Finite Group Databasebeta

← #10829277

Subgroups — #10829277

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3#82375C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71685C212yesmaximal, complemented
7#71680C212yesmaximal, complemented
8#71615C211yesmaximal, characteristic, complemented
9#71605C211yesmaximal, characteristic, complemented
10#71585C212yesmaximal, complemented
11#71576C212yesmaximal, complemented
12#24133—31nomaximal, Sylow
13C3×#555—24no—
14C3×#555—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124—22no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#117—21no—
22C2×C3×#117C2212yes—
23C22×#214C2211yescharacteristic
24C2×#924C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26#5222—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5094—24no—
33#5018—24no—
34#4983—21no—
35C2×#810C2212yes—
36C2×#810C2211yescharacteristic
37C2×#808C2212yes—
38C2×#807C2211yescharacteristic
39#4944—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#555D311yescharacteristic
47C2×#475—31no—
48C2×#440—31no—
49#2536—31no—
50#2034—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1873—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×D4—22no—
59C22×C3×D4D411yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2312yes—
63C22×Dic6C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#124—216no—
67C3×#121—28no—
68C3×#121—24no—
69C3×#121—24no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4D414yes—
74C3×#117—28no—
75C3×#117—24no—
76C3×#117—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42—22no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88C2×#217—22no—
89C2×#214—22no—
90C2×#214—22no—
91C2×#214—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94#924—24no—
95#924—42no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C4×Dic6—22no—
103C4×Dic6—22no—
104#832—22no—
105#832—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#830—24no—
110#830—42no—
111#810—22no—
112#810—24no—
113#808—24no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118#555—24no—
119#555—24no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#124—22no—
122C2×C22⋊Q8—32no—
123C2×#121—21no—
124C2×#121C2×D312yes—
125C2×C4×Q8—31no—
126C2×C4×D4C2×D311yescharacteristic
127C2×C4×D4C2×D311yescharacteristic
128C2×#117—21no—
129C2×#117C2×D312yes—
130#475—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×C4⋊C8—31no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#398—61no—
143#394—61no—
144#388—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3—21no—
151C24×C3C2×D412yes—
152C2×C3×Q8—24no—
153C2×C3×Q8—82no—
154C2×C3×D4—28no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—42no—
162C2×C3×D4—82no—
163C2×C3×D4—82no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—44no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6#4112yes—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4C2×D414yes—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C42—28no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—44no—
202#217—44no—
203#214—28no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208Dic3⋊C4—44no—
209Dic3⋊C4—42no—
210C4×Dic3—42no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—21no—
221C22×Q8—31no—
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—22no—
234C4⋊Q8#6714yes—
235#124—216no—
236C22⋊Q8—64no—
237C22⋊Q8—122no—
238#121—28no—
239#121—24no—
240#121—24no—
241C4×Q8—62no—
242C4×Q8—62no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4#6714yes—
247#117—28no—
248#117—24no—
249#117—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4#6712yes—
264C2×C42—22no—
265C2×C42—61no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—62no—
270#103—62no—
271#103—64no—
272#102—62no—
273#102—122no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—62no—
277#95—62no—
278#95—61no—
279#95—61no—
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×Q8—84no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—44no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—88no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—48no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322C2×C3×C42−1+411yescharacteristic
323C2×Dic3—22no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331Dic6—22no—
332Dic6—22no—
333Dic6—41no—
334Dic6—41no—
335C3⋊C8—22no—
336C3⋊C8—41no—
337C3⋊C8—42no—
338C24—21no—
339C24C2×#6712yes—
340C2×Q8—24no—
341C2×Q8—32no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—82no—
348C2×D4—28no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4C2×#6712yes—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4C2×#6712yes—
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C23×D311yescharacteristic
376SD16—68no—
377C2×C8—32no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—62no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—62no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—124no—
395C4⋊C4—122no—
396C4⋊C4C2×#6714yes—
397C4⋊C4C2×#6712yes—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—124no—
403C22⋊C4—124no—
404C42—28no—
405C42—122no—
406C42C2×#6712yes—
407C42C2×#6712yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#54611yescharacteristic
422C22×C3#54711yescharacteristic
423C22×C3#57512yes—
424C22×C3#57712yes—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—44no—
430C3×C4#52511yescharacteristic
431C3×C4#53711yescharacteristic
432C3×C4#54811yescharacteristic
433C3×C4#55011yescharacteristic
434Dic3—42no—
435Dic3—81no—
436Dic3—82no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23C22×#6711yescharacteristic
450Q8—62no—
451Q8—62no—
452Q8—84no—
453Q8—121no—
454Q8—121no—
455D4—22no—
456D4—22no—
457D4—44no—
458D4—41no—
459D4—41no—
460D4—88no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—62no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—121no—
484C2×C4#95312yes—
485C2×C4#95312yes—
486C2×C4D3×#4111yescharacteristic
487C2×C4D3×#4111yescharacteristic
488C2×C4#100911yescharacteristic
489C2×C4#101011yescharacteristic
490C2×C4C22×#6711yescharacteristic
491C2×C4C22×#6711yescharacteristic
492C8—62no—
493C8—121no—
494C8—122no—
495C2×C3—82no—
496C2×C3—82no—
497C2×C3#240711yescharacteristic
498C2×C3#300511yescharacteristic
499C2×C3#300911yescharacteristic
500C2×C3#332112yes—
501C2×C3#334812yes—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22—82no—
514C22#582212yes—
515C22#582411yescharacteristic
516C22#582911yescharacteristic
517C22#583612yes—
518C22C2×#95311yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—122no—
525C4—241no—
526C4—242no—
527C4#582811yescharacteristic
528C4#583111yescharacteristic
529C4#583311yescharacteristic
530C4#583411yescharacteristic
531C3#2413311yescharacteristic, complemented, Sylow
532C2—82no—
533C2—82no—
534C2#8364912yes—
535C2#8365612yes—
536C2#8408111yescharacteristic
537C2#8409711yescharacteristic
538C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#555C2
socleC23×C3C22×D4
hypercenterC2×#555D3
hypocenterC3#24133
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
3. #82375C2semidirect
4. C2×#5018C2semidirect
5. #71788C2semidirect
6. #71685C2semidirect
7. #71680C2semidirect
8. #71615C2semidirect
9. #71605C2semidirect
10. #71585C2semidirect
11. #71576C2semidirect
531. C3#24133semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×#50942
#823752
C2×#50182
#717882
#716852
#716802
#716152
#716052
#715852
#715762
#241333