Finite Group Databasebeta

← #10710657

Subgroups — #10710657

Order 768.

1160 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2C3×#1785C211yesmaximal, characteristic, complemented
3#72139C211yesmaximal, characteristic, complemented
4C2×#4476C212yesmaximal, complemented
5C2×#4475C212yesmaximal, complemented
6#12689—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C23×C4×D3C411yescharacteristic, complemented
9#5828—24no—
10#5828—24no—
11C2×#946—21no—
12C2×#938—21no—
13C2×#937C411yescharacteristic, complemented
14C2×#937C2211yescharacteristic
15C2×#935C411yescharacteristic, complemented
16C2×#935C2211yescharacteristic
17C2×C4×#67—21no—
18C2×C4×#67—21no—
19C2×#926C411yescharacteristic, complemented
20C2×C3×#101—21no—
21C2×C3×#100C2212yes—
22C3×#407—21no—
23C2×#834—21no—
24C2×#834C2212yes—
25#5211—21no—
26#4476—24no—
27#4475—24no—
28C2×#525—31no—
29C2×#352—32no—
30C2×AΣL(1,𝔽16)—32no—
31#1785D311yescharacteristic, complemented
32#1782—31no—
33C24×D3C2×C411yescharacteristic
34C2×C3×#41—22no—
35C2×C3×#41—22no—
36C2×C3×#41—42no—
37C2×C3×#41—42no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4C2×C411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×C3×D4C2311yescharacteristic
42C23×C3×C4C2×C411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C22×#67—21no—
46C22×#67—21no—
47C23×Dic3C2×C411yescharacteristic
48C22×C4×D3—22no—
49C22×C4×D3—21no—
50C22×C4×D3—21no—
51C22×C4×D3—22no—
52C22×C4×D3—42no—
53C22×C4×D3—42no—
54C2×C3×M16—22no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—22no—
58C2×#217C2×C411yescharacteristic
59C2×#217C2×C411yescharacteristic
60#946—28no—
61#938—28no—
62#937—22no—
63#937—22no—
64#937—22no—
65#937D414yes—
66#935—22no—
67#935—22no—
68#935—22no—
69#935D414yes—
70C4×#67—28no—
71C4×#67—28no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74#926—22no—
75#926—22no—
76C2×C4⋊Dic3—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—21no—
79C2×C4×Dic3—21no—
80C3×#101—42no—
81C3×#100—22no—
82C3×#100—22no—
83C3×C22⋊C8—42no—
84#834—22no—
85#834—22no—
86#834—42no—
87C3⋊#95—42no—
88C22×#41C2×D311yescharacteristic
89C24×C4—31no—
90#525—64no—
91#525—64no—
92C2×#123—31no—
93C2×#123—31no—
94C2×C22⋊Q8—61no—
95C2×#121—61no—
96C2×#120—31no—
97C2×#120—31no—
98C2×C4×D4—61no—
99C2×C4×D4—61no—
100C2×#117—31no—
101C2×#101—21no—
102C2×#100C2×D312yes—
103C2×#99—32no—
104C2×#99—61no—
105#407—21no—
106#381—61no—
107#352—64no—
108AΣL(1,𝔽16)—64no—
109C24×C3C22⋊C411yescharacteristic
110C24×C3C22×C411yescharacteristic
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—44no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3C22⋊C412yes—
118C3×#41—48no—
119C3×#41—48no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—41no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4C2×D412yes—
133C2×C3×D4C2×D412yes—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C22⋊C411yescharacteristic
145C22×C3×C4C22⋊C411yescharacteristic
146C22×C3×C4C22⋊C411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C2×#67—24no—
149C2×#67—24no—
150C2×#67—44no—
151C2×#67—44no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—41no—
157C22×Dic3—41no—
158C22×Dic3—42no—
159C22×Dic3C22⋊C412yes—
160C2×C4×D3—24no—
161C2×C4×D3—24no—
162C2×C4×D3—48no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C2×C4×D3—48no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C3×M16—42no—
169C3×M16—42no—
170C2×C3×C8—42no—
171#217—24no—
172#217—21no—
173#217—21no—
174#217—24no—
175#217—42no—
176#217—42no—
177#217C22⋊C412yes—
178#217C22⋊C412yes—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—24no—
181C4⋊Dic3—24no—
182Dic3⋊C4—24no—
183Dic3⋊C4—24no—
184C4×Dic3—24no—
185C25—31no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—42no—
189C2×#41—42no—
190C22×Q8#6711yescharacteristic
191C22×D4—61no—
192C22×D4—61no—
193C22×D4C4×D311yescharacteristic
194C22×D4D1211yescharacteristic
195C22×D4C22×D311yescharacteristic
196C23×C4—31no—
197C23×C4—62no—
198C23×C4—62no—
199C23×C4—61no—
200C23×C4—61no—
201C23×C4—122no—
202C23×C4—122no—
203C23×C4C4×D311yescharacteristic
204C23×C4D1211yescharacteristic
205C23×C4#6711yescharacteristic
206C2×M16—22no—
207#123—34no—
208#123—62no—
209#123—62no—
210#123—62no—
211C22⋊Q8—68no—
212#121—68no—
213#120—34no—
214#120—62no—
215#120—62no—
216#120—62no—
217C4×D4—68no—
218C4×D4—68no—
219#117—62no—
220#117—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C22⋊C4—31no—
225C2×C22⋊C4—31no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—62no—
231C2×C42—61no—
232#101—42no—
233#100—22no—
234#100—22no—
235#99—62no—
