Finite Group Databasebeta

← #10767839

Subgroups — #10767839

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85718C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic, complemented
3#84059C212yesmaximal, complemented
4#71552C211yesmaximal, characteristic, complemented
5#69707C212yesmaximal, complemented
6#69706C212yesmaximal, complemented
7#69705C214yesmaximal, complemented
8#69398C212yesmaximal, complemented
9#18165—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×C4⋊C4—21no—
12C22×C3×C4⋊C4C2211yescharacteristic
13C2×#942—22no—
14#5791—21no—
15#5791—21no—
16#5791C2212yes—
17#5515—21no—
18#5515C2212yes—
19#5514—22no—
20#5513—22no—
21#5511—22no—
22#5510C2211yescharacteristic
23#5507—22no—
24#5504—21no—
25#5500C2211yescharacteristic
26C3×#395—21no—
27C3×#395—21no—
28C3×#395C2212yes—
29C3×#395C2212yes—
30C2×C3×#95—21no—
31C2×C3×#95C2211yescharacteristic
32#5228—22no—
33#5228C2214yes—
34#5227—22no—
35#5227C2214yes—
36#5225—22no—
37#5207C2212yes—
38#4528C2212yes—
39#4528C2212yes—
40#4452C2214yes—
41#4451C2214yes—
42#3492—31no—
43#3004—32no—
44C2×#395D311yescharacteristic, complemented
45#1605—32no—
46#1604—32no—
47#1603—34no—
48#1504—32no—
49#1463—31no—
50C22×C3×Q8—21no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—44no—
54C22×C3×Q8C2312yes—
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×#235—21no—
61C2×#235—21no—
62C2×#235D412yes—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—44no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72#953—22no—
73C2×#216—24no—
74#942—48no—
75C2×#215—24no—
76#930—21no—
77#930—21no—
78#930—21no—
79#930—21no—
80#930D412yes—
81#930C2312yes—
82#929—22no—
83#929—22no—
84#929—22no—
85C4×#67—22no—
86C4×#67—22no—
87C4×#67—22no—
88C4×#67—22no—
89C4×#67—22no—
90C4×#67—22no—
91#926—21no—
92#926—21no—
93#926D412yes—
94#924—21no—
95#924—21no—
96#924—21no—
97#924—21no—
98#924D412yes—
99#924C2312yes—
100C2×#208—22no—
101#898—22no—
102#898—22no—
103#897—22no—
104#896—22no—
105#896—22no—
106#895—22no—
107#894—22no—
108#894—22no—
109#893—22no—
110#892—22no—
111D3×C4⋊C4—22no—
112#890—22no—
113#889—22no—
114#889—22no—
115#888—22no—
116#887—22no—
117C3×#95—22no—
118C3×#95—22no—
119C3×#95—21no—
120C3×#95C2312yes—
121#835—24no—
122#835—24no—
123#830—22no—
124#830C2314yes—
125C23×Q8C2×D311yescharacteristic
126C2×#137—62no—
127#554—62no—
128#553—62no—
129#552—62no—
130#547—62no—
131#544—31no—
132#539—32no—
133#539—61no—
134#537—31no—
135#525—32no—
136#525—61no—
137#525—61no—
138#520—61no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2×D311yescharacteristic
141#451—34no—
142#451—62no—
143#448—34no—
144#448—62no—
145#419—62no—
146#407—32no—
147#395—21no—
148#395—21no—
149#395C2×D312yes—
150#395C2×D312yes—
151C2×#95—21no—
152C2×#95C2×D311yescharacteristic
153#328—34no—
154#327—34no—
155#324—32no—
156#324—32no—
157C24×C3C2×D411yescharacteristic
158C2×C3×Q8—28no—
159C2×C3×Q8—24no—
160C2×C3×Q8—44no—
161C2×C3×Q8—44no—
162C2×C3×Q8—44no—
163C2×C3×Q8—48no—
164C2×C3×Q8—42no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—41no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4—41no—
180C22×C3×C4—41no—
181C22×C3×C4—42no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×#67—22no—
191C2×#67—22no—
192C2×#67—22no—
193#235—42no—
194#235—42no—
195#235—44no—
196#235—82no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12C2×D412yes—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6C2×D412yes—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—44no—
211C3×C4⋊C4—44no—
212#217—22no—
213#217—22no—
214#217—22no—
215#216—48no—
216#215—48no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—22no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3C2×D412yes—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—22no—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234Dic3⋊C4—22no—
235C4×Dic3—22no—
236C4×Dic3—22no—
237C4×Dic3—22no—
238#208—44no—
239C2×C3⋊C8—24no—
240C2×#41—32no—
241C2×#41—61no—
242C2×#41—61no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—44no—
247C22×Q8C22×D312yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4#6711yescharacteristic
251C23×C4#6711yescharacteristic
252C23×C4C22×D311yescharacteristic
253#137—128no—
254C2×Q16—64no—
255C2×SD16—64no—
256C2×M16—62no—
257C4⋊Q8—62no—
258#126—62no—
259#126—62no—
260#125—62no—
261#125—62no—
262#125—62no—
263#123—62no—
264#123—62no—
265#123—62no—
266#123—62no—
267#123—62no—
268C22⋊Q8—32no—
269C22⋊Q8—32no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272C22⋊Q8—62no—
