Finite Group Databasebeta

← #10848836

Subgroups — #10848836

Order 768.

1222 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×#5697C211yesmaximal, characteristic, complemented
2C2×C3×#488C211yesmaximal, characteristic, complemented
3#83001C211yesmaximal, characteristic, complemented
4C2×#5058C211yesmaximal, characteristic, complemented
5#71741C211yesmaximal, characteristic, complemented
6#70276C212yesmaximal, complemented
7#70272C212yesmaximal, complemented
8#70208C212yesmaximal, complemented
9#70204C212yesmaximal, complemented
10#69772C211yesmaximal, characteristic, complemented
11#69753C211yesmaximal, characteristic, complemented
12#25611—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×#946C2211yescharacteristic
16C22×#215C2212yes—
17C2×#938C2211yescharacteristic
18C22×#213C2212yes—
19#5697—24no—
20#5697—24no—
21C2×#895—22no—
22C2×#895C2212yes—
23C2×#894—22no—
24C2×#893C2212yes—
25C2×#874C2211yescharacteristic
26C2×#872C2211yescharacteristic
27C3×#488—24no—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31#5229C2211yescharacteristic
32#5221C2211yescharacteristic
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36#5058—24no—
37#4961—22no—
38#4948—22no—
39#4935C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C2×#540—31no—
45C2×#496—31no—
46C2×#488D311yescharacteristic, complemented
47#2620—31no—
48#2000—32no—
49#1999—32no—
50#1992—31no—
51#1957—31no—
52#1879—32no—
53#1877—32no—
54#1750—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4C2311yescharacteristic
57C22×#67—21no—
58C22×D12—22no—
59C22×D12—22no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×C4×D3C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×#125—22no—
65C3×#125D414yescomplemented
66C3×#124—22no—
67C3×#124—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—21no—
73C2×#217—22no—
74#946—42no—
75C2×#215—24no—
76C2×#215—24no—
77C2×#215—42no—
78#938—42no—
79C2×#213—24no—
80C2×#213—24no—
81C2×#213—42no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4D412yes—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4D412yes—
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#895—216no—
94#895—24no—
95#895—24no—
96#894—216no—
97#893—24no—
98#893—24no—
99#874—22no—
100#874D414yescomplemented
101#872—22no—
102#872—22no—
103C3×#103—24no—
104C3×#102—24no—
105C3×C4×C8—22no—
106C3⋊#95—22no—
107#821—42no—
108#821—42no—
109(C2×D3)⋊C8—24no—
110#803—22no—
111C22×SD16—32no—
112C22×D8—32no—
113#540—64no—
114#540—64no—
115C2×#127—31no—
116C2×#125C2×D311yescharacteristic
117C2×#124C2×D311yescharacteristic
118C2×#123—62no—
119C2×C22⋊Q8—31no—
120C2×#121—31no—
121C2×#121—32no—
122C2×#121—62no—
123C2×C4×D4—32no—
124C2×#117—31no—
125#496—64no—
126#488—24no—
127C2×#103C2×D312yes—
128C2×#102—32no—
129C2×#102—32no—
130C2×#102C2×D312yes—
131C2×C22⋊C8—32no—
132C2×#97—31no—
133C2×C4×C8C2×D311yescharacteristic
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384—31no—
139#340—32no—
140#340—32no—
141#336—32no—
142C24×C3—21no—
143C23×D3—22no—
144C23×D3C2×D411yescharacteristic
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2411yescharacteristic
157C2×#67—42no—
158C2×#67—82no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D411yescharacteristic
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—28no—
169C2×D12—28no—
170C2×D12—82no—
171C2×D12—82no—
172C2×D12—82no—
173C2×D12—82no—
174C2×D12C2×D414yes—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—28no—
180C2×C4×D3—82no—
181C2×C4×D3—82no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4C2×D414yes—
188C3×C22⋊C4—44no—
189C3×C42C2×D412yes—
190C3×C42C2×D412yes—
191#217—42no—
192#217—82no—
193#215—48no—
194#213—48no—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—28no—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4—42no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3—82no—
202Dic3⋊C4—24no—
203Dic3⋊C4—42no—
204C4×Dic3—21no—
205C4×Dic3—21no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8#4114yes—
209C22×Q8C22×D311yescharacteristic
210C22×D4—32no—
211C22×D4—62no—
212C22×D4—61no—
213C22×D4—62no—
214C22×D4C22×D311yescharacteristic
215C23×C4—31no—
216C23×C4—62no—
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—64no—
221C2×D8—64no—
222C2×D8—122no—
223C22×C8—32no—
224C22×C8C22×D312yes—
225#127—34no—
226#127—62no—
227#125—22no—
228#125#6714yescomplemented
229#124—22no—
230#124—22no—
231#123—616no—
232C22⋊Q8—122no—
233#121—64no—
234#121—616no—
235#121—64no—
236#121—122no—
237C4×D4—64no—
238C4×D4—64no—
239#117—62no—
