Finite Group Databasebeta

← #10829957

Subgroups — #10829957

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5567C211yesmaximal, characteristic, complemented
2C2×C3×C4×SD16C211yesmaximal, characteristic
3#82376C211yesmaximal, characteristic
4C2×#5019C211yesmaximal, characteristic
5#72057C212yesmaximal
6#71626C211yesmaximal, characteristic, complemented
7#70588C212yesmaximal, complemented
8#70284C212yesmaximal, complemented
9#70213C212yesmaximal
10#69788C211yesmaximal, characteristic, complemented
11#69771C211yesmaximal, characteristic, complemented
12#24187—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#944C2212yes—
17C22×#214C2212yes—
18C2×#924—21no—
19#5567—24no—
20#5567—24no—
21C2×#889—22no—
22C2×#887C2212yes—
23C2×#870—21no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2211yescharacteristic
27C3×C4×SD16—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C2×#835C2212yes—
33#5221C2212yes—
34#5212—21no—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#5019—24no—
38#4972C2212yes—
39C2×#810C2212yes—
40#4948—22no—
41#4935C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#461—31no—
48C2×C4×SD16D311yescharacteristic
49#2538—31no—
50#2043—32no—
51#2013—32no—
52#1984—31no—
53#1956—31no—
54#1880—32no—
55#1865—32no—
56#1803—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2312yes—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79#944—24no—
80#944—24no—
81C2×#214—24no—
82C2×#214—24no—
83C2×#214—42no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2312yes—
90#924—44no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4D412yes—
96C2×C4×Dic3—22no—
97C2×C4×Dic3C2312yes—
98C22×C3⋊C8C2312yes—
99#889—216no—
100#887—24no—
101#887—24no—
102#870—28no—
103#869—22no—
104#869D414yes—
105C4×Dic6—28no—
106C4×Dic6—22no—
107C4×Dic6—22no—
108C3×#106—22no—
109C3×#103—22no—
110C3×#102—22no—
111C3×C4×C8—22no—
112#835—42no—
113C3⋊#95—22no—
114#816—24no—
115#816—42no—
116#810—24no—
117C4⋊(C3⋊C8)—22no—
118C22×SD16—32no—
119C22×SD16C2×D311yescharacteristic
120#557—64no—
121#557—64no—
122C2×C4⋊Q8—32no—
123C2×C4⋊Q8—31no—
124C2×#125—61no—
125C2×#125—62no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×Q8—61no—
130C2×C4×Q8C2×D311yescharacteristic
131C2×C4×D4C2×D311yescharacteristic
132#461—64no—
133C4×SD16—24no—
134C2×#106C2×D311yescharacteristic
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#103—32no—
140C2×#103C2×D311yescharacteristic
141C2×#102C2×D311yescharacteristic
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D412yes—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4#4112yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6SD1614yescomplemented
185C2×Dic6C2×D414yes—
186C3×SD16—28no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8#4112yes—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4C2×D412yes—
198C3×C22⋊C4—42no—
199C3×C42—24no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202#217—44no—
203#217—82no—
204#214—48no—
205C4⋊Dic3—24no—
206C4⋊Dic3—22no—
207C4⋊Dic3—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—42no—
211C4⋊Dic3—81no—
212Dic3⋊C4—28no—
213Dic3⋊C4—28no—
214Dic3⋊C4—28no—
215Dic3⋊C4—24no—
216Dic3⋊C4—44no—
217C4×Dic3—28no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C22×Q8—31no—
224C22×Q8—32no—
225C22×Q8C22×D311yescharacteristic
226C22×D4C22×D311yescharacteristic
227C23×C4—21no—
228C2×SD16—22no—
229C2×SD16—22no—
230C2×SD16—22no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—32no—
235C22×C8C22×D311yescharacteristic
236C22×C8C22×D311yescharacteristic
237C4⋊Q8—34no—
238C4⋊Q8—64no—
239C4⋊Q8—64no—
240C4⋊Q8—62no—
241#125—68no—
242#125—616no—
243C22⋊Q8—124no—
244C4×Q8—22no—
245C4×Q8—68no—
246C4×Q8—64no—
247C4×Q8—64no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×Q8#6714yes—
251C4×D4—22no—
252C4×D4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—21no—
271C2×C42—32no—
272C2×C42—62no—
273C2×C42C22×D311yescharacteristic
274#106—22no—
275C4⋊C8—62no—
276#103—22no—
277#103—64no—
278#103—64no—
279#103—122no—
280#103—122no—
281#102—22no—
282C4×C8—22no—
283#95—62no—
