Finite Group Databasebeta

← #10837639

Subgroups — #10837639

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#443C211yesmaximal, characteristic, complemented
2#83901C212yesmaximal, complemented
3C2×#5142C211yesmaximal, characteristic, complemented
4#83799C211yesmaximal, characteristic, complemented
5#83785C211yesmaximal, characteristic, complemented
6#72016C211yesmaximal, characteristic, complemented
7#70715C212yesmaximal, complemented
8#70366C212yesmaximal, complemented
9#70359C212yesmaximal, complemented
10#69794C211yesmaximal, characteristic, complemented
11#69776C211yesmaximal, characteristic, complemented
12#24797—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5830C2212yes—
17#5827C2212yes—
18C2×#949C2212yes—
19C22×D24C2211yescharacteristic
20C3×#443—24no—
21C2×C3×#108C2211yescharacteristic
22C2×C3×#104C2211yescharacteristic
23C2×C3×#104C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25#5232C2212yes—
26#5185—22no—
27#5165—22no—
28#5163—22no—
29#5150—22no—
30#5142—24no—
31#5135—22no—
32#5131C2212yes—
33C2×#823C2212yes—
34#5121—22no—
35#5121C2212yes—
36#5115—21no—
37#5115—21no—
38C2×#821C2212yes—
39C2×#819C2211yescharacteristic
40#5106—21no—
41#5106C2211yescharacteristic
42C2×#818—21no—
43C2×#818C2211yescharacteristic
44#5102C2212yes—
45#4939C2212yes—
46#4554C2212yes—
47#4523C2212yes—
48C2×#472—31no—
49#3058—31no—
50#3045—32no—
51C2×#443D311yescharacteristic, complemented
52#2913—31no—
53#2063—32no—
54#2037—32no—
55#1970—31no—
56#1945—31no—
57#1883—32no—
58#1816—31no—
59C2×C3×#41C2311yescharacteristic
60C2×C3×#41C2311yescharacteristic
61C22×D12C2312yes—
62C3×#135—22no—
63C3×#135—22no—
64C2×C3×Q16D411yescharacteristic
65C2×C3×SD16D411yescharacteristic
66C2×C3×SD16D411yescharacteristic
67C2×C3×D8D411yescharacteristic
68C3×#131—22no—
69C3×#131—22no—
70C3×#131—22no—
71C3×#131D414yescomplemented
72C2×C3×M16—21no—
73C2×C3×M16—21no—
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#949—24no—
82#949—24no—
83#947—22no—
84Q8×Dic3—22no—
85C2×#215—22no—
86#940—22no—
87#937—42no—
88D4×Dic3—22no—
89D4×Dic3—42no—
90C2×#213—22no—
91#930—22no—
92#930—22no—
93C2×(C2×D3)⋊C4—42no—
94#926—22no—
95#926C2312yes—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3C2312yes—
98C2×#208C2312yes—
99C2×D24—21no—
100C2×D24—22no—
101C2×D24—21no—
102C2×D24—22no—
103C3×#108—22no—
104C3×#104—22no—
105C3×#104—22no—
106C3×C4×C8—22no—
107#835—22no—
108#832—22no—
109#823—24no—
110#821—22no—
111#821—22no—
112#821—22no—
113#821—22no—
114#819—22no—
115#818—21no—
116#818—21no—
117#818—22no—
118#818D411yescharacteristic
119#818D411yescharacteristic
120#817—22no—
121#817D412yes—
122Dic3⋊C8—22no—
123Dic3⋊C8—22no—
124C2×AGL(1,ℤ/8)—32no—
125C2×#135C2×D311yescharacteristic
126C22×D8—31no—
127C2×#131C2×D311yescharacteristic
128C2×#131C2×D311yescharacteristic
129#535—32no—
130#520—32no—
131#472—64no—
132#469—62no—
133#466—61no—
134#466—61no—
135#464—62no—
136#463—62no—
137#460—62no—
138#459—62no—
139#443—24no—
140C2×#108—31no—
141C2×#108C2×D311yescharacteristic
142#428—32no—
143#428—62no—
144#427—31no—
145#427—61no—
146C2×#107—31no—
147C2×#107—61no—
148#424—32no—
149C2×#104C2×D311yescharacteristic
150C2×#104C2×D311yescharacteristic
151#418—32no—
152#417—32no—
153C2×#102—32no—
154C2×#100—32no—
155C2×C4×C8C2×D311yescharacteristic
156#390—32no—
157#343—32no—
158#339—32no—
159C23×D3—22no—
160C3×#41—21no—
161C3×#41—21no—
162C3×#41—21no—
163C3×#41—21no—
164C3×#41—21no—
165C3×#41—21no—
166C3×#41C2×D412yes—
