Finite Group Databasebeta

← #11792734

Subgroups — #11792734

Order 1152.

817 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic, complemented
2C22×#3717C211yesmaximal, characteristic, complemented
3C2×#63825C211yesmaximal, characteristic
4C2×#63650C211yesmaximal, characteristic, complemented
5C2×#63560C211yesmaximal, characteristic, complemented
6#10591070C211yesmaximal, characteristic, complemented
7C3×#4558C211yesmaximal, characteristic, complemented
8#72107—31nomaximal
9#68943—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#909—22no—
18C3×#909—22no—
19C2×C3×Dic12—21no—
20C2×C3×#204—21no—
21C2×#3717—22no—
22C2×#3717—21no—
23C2×#3717—22no—
24C2×#3717—22no—
25#63825—22no—
26#63825—22no—
27C2×#3714—21no—
28#63650—22no—
29#63560—22no—
30(C2×D3)⋊(C3⋊C8)—22no—
31C2×C3×#137—31no—
32C2×#949—31no—
33C22×#215—31no—
34C2×#909D311yescharacteristic, complemented
35C2×#908—31no—
36C22×#204—31no—
37#5234—31no—
38C2×#836—31no—
39C2×#824—31no—
40C2×#823—31no—
41C2×#823—31no—
42C2×#823—61no—
43#5110—31no—
44C3×#341—31no—
45#4558D311yescharacteristic, complemented
46C23×C3⋊D3—21no—
47C22×C32×C4C2311yescharacteristic
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—21no—
51C2×C4⋊(C3⋊D3)—21no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)D412yes—
54C2×C3×#67—21no—
55C22×C3×Dic3—21no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—21no—
60C2×C3×D12D411yescharacteristic
61C2×C3×C4×D3—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6D411yescharacteristic
68C2×C3×Dic6D412yes—
69C32×M16—21no—
70C32×M16D412yes—
71C2×C32×C8—21no—
72#3748—21no—
73#3748—21no—
74C2×C3⋊(C3⋊C8)—21no—
75C3×#208—21no—
76C3×#208D412yes—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8D412yes—
81C3×Dic12—24no—
82C3×#204—24no—
83#3717—24no—
84#3717—28no—
85#3714—24no—
86#2019—91noSylow
87C2×C3×#41—31no—
88C22×C3×Q8—31no—
89C2×#235C2×D311yescharacteristic
90C22×D12—31no—
91C22×D12—31no—
92C22×D12—61no—
93C22×Dic6C2×D311yescharacteristic
94C3×#137—62no—
95C3×#137—62no—
96C2×C3×Q16—61no—
97C2×C3×SD16—61no—
98C2×C3×M16—21no—
99C2×C3×M16—31no—
100C2×C3×M16C2×D311yescharacteristic
101C2×C3×M16C2×D311yescharacteristic
102C22×C3×C8—31no—
103#949—62no—
104#949—62no—
105C2×#215—62no—
106C2×#215—62no—
107C2×#215—61no—
108C2×#215—62no—
109C2×#213—61no—
110C2×#208—31no—
111C2×#208—31no—
112C2×#208—61no—
113C2×#208C2×D311yescharacteristic
114C22×C3⋊C8C2×D311yescharacteristic
115#909—22no—
116#909—22no—
117#908—62no—
118#908—62no—
119C2×Dic12—21no—
120C2×D24—61no—
121C2×#204—21no—
122C2×#204—62no—
123C2×#204—62no—
124C2×#204—62no—
125C2×#204—61no—
126#836—62no—
127#830—62no—
128#824—62no—
129#823—64no—
130#823—62no—
131#823—62no—
132(C2×D3)⋊C8—62no—
133C23×C32#4111yescharacteristic
134C22×C3⋊D3—22no—
135C22×C3⋊D3—41no—
136C22×C3⋊D3—41no—
137C22×C3⋊D3—82no—
138C22×C3×D3—21no—
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C4⋊(C3⋊D3)—22no—
146C4⋊(C3⋊D3)—22no—
147C4⋊(C3⋊D3)—41no—
148C4⋊(C3⋊D3)—41no—
149C3×#67—41no—
150C3×#67—41no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—41no—
154C2×C3×Dic3—41no—
155C3×D12—21no—
156C3×D12—21no—
157C3×C4×D3—41no—
158C3×C4×D3—41no—
159C3×Dic6—22no—
160C3×Dic6—21no—
161C3×Dic6—22no—
162C3×Dic6—21no—
163C3×Dic6—41no—
164C3×Dic6—41no—
165C32×C8—22no—
166C3⋊(C3⋊C8)—41no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8—22no—
169C2×#137—31no—
170C2×AGL(1,ℤ/8)—91no—
171C22×SD16—91no—
172C2×#101—91no—
173C2×#100—91no—
174#408—91no—
175#341—31no—
176C23×D3—62no—
177C23×D3—61no—
178C23×D3—61no—
179C3×#41—61no—
180C3×#41—61no—
181C3×#41—61no—
182C3×#41—61no—
183C2×C3×Q8—31no—
184C2×C3×Q8—32no—
