Finite Group Databasebeta

← #11788078

Subgroups — #11788078

Order 1152.

1077 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64139C211yesmaximal, characteristic
2C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
3C3×#5212C211yesmaximal, characteristic, complemented
4C2×#63564C212yesmaximal, complemented
5#10589164C212yesmaximal, complemented
6#71973—31nomaximal
7#69515—31nomaximal
8C23×C32×C4C2211yescharacteristic
9C22×#638—21no—
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11#64139—22no—
12#64139C414yescomplemented
13C2×#3752—21no—
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×#217—21no—
16C2×C3×C4⋊Dic3C2212yes—
17C22×C3×C3⋊C8—22no—
18C22×C3×C3⋊C8—24no—
19C22×C3×C3⋊C8C2212yes—
20C3×C3⋊#95—21no—
21#63564—24no—
22C23×C3×C8—31no—
23C2×#930—31no—
24C2×#930—31no—
25C2×#930—61no—
26C23×C3⋊C8D311yescharacteristic, complemented
27C3×#380—31no—
28#5212D311yescharacteristic, complemented
29C2×#821—32no—
30C2×#809—32no—
31#4551—32no—
32#4529—32no—
33C24×C32D411yescharacteristic
34C23×C3⋊D3—21no—
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4C2×C412yes—
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C2×#638—24no—
42C2×#638—42no—
43C2×#638—42no—
44C22×C3⋊Dic3—21no—
45C2×C4⋊(C3⋊D3)—22no—
46C2×C4⋊(C3⋊D3)—22no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)C2×C412yes—
49C22×C3×Dic3—22no—
50#3752—24no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C2×C412yes—
53C3×#217—42no—
54C3×C4⋊Dic3—22no—
55C3×C4⋊Dic3—22no—
56C2×C3×C3⋊C8—28no—
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8—22no—
59C2×C3×C3⋊C8D414yescomplemented
60#1893—91noSylow
61C23×C3×C4—21no—
62C23×C3×C4C2×D311yescharacteristic
63C23×C3×C4C2×D311yescharacteristic
64C22×#67—62no—
65C22×#67—61no—
66C22×#67—61no—
67C22×D12—31no—
68C22×D12—31no—
69C22×D12—61no—
70C22×C3×C8—32no—
71C22×C3×C8—64no—
72C22×C3×C8—62no—
73C2×C3×C4⋊C4—32no—
74C2×C3×C22⋊C4—61no—
75C2×#217—21no—
76#930—34no—
77#930—34no—
78#930—64no—
79#930—64no—
80#930—62no—
81#930—62no—
82C2×(C2×D3)⋊C4—62no—
83C2×(C2×D3)⋊C4—61no—
84C2×(C2×D3)⋊C4—61no—
85C2×C4⋊Dic3—31no—
86C2×C4⋊Dic3—31no—
87C2×C4⋊Dic3—61no—
88C2×C4⋊Dic3C2×D312yes—
89C22×C3⋊C8—22no—
90C22×C3⋊C8—24no—
91C22×C3⋊C8C2×D312yes—
92C3×#95—61no—
93C3⋊#95—21no—
94#821—64no—
95#809—64no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32C22⋊C412yes—
99C23×C32D812yes—
100C23×C32SD1612yes—
101C23×C32C2×D411yescharacteristic
102C22×C3⋊D3—22no—
103C22×C3⋊D3—42no—
104C22×C3⋊D3—41no—
105C22×C3⋊D3—41no—
106C22×C3⋊D3—42no—
107C2×C32×C4—22no—
108C2×C32×C4—24no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4C22⋊C412yes—
112C2×C32×C4C22×C411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D412yes—
115C2×C32×C4#4112yes—
116#638—48no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—41no—
119C2×C3⋊Dic3—41no—
120C4⋊(C3⋊D3)—24no—
121C4⋊(C3⋊D3)—41no—
122C4⋊(C3⋊D3)—41no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—42no—
126C3×C3⋊C8—24no—
127C23×C8—31no—
128C2×#121—91no—
129C2×#102—92no—
130#380—31no—
131#340—92no—
132C24×C3—21no—
133C24×C3D1211yescharacteristic
134C24×C3#6711yescharacteristic
135C23×D3—62no—
136C23×D3—61no—
137C23×D3—61no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—62no—
149C22×C3×C4C4×D312yes—
150C22×C3×C4C4×D312yes—
151C22×C3×C4D1211yescharacteristic
152C22×C3×C4D1212yes—
153C22×C3×C4#6711yescharacteristic
154C22×C3×C4#6712yes—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C22×D311yescharacteristic
157C2×#67—68no—
158C2×#67—64no—
159C2×#67—64no—
160C2×#67—124no—
161C2×#67—124no—
162C2×#67—122no—
163C2×#67—122no—
164C2×#67—122no—
