Finite Group Databasebeta

← #10845404

Subgroups — #10845404

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2537C211yesmaximal, characteristic
2#82384C211yesmaximal, characteristic, complemented
3C2×#5012C211yesmaximal, characteristic, complemented
4C2×#5008C211yesmaximal, characteristic, complemented
5#71682C212yesmaximal, complemented
6#71679C212yesmaximal, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69567C211yesmaximal, characteristic, complemented
10#69513C212yesmaximal, complemented
11#69507C212yesmaximal, complemented
12#25346—31nomaximal, Sylow
13C2×C3×#121—21no—
14C2×C3×#121C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#208C2212yes—
18C2×#869C2211yescharacteristic
19C3×#394—22no—
20C3×#384C2211yescharacteristic
21C3×#380—21no—
22C3×#380C2212yes—
23#5221—21no—
24C2×#832C2211yescharacteristic
25C2×#832C2212yes—
26C2×#832C2211yescharacteristic
27#5212C2212yes—
28#5212C2212yes—
29C2×#830C2212yes—
30#5012—24no—
31#5008—24no—
32#4977—21no—
33C2×#808C2212yes—
34C2×#807C2212yes—
35#4947—22no—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2211yescharacteristic
42#4529C2212yes—
43#4529C2211yescharacteristic
44C2×#476—31no—
45C2×#436—31no—
46#2563—31no—
47#2537D311yescharacteristic
48#2077—32no—
49#2033—32no—
50#1990—31no—
51#1947—31no—
52#1895—32no—
53#1809—32no—
54C4×#102—31no—
55C22×C3×D4—22no—
56C22×C3×D4D411yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2312yes—
60C22×Dic6—21no—
61C3×#121—24no—
62C3×#121—42no—
63C3×#121—44no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#217—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4×Dic3—21no—
85C2×#208—24no—
86C2×#208—24no—
87C2×#208—42no—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91#869—22no—
92#869—22no—
93C3×#95—22no—
94#832—22no—
95#832—24no—
96#832—22no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#830—24no—
101#808—24no—
102#807—42no—
103C4⋊(C3⋊C8)D414yescomplemented
104C4×C3⋊C8—22no—
105C22×M16—32no—
106C2×C4⋊Q8—31no—
107C2×#121—21no—
108C2×#121C2×D312yes—
109C2×C4×D4C2×D311yescharacteristic
110C2×C4×D4C2×D311yescharacteristic
111#476—64no—
112#436—64no—
113C2×#107—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#102—31no—
117C2×#102—32no—
118C2×#102—31no—
119C2×C22⋊C8—32no—
120C2×C4×C8—31no—
121#394—22no—
122#393—62no—
123#390—61no—
124#384C2×D311yescharacteristic
125#382—61no—
126#380—21no—
127#380—32no—
128#380—32no—
129#380C2×D312yes—
130C4×C4⋊C4—31no—
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—32no—
136C24×C3—21no—
137C24×C3C2×D412yes—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—82no—
149C2×C3×D4#4112yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C2×Dic6—42no—
169C2×Dic6—81no—
170C2×Dic6—81no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—81no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—42no—
182C3×C42C2×D412yes—
183C3×C42#4112yes—
184#217—82no—
185#217—82no—
186C4⋊Dic3—22no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—22no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—42no—
193C4×Dic3—42no—
194#208—48no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8C2×D414yes—
203C2×C3⋊C8#4112yes—
204C22×Q8—61no—
205C22×D4—22no—
206C22×D4#6711yescharacteristic
207C22×D4C22×D311yescharacteristic
208C23×C4—21no—
209C23×C4C22×D312yes—
210C2×M16—64no—
211C2×M16—64no—
212C2×M16—122no—
213C22×C8—31no—
214C22×C8—32no—
215C22×C8—31no—
216C4⋊Q8—62no—
217C4⋊Q8—62no—
218#121—24no—
219#121—42no—
220#121—44no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4#6711yescharacteristic
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D312yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4#6712yes—
240C2×C42—61no—
241C2×C42C22×D311yescharacteristic
242#107—122no—
243#106—64no—
244C4⋊C8—34no—
245#102—62no—
246#102—64no—
247#102—62no—
248C22⋊C8—64no—
249C4×C8—62no—
250#95—22no—
251#95—61no—
252#95—62no—
253#95—62no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3C22×D411yescharacteristic
