Finite Group Databasebeta

← #10848834

Subgroups — #10848834

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×#5535C211yesmaximal, characteristic, complemented
2C3×#2620C211yesmaximal, characteristic, complemented
3C2×#5058C211yesmaximal, characteristic, complemented
4C2×#5056C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71582C211yesmaximal, characteristic, complemented
8#69687C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69583C211yesmaximal, characteristic, complemented
11#69580C211yesmaximal, characteristic, complemented
12#25611—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#215C2212yes—
18C22×#213C2212yes—
19C2×#930C2212yes—
20#5535—24no—
21#5535—24no—
22C2×#894—22no—
23C2×#884—22no—
24C2×#874C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×#870C2211yescharacteristic
27C3×#396—22no—
28C3×#394—22no—
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#384C2211yescharacteristic
32C2×#835C2212yes—
33C2×#832C2212yes—
34C2×#830C2212yes—
35#5058—24no—
36#5056—24no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#803C2211yescharacteristic
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C2×#540—31no—
45C2×#496—31no—
46C2×#488—31no—
47#2620D311yescharacteristic, complemented
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×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60C22×D12—22no—
61C22×D12C2312yes—
62C22×C4×D3—22no—
63C3×#124—22no—
64C3×#124—22no—
65C3×C22⋊Q8—42no—
66C3×#121—42no—
67C3×#117—22no—
68C3×#117—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D412yes—
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#215—24no—
82C2×#215—24no—
83C2×#215—42no—
84C2×#213—24no—
85C2×#213—24no—
86C2×#213—42no—
87#930—24no—
88#930—24no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2312yes—
95#894—216no—
96#884—216no—
97#874—22no—
98#874D414yescomplemented
99C4×D12—24no—
100C4×D12—24no—
101#870—22no—
102#870D414yescomplemented
103C3×#95—22no—
104#835—24no—
105#832—24no—
106#830—24no—
107#809—42no—
108#809—42no—
109#803—22no—
110C4×C3⋊C8—22no—
111C22×SD16—32no—
112C22×D8—32no—
113#540—64no—
114#540—64no—
115C2×#127—31no—
116C2×#125—31no—
117C2×#124C2×D311yescharacteristic
118C2×#123—62no—
119C2×C22⋊Q8C2×D311yescharacteristic
120C2×#121—32no—
121C2×#121—62no—
122C2×#121C2×D311yescharacteristic
123C2×C4×D4—32no—
124C2×#117C2×D311yescharacteristic
125#496—64no—
126#488—64no—
127C2×#103—32no—
128C2×#102—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×C22⋊C8—32no—
132C2×#97—31no—
133C2×C4×C8—31no—
134#396—22no—
135#394—22no—
136#391C2×D311yescharacteristic
137#390C2×D311yescharacteristic
138#384C2×D311yescharacteristic
139#340—32no—
140#340—32no—
141#336—32no—
142C24×C3—21no—
143C24×C3C2×D411yescharacteristic
144C23×D3—22no—
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—
153C2×C3×D4—42no—
154C2×C3×D4—82no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C2×#67—28no—
167C2×#67—82no—
168C2×#67—82no—
169C22×Dic3—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—28no—
176C2×D12—82no—
177C2×D12—82no—
178C2×D12C2×D414yes—
179C2×C4×D3—28no—
180C2×C4×D3—82no—
181C2×C4×D3—82no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—82no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—42no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—82no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#215—48no—
197#213—48no—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—28no—
200C4⋊Dic3—22no—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—28no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8#4114yes—
209C22×Q8C22×D311yescharacteristic
210C22×D4—21no—
211C22×D4—32no—
212C22×D4—62no—
213C22×D4—62no—
214C22×D4C22×D311yescharacteristic
215C23×C4—62no—
216C23×C4C22×D311yescharacteristic
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—64no—
221C2×D8—64no—
222C2×D8—122no—
223C22×C8—32no—
224C22×C8—32no—
225#127—34no—
226#127—62no—
227#125—34no—
228#125—62no—
229#124—22no—
230#124—22no—
231#123—616no—
232C22⋊Q8—42no—
233#121—42no—
234#121—616no—
235#121—64no—
236#121—64no—
237C4×D4—64no—
238C4×D4—64no—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4#6712yes—
246C2×C4⋊C4C22×D311yescharacteristic
