Finite Group Databasebeta

← #10844799

Subgroups — #10844799

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2#83244C211yesmaximal, characteristic, complemented
3#82376C211yesmaximal, characteristic
4C2×#4831C211yesmaximal, characteristic, complemented
5#72051C212yesmaximal
6#72050C212yesmaximal
7#71785C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic
9#70453C211yesmaximal, characteristic, complemented
10#70256C212yesmaximal, complemented
11#70248C212yesmaximal, complemented
12#25301—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#924—21no—
17C2×C4×Dic6C2211yescharacteristic
18C3×#461—24no—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#103C2212yes—
21#5229C2212yes—
22#5229C2212yes—
23C2×#835C2212yes—
24C2×#835C2212yes—
25#5221C2212yes—
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5212—21no—
29C2×#818C2212yes—
30C2×#815C2212yes—
31C2×Dic3⋊C8C2212yes—
32#4985—22no—
33#4977—21no—
34#4973—21no—
35#4972C2212yes—
36#4948—22no—
37#4935C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4831—24no—
41#4551C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#461D311yescharacteristic
45C2×#446—31no—
46#2568—31no—
47#2538—31no—
48#2068—32no—
49#2055—32no—
50#1973—32no—
51#1956—31no—
52#1947—31no—
53#1935—32no—
54C4×#102—31no—
55C22×C3×Q8C2312yes—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×Dic6D411yescharacteristic
59C2×C3×SD16—24no—
60C2×C3×SD16—24no—
61C2×C3×SD16—42no—
62C22×C3×C8C2312yes—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2312yes—
80#924—44no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4D412yes—
83C2×C4×Dic3—21no—
84C2×C4×Dic3C2312yes—
85C22×C3⋊C8C2311yescharacteristic
86C22×C3⋊C8C2311yescharacteristic
87C4×Dic6—22no—
88C4×Dic6—22no—
89C3×C4⋊C8D414yes—
90C3×#103—24no—
91#835—24no—
92#835—42no—
93#832—22no—
94#832—22no—
95C3⋊#95—21no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#818—42no—
102#815—24no—
103Dic3⋊C8—24no—
104C4×C3⋊C8—22no—
105C22×SD16C2×D312yes—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×C22⋊Q8—61no—
108C2×C4×Q8—31no—
109C2×C4×D4C2×D311yescharacteristic
110#461—24no—
111#446—64no—
112C2×#107—32no—
113C2×C4⋊C8—32no—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103C2×D312yes—
118C2×#102—31no—
119C2×#102—31no—
120C2×#97—32no—
121C2×C4×C8—31no—
122#400—62no—
123#396—62no—
124#391—32no—
125#391—32no—
126#391—61no—
127#390—32no—
128#389—32no—
129#384—31no—
130#382—61no—
131#380—61no—
132C4×C4⋊C4—31no—
133#340—32no—
134#340—31no—
135#340—31no—
136C24×C3—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4#4112yes—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3C2×D412yes—
161C22×Dic3C2×D412yes—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C3×SD16—48no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8C2×D414yes—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C22⋊C4—42no—
174C3×C42C2×D412yes—
175C3×C42#4112yes—
176#217—44no—
177#217—82no—
178C4⋊Dic3—22no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3—22no—
182C4⋊Dic3—21no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—42no—
186C4⋊Dic3—81no—
187Dic3⋊C4—24no—
188Dic3⋊C4—44no—
189C4×Dic3—22no—
190C4×Dic3—22no—
191C4×Dic3—42no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8#4112yes—
198C22×Q8—31no—
199C22×Q8C22×D312yes—
200C22×D4C22×D311yescharacteristic
201C23×C4—21no—
202C2×SD16—24no—
203C2×SD16—24no—
204C2×SD16—42no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8C22×D312yes—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C22⋊Q8—124no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4C22×D311yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—32no—
230C2×C42—61no—
231C2×C42C22×D311yescharacteristic
232#107—122no—
233C4⋊C8—64no—
234C4⋊C8#6714yes—
235#103—24no—
236#103—64no—
237#103—122no—
238#102—62no—
239#102—62no—
240#97—64no—
241C4×C8—62no—
