Finite Group Databasebeta

← #10830110

Subgroups — #10830110

Order 768.

1380 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic, complemented
3#82480C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic, complemented
5#71768C212yesmaximal, complemented
6#71765C211yesmaximal, characteristic, complemented
7#71762C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71612C212yesmaximal, complemented
10#71606C212yesmaximal, complemented
11#71605C211yesmaximal, characteristic, complemented
12#24200—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#550—24no—
15C3×#550—24no—
16C2×C3×#127C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121—22no—
19C2×C3×#121—21no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C22×#214C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26#5230—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5075—24no—
33#5018—24no—
34#4985—21no—
35#4978—22no—
36C2×#810C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4936C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550D311yescharacteristic
47C2×#460—31no—
48C2×#440—31no—
49#2581—31no—
50#2041—32no—
51#2014—32no—
52#1985—31no—
53#1961—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—22no—
67C23×C3×C4—21no—
68C22×Dic6C2311yescharacteristic
69C3×#127—24no—
70C3×#127—24no—
71C3×#127—22no—
72C3×#127D414yes—
73C3×#121—28no—
74C3×#121—216no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C3×C4×D4—22no—
80C3×C4×D4D414yes—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#214—22no—
89C2×#214—22no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—22no—
93C2×#214—42no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
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—24no—
106#832—24no—
107#832—42no—
108#832—42no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#810—22no—
115#807—22no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—21no—
119C22×SD16—31no—
120C22×SD16—32no—
121#550—24no—
122#550—24no—
123C2×#127C2×D312yes—
124C2×#127C2×D311yescharacteristic
125C2×#121—22no—
126C2×#121—21no—
127C2×C4×Q8—31no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4C2×D312yes—
130C2×C4×D4C2×D311yescharacteristic
131#460—64no—
132#440—64no—
133C2×#107—31no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#97—31no—
142#400—61no—
143#395—61no—
144#391—32no—
145#389—32no—
146#387—62no—
147#382—31no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—22no—
152C24×C3—21no—
153C3×#41—432no—
154C2×C3×Q8—22no—
155C2×C3×Q8—24no—
156C2×C3×D4—28no—
157C2×C3×D4—28no—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4—82no—
174C2×C3×D4—82no—
175C2×C3×D4—82no—
176C2×C3×D4C2×D414yes—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—28no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6#4112yescomplemented
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—24no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208#214—28no—
209#214—48no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214Dic3⋊C4—42no—
215Dic3⋊C4—44no—
216C4×Dic3—42no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8#4112yescomplemented
226C2×#41—28no—
227C2×#41—48no—
228C22×Q8—31no—
229C22×Q8#6711yescharacteristic
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4C22×D312yes—
235C22×D4C22×D311yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×SD16—62no—
241C2×SD16—64no—
242C2×SD16—64no—
243C2×SD16—122no—
244C22×C8—31no—
245C22×C8—32no—
246C22×C8—31no—
247#127—24no—
248#127—24no—
249#127—22no—
250#127#6714yes—
251#121—28no—
252#121—216no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—22no—
260C4×D4#6714yes—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4#6712yes—
266C2×C4⋊C4#6711yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C42—61no—
271C2×C42C22×D312yes—
272C2×C42C22×D311yescharacteristic
273#107—62no—
274C4⋊C8—62no—
275#103—62no—
276#102—64no—
277#102—62no—
278#102—64no—
279#102—122no—
280#102—122no—
281#97—62no—
282#95—61no—
283#95—61no—
284#95—62no—
285#95—62no—
