Finite Group Databasebeta

← #10845407

Subgroups — #10845407

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#436C211yesmaximal, characteristic, complemented
2#82417C211yesmaximal, characteristic
3#82382C211yesmaximal, characteristic, complemented
4C2×#4846C211yesmaximal, characteristic, complemented
5#71990C212yesmaximal, complemented
6#71988C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69939C212yesmaximal, complemented
9#69935C212yesmaximal, complemented
10#69777C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#25347—31nomaximal, Sylow
13C22×C3×M16C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#938—21no—
16C2×#936—21no—
17C2×#930C2212yes—
18C2×C4×D12C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#436—24no—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2212yes—
23C2×C3×C4×C8C2211yescharacteristic
24#5212C2212yes—
25#5212C2212yes—
26#5212C2212yes—
27C2×#821C2211yescharacteristic
28C2×#821C2212yes—
29C2×#821C2211yescharacteristic
30C2×#818C2212yes—
31C2×#817C2212yes—
32#4987—21no—
33#4973—21no—
34#4961—22no—
35#4953—22no—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4846—24no—
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4551C2212yes—
42#4551C2211yescharacteristic
43#4551C2212yes—
44C2×#474—31no—
45C2×#436D311yescharacteristic, complemented
46#2547—31no—
47#2537—31no—
48#2076—32no—
49#2034—32no—
50#1990—31no—
51#1948—31no—
52#1895—32no—
53#1809—32no—
54C4×#102—31no—
55C22×C3×D4D411yescharacteristic
56C23×C3×C4C2312yes—
57C22×#67—22no—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6—21no—
61C2×C3×M16—24no—
62C2×C3×M16—24no—
63C2×C3×M16—42no—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75#938—44no—
76#936—44no—
77#930—24no—
78#930—42no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4×Dic3—21no—
90C4×D12—22no—
91C4×D12—22no—
92#869—22no—
93#869—22no—
94C3×C4⋊C8D414yescomplemented
95C3×C22⋊C8—24no—
96C3×C4×C8—22no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#821—22no—
102#821—24no—
103#821—22no—
104#818—24no—
105#817—42no—
106C22×M16C2×D312yes—
107C2×C4⋊Q8—31no—
108C2×#124—61no—
109C2×#121—32no—
110C2×#121—61no—
111C2×C4×D4—31no—
112C2×C4×D4C2×D311yescharacteristic
113#474—64no—
114#436—24no—
115C2×#107—32no—
116C2×#106—32no—
117C2×C4⋊C8C2×D311yescharacteristic
118C2×#102—31no—
119C2×#102—32no—
120C2×#102—31no—
121C2×C22⋊C8C2×D312yes—
122C2×C4×C8C2×D311yescharacteristic
123#397—62no—
124#394—62no—
125#391—61no—
126#384—31no—
127#380—32no—
128#380—32no—
129#380—32no—
130#380—61no—
131C4×C4⋊C4—31no—
132#340—31no—
133#340—32no—
134#340—31no—
135#340—32no—
136#340—32no—
137C24×C3C2×D412yes—
138C23×D3—21no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C2×#67—44no—
153C2×#67—44no—
154C2×#67—82no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12#4112yes—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×Dic6—42no—
170C2×Dic6—42no—
171C2×Dic6—42no—
172C3×M16—48no—
173C2×C3×C8—21no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8C2×D414yes—
181C2×C3×C8#4112yes—
182C3×C4⋊C4—22no—
183C3×C22⋊C4—24no—
184C3×C42C2×D412yes—
185C3×C42#4112yes—
186#217—44no—
187#217—44no—
188#217—44no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—42no—
191C4⋊Dic3—21no—
192C4⋊Dic3—22no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—22no—
196C4⋊Dic3—21no—
197C4⋊Dic3—22no—
198C4⋊Dic3—21no—
199C4⋊Dic3—22no—
200C4⋊Dic3—21no—
201C4⋊Dic3—42no—
202C4⋊Dic3—42no—
203C4⋊Dic3—81no—
204C4×Dic3—42no—
205C22×Q8—61no—
206C22×D4—31no—
207C22×D4—62no—
208C22×D4D1211yescharacteristic
209C23×C4—61no—
210C23×C4C22×D312yes—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—42no—
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D311yescharacteristic
216C22×C8C22×D312yes—
217C4⋊Q8—62no—
218C4⋊Q8—62no—
219#124—124no—
220#121—64no—
221#121—122no—
222#121—124no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—62no—
226C4×D4—62no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4D1211yescharacteristic
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4D1212yes—
242C2×C42—61no—
243C2×C42C22×D311yescharacteristic
244#107—64no—
245#106—122no—
246C4⋊C8#6714yescomplemented
247#102—62no—
248#102—62no—
249#102—64no—
250C22⋊C8—24no—
251C4×C8—22no—
252#95—61no—
253#95—62no—
254#95—62no—
255#95—62no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3C22×D411yescharacteristic
