Finite Group Databasebeta

← #10944260

Subgroups — #10944260

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C211yesmaximal, characteristic
2#82820C211yesmaximal, characteristic
3C2×#5051C211yesmaximal, characteristic, complemented
4C2×#4813C212yesmaximal, complemented
5#73950C211yesmaximal, characteristic
6C3×#1894C211yesmaximal, characteristic, complemented
7#71989C211yesmaximal, characteristic, complemented
8#70271C212yesmaximal, complemented
9#69512C211yesmaximal, characteristic, complemented
10#68884C212yesmaximal, complemented
11#68869C212yesmaximal, complemented
12#32934—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#946C2212yes—
17C22×#216C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C2×#907—24no—
21C2×#907—28no—
22C22×ΓL(1,𝔽25)C2212yes—
23C22×D3×C8C2212yes—
24C2×#897C2212yes—
25C2×C3×#103C2212yes—
26C3×#380C2211yescharacteristic
27C2×#835C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#816C2212yes—
30#5051—24no—
31#4991—21no—
32#4987C2212yes—
33#4987C2212yes—
34#4979—21no—
35C2×#810C2211yescharacteristic
36C2×#807C2211yescharacteristic
37C2×#807C2212yes—
38#4961—22no—
39#4959—22no—
40#4813—28no—
41#4672—21no—
42#4656—21no—
43C3×#340C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#467—32no—
48C2×#467—31no—
49#2616—31no—
50#2616—31no—
51#1894—32no—
52#1894—32no—
53#1894—31no—
54#1894D311yescharacteristic, complemented
55#1808—31no—
56#1808—32no—
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67D412yes—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—24no—
63C2×#235—24no—
64C22×D12D411yescharacteristic
65C22×C4×D3C2312yes—
66C22×Dic6D411yescharacteristic
67C22×Dic6C2311yescharacteristic
68C2×C3×M16—22no—
69C2×C3×M16—24no—
70C22×C3×C8C2312yes—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×#217—21no—
79C2×#217—21no—
80#946—24no—
81#946—24no—
82C2×#216—22no—
83C2×#216—24no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91#924—22no—
92#924—22no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—22no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8D414yes—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99#907—232no—
100C2×ΓL(1,𝔽25)—24no—
101C2×ΓL(1,𝔽25)—24no—
102C2×ΓL(1,𝔽25)—24no—
103C2×D3×C8—24no—
104C2×D3×C8—24no—
105C2×D3×C8D418yes—
106#897—24no—
107#897—24no—
108C3×#103—24no—
109C3×#95—21no—
110#835—22no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#816—24no—
115#810—22no—
116#807—22no—
117#807—24no—
118C22×#41—31no—
119C22×Q16—31no—
120C2×#131—64no—
121C2×#131—68no—
122C22×M16—32no—
123C22×M16C2×D311yescharacteristic
124C23×C8—31no—
125C23×C8—32no—
126C2×C22⋊Q8—32no—
127C2×C22⋊Q8—31no—
128C2×C22⋊Q8—32no—
129C2×C22⋊Q8C2×D311yescharacteristic
130#467—68no—
131#467—64no—
132C2×#107—32no—
133C2×#107—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103C2×D312yes—
138#395—61no—
139#395—61no—
140#394—61no—
141#394—62no—
142#394—62no—
143#394—61no—
144#380—31no—
145#380—32no—
146#380—32no—
147#380C2×D311yescharacteristic
148#340—32no—
149#340—32no—
150#340C2×D312yes—
151C24×C3C2×D411yescharacteristic
152C23×D3C2×D412yes—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C2×#67—24no—
167C2×#67—24no—
168C2×#67—24no—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3C2×D412yes—
173#235—28no—
174#235—216no—
175#235—28no—
176C2×D12—24no—
177C2×D12—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3C2×D414yes—
183C2×C4×D3#4114yes—
184C2×Dic6—22no—
185C2×Dic6—24no—
186C2×Dic6—21no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C3×M16—28no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8C2×D414yes—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—81no—
201#217—42no—
202#217—81no—
203#216—44no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—82no—
207(C2×D3)⋊C4—82no—
208C4⋊Dic3—22no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—44no—
213Dic3⋊C4—42no—
214Dic3⋊C4—44no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—28no—
217C2×C3⋊C8C2×D412yes—
218C2×C3⋊C8C2×D412yes—
219C2×C3⋊C8C2×D414yes—
220ΓL(1,𝔽25)—216no—
221D3×C8—216no—
222C2×#41—64no—
223C2×#41—64no—
224C2×#41—62no—
225C2×#41—62no—
226C22×Q8—31no—
227C22×Q8—31no—
228C22×Q8C22×D311yescharacteristic
229C22×D4—32no—
230C22×D4—31no—
231C23×C4—32no—
232C23×C4C22×D311yescharacteristic
233C2×Q16—64no—
234C2×Q16—62no—
235#131—632no—
236C2×M16—22no—
237C2×M16—24no—
238C2×M16—64no—
239C2×M16—64no—
240C2×M16—64no—
241C22×C8—31no—
242C22×C8—34no—
243C22×C8—31no—
244C22×C8—38no—