185C2×C3×Q8—61no—
186C2×C3×Q8—61no—
187C2×C3×Q8—61no—
188C2×C3×Q8—61no—
189C2×C3×Q8—61no—
190C2×C3×D4—31no—
191C2×C3×D4—61no—
192C22×C3×C4—21no—
193C22×C3×C4—61no—
194C22×C3×C4—61no—
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C22×D311yescharacteristic
197C2×#67—21no—
198C22×Dic3—21no—
199#235—21no—
200#235—21no—
201#235—21no—
202#235—21no—
203C2×D12—32no—
204C2×D12—32no—
205C2×D12—62no—
206C2×D12—61no—
207C2×D12—61no—
208C2×D12—61no—
209C2×D12—61no—
210C2×D12—61no—
211C2×D12—61no—
212C2×D12—61no—
213C2×D12—61no—
214C2×D12—62no—
215C2×D12—61no—
216C2×D12—61no—
217C2×D12—121no—
218C2×D12—121no—
219C2×D12—121no—
220C2×D12—121no—
221C2×D12#6711yescharacteristic
222C2×C4×D3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#6711yescharacteristic
229C2×Dic6#6712yes—
230C3×Q16—64no—
231C3×SD16—64no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—22no—
235C3×M16—32no—
236C3×M16—41no—
237C3×M16—61no—
238C3×M16D1212yes—
239C3×M16#6712yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—32no—
243C2×C3×C8—41no—
244C2×C3×C8—61no—
245C2×C3×C8—61no—
246C2×C3×C8—61no—
247#215—68no—
248#215—64no—
249#213—64no—
250#208—21no—
251#208—62no—
252#208—61no—
253#208—61no—
254#208—61no—
255#208—61no—
256#208—121no—
257#208D1212yes—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8—61no—
263C2×C3⋊C8—121no—
264C2×C3⋊C8D1212yes—
265Dic12—24no—
266D24—64no—
267#204—24no—
268#204—68no—
269#204—64no—
270C22×C32—22no—
271C22×C32#12111yescharacteristic
272C22×C32#12111yescharacteristic
273C22×C32#12111yescharacteristic
274C2×C3⋊D3—42no—
275C2×C3⋊D3—82no—
276C2×C3⋊D3—82no—
277C2×C3⋊D3—82no—
278C2×C3×D3—41no—
279C2×C3×D3—42no—
280C32×C4#12011yescharacteristic
281C32×C4#12011yescharacteristic
282C32×C4#12011yescharacteristic
283C32×C4#12711yescharacteristic
284C3×Dic3—42no—
285C3×Dic3—41no—
286C2×#41—31no—
287C22×Q8—31no—
288C22×D4—91no—
289#137—62no—
290#137—62no—
291AGL(1,ℤ/8)—182no—
292AGL(1,ℤ/8)—182no—
293C2×Q16—61no—
294C2×SD16—61no—
295C2×SD16—182no—
296C2×SD16—182no—
297C2×SD16—182no—
298C2×SD16—181no—
299C2×D8—181no—
300C2×M16—31no—
301C2×M16—91no—
302C2×M16D3211yescharacteristic
303C22×C8—31no—
304#101—182no—
305#100—182no—
306C22⋊C8—182no—
307C23×C3—21no—
308C23×C3—61no—
309C23×C3#23711yescharacteristic
310C23×C3#23911yescharacteristic
311C22×D3—21no—
312C22×D3—64no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—122no—
316C22×D3—122no—
317C22×D3—121no—
318C22×D3—121no—
319C22×D3—121no—
320C22×D3—121no—
321C22×D3—242no—
322C22×D3—242no—
323C22×D3—244no—
324C3×Q8—62no—
325C3×Q8—61no—
326C3×Q8—62no—
327C3×Q8—61no—
328C3×Q8—121no—
329C3×Q8—121no—
330C3×D4—61no—
331C3×D4—61no—
332C3×D4—121no—
333C3×D4—121no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—61no—
341C2×C3×C4—62no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4—121no—
345C2×C3×C4—121no—
346C2×C3×C4C2×D1211yescharacteristic
347C2×C3×C4C2×D1211yescharacteristic
348C2×C3×C4C2×D1211yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C2×#6711yescharacteristic
357C2×C3×C4C2×#6711yescharacteristic
358#67—41no—
359#67—41no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364D12—21no—
365D12—21no—
366D12—62no—
367D12—62no—
368D12—62no—
369D12—62no—
370D12—122no—
371D12—122no—
372D12—122no—
373D12—122no—
374D12—121no—
375D12—121no—
376D12—121no—
377D12—121no—
378C4×D3—41no—
379C4×D3—41no—
380Dic6—22no—
381Dic6—21no—
382Dic6—22no—
383Dic6—21no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—22no—
387C3×C8—22no—
388C3×C8—42no—
389C3×C8—62no—
390C3×C8—62no—
391C3⋊C8—22no—
392C3⋊C8—22no—
393C3⋊C8—122no—
394C3⋊C8—121no—
