Finite Group Databasebeta

← #10900881

Subgroups — #10900881

Order 768.

1260 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5825C211yesmaximal, characteristic
2C2×C3×#452C211yesmaximal, characteristic
3#82787C211yesmaximal, characteristic, complemented
4C2×#5050C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic
6#72073C212yesmaximal, complemented
7#72058C212yesmaximal, complemented
8#70284C212yesmaximal, complemented
9#70216C212yesmaximal, complemented
10#69963C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#29633—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×Q16C2211yescharacteristic
16C22×C3×SD16C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18#5825—24no—
19#5825—24no—
20C2×#944C2212yes—
21C22×#216C2212yes—
22C2×#936C2212yes—
23C2×#935—22no—
24C2×#926C2211yescharacteristic
25C2×#924—21no—
26C2×#924—21no—
27C3×#452—24no—
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31#5229C2212yes—
32C2×#835C2212yes—
33C2×#832C2212yes—
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#816C2212yes—
37C2×#816C2212yes—
38#5050—24no—
39#4986—21no—
40#4981—21no—
41#4972C2212yes—
42C2×#810C2212yes—
43#4952—22no—
44#4550C2212yes—
45#4550C2212yes—
46C2×#536—31no—
47C2×#452D311yescharacteristic
48C2×#451—31no—
49#2625—31no—
50#2043—32no—
51#2042—32no—
52#2016—32no—
53#2015—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C2×C3×#41—22no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×Dic3—21no—
68C22×Dic6—21no—
69C22×Dic6—22no—
70C22×Dic6—42no—
71C22×Dic6—42no—
72C22×Dic6D414yes—
73C22×Dic6C2312yes—
74C2×C3×Q16—22no—
75C2×C3×Q16—42no—
76C2×C3×SD16—22no—
77C2×C3×SD16—42no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217—21no—
88C2×#217D412yes—
89#944—24no—
90#944—24no—
91C2×#216—24no—
92C2×#216—42no—
93C2×#216—42no—
94#936—24no—
95#936—24no—
96#935—216no—
97#926—22no—
98#926—22no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2312yes—
102#924—28no—
103#924—28no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3C2312yes—
108C22×C3⋊C8C2312yes—
109C3×#103—22no—
110C3×#102—22no—
111C3×C22⋊C8—22no—
112#835—42no—
113#832—42no—
114C3⋊#95—21no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#816—42no—
120#816—42no—
121#810—24no—
122C22×#41C2×D311yescharacteristic
123C23×Q8—31no—
124C22×Q16—32no—
125C22×Q16C2×D311yescharacteristic
126C22×SD16C2×D311yescharacteristic
127#536—64no—
128#536—64no—
129C2×C4⋊Q8—32no—
130C2×#124—32no—
131C2×#123—62no—
132C2×C22⋊Q8—61no—
133C2×C22⋊Q8—61no—
134C2×C22⋊Q8C2×D311yescharacteristic
135C2×#117—31no—
136#452—24no—
137#451—64no—
138C2×#103—32no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103C2×D311yescharacteristic
143C2×#102—32no—
144C2×#102C2×D311yescharacteristic
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#398—61no—
149#397—62no—
150#391—32no—
151#389—32no—
152#380—31no—
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C3×#41—44no—
158C3×#41—44no—
159C3×#41—48no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—24no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—44no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—24no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—44no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—44no—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D412yes—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—24no—
196C2×Dic6—22no—
197C2×Dic6—48no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C2×Dic6C2×D414yes—
202C3×Q16—44no—
203C3×SD16—44no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—42no—
211C3×C22⋊C4—42no—
212#217—28no—
213#217—28no—
214#217—24no—
215#217—81no—
216#216—48no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221Dic3⋊C4—28no—
222Dic3⋊C4—28no—
223Dic3⋊C4—82no—
224C4×Dic3—22no—
225C4×Dic3—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×#41—22no—
230C2×#41—42no—
231C2×#41—42no—
232C2×#41#6714yes—
233C22×Q8—34no—
234C22×Q8—32no—
235C22×Q8—61no—
236C22×Q8—62no—
237C22×Q8—122no—
238C22×Q8—122no—
239C22×Q8C22×D311yescharacteristic
240C22×Q8C22×D311yescharacteristic
241C22×D4—21no—
242C22×D4C22×D311yescharacteristic
243C23×C4—21no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×Q16—22no—
247C2×Q16—42no—
248C2×Q16—64no—
249C2×Q16—122no—
250C2×Q16—122no—
251C2×SD16—22no—
252C2×SD16—42no—
