Finite Group Databasebeta

← #10829246

Subgroups — #10829246

Order 768.

1184 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5543C211yesmaximal, characteristic, complemented
2C3×#2535C211yesmaximal, characteristic
3C2×#5007C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic, complemented
5#71685C212yesmaximal, complemented
6#71677C212yesmaximal, complemented
7#71615C211yesmaximal, characteristic, complemented
8#69559C211yesmaximal, characteristic, complemented
9#69555C211yesmaximal, characteristic, complemented
10#69527C212yesmaximal, complemented
11#69504C212yesmaximal, complemented
12#24130—31nomaximal, Sylow
13C2×C3×#121C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3—21no—
17C22×#214C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C2×#924C2212yes—
20#5543—24no—
21#5543—24no—
22C2×#890—21no—
23C2×#878—22no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#397—21no—
27C3×#392—21no—
28C3×#386—22no—
29C3×#382C2211yescharacteristic
30C3×#380C2212yes—
31C2×#832C2212yes—
32C2×#832C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5018—24no—
36#5007—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2212yes—
39C2×#807C2211yescharacteristic
40C2×#807C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#440—31no—
49#2535D311yescharacteristic
50#2036—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1852—32no—
56#1805—31no—
57C22×C3×D4—22no—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2312yes—
62C23×Dic3—22no—
63C22×Dic6—21no—
64C22×Dic6C2311yescharacteristic
65C3×#121—24no—
66C3×#121—42no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80D4×Dic3—28no—
81C2×#214—22no—
82C2×#214—22no—
83C2×#214—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90#924—24no—
91#924—24no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98#890—28no—
99#878—216no—
100#869—22no—
101#869D414yescomplemented
102C4×Dic6—22no—
103C4×Dic6D414yescomplemented
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—21no—
107#832—24no—
108#832—22no—
109#830—24no—
110#830—42no—
111#810—22no—
112#810—42no—
113#807—24no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118#554—64no—
119#554—64no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—62no—
124C2×#121C2×D312yes—
125C2×C4×Q8—31no—
126C2×C4×D4—61no—
127C2×C4×D4C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4—32no—
130#474—64no—
131#440—64no—
132C2×#107—31no—
133C2×#107—32no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#397—21no—
143#392—21no—
144#386—22no—
145#382C2×D311yescharacteristic
146#380C2×D312yes—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3—21no—
151C24×C3C2×D412yes—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4—82no—
162C2×C3×D4#4112yes—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—42no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—28no—
181C22×Dic3—82no—
182C22×Dic3—82no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—24no—
189C2×Dic6—82no—
190C2×Dic6C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—82no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202#217—28no—
203#214—28no—
204C4⋊Dic3—28no—
205C4⋊Dic3—24no—
206C4⋊Dic3—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3C2×D412yes—
209C4⋊Dic3C2×D414yes—
210Dic3⋊C4—28no—
211Dic3⋊C4—24no—
212C4×Dic3—24no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8#4112yes—
222C22×Q8—31no—
223C22×Q8—61no—
224C22×D4—22no—
225C22×D4#6711yescharacteristic
226C22×D4C22×D311yescharacteristic
227C23×C4—21no—
228C23×C4—62no—
229C23×C4C22×D312yes—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C22×C8—31no—
234C22×C8—31no—
235C22×C8—32no—
236C4⋊Q8—34no—
237C4⋊Q8—62no—
238#125—68no—
239C22⋊Q8—616no—
240C22⋊Q8—64no—
241C22⋊Q8—64no—
242#121—24no—
243#121—42no—
244C4×Q8—34no—
245C4×Q8—62no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—68no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4#6711yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4#6712yes—
267C2×C42—61no—
268C2×C42C22×D311yescharacteristic
269#107—62no—
270#107—64no—
271C4⋊C8—62no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—64no—
276C22⋊C8—64no—
277C22⋊C8—122no—
278#97—62no—
279#95—22no—
280#95—21no—
281#95—21no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3#13714yes—
294C23×C3C22×D411yescharacteristic
295C3×D4—28no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—88no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—84no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—88no—
333Dic6—22no—
334Dic6—22no—
335Dic6—41no—
336Dic6—41no—
337Dic6—84no—
338C3⋊C8—22no—
339C3⋊C8—41no—
340C3⋊C8—42no—
341C24—21no—
342C24C2×#6712yes—
343C2×Q8—32no—
344C2×Q8—64no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—242no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—82no—
361C2×D4#23712yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—42no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—68no—
375C22×C4—82no—
376C22×C4—242no—
377C22×C4—242no—
378C22×C4C2×#6712yes—
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—68no—
383C2×C8—32no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—62no—
388C2×C8—61no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—61no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—34no—
399C4⋊C4—68no—
400C4⋊C4—68no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4—61no—
404C4⋊C4—62no—
405C22⋊C4—24no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—68no—
409C22⋊C4—82no—
410C42—64no—
411C42D3×D412yes—
412C42D3×D412yes—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—82no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54611yescharacteristic
427C22×C3#54711yescharacteristic
428C22×C3C2×#13712yes—
429C22×C3#57612yes—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—81no—
434C3×C4—82no—
435C3×C4#52511yescharacteristic
436C3×C4#53711yescharacteristic
437C3×C4#54811yescharacteristic
438C3×C4#55011yescharacteristic
439Dic3—42no—
440Dic3—44no—
441Dic3—42no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—42no—
453C23#95114yes—
454C23C22×#6711yescharacteristic
455Q8—62no—
456Q8—62no—
457Q8—121no—
458Q8—121no—
459Q8—244no—
460D4—28no—
461D4—22no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—88no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—64no—
483C2×C4—84no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—248no—
490C2×C4#93712yes—
491C2×C4#93712yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23711yescharacteristic
495C2×C4#100411yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#101011yescharacteristic
498C8—62no—
499C8—122no—
500C8—121no—
501C2×C3—44no—
502C2×C3—82no—
503C2×C3#240711yescharacteristic
504C2×C3#300411yescharacteristic
505C2×C3#300811yescharacteristic
506C2×C3#331412yes—
507C2×C3#334612yes—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—44no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22C2×#93711yescharacteristic
521C22#580611yescharacteristic
522C22C2×#95112yes—
523C22#582411yescharacteristic
524C22#583512yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—82no—
529C4—81no—
530C4—122no—
531C4—122no—
532C4—124no—
533C4#559811yescharacteristic
534C4#559911yescharacteristic
535C4#560611yescharacteristic
536C4#560811yescharacteristic
537C3#2413011yescharacteristic, complemented, Sylow
538C2—44no—
539C2—82no—
540C2#8270111yescharacteristic
541C2#8293311yescharacteristic
542C2#8338311yescharacteristic
543C2#8349812yes—
544C2#8360912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2535C2
socleC23×C3C22×D4
hypercenter#2535D3
hypocenterC3#24130
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#5543C2semidirect
3. C2×#5007C2semidirect
4. C2×#5018C2semidirect
5. #71685C2semidirect
6. #71677C2semidirect
7. #71615C2semidirect
8. #69559C2semidirect
9. #69555C2semidirect
10. #69527C2semidirect
11. #69504C2semidirect
101. #869D4semidirect
103. C4×Dic6D4semidirect
537. C3#24130semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
C3×#25352
C2×#50072
C2×#50182
#716852
#716772
#716152
#695592
#695552
#695272
#695042
#241303