Finite Group Databasebeta

← #10824335

Subgroups — #10824335

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4758C212yesmaximal
2#76289C211yesmaximal, characteristic
3#74320C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic
5#70217C212yesmaximal
6#69661C211yesmaximal, characteristic
7#69570C211yesmaximal, characteristic, complemented
8#68912C212yesmaximal, complemented
9#68781C212yesmaximal, complemented
10#68631C211yesmaximal, characteristic, complemented
11#68602C211yesmaximal, characteristic, complemented
12#23677—31nomaximal, Sylow
13C2×D4×Dic3C2211yescharacteristic
14C22×#214C2212yes—
15C2×#887C2212yes—
16C2×#877—21no—
17C2×#869C2211yescharacteristic
18C2×C3×#102C2212yes—
19C3×#390C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5221C2211yescharacteristic
22C2×#816C2212yes—
23C2×#816C2212yes—
24C2×Dic3⋊C8C2212yes—
25C2×C8×Dic3C2212yes—
26Dic3×C4⋊C4—21no—
27Dic3×C4⋊C4—21no—
28Dic3×C4⋊C4C2211yescharacteristic
29#4973C2212yes—
30#4972—21no—
31#4972—21no—
32#4972C2211yescharacteristic
33C2×#810C2212yes—
34C2×#808C2212yes—
35#4945—21no—
36#4944—21no—
37Dic3×C22⋊C4—21no—
38C42×Dic3C2211yescharacteristic
39#4928C2211yescharacteristic
40#4758—28no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×C4×SD16—32no—
46C4×C4⋊Q8—31no—
47#2293—31no—
48#2084—31no—
49#2071D311yescharacteristic
50#2066—32no—
51#2055—32no—
52#2043—31no—
53#2013—31no—
54C4×C4⋊C8—31no—
55C4×#103—32no—
56C22×C3×D4C2311yescharacteristic
57C23×Dic3—21no—
58C22×Dic6C2312yes—
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#217—21no—
68C2×#217—21no—
69C2×#217—21no—
70D4×Dic3—22no—
71D4×Dic3D414yes—
72C2×#214—24no—
73C2×#214—24no—
74C2×#214—24no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91#887—24no—
92#887—24no—
93#877—44no—
94#869—24no—
95C3×#102—24no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#816—24no—
102#816—24no—
103Dic3⋊C8—24no—
104C8×Dic3—24no—
105#810—24no—
106#808—24no—
107C22×SD16—32no—
108C2×C4⋊Q8—31no—
109C2×C4×Q8—32no—
110C2×C4×D4—31no—
111C2×#117—61no—
112C4×SD16—68no—
113C2×#106—32no—
114C2×C4⋊C8—32no—
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#102C2×D312yes—
119C2×C4×C8—32no—
120#391—32no—
121#390—31no—
122#390C2×D311yescharacteristic
123#389—31no—
124#389—31no—
125#389—61no—
126#389—61no—
127#388—61no—
128#388—61no—
129#384C2×D311yescharacteristic
130C4×C4⋊C4—31no—
131C4×C4⋊C4—61no—
132C4×C4⋊C4—61no—
133C4×C22⋊C4—61no—
134C43—31no—
135#340—32no—
136#340C2×D312yes—
137#336—31no—
138#336C2×D311yescharacteristic
139C24×C3—21no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4C2×D412yes—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—24no—
152C22×Dic3—21no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3C2×D411yescharacteristic
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6#4114yes—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—42no—
175C3×C42—22no—
176#217—24no—
177#217—42no—
178#217—42no—
179#214—216no—
180C4⋊Dic3—24no—
181C4⋊Dic3—24no—
182C4⋊Dic3—21no—
183C4⋊Dic3—22no—
184C4⋊Dic3C2×D412yes—
185Dic3⋊C4—42no—
186Dic3⋊C4—44no—
187Dic3⋊C4—42no—
188C4×Dic3—22no—
189C4×Dic3—22no—
190C4×Dic3—22no—
191C4×Dic3—41no—
192C4×Dic3—44no—
193C4×Dic3—42no—
194C4×Dic3SD1614yescomplemented
195C4×Dic3C2×D412yes—
196C4×Dic3#4112yes—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8#4114yes—
200C22×Q8—32no—
201C22×D4C22×D311yescharacteristic
202C23×C4—61no—
203C2×SD16—64no—
204C2×SD16—64no—
205C2×SD16—64no—
206C22×C8—32no—
207C22×C8C22×D312yes—
208C4⋊Q8—64no—
209C4×Q8—64no—
210C4×Q8—64no—
211C4×D4—34no—
212C4×D4—62no—
213#117—124no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D312yes—
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—31no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—61no—
237C2×C42—62no—
238C2×C42C22×D311yescharacteristic
239#106—64no—
240C4⋊C8—64no—
241#103—64no—
242#103—64no—
243#103—64no—
244#102—24no—
245C4×C8—64no—
246#95—61no—
247#95—61no—
248#95—62no—
249#95—62no—
250#95—61no—
251C23×C3—22no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3C22×D411yescharacteristic
