Finite Group Databasebeta

← #10829986

Subgroups — #10829986

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5713C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic
3#82482C211yesmaximal, characteristic, complemented
4C2×C4×#216C211yesmaximal, characteristic, complemented
5#72058C212yesmaximal, complemented
6#72047C212yesmaximal, complemented
7#71764C211yesmaximal, characteristic
8#70485C211yesmaximal, characteristic, complemented
9#70472C211yesmaximal, characteristic, complemented
10#70216C212yesmaximal, complemented
11#70167C212yesmaximal, complemented
12#24190—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C22×#216C2211yescharacteristic
17C2×#936—21no—
18C2×#936C2212yes—
19C2×#935—22no—
20C2×D4×Dic3—21no—
21C2×D4×Dic3C2212yes—
22#5713—24no—
23#5713—24no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#460—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29#5229C2212yes—
30C2×#835C2211yescharacteristic
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×#816C2212yes—
34C4×#216—24no—
35#4982—22no—
36#4972C2212yes—
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42C2×C4×C3⋊C8C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#559—31no—
47C2×#460D311yescharacteristic
48C2×C4×Q16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1957—31no—
54#1877—32no—
55#1862—32no—
56#1803—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2312yes—
61C23×Dic3—22no—
62C22×Dic6D411yescharacteristic
63C22×Dic6C2311yescharacteristic
64C2×C3×SD16—24no—
65C2×C3×SD16—24no—
66C2×C3×SD16—42no—
67C22×C3×C8C2312yes—
68C3×#127—22no—
69C3×#127D414yes—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—22no—
79C2×#216—22no—
80C2×#216—22no—
81C2×#216—22no—
82#936—28no—
83#936—24no—
84#936—24no—
85#935—216no—
86D4×Dic3—28no—
87D4×Dic3—24no—
88D4×Dic3—24no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D412yes—
97C2×C4×Dic3—21no—
98C2×C4×Dic3C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×C4⋊C8—22no—
106C3×#102—24no—
107#835—22no—
108#832—24no—
109#832—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#816—42no—
114#810—22no—
115#807—22no—
116C4×C3⋊C8—22no—
117C22×Q16—31no—
118C22×SD16C2×D312yes—
119#559—64no—
120#559—64no—
121C2×#127C2×D311yescharacteristic
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#460—24no—
131C4×Q16—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102C2×D312yes—
140C2×C4×C8—31no—
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8#4112yes—
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—81no—
167C2×C3×D4—81no—
168C2×C3×D4C2×D414yes—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—28no—
177C22×Dic3—22no—
178C22×Dic3—82no—
179C22×Dic3—82no—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×D412yes—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6C2×D412yes—
191C3×SD16—48no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—42no—
198C3×C42—42no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—28no—
202#217—28no—
203#217—28no—
204#216—28no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4⋊Dic3—81no—
211C4⋊Dic3C2×D412yes—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—82no—
215C4×Dic3—24no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8#4112yes—
224C22×Q8—31no—
225C22×Q8—31no—
226C22×Q8C22×D311yescharacteristic
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C22×D312yes—
230C23×C4—62no—
231C2×Q16—62no—
232C2×Q16—62no—
233C2×Q16—62no—
234C2×SD16—24no—
235C2×SD16—24no—
236C2×SD16—42no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8C22×D312yes—
240#127—22no—
241#127#6714yes—
242#124—64no—
243#124—68no—
244#124—64no—
245#123—616no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—34no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×D4—64no—
253C4×D4—68no—
254C4×D4—64no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—21no—
269C2×C42—32no—
270C2×C42—61no—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273C4⋊C8—22no—
274#103—62no—
275#103—62no—
276#103—122no—
277#102—24no—
278#102—64no—
279#102—122no—
280C4×C8—62no—
281#95—62no—
