Finite Group Databasebeta

← #10829257

Subgroups — #10829257

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2C3×#2535C211yesmaximal, characteristic
3C2×#5008C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic, complemented
5#71685C212yesmaximal, complemented
6#71678C212yesmaximal, complemented
7#71615C211yesmaximal, characteristic, complemented
8#69559C211yesmaximal, characteristic, complemented
9#69555C211yesmaximal, characteristic, complemented
10#69526C212yesmaximal, complemented
11#69506C212yesmaximal, complemented
12#24131—31nomaximal, Sylow
13C2×C3×#121C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936—21no—
17C22×#214C2211yescharacteristic
18C2×#926C2212yes—
19C2×#924C2212yes—
20#5542—24no—
21#5542—24no—
22C2×#887—21no—
23C2×#879—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#5008—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2212yes—
39C2×#808C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#476—31no—
48C2×#440—31no—
49#2535D311yescharacteristic
50#2035—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1875—32no—
55#1854—32no—
56#1805—31no—
57C22×C3×D4—22no—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2312yes—
62C22×Dic6—21no—
63C22×Dic6C2311yescharacteristic
64C3×#121—24no—
65C3×#121—42no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#217—22no—
80#936—28no—
81C2×#214—22no—
82C2×#214—22no—
83C2×#214—22no—
84#926—24no—
85#926—24no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89#924—24no—
90#924—24no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—22no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98#887—28no—
99#879—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#808—24no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118#556—64no—
119#556—64no—
120C2×C4⋊Q8—31no—
121C2×#126—62no—
122C2×#124—61no—
123C2×C22⋊Q8—32no—
124C2×#121C2×D312yes—
125C2×C4×Q8—31no—
126C2×C4×Q8—61no—
127C2×C4×D4C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—32no—
130#476—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—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—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—42no—
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—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—24no—
185C2×Dic6—21no—
186C2×Dic6—44no—
187C2×Dic6C2×D412yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—82no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—28no—
200#217—28no—
201#214—28no—
202C4⋊Dic3—24no—
203C4⋊Dic3—21no—
204C4⋊Dic3—22no—
205C4⋊Dic3C2×D412yes—
206C4⋊Dic3C2×D414yes—
207Dic3⋊C4—28no—
208Dic3⋊C4—24no—
209C4×Dic3—28no—
210C4×Dic3—24no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—31no—
221C22×Q8—61no—
222C22×D4—22no—
223C22×D4#6711yescharacteristic
224C22×D4C22×D311yescharacteristic
225C23×C4—21no—
226C23×C4C22×D312yes—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8—32no—
233C4⋊Q8—34no—
234C4⋊Q8—62no—
235#126—616no—
236#124—68no—
237C22⋊Q8—64no—
238C22⋊Q8—64no—
239#121—24no—
240#121—42no—
241C4×Q8—34no—
242C4×Q8—68no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248#117—64no—
249#117—64no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4#6712yes—
264C2×C42—62no—
265C2×C42—61no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—62no—
270#103—62no—
271#103—122no—
272#102—62no—
273#102—64no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—62no—
277#95—22no—
278#95—21no—
279#95—21no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3C22×D411yescharacteristic
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—44no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—88no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×C3×C42−1+411yescharacteristic
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—24no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—44no—
328C2×Dic3—42no—
329Dic6—22no—
330Dic6—22no—
331Dic6—48no—
332Dic6—41no—
333Dic6—41no—
334C3⋊C8—22no—
335C3⋊C8—41no—
336C3⋊C8—42no—
337C24—21no—
338C24C2×#6712yes—
339C2×Q8—32no—
340C2×Q8—64no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—124no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—82no—
357C2×D4#23712yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—82no—
371C22×C4C2×#6712yes—
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C23×D311yescharacteristic
375SD16—68no—
376C2×C8—32no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—62no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—61no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—34no—
391C4⋊C4—32no—
392C4⋊C4—68no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—61no—
396C4⋊C4—62no—
397C22⋊C4—24no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C22⋊C4—82no—
403C42—68no—
404C42—64no—
405C42D3×D412yes—
406C42D3×D412yes—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—82no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3C2×#12011yescharacteristic
420C22×C3#54611yescharacteristic
421C22×C3#54711yescharacteristic
422C22×C3#57512yes—
423C22×C3#57712yes—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—81no—
428C3×C4—82no—
429C3×C4#52511yescharacteristic
430C3×C4#53711yescharacteristic
431C3×C4#54811yescharacteristic
432C3×C4#55011yescharacteristic
433Dic3—42no—
434Dic3—44no—
435Dic3—42no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—42no—
448C23C22×#6711yescharacteristic
449Q8—62no—
450Q8—62no—
451Q8—128no—
452Q8—121no—
453Q8—121no—
454D4—22no—
455D4—22no—
456D4—44no—
457D4—41no—
458D4—41no—
459D4—88no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—62no—
475C2×C4—64no—
476C2×C4—62no—
477C2×C4—84no—
478C2×C4—121no—
479C2×C4—124no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4#93712yes—
483C2×C4#93712yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23711yescharacteristic
487C2×C4#100411yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4#101011yescharacteristic
490C8—62no—
491C8—122no—
492C8—121no—
493C2×C3—82no—
494C2×C3—82no—
495C2×C3#240711yescharacteristic
496C2×C3#300511yescharacteristic
497C2×C3#301011yescharacteristic
498C2×C3#331112yes—
499C2×C3#335012yes—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22C2×#93711yescharacteristic
513C22#580611yescharacteristic
514C22#582212yes—
515C22#582411yescharacteristic
516C22#583612yes—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—82no—
521C4—81no—
522C4—122no—
523C4—122no—
524C4—124no—
525C4#559811yescharacteristic
526C4#559911yescharacteristic
527C4#560611yescharacteristic
528C4#560811yescharacteristic
529C3#2413111yescharacteristic, complemented, Sylow
530C2—82no—
531C2—82no—
532C2#8270111yescharacteristic
533C2#8292811yescharacteristic
534C2#8339011yescharacteristic
535C2#8349712yes—
536C2#8361112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2535C2
socleC23×C3C22×D4
hypercenter#2535D3
hypocenterC3#24131
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×#5542C2semidirect
3. C2×#5008C2semidirect
4. C2×#5018C2semidirect
5. #71685C2semidirect
6. #71678C2semidirect
7. #71615C2semidirect
8. #69559C2semidirect
9. #69555C2semidirect
10. #69526C2semidirect
11. #69506C2semidirect
101. #869D4semidirect
103. C4×Dic6D4semidirect
529. C3#24131semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C3×#25352
C2×#50082
C2×#50182
#716852
#716782
#716152
#695592
#695552
#695262
#695062
#241313