253C22×C8—32no—
254C22×C8C22×D311yescharacteristic
255C22×C8C22×D311yescharacteristic
256C4⋊Q8—64no—
257C4⋊Q8—64no—
258#124—64no—
259#124—64no—
260#123—616no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—68no—
264C22⋊Q8—68no—
265#117—62no—
266#117—62no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C42—32no—
281#103—22no—
282#103—64no—
283#103—122no—
284#103—122no—
285#103—122no—
286#102—22no—
287#102—122no—
288C22⋊C8—22no—
289C22⋊C8—62no—
290#95—62no—
291#95—62no—
292#95—62no—
293#95—61no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3#12012yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—42no—
308C3×Q8—42no—
309C3×Q8—41no—
310C3×Q8—41no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—48no—
314C3×D4—42no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—48no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4#12012yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4H(5,𝔽2)12yes—
331C2×C3×C4H(5,𝔽2)12yes—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—48no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344Dic6—24no—
345Dic6—24no—
346Dic6—48no—
347Dic6—42no—
348Dic6—42no—
349C3×C8—41no—
350C3×C8—41no—
351C3⋊C8—42no—
352C24—21no—
353C24D3×D411yescharacteristic
354#41—44no—
355#41—44no—
356#41—48no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—34no—
365C2×Q8—64no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—128no—
372C2×Q8—124no—
373C2×Q8—124no—
374C2×Q8—124no—
375C2×Q8C2×#6712yes—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4C2×#6712yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—44no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—124no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D412yes—
397C22×C4C2×#6712yes—
398C22×C4C23×D311yescharacteristic
399Q16—44no—
400Q16—128no—
401SD16—44no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—68no—
413C4⋊C4—68no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—242no—
419C22⋊C4—42no—
420C22⋊C4—68no—
421C22⋊C4—68no—
422C22⋊C4—64no—
423C22⋊C4—241no—
424C42—62no—
425C42—62no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#53512yes—
438C22×C3#57711yescharacteristic
439C22×C3#57811yescharacteristic
440C22×C3#57812yes—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4#53411yescharacteristic
446C3×C4#53411yescharacteristic
447C3×C4#53412yes—
448Dic3—44no—
449Dic3—42no—
450Dic3—42no—
451Dic3—82no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23#93712yes—
462C23C2×D3×D411yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—41no—
467Q8—41no—
468Q8—42no—
469Q8—64no—
470Q8—64no—
471Q8—128no—
472Q8—122no—
473Q8—122no—
474D4—22no—
475D4—22no—
476D4—48no—
477D4—42no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—128no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4#93712yes—
503C2×C4C2×D3×D411yescharacteristic
504C2×C4#100412yes—
505C2×C4#100412yes—
506C2×C4C22×#6711yescharacteristic
507C8—41no—
508C8—41no—
509C8—122no—
510C2×C3—42no—
511C2×C3—82no—
512C2×C3#261311yescharacteristic
513C2×C3#301111yescharacteristic
514C2×C3#301311yescharacteristic
515C2×C3#322312yes—
516C2×C3#322512yes—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22#559212yes—
528C22#577011yescharacteristic
529C22#577111yescharacteristic
530C22#577912yes—
531C22C2×#93711yescharacteristic
532C4—22no—
533C4—42no—
534C4—81no—
535C4—81no—
536C4—122no—
537C4—124no—
538C4—122no—
539C4—242no—
540C4#558912yes—
541C4#580411yescharacteristic
542C4#580411yescharacteristic
543C3#2963311yescharacteristic, complemented, Sylow
544C2—42no—
545C2—82no—
546C2#8022412yes—
547C2#8027112yes—
548C2#8260711yescharacteristic
549C2#8391311yescharacteristic
550C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#452C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#452D3
hypocenterC3#29633
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#452

Complements

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

NH ≅ G/Ntype
3. #82787C2semidirect
4. C2×#5050C2semidirect
6. #72073C2semidirect
7. #72058C2semidirect
8. #70284C2semidirect
9. #70216C2semidirect
10. #69963C2semidirect
11. #69911C2semidirect
543. C3#29633semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
C2×C3×#4522
#827872
C2×#50502
#721352
#720732
#720582
#702842
#702162
#699632
#699112
#296333