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—41no—
273C3×D4—44no—
274C3×D4—88no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—84no—
291C2×C3×C4#12112yes—
292C2×C3×C4#12112yes—
293C2×C3×C4AGL(1,ℤ/8)12yes—
294C2×C3×C4#13712yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4H(5,𝔽2)11yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310Dic6—84no—
311C3⋊C8—24no—
312C3⋊C8—22no—
313C3⋊C8—41no—
314C24—21no—
315C24C2×#6712yes—
316C2×Q8—122no—
317C2×Q8—241no—
318C2×Q8—241no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—82no—
330C2×D4#23712yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—62no—
345C22×C4C2×#6712yes—
346C22×C4C2×#6711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C23×D311yescharacteristic
349M16—128no—
350C2×C8—34no—
351C2×C8—32no—
352C2×C8—61no—
353C2×C8—62no—
354C2×C8—61no—
355C2×C8—62no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—62no—
366C4⋊C4—61no—
367C4⋊C4—62no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—81no—
372C4⋊C4—122no—
373C22⋊C4—24no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—242no—
378C22⋊C4—242no—
379C42—122no—
380C42D3×D412yes—
381C42#23712yes—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3C2×#12111yescharacteristic
397C22×C3#54611yescharacteristic
398C22×C3#54711yescharacteristic
399C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C22×C3C2×#13711yescharacteristic
401C22×C3#57611yescharacteristic
402C22×C3#57711yescharacteristic
403C3×C4—22no—
404C3×C4—42no—
405C3×C4—81no—
406C3×C4—82no—
407C3×C4#46814yes—
408C3×C4#52911yescharacteristic
409C3×C4#54211yescharacteristic
410C3×C4#54611yescharacteristic
411C3×C4#54811yescharacteristic
412Dic3—81no—
413Dic3—82no—
414Dic3—81no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C22×#6711yescharacteristic
430Q8—244no—
431D4—22no—
432D4—22no—
433D4—44no—
434D4—41no—
435D4—41no—
436D4—88no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—84no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4#93212yes—
463C2×C4#93912yes—
464C2×C4#93912yes—
465C2×C4#95112yes—
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×#23711yescharacteristic
468C2×C4C2×#23711yescharacteristic
469C2×C4#100411yescharacteristic
470C2×C4D3×#4111yescharacteristic
471C2×C4#101011yescharacteristic
472C8—62no—
473C8—64no—
474C8—121no—
475C2×C3—82no—
476C2×C3—82no—
477C2×C3#242711yescharacteristic
478C2×C3C2×#46811yescharacteristic
479C2×C3#306211yescharacteristic
480C2×C3#332511yescharacteristic
481C2×C3#333111yescharacteristic
482C2×C3#335411yescharacteristic
483C2×C3#335711yescharacteristic
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22C2×#93211yescharacteristic
499C22C2×#93911yescharacteristic
500C22#580611yescharacteristic
501C22C2×#95111yescharacteristic
502C22#582411yescharacteristic
503C22#583511yescharacteristic
504C22#583711yescharacteristic
505C4—22no—
506C4—42no—
507C4—82no—
508C4—81no—
509C4—241no—
510C4—241no—
511C4—242no—
512C4#504214yes—
513C4#559811yescharacteristic
514C4#560011yescharacteristic
515C4#560311yescharacteristic
516C4#561211yescharacteristic
517C3#2534611yescharacteristic, complemented, Sylow
518C2—82no—
519C2—82no—
520C2C2×#504211yescharacteristic
521C2#8264711yescharacteristic
522C2#8342211yescharacteristic
523C2#8348711yescharacteristic
524C2#8351511yescharacteristic
525C2#8352111yescharacteristic
526C2#8365111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3C22×D4
hypercenter#2537D3
hypocenterC3#25346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #82384C2semidirect
3. C2×#5012C2semidirect
4. C2×#5008C2semidirect
5. #71682C2semidirect
6. #71679C2semidirect
7. C4×#832C2semidirect
8. #69582C2semidirect
9. #69567C2semidirect
10. #69513C2semidirect
11. #69507C2semidirect
103. C4⋊(C3⋊C8)D4semidirect
517. C3#25346semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
#823842
C2×#50122
C2×#50082
#716822
#716792
C4×#8322
#695822
#695672
#695132
#695072
#253463