Finite Group Databasebeta

← #10831717

Subgroups — #10831717

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5543C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic
3#82388C211yesmaximal, characteristic, complemented
4C2×#5007C211yesmaximal, characteristic, complemented
5#72037C212yesmaximal
6#71605C211yesmaximal, characteristic, complemented
7#70566C212yesmaximal, complemented
8#70167C212yesmaximal, complemented
9#70164C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24339—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3C2212yes—
17C22×C4⋊Dic3—21no—
18C2×#924—21no—
19#5543—24no—
20#5543—24no—
21C2×#890C2212yes—
22C2×#878—22no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#442—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5221C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33#5210—22no—
34#5007—24no—
35#4972C2212yes—
36C2×#807C2212yes—
37#4949—21no—
38#4947—21no—
39#4936C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#442D311yescharacteristic
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×Dic3—22no—
62C22×Dic6—21no—
63C22×Dic6D411yescharacteristic
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C2×C3×D8Q814yescomplemented
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81D4×Dic3—24no—
82D4×Dic3—24no—
83C2×C4⋊Dic3—28no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89#924—28no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3C2312yes—
94C22×C3⋊C8C2312yes—
95#890—24no—
96#890—24no—
97#878—216no—
98#869—22no—
99#869D414yes—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×#106—22no—
103C3×#102—22no—
104C3×#102—22no—
105C3×#97—22no—
106#832—24no—
107#832—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111#807—24no—
112C4⋊(C3⋊C8)—22no—
113C22×D8C2×D311yescharacteristic
114#554—64no—
115#554—64no—
116C2×C4⋊Q8—31no—
117C2×#125—32no—
118C2×C22⋊Q8—62no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—31no—
121C2×C4×D4—32no—
122C2×C4×D4C2×D311yescharacteristic
123C2×C4×D4C2×D311yescharacteristic
124C22×C4⋊C4—61no—
125#474—64no—
126#442—24no—
127C2×#107—32no—
128C2×#106C2×D311yescharacteristic
129C2×C4⋊C8—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102C2×D311yescharacteristic
133C2×#102C2×D311yescharacteristic
134C2×#97C2×D311yescharacteristic
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C24×C3—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4C2×D412yes—
157C2×C3×D4C2×Q812yes—
158C2×C3×D4C2×Q812yes—
159C2×C3×D4#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—21no—
163C22×C3×C4—44no—
164C22×C3×C4—42no—
165C22×C3×C4—82no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—22no—
170C22×Dic3—28no—
171C22×Dic3—22no—
172C22×Dic3—82no—
173C22×Dic3—82no—
174C22×Dic3C2×D412yes—
175C22×Dic3C2×D412yes—
176C2×Dic6—22no—
177C2×Dic6—24no—
178C2×Dic6—42no—
179C2×Dic6—82no—
180C3×D8—28no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8C2×Q812yes—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—42no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195#217—28no—
196#217—44no—
197#217—82no—
198C4⋊Dic3—28no—
199C4⋊Dic3—22no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3C2×D414yes—
205Dic3⋊C4—28no—
206Dic3⋊C4—24no—
207Dic3⋊C4—82no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C22×Q8—31no—
214C22×Q8—61no—
215C22×D4C22×D311yescharacteristic
216C22×D4C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—62no—
220C2×D8—22no—
221C2×D8—22no—
222C2×D8Dic614yescomplemented
223C22×C8—32no—
224C22×C8C22×D311yescharacteristic
225C22×C8C22×D311yescharacteristic
226C4⋊Q8—34no—
227C4⋊Q8—62no—
228#125—64no—
229#125—64no—
230C22⋊Q8—616no—
231C22⋊Q8—68no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—64no—
238C4×D4—64no—
239C4×D4#6714yes—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—68no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C42—32no—
257C2×C42C22×D311yescharacteristic
258#107—64no—
259#106—22no—
260C4⋊C8—62no—
261#102—22no—
262#102—22no—
263#102—64no—
