Finite Group Databasebeta

← #10847408

Subgroups — #10847408

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×#5470C211yesmaximal, characteristic
2C3×#2855C211yesmaximal, characteristic
3C2×#5051C212yesmaximal
4C2×#4524C211yesmaximal, characteristic
5#69691C211yesmaximal, characteristic, complemented
6#69690C212yesmaximal, complemented
7#69668C211yesmaximal, characteristic, complemented
8#69633C211yesmaximal, characteristic, complemented
9#69631C212yesmaximal, complemented
10#69604C211yesmaximal, characteristic, complemented
11#69514C212yesmaximal
12#25498—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×#216C2212yes—
16C2×#926—21no—
17C2×#924—21no—
18C2×#924C2212yes—
19C23×C3⋊C8C2212yes—
20#5470—28no—
21C2×#870—21no—
22C2×#869C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6—21no—
25C2×C4×Dic6C2212yes—
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#387—21no—
31C3×#387—21no—
32C3×#385—21no—
33C3×#381—21no—
34C3×#380C2211yescharacteristic
35C2×#835C2212yes—
36#5051—28no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#807C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#467—32no—
48#2855D311yescharacteristic
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×Q8C2311yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×Dic6—21no—
62C22×Dic6C2312yes—
63C3×C22⋊Q8—24no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—24no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#216—24no—
79C2×#216—24no—
80C2×#216—42no—
81#926—28no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2312yes—
86#924—28no—
87#924—24no—
88#924D418yes—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—22no—
92C22×C3⋊C8—24no—
93C22×C3⋊C8—24no—
94C22×C3⋊C8—42no—
95C22×C3⋊C8C2312yes—
96C22×C3⋊C8C2312yes—
97#870—28no—
98#869—24no—
99#869—24no—
100C4×Dic6—28no—
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110#835—24no—
111#810—24no—
112#810—24no—
113#810—42no—
114#807—24no—
115C22×Q16—32no—
116C23×C8—32no—
117#533—68no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8—31no—
120C2×#125—61no—
121C2×C22⋊Q8—32no—
122C2×C22⋊Q8—61no—
123C2×C22⋊Q8C2×D311yescharacteristic
124C2×C4×Q8—32no—
125C2×C4×Q8—61no—
126C2×#117—61no—
127C22×C42C2×D311yescharacteristic
128#467—68no—
129C2×#107—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134#391C2×D311yescharacteristic
135#391C2×D311yescharacteristic
136#389C2×D311yescharacteristic
137#389C2×D311yescharacteristic
138#387—21no—
139#387—21no—
140#385—21no—
141#381—21no—
142#380C2×D311yescharacteristic
143#340—32no—
144#340—32no—
145#336—31no—
146#336—32no—
147#336—31no—
148C24×C3C2×D411yescharacteristic
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—41no—
165C22×C3×C4Q1614yescomplemented
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—24no—
180C2×Dic6—22no—
181C2×Dic6—42no—
182C2×Dic6—42no—
183C2×Dic6C2×D414yes—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197#217—28no—
198#216—48no—
199C4⋊Dic3—24no—
200C4⋊Dic3—24no—
201C4⋊Dic3—24no—
202C4⋊Dic3—22no—
203C4⋊Dic3C2×D414yes—
204Dic3⋊C4—28no—
205Dic3⋊C4—28no—
206C4×Dic3—28no—
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—32no—
215C22×Q8—61no—
216C22×Q8C22×D311yescharacteristic
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—64no—
229C4⋊Q8—64no—
230#125—68no—
231C22⋊Q8—24no—
232C22⋊Q8—38no—
233C22⋊Q8—68no—
234C22⋊Q8—64no—
235C4×Q8—64no—
236C4×Q8—64no—
237C4×Q8—68no—
238#117—68no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4C22×D311yescharacteristic
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—62no—
254C2×C42—24no—
255C2×C42—62no—
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—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
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—42no—
281C3×Q8—42no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—28no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—44no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4#12012yes—
310C2×C3×C4C2×Q1612yes—
311C2×C3×C4C2×Q1612yes—
312C2×C3×C4#13512yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4112yes—
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323Dic6—24no—
324Dic6—24no—
325Dic6—48no—
326Dic6—42no—
327Dic6—42no—
328C3⋊C8—24no—
329C3⋊C8—42no—
330C24C2×#6711yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—34no—
336C2×Q8—64no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—122no—
343C2×Q8—122no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4#21614yescomplemented
359C22×C4D3×D414yes—
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
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—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—34no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—64no—
383C4⋊C4—64no—
384C4⋊C4—68no—
385C4⋊C4—64no—
386C4⋊C4—68no—
387C4⋊C4—62no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—68no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—68no—
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—41no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4—81no—
419C3×C4#54612yes—
420C3×C4#54712yes—
421Dic3—44no—
422Dic3—44no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23#93712yes—
428C23C2×#21612yes—
429C23#95112yes—
430C23C22×#6711yescharacteristic
431Q8—42no—
432Q8—42no—
433Q8—64no—
434Q8—64no—
435Q8—128no—
436Q8—122no—
437Q8—122no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—28no—
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—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—64no—
466C2×C4—64no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4#93712yes—
472C2×C4C2×#21612yes—
473C2×C4C2×#21612yes—
474C2×C4#95012yes—
475C2×C4C2×D3×D412yes—
476C2×C4D3×#4112yes—
477C2×C4#100911yescharacteristic
478C2×C4#101011yescharacteristic
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#505014yes—
496C22C2×#93711yescharacteristic
497C22C22×#21611yescharacteristic
498C22C2×#95011yescharacteristic
499C22#582211yescharacteristic
500C22C2×#95111yescharacteristic
501C22#583211yescharacteristic
502C22#583311yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—124no—
513C4—124no—
514C4#565412yes—
515C4#565512yes—
516C3#2549811yescharacteristic, complemented, Sylow
517C2—24no—
518C2C2×#505011yescharacteristic
519C2#8284811yescharacteristic
520C2#8292611yescharacteristic
521C2#8313811yescharacteristic
522C2#8315511yescharacteristic
523C2#8334411yescharacteristic
524C2#8337311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2855C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2855D3
hypocenterC3#25498
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
5. #69691C2semidirect
6. #69690C2semidirect
7. #69668C2semidirect
8. #69633C2semidirect
9. #69631C2semidirect
10. #69604C2semidirect
165. 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×#54702
C3×#28552
C2×#50512
C2×#45242
#696912
#696902
#696682
#696332
#696312
#696042
#695142
#254983