Finite Group Databasebeta

← #10844788

Subgroups — #10844788

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2#83179C211yesmaximal, characteristic, complemented
3#82376C211yesmaximal, characteristic, complemented
4C2×#4831C211yesmaximal, characteristic, complemented
5#72040C212yesmaximal, complemented
6#72039C212yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic, complemented
9#70454C211yesmaximal, characteristic, complemented
10#70188C212yesmaximal, complemented
11#70179C212yesmaximal, complemented
12#25300—31nomaximal, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936—21no—
17C2×#924—21no—
18C2×C4×Dic6C2211yescharacteristic
19C3×#459—24no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2212yes—
22#5222—22no—
23#5221—21no—
24#5221C2212yes—
25#5221C2212yes—
26#5221C2212yes—
27C2×#832C2212yes—
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2212yes—
31#5212—21no—
32C2×#817C2212yes—
33C2×#815C2212yes—
34C2×Dic3⋊C8C2212yes—
35#4972C2212yes—
36#4948—22no—
37#4935C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4831—24no—
41#4551C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#459D311yescharacteristic
45C2×#446—31no—
46#2548—31no—
47#2538—31no—
48#2065—32no—
49#2054—32no—
50#1972—32no—
51#1957—31no—
52#1948—31no—
53#1936—32no—
54C4×#102—31no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C2×C3×D8—24no—
62C2×C3×D8—24no—
63C2×C3×D8—42no—
64C22×C3×C8C2312yes—
65C3×#127—22no—
66C3×#127—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—22no—
78#936—44no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85#924—44no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3—21no—
89C2×C4×Dic3C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C4×Dic6—22no—
93C4×Dic6—22no—
94C3×C4⋊C8D414yes—
95C3×#102—24no—
96#832—22no—
97#832—22no—
98#832—24no—
99#832—42no—
100C3⋊#95—21no—
101C3⋊#95—22no—
102#817—42no—
103#815—24no—
104Dic3⋊C8—24no—
105C4×C3⋊C8—22no—
106C22×D8C2×D312yes—
107C2×#127C2×D311yescharacteristic
108C2×#124—61no—
109C2×C22⋊Q8—61no—
110C2×C4×Q8—31no—
111C2×C4×D4C2×D311yescharacteristic
112#459—24no—
113#446—64no—
114C2×#106—32no—
115C2×C4⋊C8—32no—
116C2×C4⋊C8C2×D311yescharacteristic
117C2×#102—31no—
118C2×#102—32no—
119C2×#102—31no—
120C2×#102—32no—
121C2×#102C2×D312yes—
122C2×#97—32no—
123C2×C4×C8—31no—
124#396—62no—
125#394—62no—
126#390—32no—
127#390—32no—
128#390—32no—
129#390—61no—
130#389—32no—
131#384—31no—
132#380—61no—
133C4×C4⋊C4—31no—
134#340—32no—
135#340—31no—
136#340—31no—
137C24×C3—21no—
138C24×C3—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—81no—
155C2×C3×D4—81no—
156C2×C3×D4#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C3×D8—48no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8C2×D414yes—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C22⋊C4—42no—
179C3×C42C2×D412yes—
180C3×C42#4112yes—
181#217—44no—
182#217—44no—
183#217—44no—
184#217—82no—
185#217—82no—
186#217—82no—
187C4⋊Dic3—22no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—21no—
192C4⋊Dic3—22no—
193C4⋊Dic3—21no—
194C4⋊Dic3—42no—
195C4⋊Dic3—81no—
196Dic3⋊C4—24no—
197Dic3⋊C4—44no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—42no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8#4112yes—
207C22×Q8—31no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C22×D312yes—
211C22×D4C22×D311yescharacteristic
212C23×C4—21no—
213C2×D8—24no—
214C2×D8—24no—
215C2×D8—42no—
216C22×C8—31no—
217C22×C8—31no—
218C22×C8C22×D312yes—
219#127—22no—
220#127—22no—
221#124—124no—
222C22⋊Q8—124no—
223C4×Q8—62no—
224C4×Q8—62no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C42—32no—
244C2×C42—61no—
245C2×C42C22×D311yescharacteristic
246#106—122no—
247C4⋊C8—64no—
248C4⋊C8#6714yes—
249#102—24no—
250#102—62no—
251#102—64no—
252#102—62no—
253#102—122no—
254#97—64no—
255C4×C8—62no—
256#95—62no—
257#95—61no—
258C23×C3—22no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3C22×D411yescharacteristic
273C3×D4—22no—
274C3×D4—22no—
275C3×D4—42no—
