Finite Group Databasebeta

← #10799053

Subgroups — #10799053

Order 768.

1566 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
1C22×#912C211yesmaximal, characteristic
2C2×D3×#107C211yesmaximal, characteristic, complemented
3C2×#4771C212yesmaximal, complemented
4C2×#4689C211yesmaximal, characteristic, complemented
5#76294C212yesmaximal, complemented
6C3×#1970C211yesmaximal, characteristic, complemented
7#72038C211yesmaximal, characteristic, complemented
8#70627C211yesmaximal, characteristic, complemented
9#70192C212yesmaximal, complemented
10#69775C211yesmaximal, characteristic, complemented
11#69649C212yesmaximal, complemented
12#21230—31nomaximal, Sylow
13C22×#237C2212yes—
14C22×C3×D8C2211yescharacteristic
15C22×#214C2212yes—
16C2×#912—24no—
17C2×#912—24no—
18C2×#912C418yescomplemented
19C22×Dic12C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×D3×C4⋊C4C2212yes—
23C2×#872C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×C4×C8C2211yescharacteristic
27C3×#390C2212yes—
28#5221C2212yes—
29C2×#832C2212yes—
30C2×#818C2211yescharacteristic
31C2×#816C2212yes—
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4973C2212yes—
35C2×#810C2212yes—
36C2×#807C2212yes—
37#4956—24no—
38#4942—24no—
39#4929C2212yes—
40C2×#803C2211yescharacteristic
41D3×#107—24no—
42#4771—28no—
43#4689—24no—
44C22×#135—31no—
45C22×#107—31no—
46C2×#417—32no—
47C2×#405—31no—
48#2311—32no—
49#1973—32no—
50#1973—32no—
51#1970—31no—
52#1970D311yescharacteristic, complemented
53#1969—31no—
54#1969—31no—
55C22×C3×D4C2312yes—
56C22×#67—22no—
57C23×Dic3—22no—
58C2×#237—24no—
59C2×#237—24no—
60C2×#237—24no—
61C2×#237—24no—
62C2×#237—24no—
63C2×#237C2×C418yes—
64C22×C4×D3—22no—
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2312yes—
67C2×C3×D8—22no—
68C2×C3×D8—22no—
69C2×C3×D8C2×C414yescomplemented
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—24no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—24no—
76C2×#214—24no—
77C2×#214—24no—
78C2×#214C2×C418yescomplemented
79C2×(C2×D3)⋊C4D412yes—
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4D412yes—
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85C22×C3⋊C8C2311yescharacteristic
86#912—232no—
87C2×Dic12—22no—
88C2×Dic12—22no—
89C2×Dic12C2×C414yescomplemented
90C2×ΓL(1,𝔽25)—22no—
91C2×ΓL(1,𝔽25)—22no—
92C2×ΓL(1,𝔽25)D414yes—
93C2×D3×C8C2×C414yes—
94C2×D3×C8D414yes—
95C2×D3×C8D414yes—
96D3×C4⋊C4—24no—
97D3×C4⋊C4—24no—
98#872—22no—
99#872—22no—
100C3×#107—22no—
101C3×#102—24no—
102C3×C4×C8—22no—
103#832—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106#818—22no—
107#816—24no—
108#815—22no—
109C8×Dic3—22no—
110#810—24no—
111#807—24no—
112#803—22no—
113C22×#41—32no—
114C2×#135—38no—
115C2×#135—64no—
116C2×#135—64no—
117C22×Q16—31no—
118C22×SD16—32no—
119C22×D8C2×D311yescharacteristic
120C22×M16—31no—
121C23×C8—31no—
122C2×#117—31no—
123C22×C4⋊C4—32no—
124C2×#107—31no—
125C2×#107—32no—
126C2×#107—64no—
127C2×#107C2×D311yescharacteristic
128#417—68no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#102—32no—
132C2×#102C2×D312yes—
133#405—64no—
134C2×#97—31no—
135C2×#97—31no—
136C2×C4×C8—31no—
137C2×C4×C8C2×D311yescharacteristic
138#391—32no—
139#391—32no—
140#390—32no—
141#390C2×D312yes—
142#386—64no—
143#380—64no—
144C24×C3—22no—
145C23×D3C2×D411yescharacteristic
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4C22×C414yes—
152C22×C3×C4—22no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2411yescharacteristic
