C3×D3—21no—
144C22×C3×D3—84no—
145C32×D4—24no—
146C32×D4—24no—
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2411yescharacteristic
151C3×#67—216no—
152C3×#67—24no—
153C3×#67—24no—
154C3×#67—41no—
155C3×#67—41no—
156C3×#67—84no—
157C2×C3×Dic3—28no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—21no—
161C3×D12—22no—
162C3×D12—22no—
163C3×D12C2×D414yes—
164C3×C4×D3—24no—
165C3×C4×D3—24no—
166C3×C4×D3—42no—
167C3×Dic6—22no—
168C3×Dic6—22no—
169C3×Dic6C2×D414yes—
170C3⋊(C3⋊C8)—24no—
171C2×H(5,𝔽2)—31no—
172#575—92no—
173#535—31no—
174#455—94no—
175#453—94no—
176#421—92no—
177C2×#100—91no—
178C24×C3—22no—
179C24×C3—62no—
180C24×C3#6712yes—
181C24×C3#6712yes—
182C23×D3—22no—
183C3×#41—34no—
184C3×#41—34no—
185C3×#41—68no—
186C3×#41—68no—
187C3×#41—62no—
188C3×#41—62no—
189C3×#41—62no—
190C3×#41—62no—
191C2×C3×Q8—32no—
192C2×C3×Q8—31no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—32no—
200C2×C3×D4—31no—
201C2×C3×D4—42no—
202C2×C3×D4—42no—
203C2×C3×D4—68no—
204C2×C3×D4—68no—
205C2×C3×D4—64no—
206C2×C3×D4—68no—
207C2×C3×D4—62no—
208C2×C3×D4—68no—
209C2×C3×D4—61no—
210C2×C3×D4—64no—
211C2×C3×D4#6712yes—
212C2×C3×D4#6712yes—
213C2×C3×D4C22×D312yes—
214C2×C3×D4C22×D312yes—
215C22×C3×C4—21no—
216C22×C3×C4—62no—
217C22×C3×C4—62no—
218C22×C3×C4—61no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C2×#67—24no—
222C2×#67—28no—
223C2×#67—22no—
224C2×#67—28no—
225C2×#67—21no—
226C2×#67—24no—
227C22×Dic3—22no—
228#237—28no—
229#237—28no—
230D3×D4—28no—
231D3×D4—28no—
232#235—22no—
233#235—22no—
234#235—22no—
235#235—22no—
236#235#6714yescomplemented
237#235C22×D314yes—
238C2×D12#6712yescomplemented
239C2×D12C22×D311yescharacteristic, complemented
240C2×C4×D3—22no—
241C2×C4×D3—21no—
242C2×Dic6#6712yescomplemented
243C2×Dic6C22×D311yescharacteristic, complemented
244C3×C4⋊C4—61no—
245C3×C22⋊C4—61no—
246C3×C22⋊C4—64no—
247C3×C42—32no—
248#217—21no—
249#217—24no—
250#216—64no—
251#216—64no—
252#215—64no—
253#215—64no—
254#214—64no—
255#214—64no—
256#213—64no—
257#213—64no—
258C4⋊Dic3—21no—
259C4×Dic3C22×D312yescomplemented
260#208—34no—
261#208—34no—
262#208—64no—
263#208—62no—
264#208—62no—
265#208—122no—
266C2×C3⋊C8—62no—
267C2×C3⋊C8—62no—
268C2×C3⋊C8—122no—
269C22×C32—24no—
270C22×C32—41no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32—82no—
274C22×C32#12011yescharacteristic
275C22×C32#12011yescharacteristic
276C22×C32C22×D411yescharacteristic
277C2×C3×D3—24no—
278C2×C3×D3—44no—
279C2×C3×D3—42no—
280C2×C3×D3—41no—
281C2×C3×D3—88no—
282C32×C4#12012yes—
283C32×C4C22×D412yes—
284C3×Dic3—24no—
285C3×Dic3—42no—
286C3×Dic3—41no—
287H(5,𝔽2)—34no—
288H(5,𝔽2)—64no—
289H(5,𝔽2)—62no—
290C2×#41—32no—
291C2×#41—31no—
292C2×#41—62no—
293C22×D4—62no—
294C22×D4—62no—
295C22×D4D3211yescharacteristic
296#137—94no—
297#137—182no—
298AGL(1,ℤ/8)—94no—
299AGL(1,ℤ/8)—182no—
300#135—184no—
301C2×M16—92no—
302#127—32no—
303#124—32no—
