Finite Group Databasebeta

← #10877231

Subgroups — #10877231

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#384C211yesmaximal, characteristic
2C4×#897C211yesmaximal, characteristic
3C3×#1947C211yesmaximal, characteristic, complemented
4#70644C212yesmaximal, complemented
5#70632C212yesmaximal
6#70485C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic
8#69588C211yesmaximal, characteristic, complemented
9#69466C211yesmaximal, characteristic, complemented
10#68878C212yesmaximal
11#68843C212yesmaximal, complemented
12#27681—31nomaximal, Sylow
13C2×#897C2211yescharacteristic
14C2×D3×C4⋊C4—21no—
15C2×D3×C4⋊C4—21no—
16C2×D3×C4⋊C4C2212yes—
17C2×C42×D3C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C3×#107C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C8C2212yes—
24C2×#816C2212yes—
25#4987C2211yescharacteristic
26#4984C2211yescharacteristic
27#4975—21no—
28#4975—21no—
29#4975C2212yes—
30#4973C2211yescharacteristic
31C2×#810C2212yes—
32C2×#807C2212yes—
33C4×(C2×D3)⋊C4—21no—
34#4935C2211yescharacteristic
35C4×C4⋊Dic3—21no—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4668—21no—
38#4653—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×C22⋊Q8—31no—
45C2×#384—31no—
46#2100—32no—
47#2095—32no—
48#1947—31no—
49#1947D311yescharacteristic, complemented
50#1872—32no—
51#1852—32no—
52#1803—31no—
53#1803—31no—
54#1797—31no—
55C22×C4×D3—22no—
56C22×C4×D3D411yescharacteristic
57C22×C4×D3D411yescharacteristic
58C22×C4×D3C2311yescharacteristic
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85#897—22no—
86#897Q814yes—
87D3×C4⋊C4—42no—
88D3×C4⋊C4—42no—
89D3×C4⋊C4—44no—
90D3×C4⋊C4—44no—
91C42×D3—22no—
92C42×D3—22no—
93C4×Dic6—22no—
94C4×Dic6—22no—
95C3×#107—42no—
96C3×C4⋊C8—22no—
97C3×#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102(C2×D3)⋊C8—24no—
103#816—42no—
104#810—42no—
105#807—42no—
106C4⋊(C3⋊C8)—22no—
107C2×C22⋊Q8—31no—
108C2×C4×Q8—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—61no—
111C22×C4⋊C4—61no—
112C22×C42—31no—
113C2×#107—32no—
114C2×#107C2×D312yes—
115C2×C4⋊C8—31no—
116C2×C4⋊C8C2×D311yescharacteristic
117C2×#103—32no—
118C2×#103—32no—
119C2×C22⋊C8—32no—
120#391—31no—
121#389—31no—
122#387—61no—
123#385—61no—
124#384—32no—
125#384—31no—
126#384—61no—
127#384—61no—
128#384C2×D311yescharacteristic
129#380—31no—
130C4×C4⋊C4—61no—
131C4×C4⋊C4C2×D311yescharacteristic
132C4×C22⋊C4—61no—
133#340—32no—
134#340—32no—
135#340—32no—
136#340C2×D312yes—
137#336—32no—
138C23×D3C2×D411yescharacteristic
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—21no—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3—22no—
160C2×C4×D3—44no—
161C2×C4×D3—44no—
162C2×C4×D3—44no—
163C2×C4×D3Q1614yes—
164C2×C4×D3C2×Q812yes—
165C2×C4×D3#4112yes—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6C2×Q812yes—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×Q812yes—
183C3×C42—42no—
184C3×C42#4112yes—
185C3×C42#4112yes—
186(C2×D3)⋊C4—24no—
187(C2×D3)⋊C4—42no—
188C4⋊Dic3—24no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4⋊Dic3—42no—
196Dic3⋊C4—44no—
197Dic3⋊C4—44no—
198Dic3⋊C4—44no—
199Dic3⋊C4—42no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—21no—
203C4×Dic3—21no—
204C4×Dic3—42no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8—31no—
209C23×C4—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—62no—
213C22×C8—32no—
214C22×C8C22×D312yes—
215C22⋊Q8—34no—
216C22⋊Q8—62no—
217C4×Q8—62no—
218C4×Q8—62no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—122no—
232C2×C4⋊C4—122no—
233C2×C4⋊C4—124no—
234C2×C4⋊C4—124no—
235C2×C4⋊C4C22×D312yes—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C42—21no—
241C2×C42—31no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—62no—
245C2×C42—62no—
246C2×C42—61no—
247C2×C42C22×D311yescharacteristic
248#107—42no—
249#107—122no—
250C4⋊C8—22no—
251C4⋊C8—62no—
252#103—122no—
253#103—122no—
254C22⋊C8—64no—
255#95—21no—
256#95—61no—
257#95—61no—
258#95—61no—
259#95—61no—
260C23×C3C22×D411yescharacteristic
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3C22⋊Q812yes—
266C22×D3C2×Q1612yes—
267C22×D3AGL(1,ℤ/8)12yes—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4#12312yes—
280C2×C3×C4#12312yes—