264#102—64no—
265#97—22no—
266#95—62no—
267#95—61no—
268#95—61no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—41no—
286C3×D4—41no—
287C3×D4C22⋊Q814yes—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—48no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—84no—
296C2×C3×C4#12012yes—
297C2×C3×C4#12012yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×Q811yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C42−1+411yescharacteristic
303C2×C3×C42−1+411yescharacteristic
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—24no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—88no—
317Dic6—44no—
318Dic6—84no—
319C3×C8—22no—
320C3×C8—41no—
321C3⋊C8—42no—
322C24—21no—
323C24—21no—
324C2×Q8—62no—
325C2×Q8—64no—
326C2×Q8—122no—
327C2×Q8—242no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4C2×Dic612yes—
337C2×D4C2×Dic612yes—
338C2×D4#23512yes—
339C2×D4C2×#6712yes—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—62no—
348C22×C4—68no—
349C22×C4—62no—
350C22×C4—82no—
351C22×C4—242no—
352C22×C4—242no—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4C23×D311yescharacteristic
356D8—28no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8C2×Dic612yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—34no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—62no—
373C4⋊C4—64no—
374C4⋊C4—64no—
375C4⋊C4—62no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—242no—
379C4⋊C4C2×#6712yes—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—68no—
383C22⋊C4—124no—
384C22⋊C4—242no—
385C42—62no—
386C42—62no—
387C42D3×D412yes—
388C42C2×#6712yes—
389C22×C3—24no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—42no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3D4×Q811yescharacteristic
402C22×C3#57512yes—
403C22×C3#57612yes—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—81no—
408C3×C4C2×C22⋊Q811yescharacteristic
409C3×C4#53611yescharacteristic
410C3×C4#54811yescharacteristic
411C3×C4D4×Q811yescharacteristic
412Dic3—42no—
413Dic3—44no—
414Dic3—42no—
415Dic3—82no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C2×D3×D411yescharacteristic
427Q8—124no—
428Q8—244no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4#92414yes—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—64no—
451C2×C4—84no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—248no—
458C2×C4#93712yes—
459C2×C4#93712yes—
460C2×C4C22×Dic611yescharacteristic
461C2×C4C2×#23511yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4#101011yescharacteristic
464C2×C4#101011yescharacteristic
465C2×C4C22×#6711yescharacteristic
466C8—22no—
467C8—41no—
468C8—122no—
469C2×C3—44no—
470C2×C3—82no—
471C2×C3#240911yescharacteristic
472C2×C3#300511yescharacteristic
473C2×C3#300811yescharacteristic
474C2×C3#337012yes—
475C2×C3#338912yes—
476C22—24no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—44no—
483C22—81no—
484C22—81no—
485C22—82no—
486C22D4×Dic611yescharacteristic
487C22#555211yescharacteristic
488C22#577312yes—
489C22#577412yes—
490C22C2×#93711yescharacteristic
491C4—22no—
492C4—22no—
493C4—42no—
494C4—81no—
495C4—122no—
496C4—122no—
497C4—124no—
498C4—242no—
499C4D4×Dic611yescharacteristic
500C4#555211yescharacteristic
501C4C2×#92411yescharacteristic
502C4#582511yescharacteristic
503C3#2433911yescharacteristic, complemented, Sylow
504C2—44no—
505C2—82no—
506C2#7985212yes—
507C2#7985612yes—
508C2#8234911yescharacteristic
509C2#8394511yescharacteristic
510C2#8394711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#442D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#5543C2semidirect
3. #82388C2semidirect
4. C2×#5007C2semidirect
6. #71605C2semidirect
7. #70566C2semidirect
8. #70167C2semidirect
9. #70164C2semidirect
10. #69838C2semidirect
11. #69828C2semidirect
66. C2×C3×D8Q8semidirect
222. C2×D8Dic6semidirect
503. C3#24339semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
C2×C3×#4422
#823882
C2×#50072
#720372
#716052
#705662
#701672
#701642
#698382
#698282
#243393