240#117—62no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4C22×D312yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—32no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—62no—
255C2×C42—31no—
256C2×C42C22×D311yescharacteristic
257#103—24no—
258#102—24no—
259#102—122no—
260#102—122no—
261C22⋊C8—64no—
262#97—62no—
263C4×C8—22no—
264#95—62no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3C22×D411yescharacteristic
271C22×D3—21no—
272C22×D3—24no—
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C3×Q8—42no—
284C3×Q8—42no—
285C3×D4—42no—
286C3×D4—42no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4#12014yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4112yes—
299C2×C3×C4H(5,𝔽2)12yes—
300#67—84no—
301C2×Dic3—22no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311D12—24no—
312D12—24no—
313D12—42no—
314D12—42no—
315D12—88no—
316D12—88no—
317C4×D3—42no—
318C4×D3—42no—
319C4×D3—88no—
320C3×C8—42no—
321C3⋊C8—24no—
322C24—21no—
323C24—31no—
324C24—62no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—34no—
334C2×D4—68no—
335C2×D4—68no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—122no—
342C2×D4—242no—
343C2×D4—242no—
344C2×D4—242no—
345C2×D4—242no—
346C2×D4—242no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—32no—
350C22×C4—31no—
351C22×C4—68no—
352C22×C4—61no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—242no—
359C22×C4—242no—
360C22×C4D3×D412yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—34no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—64no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—122no—
376C4⋊C4—242no—
377C4⋊C4C2×#6714yes—
378C22⋊C4—44no—
379C22⋊C4—68no—
380C22⋊C4—68no—
381C22⋊C4—64no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—242no—
385C42—61no—
386C42—61no—
387C42D3×D412yes—
388C42C2×#6712yes—
389C22×C3—82no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—82no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#53211yescharacteristic
396C22×C3#53511yescharacteristic
397C22×C3#57612yes—
398C22×C3#57712yes—
399C2×D3—42no—
400C2×D3—41no—
401C2×D3—44no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—44no—
405C2×D3—42no—
406C2×D3—44no—
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C3×C4—24no—
412C3×C4—44no—
413C3×C4—81no—
414C3×C4#54612yes—
415C3×C4#55012yes—
416Dic3—41no—
417Dic3—42no—
418Dic3—81no—
419Dic3—81no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—61no—
426C23—61no—
427C23—62no—
428C23—62no—
429C23—64no—
430C23—61no—
431C23—122no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23—122no—
436C23—122no—
437C23C2×D3×D411yescharacteristic
438Q8—42no—
439Q8—42no—
440D4—42no—
441D4—42no—
442D4—64no—
443D4—64no—
444D4—122no—
445D4—122no—
446D4—248no—
447D4—248no—
448D4—244no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—61no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—248no—
470C2×C4#93714yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4D3×#4112yes—
473C2×C4#100912yes—
474C2×C4C22×#6711yescharacteristic
475C8—42no—
476C8—64no—
477C2×C3—161no—
478C2×C3#243111yescharacteristic
479C2×C3#301212yes—
480C2×C3#317912yes—
481C2×C3#322011yescharacteristic
482C2×C3#322111yescharacteristic
483D3—81no—
484D3—84no—
485C22—81no—
486C22—82no—
487C22—82no—
488C22—81no—
489C22—81no—
490C22—121no—
491C22—124no—
492C22—124no—
493C22—122no—
494C22—124no—
495C22—122no—
496C22—121no—
497C22—121no—
498C22—242no—
499C22—242no—
500C22—242no—
501C22—242no—
502C22#566811yescharacteristic
503C22#567411yescharacteristic
504C22#576912yes—
505C22#577012yes—
506C22C2×#93711yescharacteristic
507C4—24no—
508C4—44no—
509C4—81no—
510C4—121no—
511C4—122no—
512C4—241no—
513C4—241no—
514C4#567612yes—
515C4#583412yes—
516C3#2561111yescharacteristic, complemented, Sylow
517C2—161no—
518C2—241no—
519C2—244no—
520C2#8163012yes—
521C2#8164411yescharacteristic
522C2#8164511yescharacteristic
523C2#8295811yescharacteristic
524C2#8391212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#488C2
socleC23×C3C22×D4
hypercenterC2×#488D3
hypocenterC3#25611
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. C2×#5697C2semidirect
2. C2×C3×#488C2semidirect
3. #83001C2semidirect
4. C2×#5058C2semidirect
5. #71741C2semidirect
6. #70276C2semidirect
7. #70272C2semidirect
8. #70208C2semidirect
9. #70204C2semidirect
10. #69772C2semidirect
11. #69753C2semidirect
46. C2×#488D3semidirect
65. C3×#125D4semidirect
100. #874D4semidirect
228. #125#67semidirect
516. C3#25611semidirect

Maximal subgroups

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

subgroupindex
C2×#56972
C2×C3×#4882
#830012
C2×#50582
#717412
#702762
#702722
#702082
#702042
#697722
#697532
#256113