Finite Group Databasebeta

← #10845396

Subgroups — #10845396

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 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#82384C211yesmaximal, characteristic, complemented
4C2×#4842C211yesmaximal, characteristic, complemented
5#71990C212yesmaximal, complemented
6#71988C212yesmaximal, complemented
7#70453C211yesmaximal, characteristic, complemented
8#69940C212yesmaximal, complemented
9#69934C212yesmaximal, complemented
10#69777C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#25346—31nomaximal, Sylow
13C22×C3×M16C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#938—21no—
16C2×#930C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×#869C2211yescharacteristic
19C3×#436—24no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23#5221—21no—
24#5212C2212yes—
25#5212C2212yes—
26#5212C2212yes—
27C2×#821C2211yescharacteristic
28C2×#821C2212yes—
29C2×#821C2211yescharacteristic
30C2×#818C2212yes—
31C2×#817C2212yes—
32#4987—21no—
33#4977—21no—
34#4961—22no—
35#4947—22no—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4842—24no—
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4551C2212yes—
42#4551C2211yescharacteristic
43#4551C2212yes—
44C2×#476—31no—
45C2×#436D311yescharacteristic, complemented
46#2563—31no—
47#2537—31no—
48#2077—32no—
49#2033—32no—
50#1990—31no—
51#1947—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#930—24no—
77#930—42no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4×Dic3—21no—
89C4×D12—22no—
90C4×D12—22no—
91#869—22no—
92#869—22no—
93C3×C4⋊C8D414yescomplemented
94C3×C22⋊C8—24no—
95C3×C4×C8—22no—
96C3⋊#95—21no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#821—22no—
101#821—24no—
102#821—22no—
103#818—42no—
104#817—24no—
105C22×M16C2×D312yes—
106C2×C4⋊Q8—31no—
107C2×#121—32no—
108C2×#121—61no—
109C2×C4×D4—31no—
110C2×C4×D4C2×D311yescharacteristic
111#476—64no—
112#436—24no—
113C2×#107—32no—
114C2×#106—32no—
115C2×C4⋊C8C2×D311yescharacteristic
116C2×#102—31no—
117C2×#102—32no—
118C2×#102—31no—
119C2×C22⋊C8C2×D312yes—
120C2×C4×C8C2×D311yescharacteristic
121#394—62no—
122#393—62no—
123#390—61no—
124#384—31no—
125#382—61no—
126#380—32no—
127#380—32no—
128#380—32no—
129#380—61no—
130C4×C4⋊C4—31no—
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—32no—
136C24×C3C2×D412yes—
137C23×D3—21no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C2×#67—44no—
152C2×#67—44no—
153C2×#67—82no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12#4112yes—
165C2×C4×D3—42no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×Dic6—42no—
169C2×Dic6—81no—
170C2×Dic6—81no—
171C3×M16—48no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8C2×D414yes—
180C2×C3×C8#4112yes—
181C3×C4⋊C4—22no—
182C3×C22⋊C4—24no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185#217—44no—
186#217—82no—
187#217—82no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—42no—
190C4⋊Dic3—21no—
191C4⋊Dic3—22no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—22no—
195C4⋊Dic3—21no—
196C4⋊Dic3—22no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—42no—
201C4⋊Dic3—42no—
202C4⋊Dic3—81no—
203C4×Dic3—42no—
204C22×Q8—61no—
205C22×D4—31no—
206C22×D4—62no—
207C22×D4D1211yescharacteristic
208C23×C4—61no—
209C23×C4C22×D312yes—
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—42no—
213C22×C8C22×D311yescharacteristic
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D312yes—
216C4⋊Q8—62no—
217C4⋊Q8—62no—
218#121—64no—
219#121—122no—
220#121—124no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—62no—
224C4×D4—62no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4D1211yescharacteristic
234C2×C22⋊C4—62no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4D1212yes—
240C2×C42—61no—
241C2×C42C22×D311yescharacteristic
242#107—122no—
243#106—64no—
244C4⋊C8#6714yescomplemented
245#102—62no—
246#102—62no—
247#102—64no—
248C22⋊C8—24no—
249C4×C8—22no—
250#95—61no—
251#95—62no—
252#95—62no—
253#95—62no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3C22×D411yescharacteristic
262C22×D3—22no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C3×D4—44no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4#12112yes—
280C2×C3×C4#12112yes—
281C2×C3×C4AGL(1,ℤ/8)12yes—