167C3×#41#4112yes—
168C2×C3×Q8C2×D411yescharacteristic
169C2×C3×Q8C2×D411yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4C2×D411yescharacteristic
173C2×C3×D4C2×D411yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2411yescharacteristic
178C2×#67—42no—
179C2×#67—42no—
180C22×Dic3—22no—
181C22×Dic3—42no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C3×Q16—22no—
189C3×SD16—22no—
190C3×SD16—22no—
191C3×D8—22no—
192C3×M16—21no—
193C3×M16—24no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—42no—
197C3×M16C2×D412yes—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—24no—
202C2×C3×C8—21no—
203C2×C3×C8—42no—
204C2×C3×C8C2×D411yescharacteristic
205C2×C3×C8C2×D411yescharacteristic
206C2×C3×C8C2×D411yescharacteristic
207C2×C3×C8C2×D411yescharacteristic
208C2×C3×C8C2×D412yes—
209C2×C3×C8#4112yes—
210C3×C42—21no—
211C3×C42—21no—
212#217—22no—
213#217—42no—
214#215—44no—
215#213—44no—
216(C2×D3)⋊C4—42no—
217(C2×D3)⋊C4—42no—
218(C2×D3)⋊C4—84no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—42no—
222C4⋊Dic3C2×D412yes—
223C4⋊Dic3C2×D412yes—
224C4×Dic3—22no—
225C4×Dic3—42no—
226#208—22no—
227#208—22no—
228C2×C3⋊C8—22no—
229D24—42no—
230D24—42no—
231C2×#41C22×D311yescharacteristic
232C2×#41C22×D311yescharacteristic
233C22×D4—32no—
234AGL(1,ℤ/8)—64no—
235AGL(1,ℤ/8)—64no—
236#135—22no—
237#135—22no—
238C2×Q16D1211yescharacteristic
239C2×SD16—62no—
240C2×SD16D1211yescharacteristic
241C2×SD16D1211yescharacteristic
242C2×D8—62no—
243C2×D8—61no—
244C2×D8—61no—
245C2×D8—62no—
246C2×D8—62no—
247C2×D8D1211yescharacteristic
248#131—22no—
249#131—22no—
250#131—22no—
251#131#6714yescomplemented
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—32no—
255C2×M16C22×D311yescharacteristic
256C2×M16C22×D311yescharacteristic
257C22×C8—21no—
258C22×C8—21no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#127—62no—
262#124—62no—
263#121—62no—
264#121—62no—
265#120—122no—
266C4×Q8—62no—
267C4×D4—62no—
268C4×D4—122no—
269#117—32no—
270#117—62no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—62no—
273C2×C22⋊C4—122no—
274C2×C42C22×D311yescharacteristic
275#108—22no—
276#108—62no—
277#107—31no—
278#107—31no—
279#107—62no—
280#107—61no—
281#107—61no—
282#106—32no—
283#106—62no—
284C4⋊C8—62no—
285C4⋊C8—62no—
286#104—22no—
287#104—22no—
288#103—62no—
289#102—62no—
290#102—62no—
291#102—62no—
292#102—62no—
293#102—62no—
294#100—64no—
295C4×C8—22no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3H(5,𝔽2)11yescharacteristic
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—82no—
303C22×D3—82no—
304C3×Q8—21no—
305C3×Q8—21no—
306C3×Q8—21no—
307C3×Q8#12112yes—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—21no—
311C3×D4—21no—
312C3×D4—21no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4#12112yes—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331#67—162no—
332#67—162no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—42no—
336C2×Dic3—82no—
337C2×Dic3—162no—
338D12—42no—
339D12—42no—
340D12—42no—
341D12—42no—
342C3×C8—22no—
343C3×C8—21no—
344C3×C8—21no—
345C3×C8—41no—
346C3×C8#12112yes—
347C3×C8#12112yes—
348C3⋊C8—42no—
349C24—62no—
350#41—21no—
351#41—21no—
352#41—21no—
353#41—21no—
354#41—21no—
355#41—21no—
356#41#23512yes—
357#41C2×#6712yes—
358C2×Q8C2×D1211yescharacteristic
359C2×Q8C2×D1211yescharacteristic
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4C2×D1211yescharacteristic
371C2×D4C2×D1211yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—62no—