395C3⋊C8—121no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32#39211yescharacteristic
399C2×C32#47112yes—
400C3⋊D3—82no—
401C3×D3—81no—
402C24—181no—
403#41—61no—
404#41—61no—
405#41—61no—
406#41—61no—
407C2×Q8—31no—
408C2×Q8—32no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×D4—31no—
415C2×D4—61no—
416C2×D4—92no—
417C2×D4—181no—
418C2×D4—181no—
419C2×D4—181no—
420C2×D4—181no—
421C2×D4—181no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4C2×D3211yescharacteristic
425Q16—64no—
426SD16—64no—
427SD16—188no—
428SD16—184no—
429D8—184no—
430M16—21no—
431M16—32no—
432M16—61no—
433M16—181no—
434M16—181no—
435M16#62612yes—
436C2×C8—21no—
437C2×C8—32no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—181no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—42no—
448C22×C3—122no—
449C22×C3—121no—
450C22×C3#89511yescharacteristic
451C22×C3#89511yescharacteristic
452C22×C3#89511yescharacteristic
453C22×C3#93811yescharacteristic
454C22×C3#93911yescharacteristic
455C22×C3#93911yescharacteristic
456C2×D3—42no—
457C2×D3—41no—
458C2×D3—124no—
459C2×D3—122no—
460C2×D3—122no—
461C2×D3—242no—
462C2×D3—242no—
463C2×D3—244no—
464C2×D3—244no—
465C2×D3—242no—
466C2×D3—242no—
467C2×D3—242no—
468C2×D3—242no—
469C2×D3—244no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—121no—
475C3×C4—122no—
476C3×C4#87411yescharacteristic
477C3×C4#88211yescharacteristic
478C3×C4#88211yescharacteristic
479C3×C4#88211yescharacteristic
480C3×C4#93711yescharacteristic
481C3×C4#93711yescharacteristic
482C3×C4#94011yescharacteristic
483C3×C4#95311yescharacteristic
484Dic3—42no—
485Dic3—41no—
486C32#201911yescharacteristic, complemented, Sylow
487C23—61no—
488C23—182no—
489C23—361no—
490C23—361no—
491C23—722no—
492C23#379611yescharacteristic
493Q8—62no—
494Q8—61no—
495Q8—62no—
496Q8—61no—
497Q8—121no—
498Q8—121no—
499D4—61no—
500D4—61no—
501D4—121no—
502D4—121no—
503D4—182no—
504D4—182no—
505D4—361no—
506D4—361no—
507C2×C4—62no—
508C2×C4—61no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4D3×D1211yescharacteristic
514C2×C4D3×D1211yescharacteristic
515C2×C4#380211yescharacteristic
516C2×C4C2×#62611yescharacteristic
517C2×C4D3×#6711yescharacteristic
518C2×C4D3×#6711yescharacteristic
519C8—22no—
520C8—62no—
521C8—62no—
522C8—361no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—241no—
532C2×C3#467111yescharacteristic
533C2×C3#489812yes—
534C2×C3#520312yes—
535C2×C3#522311yescharacteristic
536D3—81no—
537D3—244no—
538D3—242no—
539D3—242no—
540C22—22no—
541C22—122no—
542C22—121no—
543C22—362no—
544C22—722no—
545C22—722no—
546C22—722no—
547C22#6388611yescharacteristic
548C22#6388611yescharacteristic
549C22#6391611yescharacteristic
550C4—121no—
551C4—122no—
552C4#6391111yescharacteristic
553C4#6392211yescharacteristic
554C4#6392211yescharacteristic
555C4#6397711yescharacteristic
556C3—21no—
557C3#6894311yescharacteristic, complemented
558C3#7210711yescharacteristic, complemented
559C2—21no—
560C2—21no—
561C2—241no—
562C2—722no—
563C2#1059098512yes—
564C2#1059139111yescharacteristic
subgroupG/H
centerC22#63886
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#63825C2
hypercenterC2×M16D32
hypocenterC32#2019
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#909C2semidirect
2. C22×#3717C2semidirect
4. C2×#63650C2semidirect
5. C2×#63560C2semidirect
6. #10591070C2semidirect
7. C3×#4558C2semidirect
34. C2×#909D3semidirect
45. #4558D3semidirect
486. C32#2019semidirect
557. C3#68943semidirect
558. C3#72107semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C22×#37172
C2×#638252
C2×#636502
C2×#635602
#105910702
C3×#45582
#721073
#689433