155C2×#67—28no—
156C2×#67—44no—
157C2×#67—44no—
158C22×Dic3—22no—
159C22×Dic3—28no—
160C22×Dic3—22no—
161C22×Dic3—44no—
162C22×Dic3—44no—
163C22×Dic3C2×D412yes—
164C22×Dic3C2×D411yescharacteristic
165#237—216no—
166#237—216no—
167#237—48no—
168#237—48no—
169C2×C4×D3—44no—
170C2×C4×D3—84no—
171C2×C4×D3C22×C412yes—
172C2×C4×D3C2×D412yes—
173C2×C4×D3C2×D412yes—
174C2×C4×D3C2×D412yes—
175C2×C4×D3#4112yes—
176C2×C4×D3#4112yes—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6C22×C414yes—
183C3×D8—28no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8C22×C412yes—
187C2×C3×C8C2×D412yes—
188C2×C3×C8C2×D412yes—
189C2×C3×C8C2×D412yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C22⋊C4—84no—
193C3×C42—21no—
194C3×C42—21no—
195#217—84no—
196#214—216no—
197(C2×D3)⋊C4—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200Dic3⋊C4—24no—
201Dic3⋊C4—44no—
202C4×Dic3—21no—
203C4×Dic3—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8C22×C412yes—
207C2×C3⋊C8C2×D412yes—
208C2×C3⋊C8C2×D412yes—
209C2×C3⋊C8C2×D412yes—
210Dic12—28no—
211ΓL(1,𝔽25)—28no—
212D3×C8C22⋊C4116yescomplemented
213C2×#41—38no—
214C2×#41—64no—
215C2×#41—64no—
216C2×#41—64no—
217C2×#41—64no—
218C2×#41—64no—
219C22×Q8—32no—
220C22×D4—62no—
221C22×D4C22×D312yes—
222C23×C4—31no—
223C23×C4—62no—
224C23×C4—62no—
225#135—632no—
226C2×Q16—34no—
227C2×Q16—62no—
228C2×Q16—62no—
229C2×SD16—38no—
230C2×SD16—64no—
231C2×SD16—64no—
232C2×D8—22no—
233C2×D8—22no—
234C2×D8C4×D314yescomplemented
235C2×M16—34no—
236C2×M16—62no—
237C2×M16—62no—
238C22×C8—31no—
239C22×C8—34no—
240C22×C8—34no—
241C22×C8—34no—
242C22×C8—31no—
243C22×C8C22×D311yescharacteristic
244C22×C8C22×D311yescharacteristic
245#117—62no—
246#117—62no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—32no—
255C2×C22⋊C4—64no—
256C2×C42—31no—
257C2×C42C22×D311yescharacteristic
258#107—22no—
259#107—64no—
260#107—62no—
261#103—64no—
262#103—64no—
263#102—24no—
264#102—64no—
265#97—62no—
266#97—62no—
267C4×C8—22no—
268C4×C8—62no—
269#95—64no—
270#95—64no—
271C23×C3—24no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—44no—
275C23×C3—44no—
276C23×C3C22×D411yescharacteristic
277C22×D3—22no—
278C22×D3—22no—
279C22×D3C4×D412yes—
280C22×D3#12112yes—
281C22×D3#12712yes—
282C3×D4—24no—
283C3×D4—24no—
284C3×D4—42no—
285C3×D4—42no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—42no—
290C2×C3×C4—44no—
291C2×C3×C4C23×C411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297#67—416no—
298C2×Dic3—24no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—416no—
305C2×Dic3—42no—
306C2×Dic3—44no—
307C2×Dic3C4×D412yes—
308C2×Dic3#12112yes—
309C2×Dic3#12712yes—
310C4×D3—88no—
311C4×D3C2×C22⋊C414yes—
312C4×D3C4×D414yes—
313C4×D3#12114yes—
314C4×D3#12414yes—
315Dic6—24no—
316Dic6—24no—
317Dic6—42no—
318Dic6—42no—
319C3×C8—22no—
320C3×C8C2×C22⋊C414yes—
321C3⋊C8—22no—
322C3⋊C8C2×C22⋊C414yes—
323C24—22no—
324C24—31no—
325#41—616no—
326#41—616no—
327#41—128no—
328#41—128no—
329C2×Q8—34no—
330C2×Q8—62no—
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—68no—
341C2×D4—124no—
342C2×D4—124no—
343C2×D4C2×C4×D314yes—
344C22×C4—22no—
345C22×C4—31no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—68no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—124no—
357C22×C4—124no—
358C22×C4—124no—
359C22×C4—244no—
360C22×C4D3×D412yes—
361C22×C4C23×D311yescharacteristic
362Q16—68no—
363SD16—616no—
364D8—28no—
365M16—68no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—32no—
