Finite Group Databasebeta

← #10847400

Subgroups — #10847400

Order 768.

1132 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
1C2×C3×#533C211yesmaximal, characteristic
2C2×#5051C212yesmaximal
3#82075C211yesmaximal, characteristic
4#71794C211yesmaximal, characteristic, complemented
5#71793C211yesmaximal, characteristic, complemented
6#71782C211yesmaximal, characteristic, complemented
7#71781C211yesmaximal, characteristic, complemented
8#71696C212yesmaximal
9#71632C212yesmaximal, complemented
10#71628C212yesmaximal, complemented
11C2×#4524C211yesmaximal, characteristic
12#25498—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#216C2212yes—
24C2×#924C2211yescharacteristic
25C23×C3⋊C8C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2212yes—
29#5212C2211yescharacteristic
30#5211—21no—
31#5210—21no—
32#5051—28no—
33C2×#810C2212yes—
34C2×#807C2212yes—
35#4933—21no—
36#4933—21no—
37#4929C2211yescharacteristic
38#4929C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#533D311yescharacteristic
47C2×#467—32no—
48#2855—31no—
49#2057—31no—
50#2043—31no—
51#2024—31no—
52#2016—31no—
53#1892—32no—
54#1878—32no—
55#1863—32no—
56C2×#336—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C3×C4⋊Q8—24no—
64C3×C4⋊Q8—24no—
65C3×#125—28no—
66C3×C22⋊Q8—28no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8D418yes—
69C3×C4×Q8—28no—
70C3×C4×Q8—24no—
71C3×C4×Q8—24no—
72C3×#117—28no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—24no—
81C2×C3×C42—22no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#217—21no—
88C2×#216—24no—
89C2×#216—24no—
90C2×#216—42no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94#924—24no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C22×C3⋊C8—24no—
99C22×C3⋊C8—24no—
100C22×C3⋊C8—42no—
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2312yes—
103#835—24no—
104#835—24no—
105#835—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#810—24no—
114#807—24no—
115C22×Q16—32no—
116C23×C8—32no—
117#533—28no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×#125—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—31no—
123C2×C22⋊Q8C2×D312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8C2×D312yes—
126C2×#117—21no—
127C22×C42C2×D311yescharacteristic
128#467—68no—
129C2×#107—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134#391—31no—
135#391—31no—
136#389—31no—
137#389—31no—
138#387—61no—
139#387—61no—
140#385—61no—
141#381—61no—
142#380—31no—
143#340—32no—
144#340—32no—
145#336—31no—
146#336—32no—
147#336—31no—
148C24×C3C2×D411yescharacteristic
149C2×C3×Q8—24no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—42no—
156C2×C3×Q8—42no—
157C2×C3×Q8C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4Q1614yescomplemented
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4C2×D414yes—
191C3×C22⋊C4—28no—
192C3×C42—28no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197#217—42no—
198#217—42no—
199#216—48no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204Dic3⋊C4—42no—
205Dic3⋊C4—42no—
206Dic3⋊C4—42no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—48no—
213C2×C3⋊C8#4114yes—
214C22×Q8—21no—
215C22×Q8—31no—
216C22×Q8C22×D312yes—
217C23×C4#6711yescharacteristic
218C23×C4#6711yescharacteristic
219C23×C4C22×D311yescharacteristic
220C2×Q16—64no—
221C2×Q16—64no—
222C2×Q16—122no—
223C22×C8—32no—
224C22×C8—32no—
225C22×C8—64no—
226C22×C8—64no—
227C22×C8—122no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230#125—28no—
231C22⋊Q8—28no—
232C22⋊Q8—24no—
233C22⋊Q8—64no—
234C22⋊Q8#6718yes—
235C4×Q8—28no—
236C4×Q8—24no—
237C4×Q8—24no—
238#117—28no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42#6711yescharacteristic
257C2×C42#6711yescharacteristic
258C2×C42C22×D311yescharacteristic
259C2×C42C22×D311yescharacteristic
260#107—64no—