286#95—62no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—28no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—24no—
304C3×D4—24no—
305C3×D4—48no—
306C3×D4—416no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—41no—
310C3×D4—41no—
311C3×D4—84no—
312C3×D4—88no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—48no—
322C2×C3×C4—416no—
323C2×C3×C4—42no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12012yes—
326C2×C3×C4AGL(1,ℤ/8)14yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4H(5,𝔽2)11yescharacteristic
332C2×C3×C42−1+411yescharacteristic
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341Dic6—22no—
342Dic6—22no—
343Dic6—41no—
344Dic6—41no—
345C3⋊C8—22no—
346C3⋊C8—41no—
347C3⋊C8—42no—
348C24—22no—
349C24—21no—
350#41—432no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—32no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C2×#6714yes—
380C2×D4C2×#6712yes—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4C2×#6712yes—
391C22×C4C2×#6712yes—
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C23×D311yescharacteristic
395SD16—68no—
396SD16—128no—
397C2×C8—32no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—62no—
401C2×C8—61no—
402C2×C8—62no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—62no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—124no—
414C4⋊C4—122no—
415C4⋊C4C2×#6712yes—
416C22⋊C4—28no—
417C22⋊C4—24no—
418C42—22no—
419C42—22no—
420C42—122no—
421C42C2×#6712yes—
422C42C2×#6712yes—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3C2×#12011yescharacteristic
436C22×C3#54811yescharacteristic
437C22×C3#55011yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)12yes—
439C22×C3#57712yes—
440C3×C4—22no—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4#52511yescharacteristic
446C3×C4#53711yescharacteristic
447C3×C4#54611yescharacteristic
448C3×C4#54711yescharacteristic
449Dic3—42no—
450Dic3—82no—
451Dic3—81no—
452C23—24no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C22×#6711yescharacteristic
465Q8—28no—
466Q8—62no—
467Q8—62no—
468Q8—121no—
469Q8—121no—
470D4—24no—
471D4—22no—
472D4—24no—
473D4—22no—
474D4—416no—
475D4—48no—
476D4—41no—
477D4—42no—
478D4—41no—
479D4—42no—
480D4—84no—
481D4—88no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—416no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4#93214yes—
502C2×C4#95312yes—
503C2×C4#95312yes—
504C2×C4D3×#4111yescharacteristic
505C2×C4D3×#4111yescharacteristic
506C2×C4#100911yescharacteristic
507C2×C4#101011yescharacteristic
508C2×C4C22×#6711yescharacteristic
509C2×C4C22×#6711yescharacteristic
510C8—62no—
511C8—122no—
512C8—121no—
513C2×C3—82no—
514C2×C3—84no—
515C2×C3#240711yescharacteristic
516C2×C3#300311yescharacteristic
517C2×C3#300911yescharacteristic
518C2×C3#336812yes—
519C2×C3#339712yes—
520C22—44no—
521C22—42no—
522C22—44no—
523C22—42no—
524C22—44no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C2×#93212yes—
533C22#583111yescharacteristic
534C22#583411yescharacteristic
535C22#583612yes—
536C22C2×#95311yescharacteristic
537C4—24no—
538C4—22no—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—122no—
543C4—241no—
544C4—242no—
545C4#582411yescharacteristic
546C4#582811yescharacteristic
547C4#582911yescharacteristic
548C4#583311yescharacteristic
549C3#2420011yescharacteristic, complemented, Sylow
550C2—84no—
551C2—82no—
552C2#8357612yes—
553C2#8358712yes—
554C2#8403811yescharacteristic
555C2#8409711yescharacteristic
556C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#550C2
socleC23×C3C22×D4
hypercenterC2×#550D3
hypocenterC3#24200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#5075C2semidirect
3. #82480C2semidirect
4. C2×#5018C2semidirect
5. #71768C2semidirect
6. #71765C2semidirect
7. #71762C2semidirect
8. #71635C2semidirect
9. #71612C2semidirect
10. #71606C2semidirect
11. #71605C2semidirect
197. C2×Dic6#41semidirect
225. C2×C3⋊C8#41semidirect
549. C3#24200semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×#50752
#824802
C2×#50182
#717682
#717652
#717622
#716352
#716122
#716062
#716052
#242003