259C3×D4—24no—
260C3×D4—42no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4C2×SD1612yes—
272C2×C3×C4#13512yes—
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4112yes—
275C2×C3×C4C2×#4112yes—
276C2×C3×C4C2×#4111yescharacteristic
277C2×Dic3—21no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—22no—
282C2×Dic3—24no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—48no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3#12112yes—
295C2×Dic3#12112yes—
296C2×Dic3C2×SD1612yes—
297C2×Dic3C2×SD1612yes—
298C2×Dic3#13712yes—
299Dic6—24no—
300Dic6—24no—
301Dic6—42no—
302Dic6—42no—
303C3×C8—42no—
304C3⋊C8—24no—
305C24—21no—
306C2×Q8—34no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4D3×D412yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—64no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—616no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—34no—
336C2×C8—62no—
337C2×C8—62no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—32no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—64no—
345C4⋊C4—64no—
346C4⋊C4—61no—
347C4⋊C4—62no—
348C4⋊C4—122no—
349C4⋊C4—124no—
350C4⋊C4—122no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—64no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C42—22no—
357C42—34no—
358C42—32no—
359C42—32no—
360C42—62no—
361C42—62no—
362C42—62no—
363C42—121no—
364C42—122no—
365C42—124no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—81no—
370C22×C3—81no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3C2×#12111yescharacteristic
374C22×C3#52511yescharacteristic
375C22×C3#53211yescharacteristic
376C22×C3C22×SD1611yescharacteristic
377C22×C3C2×#13511yescharacteristic
378C22×C3C2×#13711yescharacteristic
379C22×C3#57511yescharacteristic
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—81no—
383C3×C4—82no—
384C3×C4#52512yes—
385C3×C4#52912yes—
386Dic3—22no—
387Dic3—22no—
388Dic3—44no—
389Dic3—41no—
390Dic3—42no—
391Dic3—81no—
392Dic3#46114yes—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23C2×D3×D411yescharacteristic
401Q8—64no—
402Q8—64no—
403Q8—122no—
404Q8—122no—
405D4—24no—
406D4—42no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—64no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—128no—
432C2×C4—122no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—122no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4#91212yes—
440C2×C4D3×SD1612yes—
441C2×C4C2×D3×D411yescharacteristic
442C2×C4D3×#4112yes—
443C2×C4D3×#4112yes—
444C2×C4D3×#4111yescharacteristic
445C8—42no—
446C8—64no—
447C2×C3—82no—
448C2×C3#239211yescharacteristic
449C2×C3C2×#46111yescharacteristic
450C2×C3#303511yescharacteristic
451C2×C3#310311yescharacteristic
452C2×C3#310811yescharacteristic
453C2×C3#316711yescharacteristic
454C2×C3#317211yescharacteristic
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22D3×#12111yescharacteristic
463C22#565611yescharacteristic
464C22#566811yescharacteristic
465C22C2×#91211yescharacteristic
466C22C2×D3×SD1611yescharacteristic
467C22#577311yescharacteristic
468C22D3×#13711yescharacteristic
469C4—22no—
470C4—22no—
471C4—34no—
472C4—62no—
473C4—62no—
474C4—81no—
475C4—82no—
476C4—121no—
477C4—122no—
478C4—124no—
479C4—241no—
480C4#565612yes—
481C4#570612yes—
482C3#2367711yescharacteristic, complemented, Sylow
483C2—82no—
484C2#7745711yescharacteristic
485C2#8033111yescharacteristic
486C2D3×#46111yescharacteristic
487C2#8126911yescharacteristic
488C2#8130811yescharacteristic
489C2#8155611yescharacteristic
490C2#8166511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2071D3
hypocenterC3#23677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #74320C2semidirect
7. #69570C2semidirect
8. #68912C2semidirect
9. #68781C2semidirect
10. #68631C2semidirect
11. #68602C2semidirect
194. C4×Dic3SD16semidirect
482. C3#23677semidirect

Maximal subgroups

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

subgroupindex
C2×#47582
#762892
#743202
C3×#20712
#702172
#696612
#695702
#689122
#687812
#686312
#686022
#236773