Finite Group Databasebeta

← #10852220

Subgroups — #10852220

Order 768.

1490 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
1#85766C211yesmaximal, characteristic, complemented
2C3×#2408C211yesmaximal, characteristic
3#84050C212yesmaximal, complemented
4#71669C214yesmaximal, complemented
5C4×#834C211yesmaximal, characteristic, complemented
6#69529C212yesmaximal, complemented
7#69497C214yesmaximal, complemented
8#25947—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#121—21no—
11C2×C3×#120C2212yes—
12C2×C3×C4×D4C2211yescharacteristic
13#5798C2212yes—
14C2×#936—22no—
15C2×#924—21no—
16C2×#924C2212yes—
17#5543—22no—
18#5539—24no—
19D4×Dic6—22no—
20#5500—21no—
21#5483—21no—
22#5483C2212yes—
23#5470—21no—
24C3×#394—21no—
25C3×#392—21no—
26C3×#390—21no—
27C3×#380C2211yescharacteristic
28C3×#379C2212yes—
29#5247C2218yes—
30#5222—24no—
31#5221—22no—
32C2×#834C2212yes—
33C4×#217C2212yes—
34#4527C2218yes—
35#4526C2212yes—
36#3501—31no—
37#2435—32no—
38#2408D311yescharacteristic
39#2029—34no—
40#1996—32no—
41#1889—34no—
42C4×#99—31no—
43C25×C3C2312yes—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—21no—
47C22×C3×D4—22no—
48C22×C3×D4—21no—
49C22×C3×D4—44no—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×Dic6—21no—
54C22×Dic6D412yescomplemented
55C3×#121—44no—
56C3×#120—28no—
57C3×C4×D4—22no—
58C3×C4×D4—22no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4D411yescharacteristic
65C2×C3×C22⋊C4C2312yes—
66C2×C3×C22⋊C4C2311yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×#217—22no—
69C2×#217—28no—
70C2×#217—28no—
71C2×#217C2314yes—
72C2×#217C2314yes—
73#936—48no—
74D4×Dic3—24no—
75D4×Dic3—24no—
76#926—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3D412yescomplemented
79#924—22no—
80#924—22no—
81#924—22no—
82#924—22no—
83#924—22no—
84#924—22no—
85#924—22no—
86#924—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4D414yescomplemented
89C2×C4×Dic3—22no—
90#890—24no—
91#888—24no—
92#878—24no—
93#878—24no—
94#878—24no—
95#878—24no—
96#870—22no—
97#869—21no—
98#869—21no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C3×#95—21no—
102#834—24no—
103C3⋊#95—22no—
104C23×D4C2×D311yescharacteristic
105#554—64no—
106#554—62no—
107D4×Q8—62no—
108#544—61no—
109#543—32no—
110#543—61no—
111#533—61no—
112C2×#124—62no—
113C2×C22⋊Q8—32no—
114C2×C22⋊Q8—61no—
115C2×#121—21no—
116C2×#120C2×D312yes—
117#518—32no—
118C2×C4×D4C2×D311yescharacteristic
119C2×#99—32no—
120#394—21no—
121#394—64no—
122#392—21no—
123#390—21no—
124#390—62no—
125#380C2×D311yescharacteristic
126#379—38no—
127#379C2×D312yes—
128C4×C22⋊C4—32no—
129#342—38no—
130#342—32no—
131C24×C3—24no—
132C24×C3—22no—
133C24×C3—24no—
134C24×C3—24no—
135C24×C3—44no—
136C24×C3—82no—
137C24×C3C2×D411yescharacteristic
138C24×C3C2411yescharacteristic
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—28no—
142C2×C3×D4—24no—
143C2×C3×D4—44no—
144C2×C3×D4—42no—
145C2×C3×D4—48no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—41no—
150C2×C3×D4—42no—
151C2×C3×D4—41no—
152C2×C3×D4—82no—
153C2×C3×D4#4114yes—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—24no—
169C22×Dic3C2×D414yes—
170C22×Dic3C2×D414yes—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—44no—
176C2×Dic6—82no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—44no—
184C3×C22⋊C4—82no—
185C3×C42—21no—
186C3×C42—21no—
187#217—24no—
188#217—24no—
189#217—24no—
190#217—24no—
191#217—24no—
192#217—416no—
193#217—88no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200Dic3⋊C4—24no—
201Dic3⋊C4—24no—
202Dic3⋊C4—24no—
203Dic3⋊C4—24no—
204C4×Dic3—24no—
205C4×Dic3—44no—
206C25C22×D312yes—
207C22×Q8—32no—
208C22×Q8—61no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—44no—
215C22×D4C22×D311yescharacteristic
216C23×C4#6711yescharacteristic
217C23×C4C22×D311yescharacteristic
218C4⋊Q8—61no—
219C4⋊Q8—61no—
220C4⋊Q8—64no—
221#125—62no—
222#125—64no—
223#124—128no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—64no—
227C22⋊Q8—64no—
228C22⋊Q8—62no—
229C22⋊Q8—62no—
230C22⋊Q8—62no—
231C22⋊Q8—62no—
232C22⋊Q8—62no—
233C22⋊Q8—62no—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236#121—44no—
237#120—28no—
238C4×Q8—62no—
239C4×Q8—62no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—64no—
243C4×D4—64no—
244#117—62no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4#6711yescharacteristic
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—34no—
255C2×C22⋊C4—34no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—68no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4#6711yescharacteristic
260C2×C22⋊C4C22×D312yes—
261C2×C22⋊C4C22×D311yescharacteristic
262C2×C42—62no—
263C2×C42#6711yescharacteristic
