Finite Group Databasebeta

← #10876187

Subgroups — #10876187

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2#82060C211yesmaximal, characteristic
3#72132C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic
5#71556C211yesmaximal, characteristic
6#71555C211yesmaximal, characteristic
7#71531C211yesmaximal, characteristic
8#69686C212yesmaximal, complemented
9#69671C212yesmaximal, complemented
10#69667C212yesmaximal, complemented
11#69666C212yesmaximal, complemented
12#27589—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C3×#391—21no—
17C3×#391—21no—
18C3×#391C2212yes—
19C3×#391C2212yes—
20C2×C3×#95—21no—
21C2×C3×#95—21no—
22C2×#835C2212yes—
23C2×#835C2212yes—
24C2×#835C2212yes—
25C2×#835C2212yes—
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28#5211—21no—
29C4×#217—21no—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#808C2212yes—
33C2×#807C2212yes—
34#4937—21no—
35#4936C2211yescharacteristic
36#4936C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4932—21no—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43#4524C2212yes—
44C2×#391D311yescharacteristic
45#2283—31no—
46#2097—32no—
47#2074—32no—
48#1928—32no—
49#1927—32no—
50#1853—31no—
51#1852—31no—
52#1817—31no—
53#1787—31no—
54#1787—31no—
55C22×C3×Q8—24no—
56C22×C3×Q8—21no—
57C22×C3×Q8—22no—
58C22×C3×Q8—22no—
59C22×C3×Q8—42no—
60C22×C3×Q8C2312yes—
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4×Dic3—21no—
82C2×C4×Dic3—21no—
83C22×C3⋊C8C2312yes—
84C22×C3⋊C8C2312yes—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89#835—24no—
90#835—24no—
91#835—42no—
92#835—42no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101#830—22no—
102#830—22no—
103#808—42no—
104#807—42no—
105C23×Q8C2×D311yescharacteristic
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×#103—32no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116#391—21no—
117#391—21no—
118#391C2×D312yes—
119#391C2×D312yes—
120#383—61no—
121#382—31no—
122#382—31no—
123#381—61no—
124#380—31no—
125#380—31no—
126C4×C4⋊C4—31no—
127C4×C4⋊C4—31no—
128C4×C22⋊C4—61no—
129#376—61no—
130C2×#95—21no—
131C2×#95—21no—
132#340—32no—
133#340—32no—
134#336—32no—
135#336—32no—
136C24×C3C2×D411yescharacteristic
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—48no—
144C2×C3×Q8—48no—
145C2×C3×Q8—42no—
146C2×C3×Q8—42no—
147C2×C3×Q8#4114yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42—21no—
185#217—42no—
186#217—42no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4×Dic3—42no—
192C4×Dic3—42no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C22×Q8—24no—
200C22×Q8—21no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—42no—
204C22×Q8C22×D312yes—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4#6711yescharacteristic
208C23×C4#6711yescharacteristic
209C23×C4C22×D311yescharacteristic
210C22×C8—32no—
211C22×C8—32no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22×D312yes—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—61no—
224C2×C42—61no—
225C2×C42#6711yescharacteristic
226C2×C42#6711yescharacteristic
227C2×C42C22×D311yescharacteristic
228C2×C42C22×D311yescharacteristic
229#107—122no—
230#106—122no—
231#103—64no—
232#103—64no—
233#103—122no—
234#103—122no—
235C22⋊C8—62no—
236C22⋊C8—62no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—61no—
242#95—61no—
243#95—61no—
244#95—61no—
245#95—61no—
246#95—61no—
247#95—61no—
248#95—61no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3#12312yes—
254C23×C3#13512yes—
255C23×C3#13712yes—
256C23×C3C22×D411yescharacteristic
257C3×Q8—24no—
258C3×Q8—24no—
259C3×Q8—48no—
260C3×Q8—42no—
261C3×Q8—42no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—44no—
279C2×C3×C4—48no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—44no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—84no—
287C2×C3×C4#12312yes—
288C2×C3×C4#13712yes—
289C2×C3×C4C2×#4112yes—
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4H(5,𝔽2)12yes—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C3⋊C8—42no—
306C3⋊C8—42no—
307C24C2×#6711yescharacteristic
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8#23514yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4#23712yes—
340C22×C4#23712yes—
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346C22×C4C23×D311yescharacteristic
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C2×C8—62no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C42—22no—
363C42—22no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—122no—
369C42—122no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3#48514yes—
377C22×C3C2×#12311yescharacteristic
378C22×C3#52511yescharacteristic
379C22×C3#53511yescharacteristic
380C22×C3C2×#13511yescharacteristic
381C22×C3C2×#13711yescharacteristic
382C22×C3C2×#13711yescharacteristic
383C22×C3#57511yescharacteristic
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—41no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—44no—
390C3×C4—82no—
391C3×C4#55112yes—
392C3×C4#55112yes—
393Dic3—81no—
394Dic3—81no—
395Dic3—81no—
396Dic3—81no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23#93512yes—
402C23#94212yes—
403C23#95012yes—
404C23C22×#6711yescharacteristic
405Q8—24no—
406Q8—24no—
407Q8—48no—
408Q8—42no—
409Q8—42no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4#93512yes—
448C2×C4#94212yes—
449C2×C4C2×#23512yes—
450C2×C4C2×#23711yescharacteristic
451C2×C4C2×#23711yescharacteristic
452C2×C4#100412yes—
453C8—122no—
454C8—122no—
455C2×C3—24no—
456C2×C3#248211yescharacteristic
457C2×C3C2×#48511yescharacteristic
458C2×C3#309511yescharacteristic
459C2×C3#310811yescharacteristic
460C2×C3#311711yescharacteristic
461C2×C3#321111yescharacteristic
462C2×C3#321811yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22#504514yes—
470C22#579111yescharacteristic
471C22C2×#93511yescharacteristic
472C22#580511yescharacteristic
473C22C2×#94211yescharacteristic
474C22C2×#94211yescharacteristic
475C22C2×#95011yescharacteristic
476C22#582211yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—44no—
483C4—82no—
484C4—241no—
485C4—241no—
486C4—241no—
487C4—241no—
488C4#549512yes—
489C4#549512yes—
490C3#2758911yescharacteristic, complemented, Sylow
491C2—24no—
492C2#8211311yescharacteristic
493C2#8244311yescharacteristic
494C2#8244511yescharacteristic
495C2C2×#504511yescharacteristic
496C2#8272711yescharacteristic
497C2#8294611yescharacteristic
498C2#8321411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#391C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#391D3
hypocenterC3#27589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
8. #69686C2semidirect
9. #69671C2semidirect
10. #69667C2semidirect
11. #69666C2semidirect
490. C3#27589semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
#820602
#721322
#715562
#715552
#715312
#696862
#696712
#696672
#696662
#275893