Finite Group Databasebeta

← #11788692

Subgroups — #11788692

Order 1152.

753 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64118C211yesmaximal, characteristic
2C2×#63817C211yesmaximal, characteristic
3#10591106C211yesmaximal, characteristic
4C2×#63567C211yesmaximal, characteristic, complemented
5C2×#63561C211yesmaximal, characteristic, complemented
6C3×#4558C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8#69007—31nomaximal
9#68949—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×#634C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13#64118—22no—
14#64118C414yescomplemented
15C2×#3746—21no—
16C2×#3744—21no—
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C2×#3712—21no—
22#63817—22no—
23#63817—22no—
24C2×#3710—21no—
25#63567—21no—
26#63567—21no—
27#63567—21no—
28#63564—21no—
29#63561—22no—
30C2×#909—31no—
31C2×#909—31no—
32C2×#909—61no—
33C2×#907—31no—
34C2×#901—31no—
35C2×#824—31no—
36C2×#824—31no—
37#5133—31no—
38C2×#816—31no—
39C2×#810—31no—
40#4966—31no—
41C3×#343—31no—
42C3×#341—31no—
43#4558D311yescharacteristic, complemented
44#4554D311yescharacteristic, complemented
45C22×C32×C4C2311yescharacteristic
46C2×#638—21no—
47C22×C3⋊Dic3—21no—
48#3828—21no—
49#3828—21no—
50#3828C2×C412yes—
51#3828D412yes—
52C2×C4⋊(C3⋊D3)D411yescharacteristic
53C2×C4×C3⋊D3—21no—
54C2×#634—21no—
55C2×#634—21no—
56C2×#634—21no—
57C2×#634—21no—
58C2×#634—21no—
59C2×#634C2×C412yes—
60C2×#634D411yescharacteristic
61C22×C3×Dic3—21no—
62C32×M16—21no—
63C32×M16C2×C412yes—
64C2×C32×C8—21no—
65#3746—24no—
66#3744—24no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C3×#208—21no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208D412yes—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8D411yescharacteristic
78C2×C3×C3⋊C8D411yescharacteristic
79#3712—24no—
80#3710—24no—
81#1908—91noSylow
82C2×#235—31no—
83C2×#235—31no—
84C2×#235—61no—
85C22×Dic6—31no—
86C22×Dic6—31no—
87C22×Dic6—61no—
88C2×C3×M16—21no—
89C2×C3×M16—31no—
90C2×C3×M16—31no—
91C2×C3×M16C2×D311yescharacteristic
92C2×C3×M16C2×D311yescharacteristic
93C22×C3×C8—31no—
94C2×C3×C4⋊C4—31no—
95C2×C4⋊Dic3C2×D311yescharacteristic
96C2×#208C2×D311yescharacteristic
97C2×#208C2×D311yescharacteristic
98C22×C3⋊C8C2×D311yescharacteristic
99#909—34no—
100#909—34no—
101#909—64no—
102#909—62no—
103#909—62no—
104#909—64no—
105#907—62no—
106#907—62no—
107C2×Dic12—61no—
108C2×Dic12—61no—
109C2×Dic12—62no—
110C2×#204—61no—
111C2×#204—61no—
112C2×#204—62no—
113#901—62no—
114#901—62no—
115C2×ΓL(1,𝔽25)—61no—
116C2×ΓL(1,𝔽25)—61no—
117C2×D3×C8—61no—
118C2×D3×C8—61no—
119#824—62no—
120#824—62no—
121#821—61no—
122#816—61no—
123#816—61no—
124#816—61no—
125#810—61no—
126#810—61no—
127#810—61no—
128#809—61no—
129C23×C32#4111yescharacteristic
130C22×C3⋊D3—21no—
131C2×C32×C4C22×C411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4#4111yescharacteristic
137#638—22no—
138#638—22no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—41no—
142C2×C3⋊Dic3—41no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)C22⋊C412yes—
145C4×C3⋊D3—22no—
146C4×C3⋊D3—22no—
147#634—22no—
148#634—21no—
149#634—22no—
150#634—41no—
151#634—41no—
152#634C22⋊C412yes—
153C2×C3×Dic3—41no—
154C2×C3×Dic3—81no—
155C32×C8—22no—
156C3×C3⋊C8—21no—
157C3×C3⋊C8—22no—
158C3×C3⋊C8—21no—
159C3×C3⋊C8—41no—
160C2×#137—91no—
161C2×#131—91no—
162#417—91no—
163C2×#103—91no—
164C2×#101—91no—
165#343—31no—
166#341—31no—
167C22×C3×C4—21no—
168C22×C3×C4—61no—
169C22×C3×C4C22×D311yescharacteristic
170C22×C3×C4C22×D311yescharacteristic
171C2×#67—61no—
172C2×#67—61no—
173C2×#67—121no—
174C22×Dic3—21no—
175C22×Dic3—61no—
176C22×Dic3—61no—
