Finite Group Databasebeta

← #10829980

Subgroups — #10829980

Order 768.

1082 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
2C2×C4×#214C211yesmaximal, characteristic
3C2×#4849C211yesmaximal, characteristic, complemented
4#74343C211yesmaximal, characteristic, complemented
5C3×#1802C211yesmaximal, characteristic
6#70472C211yesmaximal, characteristic, complemented
7#70215C212yesmaximal
8#70178C212yesmaximal
9#69579C211yesmaximal, characteristic, complemented
10#68908C212yesmaximal, complemented
11#68907C212yesmaximal, complemented
12#24189—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936C2212yes—
15C22×#214C2211yescharacteristic
16C2×#926—21no—
17C2×#924—21no—
18C22×Dic12C2212yes—
19#5542—24no—
20#5542—24no—
21C2×#887C2212yes—
22C2×#879—22no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#384C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#816C2212yes—
31C4×#214—24no—
32#4984—21no—
33#4984C2212yes—
34#4981—22no—
35#4973—21no—
36#4973C2212yes—
37C2×#810C2212yes—
38C2×#810C2211yescharacteristic
39C2×#810C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4849—24no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×C4×SD16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1956—31no—
54#1878—32no—
55#1865—32no—
56#1802D311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×Dic6D411yescharacteristic
60C22×Dic6C2312yes—
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2312yes—
63C3×C4×D4—22no—
64C3×C4×D4D414yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#217—22no—
73#936—24no—
74#936—24no—
75C2×#214—22no—
76C2×#214—22no—
77C2×#214—22no—
78#926—28no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84#924—28no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C2×Dic12—24no—
94C2×Dic12—24no—
95C2×Dic12—42no—
96#887—24no—
97#887—24no—
98#879—216no—
99#869—22no—
100#869D414yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×C4⋊C8—22no—
106C3×#102—42no—
107#832—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#816—24no—
112#810—22no—
113#810—24no—
114#810—42no—
115#808—22no—
116C4×C3⋊C8—22no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#462—64no—
131C4×SD16—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102C2×D312yes—
140C2×C4×C8—31no—
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384C2×D311yescharacteristic
147#340C2×D312yes—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4C2×D412yes—
157C22×C3×C4—21no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3C2×D412yes—
169C22×Dic3C2×D412yes—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—21no—
175C2×Dic6—22no—
176C2×Dic6—21no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—21no—
183C2×Dic6C2×D414yes—
184C2×Dic6#4112yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—81no—
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—24no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197#217—28no—
198#217—28no—
199#214—28no—
200C4⋊Dic3—22no—
201C4⋊Dic3—24no—
202C4⋊Dic3—24no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—82no—
206Dic3⋊C4—28no—
207Dic3⋊C4—24no—
208Dic3⋊C4—42no—
209C4×Dic3—28no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C4×Dic3—42no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219Dic12—48no—
220C22×Q8—31no—
221C22×Q8—32no—
222C22×Q8—31no—
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—62no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D312yes—
234C4⋊Q8—34no—
235C4⋊Q8—62no—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8—62no—
246C4×D4—22no—
247C4×D4D1214yes—
248#117—68no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4C22×D312yes—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—32no—
265C2×C42—62no—
266C2×C42—61no—
267C2×C42C22×D311yescharacteristic
268#106—62no—
269C4⋊C8—22no—
270#103—64no—
271#103—64no—
272#103—62no—
273#103—122no—
274#102—42no—
275#102—62no—
276C4×C8—62no—
277#95—62no—
278#95—62no—
279#95—62no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3C22×D411yescharacteristic
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—41no—
291C3×D4—41no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—48no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12012yes—
303C2×C3×C4#13512yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326Dic6—22no—
327Dic6—22no—
328Dic6—24no—
329Dic6—24no—
330Dic6—42no—
331Dic6—41no—
332Dic6—42no—
333Dic6—41no—
334Dic6—44no—
335C3×C8—42no—
336C3⋊C8—22no—
337C3⋊C8—41no—
338C24—21no—
339C2×Q8—34no—
340C2×Q8—32no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—62no—
349C2×Q8—61no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—61no—
353C2×Q8—62no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4C2×D1212yes—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4D3×D411yescharacteristic
371C22×C4D3×D411yescharacteristic
372C22×C4C23×D311yescharacteristic
373Q16—128no—
374SD16—68no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—32no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—61no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—68no—
389C4⋊C4—62no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—81no—
396C4⋊C4—122no—
397C4⋊C4—242no—
398C4⋊C4C2×D1212yes—
399C22⋊C4—24no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C42—68no—
403C42—62no—
404C42—62no—
405C42—122no—
406C42C2×D1212yes—
407C42D3×D412yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#54811yescharacteristic
417C22×C3#55011yescharacteristic
418C22×C3C2×#13511yescharacteristic
419C22×C3C2×#13711yescharacteristic
420C22×C3#57711yescharacteristic
421C22×C3#57811yescharacteristic
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—82no—
425C3×C4#45214yes—
426C3×C4#52511yescharacteristic
427C3×C4#53711yescharacteristic
428C3×C4#54611yescharacteristic
429C3×C4#54711yescharacteristic
430Dic3—42no—
431Dic3—44no—
432Dic3—42no—
433Dic3—42no—
434Dic3—81no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C2×D3×D411yescharacteristic
443Q8—62no—
444Q8—62no—
445Q8—64no—
446Q8—64no—
447Q8—122no—
448Q8—121no—
449Q8—122no—
450Q8—124no—
451Q8—121no—
452D4—22no—
453D4—22no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—48no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—64no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4#88212yes—
481C2×C4#88212yes—
482C2×C4#91212yes—
483C2×C4#91512yes—
484C2×C4C22×D1211yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4#100411yescharacteristic
487C2×C4#100711yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4D3×#4111yescharacteristic
490C8—42no—
491C8—62no—
492C8—121no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3C2×#45211yescharacteristic
496C2×C3#301311yescharacteristic
497C2×C3#336411yescharacteristic
498C2×C3#336911yescharacteristic
499C2×C3#339411yescharacteristic
500C2×C3#339811yescharacteristic
501C22—42no—
502C22—42no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C2×#88211yescharacteristic
509C22#559911yescharacteristic
510C22#563311yescharacteristic
511C22C2×#91211yescharacteristic
512C22C2×#91511yescharacteristic
513C22#577511yescharacteristic
514C22#577911yescharacteristic
515C4—22no—
516C4—42no—
517C4—82no—
518C4—122no—
519C4—124no—
520C4—122no—
521C4—122no—
522C4—241no—
523C4#478314yes—
524C4#551011yescharacteristic
525C4#552411yescharacteristic
526C4#561211yescharacteristic
527C4#563111yescharacteristic
528C3#2418911yescharacteristic, complemented, Sylow
529C2—82no—
530C2#7401511yescharacteristic
531C2C2×#478311yescharacteristic
532C2#7888711yescharacteristic
533C2#8032511yescharacteristic
534C2#8042211yescharacteristic
535C2#8099411yescharacteristic
536C2#8109111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1802D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. C2×#4849C2semidirect
4. #74343C2semidirect
6. #70472C2semidirect
9. #69579C2semidirect
10. #68908C2semidirect
11. #68907C2semidirect
528. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C4×#2142
C2×#48492
#743432
C3×#18022
#704722
#702152
#701782
#695792
#689082
#689072
#241893