376C22×C4—122no—
377C22×C4C23×D311yescharacteristic
378Q16—22no—
379SD16—22no—
380SD16—22no—
381SD16—124no—
382D8—22no—
383D8—124no—
384D8—122no—
385D8—122no—
386M16—21no—
387M16—24no—
388M16—21no—
389M16—21no—
390M16—42no—
391M16—62no—
392M16—62no—
393M16C2×#6712yes—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—24no—
398C2×C8—21no—
399C2×C8—42no—
400C2×C8—62no—
401C2×C8C2×D1211yescharacteristic
402C2×C8C2×D1211yescharacteristic
403C2×C8D3×D411yescharacteristic
404C2×C8D3×D411yescharacteristic
405C2×C8#23712yes—
406C2×C8C2×#6712yes—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—122no—
412C22⋊C4—62no—
413C22⋊C4—122no—
414C22⋊C4—122no—
415C22⋊C4—122no—
416C22⋊C4—244no—
417C42—21no—
418C42—21no—
419C42—62no—
420C42—122no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—22no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3#53711yescharacteristic
430C22×C3D4211yescharacteristic
431C22×C3#55311yescharacteristic
432C2×D3—84no—
433C2×D3—82no—
434C2×D3—82no—
435C2×D3—82no—
436C2×D3—162no—
437C3×C4—22no—
438C3×C4—42no—
439C3×C4—41no—
440C3×C4C2×#12111yescharacteristic
441C3×C4C2×#12111yescharacteristic
442C3×C4#54711yescharacteristic
443C3×C4#54811yescharacteristic
444Dic3—42no—
445Dic3—42no—
446Dic3—82no—
447C23—21no—
448C23—21no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—242no—
453C23—242no—
454C23#100911yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8#93012yes—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4—122no—
467D4—122no—
468D4—122no—
469D4—122no—
470D4—482no—
471D4—482no—
472D4#93012yes—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—122no—
485C2×C4—242no—
486C2×C4—482no—
487C2×C4C22×D1211yescharacteristic
488C2×C4C2×#23511yescharacteristic
489C2×C4C2×D3×D411yescharacteristic
490C2×C4C2×#23711yescharacteristic
491C2×C4#101011yescharacteristic
492C2×C4C22×#6711yescharacteristic
493C8—21no—
494C8—21no—
495C8—22no—
496C8—41no—
497C8—122no—
498C8#93912yes—
499C8#93912yes—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—42no—
503C2×C3—81no—
504C2×C3#241611yescharacteristic
505C2×C3#329011yescharacteristic
506C2×C3#329211yescharacteristic
507D3—162no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—244no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22—482no—
521C22D4×D1211yescharacteristic
522C22#556511yescharacteristic
523C22#583311yescharacteristic
524C4—22no—
525C4—42no—
526C4—41no—
527C4—122no—
528C4—122no—
529C4—242no—
530C4#555211yescharacteristic
531C4#555511yescharacteristic
532C4C2×#93011yescharacteristic
533C4C2×#93911yescharacteristic
534C3#2479711yescharacteristic, complemented, Sylow
535C2—21no—
536C2—21no—
537C2—42no—
538C2—81no—
539C2—482no—
540C2#7989111yescharacteristic
541C2#7989411yescharacteristic
542C2#8235211yescharacteristic
subgroupG/H
centerC22D4×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#443C2
socleC22×C3D42
hypercenterC2×#443D3
hypocenterC3#24797
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×C3×#443C2semidirect
2. #83901C2semidirect
3. C2×#5142C2semidirect
4. #83799C2semidirect
5. #83785C2semidirect
6. #72016C2semidirect
7. #70715C2semidirect
8. #70366C2semidirect
9. #70359C2semidirect
10. #69794C2semidirect
11. #69776C2semidirect
51. C2×#443D3semidirect
71. C3×#131D4semidirect
251. #131#67semidirect
534. C3#24797semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4432
#839012
C2×#51422
#837992
#837852
#720162
#707152
#703662
#703592
#697942
#697762
#247973