304#121—62no—
305#120—62no—
306#117—31no—
307#104—98no—
308#100—94no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—21no—
314C23×C3—24no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—68no—
318C23×C3—62no—
319C23×C3—61no—
320C23×C3—82no—
321C23×C3—82no—
322C23×C3—84no—
323C23×C3—244no—
324C23×C3C2×#6714yes—
325C23×C3C2×#6714yes—
326C23×C3C2×#6711yescharacteristic
327C23×C3C2×#6711yescharacteristic
328C22×D3—22no—
329C22×D3—28no—
330C22×D3—21no—
331C22×D3—84no—
332C3×Q8—34no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×D4—24no—
336C3×D4—24no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—34no—
340C3×D4—44no—
341C3×D4—44no—
342C3×D4—616no—
343C3×D4—64no—
344C3×D4—64no—
345C3×D4—62no—
346C3×D4—62no—
347C3×D4—121no—
348C3×D4—121no—
349C3×D4—244no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—68no—
355C2×C3×C4—62no—
356C2×C3×C4—64no—
357C2×C3×C4—64no—
358C2×C3×C4—62no—
359C2×C3×C4—61no—
360C2×C3×C4—122no—
361C2×C3×C4D3×D412yes—
362C2×C3×C4D3×D412yes—
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C2×#6711yescharacteristic
365C2×C3×C4C2×#6712yes—
366C2×C3×C4C2×#6712yes—
367C2×C3×C4C23×D311yescharacteristic
368C2×C3×C4C23×D311yescharacteristic
369#67—216no—
370#67—24no—
371#67—24no—
372#67—41no—
373#67—41no—
374#67—84no—
375C2×Dic3—28no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379D12—22no—
380D12—22no—
381D12C2×#6714yes—
382C4×D3—24no—
383C4×D3—24no—
384C4×D3—42no—
385Dic6—22no—
386Dic6—22no—
387Dic6C2×#6714yes—
388C3⋊C8—64no—
389C3⋊C8—64no—
390C3⋊C8—124no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—44no—
395C2×C32C2×#12011yescharacteristic
396C3×D3—44no—
397C3×D3—81no—
398C24—62no—
399C24#62512yes—
400#41—34no—
401#41—34no—
402#41—68no—
403#41—68no—
404#41—62no—
405#41—62no—
406#41—62no—
407#41—62no—
408C2×Q8—31no—
409C2×Q8—32no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—31no—
413C2×D4—32no—
414C2×D4—68no—
415C2×D4—68no—
416C2×D4—64no—
417C2×D4—68no—
418C2×D4—62no—
419C2×D4—68no—
420C2×D4—61no—
421C2×D4—64no—
422C2×D4#62512yes—
423C2×D4C2×D3212yes—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4C2×D3211yescharacteristic
428Q16—184no—
429SD16—184no—
430SD16—184no—
431D8—184no—
432M16—94no—
433M16—182no—
434C2×C8—182no—
435C4⋊C4—61no—
436C22⋊C4—61no—
437C22⋊C4—64no—
438C42—32no—
439C22×C3—24no—
440C22×C3—24no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—44no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—42no—
451C22×C3—44no—
452C22×C3—64no—
453C22×C3—84no—
454C22×C3—84no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—84no—
458C22×C3—124no—
459C22×C3—122no—
460C22×C3—121no—
461C22×C3—248no—
462C22×C3#93711yescharacteristic
463C22×C3#93711yescharacteristic
464C22×C3#95311yescharacteristic
465C22×C3#95311yescharacteristic
466C22×C3C22×#6711yescharacteristic
467C22×C3C22×#6711yescharacteristic
468C2×D3—24no—
469C2×D3—44no—