177C22×Dic3—121no—
178#235—32no—
179#235—32no—
180#235—32no—
181#235—32no—
182#235—62no—
183#235—62no—
184#235—61no—
185#235—61no—
186#235—61no—
187#235—61no—
188#235—121no—
189#235—121no—
190C2×D12—31no—
191C2×D12—31no—
192C2×D12—61no—
193C2×C4×D3—61no—
194C2×C4×D3—61no—
195C2×C4×D3—121no—
196C2×Dic6—31no—
197C2×Dic6—31no—
198C2×Dic6—32no—
199C2×Dic6—32no—
200C2×Dic6—61no—
201C2×Dic6—62no—
202C2×Dic6—62no—
203C2×Dic6—62no—
204C2×Dic6—61no—
205C2×Dic6—61no—
206C2×Dic6—61no—
207C2×Dic6—61no—
208C2×Dic6—61no—
209C2×Dic6—61no—
210C2×Dic6—61no—
211C2×Dic6—61no—
212C2×Dic6—61no—
213C2×Dic6—61no—
214C2×Dic6—121no—
215C2×Dic6—121no—
216C2×Dic6—121no—
217C3×M16—22no—
218C3×M16—21no—
219C3×M16—21no—
220C3×M16—22no—
221C3×M16—32no—
222C3×M16—61no—
223C3×M16—61no—
224C3×M16—61no—
225C3×M16C4×D312yes—
226C3×M16C4×D312yes—
227C2×C3×C8—22no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—31no—
231C2×C3×C8—31no—
232C2×C3×C8—61no—
233C2×C3×C8—61no—
234C2×C3×C8—61no—
235C2×C3×C8—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—61no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240#208—21no—
241#208—21no—
242#208—21no—
243#208D1212yes—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8D1211yescharacteristic
249C2×C3⋊C8D1211yescharacteristic
250Dic12—64no—
251Dic12—64no—
252Dic12—68no—
253#204—64no—
254#204—64no—
255#204—68no—
256ΓL(1,𝔽25)—64no—
257ΓL(1,𝔽25)—64no—
258D3×C8—64no—
259D3×C8—64no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32C4×D411yescharacteristic
263C22×C32#12111yescharacteristic
264C22×C32#12311yescharacteristic
265C2×C3⋊D3—22no—
266C2×C3⋊D3—41no—
267C2×C3⋊D3—41no—
268C32×C4C2×C22⋊C411yescharacteristic
269C32×C4C4×D411yescharacteristic
270C32×C4#12011yescharacteristic
271C32×C4#12111yescharacteristic
272C3⋊Dic3—22no—
273C3⋊Dic3—42no—
274C3×Dic3—81no—
275C2×#41—91no—
276C22×Q8—91no—
277#137—94no—
278#137—182no—
279C2×Q16—181no—
280C2×SD16—181no—
281#131—182no—
282#131—182no—
283C2×M16—31no—
284C2×M16—31no—
285C2×M16—181no—
286C2×M16D3211yescharacteristic
287C22×C8—31no—
288C22×C8—181no—
289C2×C4⋊C4—31no—
290#103—181no—
291#103—181no—
292#103—181no—
293#102—181no—
294#101—182no—
295C23×C3—21no—
296C23×C3#23511yescharacteristic
297C23×C3#23911yescharacteristic
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—121no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—121no—
308C2×C3×C4—241no—
309C2×C3×C4C2×C4×D311yescharacteristic
310C2×C3×C4C2×C4×D311yescharacteristic
311C2×C3×C4C2×D1211yescharacteristic
312C2×C3×C4C2×D1211yescharacteristic
313C2×C3×C4#23511yescharacteristic
314C2×C3×C4D3×D411yescharacteristic
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4D3×D411yescharacteristic
318C2×C3×C4D3×D411yescharacteristic
319C2×C3×C4D3×D411yescharacteristic
320C2×C3×C4#23911yescharacteristic
321#67—62no—
322#67—62no—
323#67—62no—
324#67—62no—
325#67—122no—
326#67—122no—
327C2×Dic3—41no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—81no—
333C2×Dic3—122no—
334C2×Dic3—121no—
335C2×Dic3—122no—
336C2×Dic3—122no—
337C2×Dic3—121no—
338C2×Dic3—121no—
339C2×Dic3—121no—
340C2×Dic3—121no—
341D12—32no—
342D12—32no—
343D12—62no—
344D12—61no—
345D12—61no—
346D12—121no—
347C4×D3—62no—
348C4×D3—62no—
349C4×D3—62no—
350C4×D3—62no—
351C4×D3—122no—
352C4×D3—122no—
353Dic6—32no—
354Dic6—32no—
355Dic6—62no—
356Dic6—61no—
357Dic6—61no—
358Dic6—64no—
359Dic6—62no—
360Dic6—62no—
361Dic6—62no—
362Dic6—62no—
363Dic6—122no—
364Dic6—121no—
365Dic6—122no—
366Dic6—122no—
367Dic6—121no—
368Dic6—121no—
369Dic6—121no—
370Dic6—121no—
371C3×C8—24no—
372C3×C8—22no—
373C3×C8—22no—
374C3×C8—61no—
375C3×C8—62no—
376C3×C8—61no—
