Finite Group Databasebeta

← #10900882

Subgroups — #10900882

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×#5239C211yesmaximal, characteristic
3C2×#4739C211yesmaximal, characteristic, complemented
4#73951C211yesmaximal, characteristic, complemented
5C3×#1788C211yesmaximal, characteristic
6#70283C212yesmaximal, complemented
7#70246C212yesmaximal, complemented
8#70215C212yesmaximal, complemented
9#70178C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic, complemented
11#69525C211yesmaximal, characteristic, complemented
12#29633—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15#5825—24no—
16#5825—24no—
17C2×#944C2212yes—
18C22×#216C2211yescharacteristic
19C2×#936C2212yes—
20C2×#935—22no—
21C22×#214C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C2×#924—21no—
24C2×#924—21no—
25C2×#924C2211yescharacteristic
26C22×Dic12C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#380C2211yescharacteristic
31#5239—24no—
32C2×#835C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#816C2212yes—
36#4984C2212yes—
37#4973C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40#4953—22no—
41#4739—24no—
42#4661—21no—
43#4659—21no—
44#4550C2212yes—
45#4550C2212yes—
46C2×#536—31no—
47C2×#452—31no—
48C2×#451—31no—
49#2625—31no—
50#2043—32no—
51#2042—32no—
52#2016—32no—
53#2015—32no—
54#1818—31no—
55#1788D311yescharacteristic
56#1787—31no—
57C2×C3×#41—22no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C22×Dic6—21no—
68C22×Dic6—22no—
69C22×Dic6—42no—
70C22×Dic6—42no—
71C22×Dic6D414yes—
72C22×Dic6C2311yescharacteristic
73C22×Dic6C2312yes—
74C22×C3×C8C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—21no—
80C2×#217D412yes—
81#944—24no—
82#944—24no—
83C2×#216—22no—
84C2×#216—42no—
85#936—24no—
86#936—24no—
87#935—216no—
88C2×#214—22no—
89C2×#214—42no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3D411yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—28no—
97#924—28no—
98#924—22no—
99#924—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3C2312yes—
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C2×Dic12—24no—
107C2×Dic12—42no—
108C2×Dic12—42no—
109C3×#103—42no—
110C3×#102—42no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113#835—22no—
114#832—22no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#816—24no—
120#810—42no—
121#810—42no—
122C22×#41C2×D311yescharacteristic
123C23×Q8—31no—
124C22×Q16—31no—
125C22×Q16—32no—
126C22×SD16—31no—
127#536—64no—
128#536—64no—
129C2×C4⋊Q8—32no—
130C2×#124—32no—
131C2×#123—62no—
132C2×C22⋊Q8—31no—
133C2×C22⋊Q8—61no—
134C2×C22⋊Q8—61no—
135C2×#117—31no—
136#452—64no—
137#451—64no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103C2×D312yes—
143C2×#102—31no—
144C2×#102C2×D312yes—
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#398—61no—
149#397—62no—
150#391—32no—
151#389—32no—
152#380C2×D311yescharacteristic
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—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—24no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—44no—
169C2×C3×D4C2×D412yes—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—44no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2411yescharacteristic
180C22×Dic3—24no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—44no—
185C22×Dic3C2×D412yes—
186C22×Dic3C2×D412yes—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—24no—
192C2×Dic6—22no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—48no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C2×Dic6C2×D414yes—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C22⋊C4—81no—
208#217—28no—
209#217—28no—
210#217—24no—
211#217—42no—
212#216—44no—
213#214—44no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—82no—
219Dic3⋊C4—28no—
220Dic3⋊C4—28no—
221Dic3⋊C4—42no—
222C4×Dic3—22no—
223C4×Dic3—22no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228Dic12—48no—
229C2×#41—22no—
230C2×#41—42no—
231C2×#41—42no—
232C2×#41D1214yes—
233C22×Q8—34no—
234C22×Q8—32no—
235C22×Q8—31no—
236C22×Q8—62no—
237C22×Q8—61no—
238C22×Q8—122no—
239C22×Q8—122no—
240C22×Q8C22×D311yescharacteristic
241C22×D4—21no—
242C22×D4C22×D311yescharacteristic