247C2×C4⋊C4C22×D311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4#6712yes—
255C2×C42C22×D311yescharacteristic
256C2×C42C22×D311yescharacteristic
257#103—64no—
258#102—64no—
259#102—122no—
260#102—122no—
261C22⋊C8—64no—
262#97—62no—
263C4×C8—62no—
264#95—22no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3C22×D411yescharacteristic
276C22×D3—24no—
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—
287C3×D4—84no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4#12014yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4112yes—
309C2×C3×C4H(5,𝔽2)12yes—
310#67—88no—
311C2×Dic3—24no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314D12—24no—
315D12—24no—
316D12—42no—
317D12—42no—
318D12—88no—
319C4×D3—88no—
320C3⋊C8—24no—
321C3⋊C8—42no—
322C24—21no—
323C24—62no—
324C24C2×#6711yescharacteristic
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—34no—
334C2×D4—42no—
335C2×D4—68no—
336C2×D4—68no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—82no—
343C2×D4—242no—
344C2×D4—242no—
345C2×D4—242no—
346C2×D4—242no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—68no—
355C22×C4—62no—
356C22×C4—242no—
357C22×C4—242no—
358C22×C4C2×#6712yes—
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6712yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364C2×C8—34no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—34no—
374C4⋊C4—42no—
375C4⋊C4—68no—
376C4⋊C4—62no—
377C4⋊C4—82no—
378C22⋊C4—24no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—68no—
383C22⋊C4—68no—
384C22⋊C4—82no—
385C42—21no—
386C42—21no—
387C42D3×D412yes—
388C42D3×D412yes—
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3—82no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—82no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#53211yescharacteristic
401C22×C3#53511yescharacteristic
402C22×C3#57612yes—
403C22×C3#57712yes—
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C3×C4—24no—
412C3×C4—41no—
413C3×C4—42no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4#54612yes—
418C3×C4#55012yes—
419Dic3—44no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—64no—
431C23—122no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23—122no—
436C23—122no—
437C23C22×#6711yescharacteristic
438Q8—42no—
439Q8—42no—
440D4—42no—
441D4—42no—
442D4—64no—
443D4—64no—
444D4—84no—
445D4—122no—
446D4—122no—
447D4—248no—
448D4—248no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—64no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—248no—
470C2×C4#93714yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4#100412yes—
474C2×C4D3×#4112yes—
475C8—64no—
476C8—122no—
477C2×C3—81no—
478C2×C3—161no—
479C2×C3#243111yescharacteristic
480C2×C3#301212yes—
481C2×C3#317912yes—
482C2×C3#322011yescharacteristic
483C2×C3#322111yescharacteristic
484D3—84no—
485C22—42no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—42no—
490C22—82no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—82no—
495C22—124no—
496C22—124no—
497C22—124no—
498C22—242no—
499C22—242no—
500C22—242no—
501C22—242no—
502C22C2×#93711yescharacteristic
503C22#580511yescharacteristic
504C22#583211yescharacteristic
505C22#583512yes—
506C22#583712yes—
507C4—24no—
508C4—41no—
509C4—42no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—124no—
514C4#559912yes—
515C4#561212yes—
516C3#2561111yescharacteristic, complemented, Sylow
517C2—81no—
518C2—161no—
519C2—244no—
520C2#8265211yescharacteristic
521C2#8338811yescharacteristic
522C2#8340911yescharacteristic
523C2#8342412yes—
524C2#8345912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2620C2
socleC23×C3C22×D4
hypercenter#2620D3
hypocenterC3#25611
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. C2×#5535C2semidirect
2. C3×#2620C2semidirect
3. C2×#5058C2semidirect
4. C2×#5056C2semidirect
5. #71699C2semidirect
6. #71684C2semidirect
7. #71582C2semidirect
8. #69687C2semidirect
9. #69657C2semidirect
10. #69583C2semidirect
11. #69580C2semidirect
47. #2620D3semidirect
98. #874D4semidirect
102. #870D4semidirect
516. C3#25611semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C3×#26202
C2×#50582
C2×#50562
#716992
#716842
#715822
#696872
#696572
#695832
#695802
#256113