281C2×C3×C4C2×Q1612yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4C22×Q811yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4H(5,𝔽2)11yescharacteristic
288C2×C3×C4H(5,𝔽2)11yescharacteristic
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—22no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3C22⋊Q812yes—
305C2×Dic3C2×Q1612yes—
306C2×Dic3#13712yes—
307C4×D3—28no—
308C4×D3—22no—
309C4×D3—22no—
310C4×D3—22no—
311C4×D3—22no—
312C4×D3—28no—
313C4×D3—88no—
314Dic6—22no—
315Dic6—22no—
316Dic6—41no—
317Dic6—41no—
318C3×C8—42no—
319C3⋊C8—42no—
320C24—31no—
321C2×Q8—32no—
322C2×Q8—61no—
323C2×Q8—61no—
324C2×Q8—61no—
325C2×Q8—61no—
326C2×Q8—61no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—34no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—124no—
349C22×C4—124no—
350C22×C4—124no—
351C22×C4D3×D411yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4C23×D311yescharacteristic
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—62no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—64no—
368C4⋊C4—61no—
369C4⋊C4—62no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C4⋊C4—124no—
375C4⋊C4—124no—
376C4⋊C4—124no—
377C4⋊C4—122no—
378C4⋊C4—122no—
379C4⋊C4D3×Q812yes—
380C22⋊C4—64no—
381C22⋊C4—122no—
382C42—42no—
383C42—62no—
384C42—62no—
385C42—61no—
386C42—61no—
387C42—122no—
388C42#23512yes—
389C42#23912yes—
390C22×C3C2×C22⋊Q811yescharacteristic
391C22×C3C2×#12311yescharacteristic
392C22×C3#53511yescharacteristic
393C22×C3C22×Q1611yescharacteristic
394C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3C2×#13711yescharacteristic
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—22no—
403C2×D3#47714yes—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—82no—
407C3×C4—81no—
408C3×C4#48414yes—
409C3×C4#55311yescharacteristic
410C3×C4#55311yescharacteristic
411C3×C4#55411yescharacteristic
412C3×C4#55411yescharacteristic
413Dic3—22no—
414Dic3—22no—
415Dic3—22no—
416Dic3—42no—
417Dic3—81no—
418Dic3—82no—
419C23—32no—
420C23—32no—
421C23—32no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23C2×D3×D411yescharacteristic
427Q8—62no—
428Q8—62no—
429Q8—121no—
430Q8—121no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—68no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—68no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—122no—
466C2×C4—248no—
467C2×C4#89412yes—
468C2×C4#89412yes—
469C2×C4#91112yes—
470C2×C4D3×Q1612yes—
471C2×C4C2×#23511yescharacteristic
472C2×C4#100411yescharacteristic
473C2×C4C2×D3×Q811yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#100911yescharacteristic
477C8—42no—
478C8—122no—
479C2×C3#248711yescharacteristic
480C2×C3C2×#47711yescharacteristic
481C2×C3#307911yescharacteristic
482C2×C3C2×#48411yescharacteristic
483C2×C3#309211yescharacteristic
484C2×C3#320911yescharacteristic
485C2×C3#321711yescharacteristic
486D3—24no—
487C22—34no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22C2×#89411yescharacteristic
495C22#560711yescharacteristic
496C22D3×C22⋊Q811yescharacteristic
497C22C2×#91111yescharacteristic
498C22C2×D3×Q1611yescharacteristic
499C22D3×AGL(1,ℤ/8)11yescharacteristic
500C22D3×#13711yescharacteristic
501C4—22no—
502C4—42no—
503C4—62no—
504C4—62no—
505C4—62no—
506C4—82no—
507C4—81no—
508C4—122no—
509C4—241no—
510C4—242no—
511C4#488414yes—
512C4#555811yescharacteristic
513C4#557911yescharacteristic
514C4#569511yescharacteristic
515C4#573011yescharacteristic
516C3#2768111yescharacteristic, complemented, Sylow
517C2—64no—
518C2#7623211yescharacteristic
519C2C2×#488411yescharacteristic
520C2#7921511yescharacteristic
521C2#8033911yescharacteristic
522C2#8048011yescharacteristic
523C2#8096611yescharacteristic
524C2D3×#47711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1947D3
hypocenterC3#27681
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. C3×#1947C2semidirect
4. #70644C2semidirect
6. #70485C2semidirect
8. #69588C2semidirect
9. #69466C2semidirect
11. #68843C2semidirect
49. #1947D3semidirect
516. C3#27681semidirect

Maximal subgroups

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

subgroupindex
D3×#3842
C4×#8972
C3×#19472
#706442
#706322
#704852
#704752
#695882
#694662
#688782
#688432
#276813