Finite Group Databasebeta

← #10823386

Subgroups — #10823386

Order 768.

1156 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic
2C3×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×#216C212yesmaximal, complemented
4#71700C212yesmaximal
5C4×#830C211yesmaximal, characteristic
6#69691C211yesmaximal, characteristic, complemented
7#69668C211yesmaximal, characteristic, complemented
8#69633C211yesmaximal, characteristic, complemented
9#69604C211yesmaximal, characteristic, complemented
10#69527C212yesmaximal
11C4×#810C212yesmaximal, complemented
12#23571—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×#216C2212yes—
17C2×#926—21no—
18C2×#924—21no—
19C2×#924C2212yes—
20#5470—28no—
21C2×#870—21no—
22C2×#869C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6—21no—
25C2×C4×Dic6C2212yes—
26C3×#391C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#387—21no—
29C3×#385—21no—
30C3×#380C2211yescharacteristic
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C2×#835C2212yes—
35C2×#830C2212yes—
36C4×#216—28no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#807C2212yes—
41C2×C4×C3⋊C8C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×C4×Q16—32no—
48C4×C22⋊Q8D311yescharacteristic
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2024—31no—
53#2016—31no—
54#2001—32no—
55C4×#103—32no—
56C4×C22⋊C8—31no—
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×Dic6—21no—
62C22×Dic6C2312yes—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×C4×Q8—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—24no—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217—22no—
82C2×#216—24no—
83C2×#216—24no—
84C2×#216—24no—
85#926—28no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90#924—28no—
91#924—24no—
92#924—24no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×C4×Dic3—22no—
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2312yes—
98#870—28no—
99#869—24no—
100#869—24no—
101C4×Dic6—28no—
102C4×Dic6—24no—
103C4×Dic6D418yes—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109#835—24no—
110#830—24no—
111#810—24no—
112#810—24no—
113#810—24no—
114#807—24no—
115C4×C3⋊C8—24no—
116C22×Q16—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×C4×Q8C2×D311yescharacteristic
127C2×#117—61no—
128C22×C42C2×D311yescharacteristic
129C4×Q16—68no—
130C2×#107—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×C22⋊C8—32no—
136C2×C4×C8—32no—
137#391C2×D311yescharacteristic
138#389C2×D311yescharacteristic
139#387—21no—
140#385—21no—
141#380C2×D311yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2×D311yescharacteristic
144C4×C22⋊C4—21no—
145#340—32no—
146#340—32no—
147#336—31no—
148#336—31no—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8#4112yes—
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—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—41no—
167C22×C3×C4Q1614yescomplemented
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—22no—
175C22×Dic3—22no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—24no—
181C2×Dic6—22no—
182C2×Dic6—42no—
183C2×Dic6—42no—
184C2×Dic6C2×D414yes—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4#4112yes—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—42no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—42no—
198C3×C42—42no—
199C3×C42C2×D414yes—
200#217—28no—
201#216—216no—
202C4⋊Dic3—24no—
203C4⋊Dic3—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3C2×D414yes—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209C4×Dic3—28no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8#4114yes—
216C22×Q8—32no—
217C22×Q8—61no—
218C22×Q8C22×D311yescharacteristic
219C23×C4#6711yescharacteristic
220C23×C4#6711yescharacteristic
221C23×C4C22×D311yescharacteristic
222C2×Q16—64no—
223C2×Q16—64no—
224C2×Q16—64no—
225C22×C8—32no—
226C22×C8—32no—
227C4⋊Q8—64no—
228C4⋊Q8—64no—
229#125—68no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—64no—
233C22⋊Q8—64no—
234C22⋊Q8—68no—
235C4×Q8—24no—
236C4×Q8—38no—
237C4×Q8—68no—
238C4×Q8—64no—
239#117—68no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—62no—
255C2×C42—24no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—62no—