282#95—62no—
283#95—62no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×D4—24no—
297C3×D4—24no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—48no—
301C3×D4—84no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4#12012yes—
311C2×C3×C4#12012yes—
312C2×C3×C4#13512yes—
313C2×C3×C4#13712yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—88no—
336Dic6—22no—
337Dic6—22no—
338Dic6—41no—
339Dic6—41no—
340Dic6—44no—
341C3×C8—42no—
342C3⋊C8—22no—
343C3⋊C8—41no—
344C24—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—32no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8#23712yes—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4C2×#6714yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—68no—
379C22×C4—62no—
380C22×C4—242no—
381C22×C4—242no—
382C22×C4D3×D411yescharacteristic
383C22×C4D3×D411yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—68no—
386SD16—48no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—32no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—61no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—32no—
399C4⋊C4—42no—
400C4⋊C4—62no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4—61no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C4⋊C4—241no—
408C4⋊C4—242no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—68no—
412C42—42no—
413C42—64no—
414C42—62no—
415C42—62no—
416C42D3×D412yes—
417C42C2×#6712yes—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54811yescharacteristic
427C22×C3#55011yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3#57611yescharacteristic
431C22×C3#57711yescharacteristic
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—81no—
435C3×C4#45214yes—
436C3×C4#52511yescharacteristic
437C3×C4#53711yescharacteristic
438C3×C4#54611yescharacteristic
439C3×C4#54711yescharacteristic
440Dic3—42no—
441Dic3—42no—
442Dic3—42no—
443Dic3—42no—
444Dic3—82no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C2×D3×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457Q8—62no—
458Q8—62no—
459Q8—124no—
460Q8—121no—
461Q8—121no—
462D4—24no—
463D4—24no—
464D4—48no—
465D4—42no—
466D4—42no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—248no—
492C2×C4#91512yes—
493C2×C4#91612yes—
494C2×C4#93712yes—
495C2×C4#93712yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×#23711yescharacteristic
498C2×C4#100411yescharacteristic
499C2×C4D3×#4111yescharacteristic
500C2×C4#101011yescharacteristic
501C2×C4C22×#6711yescharacteristic
502C8—42no—
503C8—62no—
504C8—121no—
505C2×C3—84no—
506C2×C3#240711yescharacteristic
507C2×C3C2×#45211yescharacteristic
508C2×C3#301211yescharacteristic
509C2×C3#336511yescharacteristic
510C2×C3#337511yescharacteristic
511C2×C3#339611yescharacteristic
512C2×C3#340011yescharacteristic
513C22—44no—
514C22—44no—
515C22—44no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22#559811yescharacteristic
521C22#559911yescharacteristic
522C22C2×#91511yescharacteristic
523C22C2×#91611yescharacteristic
524C22#577411yescharacteristic
525C22#577511yescharacteristic
526C22C2×#93711yescharacteristic
527C4—22no—
528C4—42no—
529C4—81no—
530C4—122no—
531C4—122no—
532C4—122no—
533C4—122no—
534C4—242no—
535C4#517314yes—
536C4#560611yescharacteristic
537C4#560811yescharacteristic
538C4#580611yescharacteristic
539C4#582411yescharacteristic
540C3#2419011yescharacteristic, complemented, Sylow
541C2—84no—
542C2#8037011yescharacteristic
543C2#8037411yescharacteristic
544C2#8037511yescharacteristic
545C2#8038011yescharacteristic
546C2#8270111yescharacteristic
547C2C2×#517311yescharacteristic
548C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#460D3
hypocenterC3#24190
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
3. #82482C2semidirect
4. C2×C4×#216C2semidirect
5. #72058C2semidirect
6. #72047C2semidirect
8. #70485C2semidirect
9. #70472C2semidirect
10. #70216C2semidirect
11. #70167C2semidirect
540. C3#24190semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×C3×#4602
#824822
C2×C4×#2162
#720582
#720472
#717642
#704852
#704722
#702162
#701672
#241903