264C22×D3—22no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
271C3×D4—44no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4#12112yes—
282C2×C3×C4#12112yes—
283C2×C3×C4AGL(1,ℤ/8)12yes—
284C2×C3×C4AGL(1,ℤ/8)12yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4H(5,𝔽2)11yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291#67—88no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307D12—22no—
308D12—22no—
309D12—41no—
310D12—41no—
311C4×D3—84no—
312Dic6—84no—
313C3×C8—24no—
314C3×C8—22no—
315C3×C8—41no—
316C24—61no—
317C24C2×D1212yes—
318C2×Q8—122no—
319C2×Q8—122no—
320C2×Q8—122no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—32no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—124no—
331C2×D4—124no—
332C2×D4—242no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4—122no—
347C22×C4C2×D1212yes—
348C22×C4C2×D1211yescharacteristic
349C22×C4C2×D1211yescharacteristic
350C22×C4C23×D311yescharacteristic
351M16—48no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8#23712yes—
360C2×C8C2×#6714yes—
361C4⋊C4—22no—
362C4⋊C4—61no—
363C4⋊C4—62no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—62no—
367C4⋊C4—61no—
368C4⋊C4—62no—
369C4⋊C4—61no—
370C4⋊C4—62no—
371C4⋊C4—61no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C4⋊C4—241no—
375C22⋊C4—24no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—122no—
381C42—122no—
382C42#23512yes—
383C42C2×#6712yes—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3C2×#12111yescharacteristic
392C22×C3#54611yescharacteristic
393C22×C3#54711yescharacteristic
394C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3#57711yescharacteristic
397C22×C3#57811yescharacteristic
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—81no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4#46914yes—
408C3×C4#52911yescharacteristic
409C3×C4#54211yescharacteristic
410C3×C4#54611yescharacteristic
411C3×C4#54811yescharacteristic
412Dic3—81no—
413Dic3—81no—
414Dic3—82no—
415Dic3—81no—
416Dic3—82no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—62no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23C22×D1211yescharacteristic
432Q8—244no—
433D4—44no—
434D4—62no—
435D4—62no—
436D4—121no—
437D4—121no—
438D4—248no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—244no—
464C2×C4#90812yes—
465C2×C4#90812yes—
466C2×C4#93012yes—
467C2×C4#93012yes—
468C2×C4C2×#23511yescharacteristic
469C2×C4C2×#23711yescharacteristic
470C2×C4#100411yescharacteristic
471C2×C4D3×#4111yescharacteristic
472C2×C4#101011yescharacteristic
473C2×C4C22×#6711yescharacteristic
474C8—22no—
475C8—24no—
476C8—41no—
477C2×C3—82no—
478C2×C3#242711yescharacteristic
479C2×C3C2×#46911yescharacteristic
480C2×C3#306311yescharacteristic
481C2×C3#332411yescharacteristic
482C2×C3#333011yescharacteristic
483C2×C3#335311yescharacteristic
484C2×C3#335411yescharacteristic
485D3—82no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—241no—
497C22—241no—
498C22—241no—
499C22—241no—
500C22#555411yescharacteristic
501C22#555511yescharacteristic
502C22C2×#90811yescharacteristic
503C22C2×#90811yescharacteristic
504C22#575111yescharacteristic
505C22#575311yescharacteristic
506C22C2×#93011yescharacteristic
507C4—22no—
508C4—42no—
509C4—241no—
510C4—242no—
511C4—241no—
512C4—242no—
513C4—241no—
514C4#513514yes—
515C4#555711yescharacteristic
516C4#556111yescharacteristic
517C4#580611yescharacteristic
518C4#583111yescharacteristic
519C3#2534711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2#7953111yescharacteristic
523C2#7953211yescharacteristic
524C2#7955511yescharacteristic
525C2#7955711yescharacteristic
526C2#8240811yescharacteristic
527C2C2×#513511yescharacteristic
528C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#436C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC2×#436D3
hypocenterC3#25347
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×C3×#436C2semidirect
3. #82382C2semidirect
4. C2×#4846C2semidirect
5. #71990C2semidirect
6. #71988C2semidirect
7. #70454C2semidirect
8. #69939C2semidirect
9. #69935C2semidirect
10. #69777C2semidirect
11. C4×#821C2semidirect
45. C2×#436D3semidirect
94. C3×C4⋊C8D4semidirect
246. C4⋊C8#67semidirect
519. C3#25347semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4362
#824172
#823822
C2×#48462
#719902
#719882
#704542
#699392
#699352
#697772
C4×#8212
#253473