259C2×C42#6711yescharacteristic
260C2×C42#6711yescharacteristic
261C2×C42C22×D311yescharacteristic
262C2×C42C22×D311yescharacteristic
263#107—64no—
264#103—64no—
265#103—64no—
266#103—64no—
267#103—64no—
268C22⋊C8—64no—
269C4×C8—64no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3C2×Q1612yes—
282C23×C3C22×D411yescharacteristic
283C3×Q8—24no—
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—28no—
292C2×C3×C4—21no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—44no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4#12012yes—
311C2×C3×C4#12012yes—
312C2×C3×C4C2×Q1612yes—
313C2×C3×C4C2×Q1612yes—
314C2×C3×C4#13512yes—
315C2×C3×C4#13712yes—
316C2×C3×C4C22×D412yes—
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326Dic6—24no—
327Dic6—24no—
328Dic6—48no—
329Dic6—42no—
330Dic6—42no—
331C3⋊C8—24no—
332C3⋊C8—42no—
333C24C2×#6711yescharacteristic
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—34no—
338C2×Q8—64no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—122no—
345C2×Q8—122no—
346C2×Q8#23512yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4#21614yescomplemented
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C23×D311yescharacteristic
369Q16—616no—
370C2×C8—34no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8—62no—
375C2×C8—62no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—34no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—68no—
386C4⋊C4—64no—
387C4⋊C4—68no—
388C4⋊C4—62no—
389C4⋊C4#23512yes—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—68no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42—68no—
399C42D3×D414yes—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3C2×#12011yescharacteristic
408C22×C3#52511yescharacteristic
409C22×C3#53211yescharacteristic
410C22×C3C22×Q1611yescharacteristic
411C22×C3C2×#13511yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57511yescharacteristic
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—41no—
419C3×C4—42no—
420C3×C4—81no—
421C3×C4—81no—
422C3×C4#45114yes—
423C3×C4#52512yes—
424C3×C4#53212yes—
425Dic3—44no—
426Dic3—44no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C2×#21612yes—
434C23C22×#6711yescharacteristic
435Q8—24no—
436Q8—42no—
437Q8—64no—
438Q8—64no—
439Q8—128no—
440Q8—122no—
441Q8—122no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—64no—
468C2×C4—64no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4#93712yes—
474C2×C4#93712yes—
475C2×C4C2×#21612yes—
476C2×C4C2×#21612yes—
477C2×C4#95012yes—
478C2×C4#95112yes—
479C2×C4C2×#23511yescharacteristic
480C2×C4C2×#23511yescharacteristic
481C2×C4C2×D3×D412yes—
482C2×C4D3×#4112yes—
483C8—64no—
484C8—122no—
485C2×C3—42no—
486C2×C3#238811yescharacteristic
487C2×C3C2×#45111yescharacteristic
488C2×C3#300211yescharacteristic
489C2×C3#310411yescharacteristic
490C2×C3#310711yescharacteristic
491C2×C3#317011yescharacteristic
492C2×C3#317211yescharacteristic
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22#579111yescharacteristic
501C22C2×#93711yescharacteristic
502C22C22×#21611yescharacteristic
503C22C2×#95011yescharacteristic
504C22#582211yescharacteristic
505C22C2×#95111yescharacteristic
506C22#583211yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—42no—
511C4—41no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—124no—
516C4—124no—
517C4#505014yes—
518C4#554712yes—
519C4#554812yes—
520C3#2357111yescharacteristic, complemented, Sylow
521C2—42no—
522C2#8240011yescharacteristic
523C2#8244711yescharacteristic
524C2#8253411yescharacteristic
525C2C2×#505011yescharacteristic
526C2#8292611yescharacteristic
527C2#8313611yescharacteristic
528C2#8334411yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC4×C22⋊Q8D3
hypocenterC3#23571
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×C4×#216C2semidirect
6. #69691C2semidirect
7. #69668C2semidirect
8. #69633C2semidirect
9. #69604C2semidirect
11. C4×#810C2semidirect
167. C22×C3×C4Q16semidirect
362. C22×C4#216semidirect
520. C3#23571semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×C4×C22⋊Q82
C2×C4×#2162
#717002
C4×#8302
#696912
#696682
#696332
#696042
#695272
C4×#8102
#235713