165C2×#67—122no—
166C22×Dic3—22no—
167C22×Dic3—62no—
168C22×Dic3—61no—
169C22×Dic3—61no—
170C2×D12—32no—
171C2×D12—32no—
172C2×D12—62no—
173C2×D12—62no—
174C2×D12—62no—
175C2×D12—62no—
176C2×D12—62no—
177C2×D12—62no—
178C2×D12—61no—
179C2×D12—61no—
180C2×D12—122no—
181C2×D12—122no—
182C2×C3×C8—34no—
183C2×C3×C8—68no—
184C2×C3×C8—62no—
185C2×C3×C8—62no—
186C3×C4⋊C4—62no—
187C3×C4⋊C4—62no—
188C3×C22⋊C4—122no—
189#217—42no—
190(C2×D3)⋊C4—68no—
191(C2×D3)⋊C4—64no—
192(C2×D3)⋊C4—64no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—32no—
196C4⋊Dic3—32no—
197C4⋊Dic3—62no—
198C4⋊Dic3—62no—
199C4⋊Dic3—61no—
200C4⋊Dic3—61no—
201C2×C3⋊C8—28no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8#6714yescomplemented
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—24no—
208C22×C32—24no—
209C22×C32#10214yescomplemented
210C22×C32C2×C22⋊C411yescharacteristic
211C22×C32#12112yes—
212C22×C32C2×D811yescharacteristic
213C22×C32C2×SD1611yescharacteristic
214C22×C32#13512yes—
215C2×C3⋊D3—42no—
216C2×C3⋊D3—42no—
217C2×C3⋊D3—42no—
218C2×C3⋊D3—82no—
219C2×C3⋊D3—82no—
220C32×C4—22no—
221C32×C4C4×D412yes—
222C32×C4#12011yescharacteristic
223C32×C4#12311yescharacteristic
224C3⋊Dic3—42no—
225C3×Dic3—82no—
226C22×D4—91no—
227C22×D4—181no—
228C23×C4D3211yescharacteristic
229C22×C8—32no—
230C22×C8—64no—
231C22×C8—62no—
232#121—94no—
233#121—182no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—91no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—181no—
238#102—184no—
239#95—61no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—21no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3#20412yes—
251C23×C3D2412yes—
252C23×C3(C2×D3)⋊C412yes—
253C23×C3(C2×D3)⋊C412yes—
254C23×C3#21312yes—
255C23×C3#21512yes—
256C23×C3C2×D1211yescharacteristic
257C23×C3C2×#6711yescharacteristic
258C22×D3—64no—
259C22×D3—62no—
260C22×D3—62no—
261C22×D3—124no—
262C22×D3—122no—
263C22×D3—122no—
264C22×D3—124no—
265C22×D3—121no—
266C22×D3—121no—
267C22×D3—122no—
268C22×D3—122no—
269C22×D3—121no—
270C22×D3—122no—
271C22×D3—121no—
272C22×D3—122no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—44no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—122no—
291C2×C3×C4—122no—
292C2×C3×C4—122no—
293C2×C3×C4(C2×D3)⋊C412yes—
294C2×C3×C4(C2×D3)⋊C412yes—
295C2×C3×C4C2×C4×D311yescharacteristic
296C2×C3×C4C2×C4×D311yescharacteristic
297C2×C3×C4C2×D1211yescharacteristic
298C2×C3×C4#23512yes—
299C2×C3×C4D3×D412yes—
300C2×C3×C4#23712yes—
301C2×C3×C4C2×#6711yescharacteristic
302C2×C3×C4C2×#6712yes—
303#67—1216no—
304#67—128no—
305#67—128no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—64no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—122no—
313C2×Dic3—122no—
314C2×Dic3—121no—
315C2×Dic3—121no—
316C2×Dic3—121no—
317C2×Dic3—121no—
318D12—64no—
319D12—64no—
320D12—124no—
321D12—122no—
322D12—122no—
323D12—121no—
324D12—121no—
325D12—121no—
326D12—121no—
327C3×C8—64no—
328C3⋊C8—24no—
329C2×C32—24no—
330C2×C32#38611yescharacteristic
331C2×C32C2×#10211yescharacteristic
332C2×C32#41611yescharacteristic
333C2×C32#46512yes—
334C2×C32#46612yes—
335C3⋊D3—82no—
336C24—181no—
337C24#62611yescharacteristic
338C2×D4—92no—
339C2×D4—184no—
340C2×D4—182no—
341C2×D4—181no—
342C2×D4—182no—
343C2×D4—362no—
344C2×D4—362no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—62no—
348C22×C4—181no—
349C22×C4#62412yes—
350C22×C4#62611yescharacteristic
351C22×C4#62612yes—
352C22×C4C2×D3211yescharacteristic
353C2×C8—34no—
354C2×C8—68no—
355C2×C8—62no—
356C2×C8—62no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—92no—
360C4⋊C4—181no—