284C23×C3—22no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—84no—
312C2×C3×C4#12012yes—
313C2×C3×C4#12012yes—
314C2×C3×C4C2×SD1612yes—
315C2×C3×C4#13712yes—
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—22no—
325C2×Dic3—24no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3C2×SD1614yes—
334Dic6—28no—
335Dic6—24no—
336Dic6—24no—
337Dic6—42no—
338Dic6—42no—
339C3×C8—22no—
340C3×C8—41no—
341C3⋊C8—42no—
342C24—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—34no—
349C2×Q8—34no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8C2×#6712yes—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4#23512yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4D3×D411yescharacteristic
375C22×C4D3×D411yescharacteristic
376C22×C4C23×D311yescharacteristic
377SD16—28no—
378SD16—128no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8#23512yes—
388C4⋊C4—24no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—68no—
393C4⋊C4—68no—
394C4⋊C4—68no—
395C4⋊C4—64no—
396C4⋊C4—62no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—124no—
402C4⋊C4—122no—
403C4⋊C4—241no—
404C4⋊C4C2×#6712yes—
405C22⋊C4—42no—
406C22⋊C4—124no—
407C22⋊C4—242no—
408C42—24no—
409C42—68no—
410C42—62no—
411C42—62no—
412C42D3×D412yes—
413C42C2×#6712yes—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54811yescharacteristic
423C22×C3#55011yescharacteristic
424C22×C3C22×SD1611yescharacteristic
425C22×C3C2×#13711yescharacteristic
426C22×C3#57711yescharacteristic
427C22×C3#57811yescharacteristic
428C3×C4—22no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—81no—
432C3×C4#44814yes—
433C3×C4#52511yescharacteristic
434C3×C4#53711yescharacteristic
435C3×C4#54611yescharacteristic
436C3×C4#54711yescharacteristic
437Dic3—24no—
438Dic3—44no—
439Dic3—42no—
440Dic3—82no—
441C23—22no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C2×D3×D411yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453Q8—68no—
454Q8—64no—
455Q8—64no—
456Q8—122no—
457Q8—122no—
458D4—22no—
459D4—22no—
460D4—41no—
461D4—41no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—34no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—64no—
480C2×C4—84no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4D3×SD1612yes—
487C2×C4#91512yes—
488C2×C4#93712yes—
489C2×C4#93712yes—
490C2×C4C2×#23511yescharacteristic
491C2×C4C2×#23511yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4#100911yescharacteristic
494C2×C4#101011yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—22no—
497C8—41no—
498C8—122no—
499C2×C3—82no—
500C2×C3#240711yescharacteristic
501C2×C3C2×#44811yescharacteristic
502C2×C3#301311yescharacteristic
503C2×C3#335911yescharacteristic
504C2×C3#337211yescharacteristic
505C2×C3#339211yescharacteristic
506C2×C3#339811yescharacteristic
507C22—42no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22#555211yescharacteristic
515C22#555311yescharacteristic
516C22C2×D3×SD1611yescharacteristic
517C22C2×#91511yescharacteristic
518C22#577011yescharacteristic
519C22#577111yescharacteristic
520C22C2×#93711yescharacteristic
521C4—22no—
522C4—42no—
523C4—42no—
524C4—64no—
525C4—81no—
526C4—122no—
527C4—124no—
528C4—242no—
529C4#517014yes—
530C4#555011yescharacteristic
531C4#555111yescharacteristic
532C4#579111yescharacteristic
533C4#583311yescharacteristic
534C3#2418711yescharacteristic, complemented, Sylow
535C2—82no—
536C2#7961911yescharacteristic
537C2#7962811yescharacteristic
538C2#7963711yescharacteristic
539C2#7964011yescharacteristic
540C2#8240211yescharacteristic
541C2C2×#517011yescharacteristic
542C2#8391311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×C4×SD16D3
hypocenterC3#24187
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#5567C2semidirect
6. #71626C2semidirect
7. #70588C2semidirect
8. #70284C2semidirect
10. #69788C2semidirect
11. #69771C2semidirect
184. C2×Dic6SD16semidirect
534. C3#24187semidirect

Maximal subgroups

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

subgroupindex
C2×#55672
C2×C3×C4×SD162
#823762
C2×#50192
#720572
#716262
#705882
#702842
#702132
#697882
#697712
#241873