236#99—62no—
237#99—122no—
238C22⋊C8—42no—
239#95—122no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3C2×C22⋊C411yescharacteristic
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3C2×C22⋊C411yescharacteristic
253C23×C3#12012yes—
254C22×D3—24no—
255C22×D3—22no—
256C22×D3—44no—
257C22×D3—44no—
258C22×D3—42no—
259C22×D3—42no—
260C22×D3—48no—
261C22×D3—44no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—42no—
265C22×D3—44no—
266C3×Q8—44no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—44no—
270C3×D4—42no—
271C3×D4—42no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4#9912yes—
286C2×C3×C4#9912yes—
287C2×C3×C4#12012yes—
288#67—48no—
289#67—48no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—82no—
300C2×Dic3—84no—
301C4×D3—48no—
302C4×D3—48no—
303C3×C8—82no—
304C24—32no—
305C24—61no—
306C24—61no—
307C24—121no—
308C24—121no—
309C24—124no—
310C24(C2×D3)⋊C411yescharacteristic
311C24C2×C4×D311yescharacteristic
312C24C2×D1211yescharacteristic
313#41—48no—
314#41—48no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—64no—
328C2×D4—64no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4D3×D412yes—
332C2×D4D3×D412yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—32no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—64no—
345C22×C4—64no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—128no—
351C22×C4—128no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4—122no—
355C22×C4—122no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4—122no—
359C22×C4(C2×D3)⋊C411yescharacteristic
360C22×C4(C2×D3)⋊C411yescharacteristic
361C22×C4(C2×D3)⋊C411yescharacteristic
362C22×C4C2×#6711yescharacteristic
363M16—42no—
364M16—42no—
365C2×C8—42no—
366C4⋊C4—64no—
367C4⋊C4—64no—
368C4⋊C4—64no—
369C22⋊C4—32no—
370C22⋊C4—32no—
371C22⋊C4—64no—
372C22⋊C4—64no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—64no—
376C22⋊C4—64no—
377C22⋊C4—122no—
378C22⋊C4—122no—
379C42—64no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—42no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—41no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3#37911yescharacteristic
396C22×C3C2×#9911yescharacteristic
397C22×C3C2×#9911yescharacteristic
398C2×D3—24no—
399C2×D3—42no—
400C2×D3—44no—
401C2×D3—48no—
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—42no—
405C2×D3—44no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—41no—
410Dic3—42no—
411Dic3—42no—
412Dic3—42no—
413Dic3—162no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—64no—
425C23—62no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23—124no—
430C23—124no—
431C23—124no—
432C23—128no—
433C23—124no—
434C23—121no—
435C23—121no—
436C23D3×C22⋊C411yescharacteristic
437C23D3×C22⋊C411yescharacteristic
438C23#88212yes—
439C23C2×(C2×D3)⋊C411yescharacteristic
440Q8—44no—
441D4—42no—
442D4—42no—
443D4—44no—
444D4—42no—
445D4—42no—
446D4—128no—
447D4—128no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—128no—
465C2×C4—128no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—122no—
472C2×C4—242no—
473C2×C4—244no—
474C2×C4#80612yes—
475C2×C4#80612yes—
476C2×C4#93712yes—
477C8—82no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—42no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—81no—
484C2×C3#178111yescharacteristic
485C2×C3#212012yes—
486D3—44no—
487C22—22no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—42no—
492C22—41no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—41no—
500C22—64no—
501C22—81no—
502C22—81no—
503C22—122no—
504C22—124no—
505C22—124no—
506C22—124no—
507C22—128no—
508C22—124no—
509C22—122no—
510C22C2×#80611yescharacteristic
511C22C2×#80611yescharacteristic
512C22#495511yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—41no—
517C4—122no—
518C4—122no—
519C4—122no—
520C4—482no—
521C3#1268911yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—42no—
525C2—41no—
526C2—41no—
527C2—81no—
528C2—124no—
529C2#6949811yescharacteristic
530C2#7014512yes—
subgroupG/H
centerC22C2×#806
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1785C2
socleC22×C3C2×#99
prime radicalC2×#5828C2
cosocleC22×C3×D4C23
hypercenter#1785D3
hypocenterC3#12689
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. C3×#1785C2semidirect
3. #72139C2semidirect
4. C2×#4476C2semidirect
5. C2×#4475C2semidirect
8. C23×C4×D3C4semidirect
13. C2×#937C4semidirect
15. C2×#935C4semidirect
19. C2×#926C4semidirect
31. #1785D3semidirect
521. C3#12689semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
C3×#17852
#721392
C2×#44762
C2×#44752
#126893