242#95—62no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—61no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3C22×D411yescharacteristic
256C3×Q8—42no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—41no—
263C3×D4—41no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—84no—
273C2×C3×C4#12112yes—
274C2×C3×C4#12112yes—
275C2×C3×C4AGL(1,ℤ/8)12yes—
276C2×C3×C4#13712yes—
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4H(5,𝔽2)11yescharacteristic
282C2×C3×C42−1+411yescharacteristic
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298Dic6—44no—
299C3×C8—24no—
300C3⋊C8—22no—
301C3⋊C8—41no—
302C24—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4#23712yes—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—62no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—48no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—32no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8C2×#6714yes—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—64no—
346C4⋊C4—62no—
347C4⋊C4—61no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—62no—
352C4⋊C4—61no—
353C4⋊C4—124no—
354C4⋊C4—122no—
355C4⋊C4—241no—
356C22⋊C4—42no—
357C22⋊C4—124no—
358C22⋊C4—242no—
359C42—62no—
360C42—62no—
361C42—122no—
362C42#23712yes—
363C42C2×#6712yes—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3—81no—
370C22×C3—81no—
371C22×C3C2×#12111yescharacteristic
372C22×C3#54811yescharacteristic
373C22×C3#55011yescharacteristic
374C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
375C22×C3C2×#13711yescharacteristic
376C22×C3#57711yescharacteristic
377C22×C3#57811yescharacteristic
378C3×C4—22no—
379C3×C4—82no—
380C3×C4—81no—
381C3×C4#46814yes—
382C3×C4#52911yescharacteristic
383C3×C4#54211yescharacteristic
384C3×C4#54611yescharacteristic
385C3×C4#54811yescharacteristic
386Dic3—42no—
387Dic3—42no—
388Dic3—81no—
389Dic3—81no—
390Dic3—82no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23C2×D3×D411yescharacteristic
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—124no—
404D4—22no—
405D4—22no—
406D4—41no—
407D4—41no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—84no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4#91412yes—
433C2×C4#91512yes—
434C2×C4#93812yes—
435C2×C4#93812yes—
436C2×C4C2×#23711yescharacteristic
437C2×C4C2×#23711yescharacteristic
438C2×C4#100411yescharacteristic
439C2×C4D3×#4111yescharacteristic
440C2×C4#101011yescharacteristic
441C2×C4C22×#6711yescharacteristic
442C8—24no—
443C8—62no—
444C8—121no—
445C2×C3—82no—
446C2×C3#242611yescharacteristic
447C2×C3C2×#46811yescharacteristic
448C2×C3#306311yescharacteristic
449C2×C3#336711yescharacteristic
450C2×C3#337211yescharacteristic
451C2×C3#339711yescharacteristic
452C2×C3#339811yescharacteristic
453C22—42no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#559811yescharacteristic
461C22#559911yescharacteristic
462C22C2×#91411yescharacteristic
463C22C2×#91511yescharacteristic
464C22#577811yescharacteristic
465C22#577911yescharacteristic
466C22C2×#93811yescharacteristic
467C4—22no—
468C4—81no—
469C4—82no—
470C4—122no—
471C4—122no—
472C4—241no—
473C4—241no—
474C4—242no—
475C4#517414yes—
476C4#560011yescharacteristic
477C4#560311yescharacteristic
478C4#580611yescharacteristic
479C4#583111yescharacteristic
480C3#2530111yescharacteristic, complemented, Sylow
481C2—82no—
482C2#8041611yescharacteristic
483C2#8041711yescharacteristic
484C2#8042211yescharacteristic
485C2#8042511yescharacteristic
486C2#8264611yescharacteristic
487C2C2×#517411yescharacteristic
488C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#461D3
hypocenterC3#25301
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #83244C2semidirect
4. C2×#4831C2semidirect
7. #71785C2semidirect
9. #70453C2semidirect
10. #70256C2semidirect
11. #70248C2semidirect
480. C3#25301semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
#832442
#823762
C2×#48312
#720512
#720502
#717852
C4×#8322
#704532
#702562
#702482
#253013