369C2×C8—32no—
370C2×C8—32no—
371C2×C8—316no—
372C2×C8—32no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8C2×C4×D312yes—
376C2×C8D3×D412yes—
377C2×C8D3×D412yes—
378C2×C8D3×D412yes—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—64no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—124no—
385C22⋊C4—64no—
386C22⋊C4—84no—
387C22⋊C4—244no—
388C42—21no—
389C42—21no—
390C42—61no—
391C42—61no—
392C22×C3—48no—
393C22×C3—44no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—84no—
397C22×C3C2×C4×D411yescharacteristic
398C22×C3C2×#12111yescharacteristic
399C22×C3C2×#12711yescharacteristic
400C22×C3#57612yes—
401C22×C3#57812yes—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—24no—
405C2×D3—24no—
406C2×D3#39014yes—
407C3×C4—42no—
408C3×C4—82no—
409C3×C4C22×C22⋊C411yescharacteristic
410C3×C4C2×C4×D411yescharacteristic
411C3×C4C2×#12111yescharacteristic
412C3×C4C2×#12411yescharacteristic
413Dic3—44no—
414Dic3—42no—
415Dic3—82no—
416Dic3#39014yes—
417C23—24no—
418C23—32no—
419C23—32no—
420C23—32no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—62no—
426C23—62no—
427C23C2×D3×D411yescharacteristic
428Q8—64no—
429Q8—64no—
430Q8—122no—
431Q8—122no—
432D4—24no—
433D4—24no—
434D4—42no—
435D4—42no—
436D4—1216no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—34no—
444C2×C4—34no—
445C2×C4—34no—
446C2×C4—34no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—64no—
452C2×C4—62no—
453C2×C4—1216no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—124no—
458C2×C4—248no—
459C2×C4C22×C4×D311yescharacteristic
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4D3×#4111yescharacteristic
464C2×C4D3×#4111yescharacteristic
465C8—22no—
466C8—34no—
467C8—62no—
468C8D3×C22⋊C414yes—
469C2×C3—84no—
470C2×C3C2×#39011yescharacteristic
471C2×C3#295112yes—
472C2×C3#306112yes—
473C2×C3#315511yescharacteristic
474C2×C3#315611yescharacteristic
475D3—24no—
476C22—34no—
477C22—44no—
478C22—48no—
479C22—64no—
480C22—62no—
481C22—62no—
482C22—64no—
483C22—84no—
484C22—82no—
485C22—82no—
486C22C4×D3×D411yescharacteristic
487C22D3×#12111yescharacteristic
488C22D3×#12711yescharacteristic
489C22#575712yes—
490C22#577112yes—
491C4—34no—
492C4—42no—
493C4—82no—
494C4—122no—
495C4—124no—
496C4—242no—
497C4C2×D3×C22⋊C411yescharacteristic
498C4C4×D3×D411yescharacteristic
499C4D3×#12111yescharacteristic
500C4D3×#12411yescharacteristic
501C3#2123011yescharacteristic, complemented, Sylow
502C2—64no—
503C2—84no—
504C2D3×#39011yescharacteristic
505C2#7958912yes—
506C2#8071712yes—
507C2#8178411yescharacteristic
508C2#8182411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1970C2
socleC23×C3C22×D4
prime radicalC22×#912C2
cosocleC22×C3×C4C24
hypercenter#1970D3
hypocenterC3#21230
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×D3×#107C2semidirect
3. C2×#4771C2semidirect
4. C2×#4689C2semidirect
5. #76294C2semidirect
6. C3×#1970C2semidirect
7. #72038C2semidirect
8. #70627C2semidirect
9. #70192C2semidirect
10. #69775C2semidirect
11. #69649C2semidirect
18. C2×#912C4semidirect
52. #1970D3semidirect
69. C2×C3×D8C2×C4semidirect
78. C2×#214C2×C4semidirect
89. C2×Dic12C2×C4semidirect
212. D3×C8C22⋊C4semidirect
234. C2×D8C4×D3semidirect
501. C3#21230semidirect

Maximal subgroups

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

subgroupindex
C22×#9122
C2×D3×#1072
C2×#47712
C2×#46892
#762942
C3×#19702
#720382
#706272
#701922
#697752
#696492
#212303