264#99—64no—
265#95—21no—
266#95—62no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—28no—
270C23×C3—24no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—48no—
276C23×C3—42no—
277C23×C3—44no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—48no—
284C23×C3—88no—
285C23×C3—88no—
286C23×C3#12412yescomplemented
287C23×C3C22×D412yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C2×#4112yes—
290C23×C32−1+412yes—
291C3×D4—28no—
292C3×D4—48no—
293C3×D4—44no—
294C3×D4—44no—
295C3×D4—88no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—81no—
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#12012yes—
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4#12412yescomplemented
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—44no—
318C2×Dic3—44no—
319C2×Dic3—44no—
320C2×Dic3—44no—
321C2×Dic3—44no—
322C2×Dic3—82no—
323C2×Dic3—82no—
324Dic6—44no—
325Dic6—82no—
326Dic6—82no—
327Dic6—82no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—44no—
333C24—82no—
334C24C2×#6711yescharacteristic
335C24C23×D311yescharacteristic
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—124no—
341C2×Q8—242no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—44no—
347C2×D4—42no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—82no—
356C2×D4#23714yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—34no—
361C22×C4—34no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—64no—
367C22×C4D3×D412yes—
368C22×C4D3×D412yes—
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—62no—
377C4⋊C4—64no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—64no—
381C4⋊C4—62no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—64no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—64no—
392C22⋊C4—64no—
393C22⋊C4—64no—
394C22⋊C4—64no—
395C22⋊C4—64no—
396C22⋊C4—82no—
397C22⋊C4—1216no—
398C22⋊C4—248no—
399C42—21no—
400C42—21no—
401C42—64no—
402C42—124no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—44no—
412C22×C3—416no—
413C22×C3—44no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—44no—
417C22×C3—81no—
418C22×C3—88no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3C2×#12011yescharacteristic
424C22×C3C2×#12411yescharacteristic
425C22×C3#53611yescharacteristic
426C22×C3#54814yes—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—82no—
433C3×C4B(4,𝔽2)12yes—
434Dic3—44no—
435Dic3—44no—
436Dic3—84no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—28no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—48no—
447C23—42no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—48no—
454C23—88no—
455C23—88no—
456C23#93612yescomplemented
457C23C2×D3×D412yes—
458C23C2×#23712yes—
459C23#101012yes—
460C23C22×#6711yescharacteristic
461Q8—124no—
462Q8—242no—
463Q8—242no—
464Q8—242no—
465D4—28no—
466D4—44no—
467D4—48no—
468D4—44no—
469D4—88no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—81no—
487C2×C4—124no—
488C2×C4—124no—
489C2×C4—124no—
490C2×C4—124no—
491C2×C4—124no—
492C2×C4—242no—
493C2×C4—242no—
494C2×C4#93612yescomplemented
495C2×C4#93711yescharacteristic
496C2×C4#93712yes—
497C2×C4#93711yescharacteristic
498C2×C3—24no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—42no—
502C2×C3—84no—
503C2×C3#243611yescharacteristic
504C2×C3C2×B(4,𝔽2)11yescharacteristic
505C2×C3#302711yescharacteristic
506C22—21no—
507C22—22no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—44no—
515C22—416no—
516C22—44no—
517C22—41no—
518C22—41no—
519C22—44no—
520C22—81no—
521C22—82no—
522C22—88no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22#559814yes—
527C22C2×#93611yescharacteristic
528C22C2×#93711yescharacteristic
529C22#582511yescharacteristic
530C4—21no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—82no—
536C4—124no—
537C4—124no—
538C4—244no—
539C4#503212yes—
540C3#2594711yescharacteristic, complemented, Sylow
541C2—21no—
542C2—24no—
543C2—21no—
544C2—42no—
545C2—84no—
546C2C2×#503211yescharacteristic
547C2#8265411yescharacteristic
548C2#8339611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC3×#2408C2
socleC22×C3C2×#120
hypercenter#2408D3
hypocenterC3#25947
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #85766C2semidirect
3. #84050C2semidirect
4. #71669C2semidirect
5. C4×#834C2semidirect
6. #69529C2semidirect
7. #69497C2semidirect
54. C22×Dic6D4semidirect
78. C2×C4⋊Dic3D4semidirect
88. C2×Dic3⋊C4D4semidirect
286. C23×C3#124semidirect
314. C2×C3×C4#124semidirect
456. C23#936semidirect
494. C2×C4#936semidirect
540. C3#25947semidirect

Maximal subgroups

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

subgroupindex
#857662
C3×#24082
#840502
#716692
C4×#8342
#695292
#694972
#259473