273C22⋊Q8—61no—
274C22⋊Q8—61no—
275C22⋊Q8—61no—
276C22⋊Q8—61no—
277#121—32no—
278#121—32no—
279#121—62no—
280#121—61no—
281#121—61no—
282#121—61no—
283#121—61no—
284#120—62no—
285C4×Q8—62no—
286C4×D4—62no—
287C4×D4—62no—
288C4×D4—62no—
289C4×D4—62no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—62no—
293#117—32no—
294#117—62no—
295#117—61no—
296#117—61no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307#103—64no—
308#103—64no—
309C22⋊C8—34no—
310C22⋊C8—62no—
311#95—22no—
312#95—22no—
313#95—21no—
314#95C22×D312yes—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3#12011yescharacteristic
320C23×C3#12011yescharacteristic
321C23×C3C22×D411yescharacteristic
322C22×D3—22no—
323C3×Q8—28no—
324C3×Q8—44no—
325C3×Q8—48no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—48no—
343C2×C3×C4—42no—
344C2×C3×C4—44no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—44no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—81no—
352C2×C3×C4#12012yes—
353C2×C3×C4#13712yescomplemented
354C2×C3×C4C22×D412yes—
355#67—42no—
356#67—42no—
357#67—44no—
358#67—44no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362D12—42no—
363D12—42no—
364D12—81no—
365C4×D3—44no—
366C4×D3—82no—
367C4×D3—82no—
368Dic6—42no—
369Dic6—42no—
370Dic6—81no—
371C3⋊C8—44no—
372C24C2×#6711yescharacteristic
373#41—122no—
374#41—124no—
375#41—122no—
376#41—242no—
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—32no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—42no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×D4—32no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—62no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4D3×D412yes—
414C22×C4D3×D412yes—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421Q16—128no—
422SD16—128no—
423M16—124no—
424C2×C8—64no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—32no—
430C4⋊C4—32no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—61no—
434C4⋊C4—62no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C22⋊C4—62no—
444C22⋊C4—62no—
445C22⋊C4—62no—
446C22⋊C4—62no—
447C22⋊C4—62no—
448C22⋊C4—62no—
449C22⋊C4—62no—
450C22⋊C4—62no—
451C22⋊C4—62no—
452C42—62no—
453C42—62no—
454C42—62no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3#45612yes—
464C22×C3B(4,𝔽2)12yes—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×#13711yescharacteristic
467C22×C3#57511yescharacteristic
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—42no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—44no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—44no—
479Dic3—42no—
480Dic3—42no—
481Dic3—42no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—62no—
487C23#93711yescharacteristic
488C23#93711yescharacteristic
489C23C22×#6711yescharacteristic
490Q8—28no—
491Q8—44no—
492Q8—48no—
493Q8—122no—
494Q8—122no—
495Q8—241no—
496D4—122no—
497D4—122no—
498D4—124no—
499D4—122no—
500D4—122no—
501D4—124no—
502D4—241no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—81no—
532C2×C4—124no—
533C2×C4—242no—
534C2×C4—242no—
535C2×C4#93712yes—
536C2×C4#94212yescomplemented
537C2×C4C2×D3×D412yes—
538C8—124no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3#300711yescharacteristic
544C2×C3C2×#45611yescharacteristic
545C2×C3C2×B(4,𝔽2)11yescharacteristic
546D3—82no—
547C22—24no—
548C22—24no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—122no—
556C22—122no—
557C22—122no—
558C22#503212yes—
559C22#509512yes—
560C22C2×#93711yescharacteristic
561C22C2×#94211yescharacteristic
562C22#582211yescharacteristic
563C4—22no—
564C4—22no—
565C4—22no—
566C4—44no—
567C4—41no—
568C4—44no—
569C4—41no—
570C4—41no—
571C4—122no—
572C4—122no—
573C4—122no—
574C3#1816511yescharacteristic, complemented, Sylow
575C2—21no—
576C2—22no—
577C2—21no—
578C2—22no—
579C2—242no—
580C2C2×#503211yescharacteristic
581C2#8273511yescharacteristic
582C2C2×#509511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#395C2
socleC22×C3C2×#120
prime radical#85718C2
hypercenterC2×#395D3
hypocenterC3#18165
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×C3×#395C2semidirect
3. #84059C2semidirect
4. #71552C2semidirect
5. #69707C2semidirect
6. #69706C2semidirect
7. #69705C2semidirect
8. #69398C2semidirect
44. C2×#395D3semidirect
353. C2×C3×C4#137semidirect
536. C2×C4#942semidirect
574. C3#18165semidirect

Maximal subgroups

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

subgroupindex
#857182
C2×C3×#3952
#840592
#715522
#697072
#697062
#697052
#693982
#181653