377C3×C8—121no—
378C3⋊C8—21no—
379C3⋊C8—22no—
380C3⋊C8—21no—
381C3⋊C8—41no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32#38611yescharacteristic
385C2×C32#47111yescharacteristic
386C2×C32#47311yescharacteristic
387C3⋊D3—42no—
388#41—92no—
389#41—92no—
390#41—181no—
391#41—181no—
392C2×Q8—92no—
393C2×Q8—91no—
394C2×Q8—181no—
395C2×Q8—181no—
396C2×Q8—181no—
397C2×Q8—181no—
398C2×Q8—181no—
399C2×D4—91no—
400C2×D4—181no—
401C22×C4—61no—
402C22×C4—181no—
403C22×C4—181no—
404C22×C4C2×D3211yescharacteristic
405Q16—184no—
406SD16—184no—
407M16—21no—
408M16—32no—
409M16—61no—
410M16—61no—
411M16—61no—
412M16—184no—
413M16#62412yes—
414C2×C8—21no—
415C2×C8—31no—
416C2×C8—31no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—184no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3C4×D1211yescharacteristic
434C22×C3#88411yescharacteristic
435C22×C3#89311yescharacteristic
436C22×C3#89411yescharacteristic
437C22×C3#89411yescharacteristic
438C22×C3#89511yescharacteristic
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—122no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—121no—
446C2×D3—241no—
447C2×D3—241no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—241no—
453C3×C4C4×D1211yescharacteristic
454C3×C4D3×C22⋊C411yescharacteristic
455C3×C4D3×C22⋊C411yescharacteristic
456C3×C4#88211yescharacteristic
457C3×C4#88211yescharacteristic
458C3×C4#88411yescharacteristic
459C3×C4#89311yescharacteristic
460C3×C4#89511yescharacteristic
461Dic3—62no—
462Dic3—62no—
463Dic3—81no—
464Dic3—122no—
465Dic3—124no—
466Dic3—122no—
467Dic3—122no—
468C32#190811yescharacteristic, complemented, Sylow
469C23—181no—
470C23#379911yescharacteristic
471Q8—92no—
472Q8—181no—
473Q8—182no—
474Q8—182no—
475Q8—361no—
476Q8—361no—
477D4—92no—
478D4—182no—
479D4—182no—
480D4—181no—
481C2×C4—121no—
482C2×C4—181no—
483C2×C4—182no—
484C2×C4—182no—
485C2×C4—182no—
486C2×C4—241no—
487C2×C4—361no—
488C2×C4—361no—
489C2×C4#379911yescharacteristic
490C2×C4D3×D1211yescharacteristic
491C2×C4D3×D1211yescharacteristic
492C2×C4C2×#62411yescharacteristic
493C2×C4#381111yescharacteristic
494C2×C4#381111yescharacteristic
495C8—22no—
496C8—61no—
497C8—61no—
498C8—62no—
499C8—121no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3#466511yescharacteristic
510C2×C3#466511yescharacteristic
511C2×C3#474811yescharacteristic
512C2×C3#475511yescharacteristic
513C2×C3#489611yescharacteristic
514C2×C3#489811yescharacteristic
515D3—122no—
516D3—122no—
517D3—242no—
518C22—21no—
519C22—21no—
520C22—182no—
521C22—361no—
522C22—361no—
523C22#6389411yescharacteristic
524C22#6389711yescharacteristic
525C22#6391011yescharacteristic
526C4—182no—
527C4—241no—
528C4—362no—
529C4#6389411yescharacteristic
530C4#6391011yescharacteristic
531C4#6392311yescharacteristic
532C4#6397411yescharacteristic
533C3—21no—
534C3#6894911yescharacteristic, complemented
535C3#6900711yescharacteristic, complemented
536C2—21no—
537C2—21no—
538C2—362no—
539C2#1059098911yescharacteristic
540C2#1059099211yescharacteristic
541C2#1059138011yescharacteristic
subgroupG/H
centerC22#63910
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1908
LCSG ▷ C2×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
4. C2×#63567C2semidirect
5. C2×#63561C2semidirect
6. C3×#4558C2semidirect
7. C3×#4554C2semidirect
14. #64118C4semidirect
43. #4558D3semidirect
44. #4554D3semidirect
468. C32#1908semidirect
534. C3#68949semidirect
535. C3#69007semidirect

Maximal subgroups

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

subgroupindex
C2×#641182
C2×#638172
#105911062
C2×#635672
C2×#635612
C3×#45582
C3×#45542
#690073
#689493