261#103—64no—
262#103—64no—
263#103—64no—
264#103—122no—
265#95—61no—
266#95—61no—
267#95—61no—
268#95—61no—
269#95—61no—
270#95—61no—
271#95—61no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3#12012yes—
277C23×C3C2×Q1612yes—
278C23×C3#13712yes—
279C23×C3C22×D411yescharacteristic
280C3×Q8—24no—
281C3×Q8—24no—
282C3×Q8—48no—
283C3×Q8—42no—
284C3×Q8—42no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—28no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4#12012yes—
307C2×C3×C4C2×Q1612yes—
308C2×C3×C4C2×Q1612yes—
309C2×C3×C4#13512yes—
310C2×C3×C4C22×D412yes—
311C2×C3×C4C2×#4112yes—
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326Dic6—42no—
327Dic6—42no—
328C3⋊C8—24no—
329C3⋊C8—42no—
330C24C2×#6711yescharacteristic
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—62no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—62no—
343C2×Q8C2×#6714yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4#21614yescomplemented
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6714yes—
365C22×C4C23×D311yescharacteristic
366Q16—128no—
367C2×C8—34no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8—128no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—22no—
380C4⋊C4—62no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—62no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C4⋊C4—122no—
387C4⋊C4C2×#6714yes—
388C22⋊C4—28no—
389C22⋊C4—122no—
390C22⋊C4—122no—
391C42—28no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3#45114yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#53211yescharacteristic
405C22×C3#53711yescharacteristic
406C22×C3C22×Q1611yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×#13711yescharacteristic
409C22×C3#57511yescharacteristic
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—44no—
415C3×C4—41no—
416C3×C4—44no—
417C3×C4#54612yes—
418C3×C4#54712yes—
419Dic3—81no—
420Dic3—81no—
421Dic3—81no—
422Dic3—81no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23#94212yes—
428C23C2×#21612yes—
429C23#95312yes—
430C23C22×#6711yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—48no—
434Q8—42no—
435Q8—42no—
436Q8—122no—
437Q8—122no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4C2×#21612yes—
472C2×C4C2×#21612yes—
473C2×C4#95012yes—
474C2×C4#95312yes—
475C2×C4D3×#4112yes—
476C2×C4#100911yescharacteristic
477C2×C4#101011yescharacteristic
478C2×C4C22×#6712yes—
479C8—64no—
480C8—122no—
481C2×C3—24no—
482C2×C3#243011yescharacteristic
483C2×C3C2×#45111yescharacteristic
484C2×C3#300211yescharacteristic
485C2×C3#317011yescharacteristic
486C2×C3#318111yescharacteristic
487C2×C3#324811yescharacteristic
488C2×C3#324911yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22#522814yes—
496C22C2×#94211yescharacteristic
497C22C22×#21611yescharacteristic
498C22C2×#95011yescharacteristic
499C22#582211yescharacteristic
500C22#583211yescharacteristic
501C22#583311yescharacteristic
502C22C2×#95311yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—44no—
509C4—44no—
510C4—241no—
511C4—241no—
512C4—241no—
513C4—241no—
514C4#582412yes—
515C4#582912yes—
516C3#2549811yescharacteristic, complemented, Sylow
517C2—24no—
518C2#8334411yescharacteristic
519C2#8334611yescharacteristic
520C2#8336411yescharacteristic
521C2#8337311yescharacteristic
522C2C2×#522811yescharacteristic
523C2#8408011yescharacteristic
524C2#8409111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#25498
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #71794C2semidirect
5. #71793C2semidirect
6. #71782C2semidirect
7. #71781C2semidirect
9. #71632C2semidirect
10. #71628C2semidirect
168. C22×C3×C4Q16semidirect
358. C22×C4#216semidirect
516. C3#25498semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50512
#820752
#717942
#717932
#717822
#717812
#716962
#716322
#716282
C2×#45242
#254983