243C23×C4—21no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×Q16—64no—
247C2×Q16—62no—
248C2×Q16—122no—
249C2×Q16—122no—
250C2×Q16—122no—
251C2×SD16—62no—
252C2×SD16—122no—
253C22×C8—31no—
254C22×C8—31no—
255C22×C8C22×D312yes—
256C4⋊Q8—64no—
257C4⋊Q8—64no—
258#124—64no—
259#124—64no—
260#123—616no—
261C22⋊Q8—68no—
262C22⋊Q8—68no—
263C22⋊Q8—62no—
264C22⋊Q8—62no—
265#117—62no—
266#117—62no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4C22×D312yes—
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—42no—
282#103—64no—
283#103—62no—
284#103—122no—
285#103—122no—
286#102—42no—
287#102—62no—
288C22⋊C8—22no—
289C22⋊C8—62no—
290#95—21no—
291#95—62no—
292#95—62no—
293#95—62no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3#12012yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—42no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—48no—
311C3×D4—42no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—48no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4#12012yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4H(5,𝔽2)12yes—
326C2×C3×C4H(5,𝔽2)12yes—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—48no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341Dic6—24no—
342Dic6—24no—
343Dic6—48no—
344Dic6—42no—
345Dic6—42no—
346Dic6—42no—
347Dic6—41no—
348Dic6—41no—
349C3×C8—42no—
350C3⋊C8—41no—
351C3⋊C8—41no—
352C24—21no—
353C24D3×D411yescharacteristic
354#41—44no—
355#41—44no—
356#41—48no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—34no—
361C2×Q8—64no—
362C2×Q8—62no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—62no—
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×D1212yes—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4C2×D1212yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—44no—
390C22×C4—64no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—124no—
395C22×C4C2×D1212yes—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D412yes—
398C22×C4C23×D311yescharacteristic
399Q16—128no—
400Q16—124no—
401SD16—124no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—68no—
412C4⋊C4—68no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—122no—
418C4⋊C4—242no—
419C22⋊C4—68no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—81no—
423C22⋊C4—122no—
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#57812yes—
440C22×C3#57811yescharacteristic
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—82no—
444C3×C4#53411yescharacteristic
445C3×C4#53411yescharacteristic
446C3×C4#53412yes—
447Dic3—44no—
448Dic3—42no—
449Dic3—42no—
450Dic3—81no—
451Dic3—81no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23#88212yes—
462C23C2×D3×D411yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—64no—
467Q8—64no—
468Q8—128no—
469Q8—122no—
470Q8—122no—
471Q8—121no—
472Q8—121no—
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—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—128no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4#88212yes—
503C2×C4C22×D1211yescharacteristic
504C2×C4C2×D3×D411yescharacteristic
505C2×C4#100412yes—
506C2×C4#100412yes—
507C8—42no—
508C8—121no—
509C8—121no—
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—
527C22C2×#88211yescharacteristic
528C22#559212yes—
529C22#577112yes—
530C22#577511yescharacteristic
531C22#577911yescharacteristic
532C4—22no—
533C4—42no—
534C4—82no—
535C4—122no—
536C4—124no—
537C4—122no—
538C4—241no—
539C4—241no—
540C4#549311yescharacteristic
541C4#549311yescharacteristic
542C4#558912yes—
543C3#2963311yescharacteristic, complemented, Sylow
544C2—42no—
545C2—82no—
546C2#7385511yescharacteristic
547C2#7883711yescharacteristic
548C2#7888711yescharacteristic
549C2#8008012yes—
550C2#8027112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#1788D3
hypocenterC3#29633
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
3. C2×#4739C2semidirect
4. #73951C2semidirect
6. #70283C2semidirect
7. #70246C2semidirect
8. #70215C2semidirect
9. #70178C2semidirect
10. #69911C2semidirect
11. #69525C2semidirect
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×#52392
C2×#47392
#739512
C3×#17882
#702832
#702462
#702152
#701782
#699112
#695252
#296333