361C22⋊C4—122no—
362C22⋊C4—184no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—24no—
367C22×C3—24no—
368C22×C3—22no—
369C22×C3—24no—
370C22×C3—24no—
371C22×C3—24no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3#80914yescomplemented
382C22×C3#82114yescomplemented
383C22×C3C2×#20411yescharacteristic
384C22×C3C2×D2411yescharacteristic
385C22×C3#90412yes—
386C22×C3C2×(C2×D3)⋊C411yescharacteristic
387C22×C3C2×(C2×D3)⋊C411yescharacteristic
388C22×C3#93012yes—
389C22×C3C2×#21311yescharacteristic
390C22×C3#93912yes—
391C22×C3C2×#21511yescharacteristic
392C22×C3#95012yes—
393C2×D3—124no—
394C2×D3—124no—
395C2×D3—124no—
396C2×D3—122no—
397C2×D3—122no—
398C2×D3—122no—
399C2×D3—122no—
400C2×D3—122no—
401C2×D3—122no—
402C2×D3—244no—
403C2×D3—244no—
404C2×D3—242no—
405C2×D3—242no—
406C2×D3—242no—
407C2×D3—242no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—42no—
414C3×C4—242no—
415C3×C4#67⋊C412yes—
416C3×C4#88211yescharacteristic
417C3×C4#88611yescharacteristic
418C3×C4C4×#6712yes—
419C3×C4#92911yescharacteristic
420C3×C4#95311yescharacteristic
421Dic3—82no—
422Dic3—124no—
423Dic3—122no—
424Dic3—122no—
425C32#189311yescharacteristic, complemented, Sylow
426C23—22no—
427C23—22no—
428C23—182no—
429C23—361no—
430C23—362no—
431C23—361no—
432C23—362no—
433C23#371412yes—
434C23#371712yes—
435C23#372212yes—
436C23C2×#62611yescharacteristic
437D4—184no—
438D4—368no—
439D4—361no—
440D4—361no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—182no—
449C2×C4—361no—
450C2×C4—361no—
451C2×C4#372212yes—
452C2×C4#380412yes—
453C2×C4C2×#62411yescharacteristic
454C2×C4C2×#62611yescharacteristic
455C2×C4D3×#6712yes—
456C8—64no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—22no—
463C2×C3—21no—
464C2×C3—44no—
465C2×C3#495811yescharacteristic
466C2×C3C2×#80911yescharacteristic
467C2×C3#500211yescharacteristic
468C2×C3#503812yes—
469C2×C3#504812yes—
470C2×C3C2×#82111yescharacteristic
471C2×C3#511311yescharacteristic
472C2×C3#511412yes—
473C2×C3#511512yes—
474C2×C3#521311yescharacteristic
475D3—244no—
476D3—242no—
477D3—242no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—22no—
482C22—362no—
483C22—362no—
484C22—362no—
485C22—722no—
486C22—722no—
487C22#6356414yescomplemented
488C22C2×#371411yescharacteristic
489C22#6382912yes—
490C22C2×#371711yescharacteristic
491C22C2×#372211yescharacteristic
492C22#6397812yes—
493C4—22no—
494C4—242no—
495C4—362no—
496C4#6395212yes—
497C4#6396611yescharacteristic
498C4#6397711yescharacteristic
499C3—21no—
500C3#6951511yescharacteristic, complemented
501C3#7197311yescharacteristic, complemented
502C2—24no—
503C2—722no—
504C2C2×#6356411yescharacteristic
505C2#1059110511yescharacteristic
506C2#1059116412yes—
507C2#1059117712yes—
508C2#1059171111yescharacteristic
subgroupG/H
centerC23C2×#626
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC23×C32×C4C22
socleC23×C32C2×D4
prime radicalC2×#64139C2
cosocleC22×C32×C4C23
hypercenterC23×C4D32
hypocenterC32#1893
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊C8C2semidirect
3. C3×#5212C2semidirect
4. C2×#63564C2semidirect
5. #10589164C2semidirect
12. #64139C4semidirect
26. C23×C3⋊C8D3semidirect
28. #5212D3semidirect
59. C2×C3×C3⋊C8D4semidirect
204. C2×C3⋊C8#67semidirect
209. C22×C32#102semidirect
381. C22×C3#809semidirect
382. C22×C3#821semidirect
425. C32#1893semidirect
487. C22#63564semidirect
500. C3#69515semidirect
501. C3#71973semidirect

Maximal subgroups

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

subgroupindex
C2×#641392
C23×C3×C3⋊C82
C3×#52122
C2×#635642
#105891642
#719733
#695153