245C22×C8—64no—
246C22×C8—64no—
247C22×C8—64no—
248C22×C8C22×D312yes—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—64no—
252C22⋊Q8—64no—
253C22⋊Q8—64no—
254C22⋊Q8—64no—
255C22⋊Q8—62no—
256C22⋊Q8—62no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4C22×D312yes—
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—61no—
273#107—64no—
274#107—62no—
275#103—24no—
276#103—62no—
277#103—62no—
278#103—64no—
279#95—21no—
280#95—62no—
281#95—62no—
282#95—61no—
283C23×C3—22no—
284C23×C3—21no—
285C23×C3—22no—
286C23×C3—21no—
287C23×C3#12112yes—
288C23×C3C22×D411yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3#12114yes—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—42no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4#12112yes—
312C2×C3×C4C22×D412yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4112yes—
316#67—216no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3#12114yes—
328D12—28no—
329C4×D3—24no—
330C4×D3—24no—
331C4×D3—24no—
332C4×D3—24no—
333Dic6—28no—
334Dic6—41no—
335Dic6—41no—
336Dic6—42no—
337C3×C8—24no—
338C3⋊C8—22no—
339C3⋊C8#12014yes—
340C24—32no—
341C24D3×D411yescharacteristic
342#41—68no—
343#41—68no—
344#41—616no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—64no—
350C2×Q8—62no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—62no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—32no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4#23512yes—
376C22×C4D3×D411yescharacteristic
377C22×C4D3×D412yes—
378C22×C4C23×D311yescharacteristic
379Q16—124no—
380M16—28no—
381M16—616no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—32no—
385C2×C8—34no—
386C2×C8—32no—
387C2×C8—616no—
388C2×C8—68no—
389C2×C8—62no—
390C2×C8D3×D414yes—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—124no—
401C4⋊C4—124no—
402C4⋊C4—122no—
403C22⋊C4—42no—
404C22⋊C4—81no—
405C22⋊C4—124no—
406C22⋊C4—124no—
407C22⋊C4—122no—
408C22⋊C4—242no—
409C22⋊C4—242no—
410C22⋊C4—241no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3C2×#12111yescharacteristic
417C22×C3C2×#12112yes—
418C22×C3#57611yescharacteristic
419C22×C3#57811yescharacteristic
420C22×C3#57812yes—
421C2×D3—24no—
422C2×D3—24no—
423C2×D3—24no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C3×C4—22no—
429C3×C4—81no—
430C3×C4—81no—
431C3×C4—82no—
432C3×C4C2×#12011yescharacteristic
433C3×C4#52511yescharacteristic
434C3×C4#52512yes—
435Dic3—24no—
436Dic3—81no—
437Dic3—81no—
438Dic3—82no—
439C23—21no—
440C23—21no—
441C23—22no—
442C23—22no—
443C23—34no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23#88412yes—
451C23C2×D3×D411yescharacteristic
452Q8—41no—
453Q8—42no—
454Q8—41no—
455Q8—68no—
456Q8—121no—
457Q8—122no—
458Q8—121no—
459D4—68no—
460D4—616no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—34no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4#88412yes—
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×D3×D412yes—
492C2×C4D3×#4112yes—
493C8—24no—
494C8—34no—
495C8—62no—
496C2×C3—42no—
497C2×C3#284711yescharacteristic
498C2×C3#305912yes—
499C2×C3#305911yescharacteristic
500C2×C3#306111yescharacteristic
501C2×C3#306112yes—
502D3—44no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—42no—
507C22—42no—
508C22—64no—
509C22—64no—
510C22—64no—
511C22—122no—
512C22—122no—
513C22—122no—
514C22—122no—
515C22C2×#88411yescharacteristic
516C22D3×#12112yes—
517C22#577411yescharacteristic
518C22#577911yescharacteristic
519C22#577912yes—
520C4—22no—
521C4—64no—
522C4—81no—
523C4—82no—
524C4—81no—
525C4—241no—
526C4—242no—
527C4—241no—
528C4#554712yes—
529C4D3×#12011yescharacteristic
530C4#565611yescharacteristic
531C3#3293411yescharacteristic, complemented, Sylow
532C2—42no—
533C2—124no—
534C2#7493511yescharacteristic
535C2#7889111yescharacteristic
536C2#7896411yescharacteristic
537C2#8080812yes—
538C2#8090512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1894C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1894D3
hypocenterC3#32934
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
3. C2×#5051C2semidirect
4. C2×#4813C2semidirect
6. C3×#1894C2semidirect
7. #71989C2semidirect
8. #70271C2semidirect
9. #69512C2semidirect
10. #68884C2semidirect
11. #68869C2semidirect
54. #1894D3semidirect
531. C3#32934semidirect

Maximal subgroups

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

subgroupindex
C22×#9072
#828202
C2×#50512
C2×#48132
#739502
C3×#18942
#719892
#702712
#695122
#688842
#688692
#329343