282C2×C3×C4#13712yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4H(5,𝔽2)11yescharacteristic
288C2×C3×C42−1+411yescharacteristic
289#67—88no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305D12—22no—
306D12—22no—
307D12—41no—
308D12—41no—
309C4×D3—84no—
310Dic6—84no—
311C3×C8—24no—
312C3×C8—22no—
313C3×C8—41no—
314C24—61no—
315C24C2×D1212yes—
316C2×Q8—122no—
317C2×Q8—241no—
318C2×Q8—241no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—32no—
323C2×D4—61no—
324C2×D4—61no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—124no—
329C2×D4—124no—
330C2×D4—242no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—122no—
343C22×C4—122no—
344C22×C4—122no—
345C22×C4C2×D1212yes—
346C22×C4C2×D1211yescharacteristic
347C22×C4C2×D1211yescharacteristic
348C22×C4C23×D311yescharacteristic
349M16—48no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8#23712yes—
358C2×C8C2×#6714yes—
359C4⋊C4—22no—
360C4⋊C4—61no—
361C4⋊C4—62no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—62no—
365C4⋊C4—61no—
366C4⋊C4—62no—
367C4⋊C4—61no—
368C4⋊C4—62no—
369C4⋊C4—61no—
370C4⋊C4—122no—
371C4⋊C4—122no—
372C4⋊C4—241no—
373C22⋊C4—24no—
374C22⋊C4—124no—
375C22⋊C4—124no—
376C22⋊C4—122no—
377C22⋊C4—242no—
378C22⋊C4—242no—
379C42—122no—
380C42#23512yes—
381C42C2×#6712yes—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3C2×#12111yescharacteristic
390C22×C3#54611yescharacteristic
391C22×C3#54711yescharacteristic
392C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
393C22×C3C2×#13711yescharacteristic
394C22×C3#57611yescharacteristic
395C22×C3#57711yescharacteristic
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—42no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—81no—
403C3×C4—22no—
404C3×C4—42no—
405C3×C4#46814yes—
406C3×C4#52911yescharacteristic
407C3×C4#54211yescharacteristic
408C3×C4#54611yescharacteristic
409C3×C4#54811yescharacteristic
410Dic3—81no—
411Dic3—81no—
412Dic3—82no—
413Dic3—81no—
414Dic3—82no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—62no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23C22×D1211yescharacteristic
430Q8—244no—
431D4—44no—
432D4—62no—
433D4—62no—
434D4—121no—
435D4—121no—
436D4—248no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—122no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—244no—
462C2×C4#90812yes—
463C2×C4#90912yes—
464C2×C4#93012yes—
465C2×C4#93012yes—
466C2×C4C2×#23511yescharacteristic
467C2×C4C2×#23711yescharacteristic
468C2×C4#100411yescharacteristic
469C2×C4D3×#4111yescharacteristic
470C2×C4#101011yescharacteristic
471C2×C4C22×#6711yescharacteristic
472C8—22no—
473C8—24no—
474C8—41no—
475C2×C3—82no—
476C2×C3#242711yescharacteristic
477C2×C3C2×#46811yescharacteristic
478C2×C3#306211yescharacteristic
479C2×C3#332511yescharacteristic
480C2×C3#333111yescharacteristic
481C2×C3#335411yescharacteristic
482C2×C3#335711yescharacteristic
483D3—82no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22#555411yescharacteristic
499C22#555511yescharacteristic
500C22C2×#90811yescharacteristic
501C22C2×#90911yescharacteristic
502C22#575111yescharacteristic
503C22#575211yescharacteristic
504C22C2×#93011yescharacteristic
505C4—22no—
506C4—42no—
507C4—241no—
508C4—242no—
509C4—241no—
510C4—242no—
511C4—241no—
512C4#513414yes—
513C4#555711yescharacteristic
514C4#556111yescharacteristic
515C4#580611yescharacteristic
516C4#583111yescharacteristic
517C3#2534611yescharacteristic, complemented, Sylow
518C2—82no—
519C2—242no—
520C2#7953211yescharacteristic
521C2#7953511yescharacteristic
522C2#7955611yescharacteristic
523C2#7955811yescharacteristic
524C2#8240811yescharacteristic
525C2C2×#513411yescharacteristic
526C2#8380211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#436C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC2×#436D3
hypocenterC3#25346
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. #82384C2semidirect
4. C2×#4842C2semidirect
5. #71990C2semidirect
6. #71988C2semidirect
7. #70453C2semidirect
8. #69940C2semidirect
9. #69934C2semidirect
10. #69777C2semidirect
11. C4×#821C2semidirect
45. C2×#436D3semidirect
93. C3×C4⋊C8D4semidirect
244. C4⋊C8#67semidirect
517. C3#25346semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4362
#824172
#823842
C2×#48422
#719902
#719882
#704532
#699402
#699342
#697772
C4×#8212
#253463