470C2×D3—42no—
471C2×D3—41no—
472C2×D3—88no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—64no—
476C3×C4—122no—
477C3×C4—121no—
478C3×C4#93712yes—
479C3×C4#95312yes—
480C3×C4C2×D3×D412yes—
481C3×C4C22×#6712yes—
482Dic3—24no—
483Dic3—42no—
484Dic3—41no—
485C32#301911yescharacteristic, complemented, Sylow
486C23—22no—
487C23—22no—
488C23—68no—
489C23—62no—
490C23—61no—
491C23—82no—
492C23—244no—
493C23C2×#62514yes—
494C23C2×#62511yescharacteristic
495Q8—34no—
496Q8—62no—
497Q8—62no—
498D4—24no—
499D4—24no—
500D4—34no—
501D4—616no—
502D4—64no—
503D4—64no—
504D4—62no—
505D4—62no—
506D4—121no—
507D4—121no—
508D4—244no—
509C2×C4—68no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—122no—
516C2×C4C2×#62511yescharacteristic
517C2×C4D3×#6712yes—
518C2×C4D3×#6712yes—
519C2×C4C22×D3211yescharacteristic
520C8—184no—
521C2×C3—22no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—44no—
532C2×C3—44no—
533C2×C3—84no—
534C2×C3—124no—
535C2×C3—241no—
536C2×C3C2×#93711yescharacteristic
537C2×C3C2×#95311yescharacteristic
538D3—44no—
539D3—81no—
540C22—24no—
541C22—41no—
542C22—44no—
543C22—44no—
544C22—64no—
545C22—82no—
546C22—124no—
547C22—122no—
548C22—121no—
549C22—248no—
550C22#6397611yescharacteristic
551C22#6397611yescharacteristic
552C22C22×#62511yescharacteristic
553C4—64no—
554C4—122no—
555C4—121no—
556C4#6397612yes—
557C4C2×D3×#6712yes—
558C3—21no—
559C3#8339411yescharacteristic, complemented
560C3#8410411yescharacteristic, complemented
561C2—21no—
562C2—21no—
563C2—21no—
564C2—44no—
565C2—124no—
566C2—241no—
567C2C2×#6397611yescharacteristic
subgroupG/H
centerC2C2×#63976
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×D4C22
socleC2×C32C2×#120
hypercenterC22×D4D32
hypocenterC32#3019
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1004C2semidirect
2. C3×#5805C2semidirect
3. #10594786C2semidirect
4. C2×#63660C2semidirect
5. #10591512C2semidirect
6. #10591509C2semidirect
7. #10591111C2semidirect
18. C2×C3×#235C22semidirect
19. C2×#3748C22semidirect
21. C3×#940C22semidirect
23. C3×#936C22semidirect
24. C3×#926C22semidirect
26. #63827C22semidirect
28. #63823C22semidirect
30. #63660C22semidirect
31. #63569C22semidirect
33. C2×#1004D3semidirect
37. #5805D3semidirect
69. C3×#235D4semidirect
71. C2×C3×D12D4semidirect
72. C2×C3×D12C23semidirect
75. C2×C3×Dic6D4semidirect
76. C2×C3×Dic6C23semidirect
78. #3748C23semidirect
83. C3×C4×Dic3C23semidirect
107. C2×#235C2×D3semidirect
122. #940C2×D3semidirect
124. #936C2×D3semidirect
127. #926C2×D3semidirect
236. #235#67semidirect
238. C2×D12#67semidirect
239. C2×D12C22×D3semidirect
242. C2×Dic6#67semidirect
243. C2×Dic6C22×D3semidirect
259. C4×Dic3C22×D3semidirect
485. C32#3019semidirect
559. C3#83394semidirect
560. C3#84104semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10042
C3×#58052
#105947862
C2×#636602
#105915122
#105915092
#105911112
#841043
#833943