276C3×D4—41no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—41no—
280C3×D4—42no—
281C3×D4—84no—
282C3×D4—84no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—84no—
289C2×C3×C4#12112yes—
290C2×C3×C4#12112yes—
291C2×C3×C4AGL(1,ℤ/8)12yes—
292C2×C3×C4AGL(1,ℤ/8)12yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314Dic6—44no—
315C3×C8—24no—
316C3⋊C8—22no—
317C3⋊C8—41no—
318C24—21no—
319C24—22no—
320C2×Q8—62no—
321C2×Q8—62no—
322C2×Q8—62no—
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—81no—
339C2×D4—81no—
340C2×D4#23712yes—
341C22×C4—21no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—62no—
350C22×C4D3×D411yescharacteristic
351C22×C4D3×D411yescharacteristic
352C22×C4C23×D311yescharacteristic
353D8—48no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—32no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8C2×#6714yes—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—64no—
366C4⋊C4—62no—
367C4⋊C4—61no—
368C4⋊C4—62no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—61no—
373C4⋊C4—124no—
374C4⋊C4—122no—
375C4⋊C4—241no—
376C22⋊C4—42no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—242no—
381C22⋊C4—242no—
382C22⋊C4—242no—
383C42—62no—
384C42—62no—
385C42—122no—
386C42#23712yes—
387C42C2×#6712yes—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3C2×#12111yescharacteristic
403C22×C3#54811yescharacteristic
404C22×C3#55011yescharacteristic
405C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
407C22×C3#57611yescharacteristic
408C22×C3#57711yescharacteristic
409C3×C4—22no—
410C3×C4—81no—
411C3×C4#46914yes—
412C3×C4#52911yescharacteristic
413C3×C4#54211yescharacteristic
414C3×C4#54611yescharacteristic
415C3×C4#54811yescharacteristic
416Dic3—42no—
417Dic3—42no—
418Dic3—81no—
419Dic3—81no—
420Dic3—82no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C2×D3×D411yescharacteristic
436Q8—124no—
437D4—22no—
438D4—22no—
439D4—42no—
440D4—41no—
441D4—42no—
442D4—42no—
443D4—41no—
444D4—42no—
445D4—84no—
446D4—84no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—84no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4#91112yes—
469C2×C4#91112yes—
470C2×C4#93812yes—
471C2×C4#93812yes—
472C2×C4C2×#23711yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4#100411yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#101011yescharacteristic
477C2×C4C22×#6711yescharacteristic
478C8—24no—
479C8—62no—
480C8—121no—
481C2×C3—82no—
482C2×C3—162no—
483C2×C3#242611yescharacteristic
484C2×C3C2×#46911yescharacteristic
485C2×C3#306211yescharacteristic
486C2×C3#336611yescharacteristic
487C2×C3#337311yescharacteristic
488C2×C3#339711yescharacteristic
489C2×C3#339911yescharacteristic
490C22—42no—
491C22—42no—
492C22—42no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22#559811yescharacteristic
505C22#559911yescharacteristic
506C22C2×#91111yescharacteristic
507C22C2×#91111yescharacteristic
508C22#577411yescharacteristic
509C22#577511yescharacteristic
510C22C2×#93811yescharacteristic
511C4—22no—
512C4—81no—
513C4—122no—
514C4—122no—
515C4—241no—
516C4—241no—
517C4—242no—
518C4#515914yes—
519C4#560011yescharacteristic
520C4#560311yescharacteristic
521C4#580611yescharacteristic
522C4#583111yescharacteristic
523C3#2530011yescharacteristic, complemented, Sylow
524C2—82no—
525C2—162no—
526C2#8032011yescharacteristic
527C2#8032311yescharacteristic
528C2#8032611yescharacteristic
529C2#8032711yescharacteristic
530C2#8264611yescharacteristic
531C2C2×#515911yescharacteristic
532C2#8395111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#25300
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. #83179C2semidirect
3. #82376C2semidirect
4. C2×#4831C2semidirect
5. #72040C2semidirect
6. #72039C2semidirect
7. #71764C2semidirect
8. C4×#832C2semidirect
9. #70454C2semidirect
10. #70188C2semidirect
11. #70179C2semidirect
523. C3#25300semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
#831792
#823762
C2×#48312
#720402
#720392
#717642
C4×#8322
#704542
#701882
#701792
#253003