Finite Group Databasebeta

← #10822572

Subgroups — #10822572

Order 768.

1306 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic, complemented
2C2×#5131C211yesmaximal, characteristic
3#82269C211yesmaximal, characteristic
4C2×#5001C211yesmaximal, characteristic
5C3×#1733C211yesmaximal, characteristic, complemented
6#71987C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#68763C218yesmaximal, complemented
9#23463—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17#5519—22no—
18#5519C2214yescomplemented
19#5517—22no—
20#5517—22no—
21#5500—22no—
22#5483—22no—
23#5470—22no—
24#5131—22no—
25#5131C2214yes—
26C2×#821—22no—
27#5001—22no—
28#5001C2214yes—
29#4988—22no—
30C2×#809—22no—
31#4955—22no—
32#4939C2214yes—
33C3×#339C2214yescomplemented
34#4554C2214yescomplemented
35#4523C2214yescomplemented
36#3458—31no—
37C2×#418—31no—
38C2×#418—31no—
39#2309—31no—
40#2027—38no—
41#1742—31no—
42#1733—31no—
43#1733D311yescharacteristic, complemented
44C24×D3—21no—
45C23×C3×C4C2311yescharacteristic
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—22no—
51C22×D12—22no—
52C22×D12—21no—
53C22×D12—22no—
54C22×D12Q814yescomplemented
55C22×D12C2312yes—
56C2×C3×M16—21no—
57C2×C3×M16—22no—
58C2×C3×M16—22no—
59C2×C3×M16C2312yes—
60C22×C3×C8—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117—22no—
64C3×#117—22no—
65C3×#117D412yes—
66C3×#117C2312yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42—21no—
72C2×C3×C42C2312yes—
73C2×#217—21no—
74C2×#217D412yes—
75C2×(C2×D3)⋊C4—28no—
76#926—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926D412yes—
83#926C2312yes—
84C2×C4⋊Dic3D412yes—
85C2×C4⋊Dic3C2312yes—
86#924—28no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3D412yes—
89C2×#208—21no—
90C2×#208—22no—
91C2×#208—22no—
92C2×#208C2312yes—
93C22×C3⋊C8—21no—
94#890—24no—
95#890—24no—
96#888—24no—
97#888—24no—
98#878—28no—
99#878—28no—
100#870—24no—
101#869—24no—
102C4×Dic6—28no—
103#821—28no—
104#809—28no—
105C23×D4—31no—
106C22×M16—31no—
107C22×M16C2×D311yescharacteristic
108#544—34no—
109#544—62no—
110#544—62no—
111#543—62no—
112#533—62no—
113#533—62no—
114#533—62no—
115C2×#117—31no—
116C2×#117—31no—
117C2×#117C2×D311yescharacteristic
118C2×#117C2×D311yescharacteristic
119#418—34no—
120#418—34no—
121#418—62no—
122#418—62no—
123C2×#102—62no—
124C2×#102—62no—
125#390—34no—
126#390—62no—
127#379—62no—
128#343—34no—
129#339—34no—
130#339C2×D314yescomplemented
131C24×C3C2×D411yescharacteristic
132C23×D3—22no—
133C23×D3—24no—
134C23×D3—41no—
135C23×D3—41no—
136C23×D3—42no—
137C23×D3—84no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×Q812yes—
146C22×C3×C4#4112yes—
147C22×C3×C4#4112yes—
148C22×C3×C4#4112yes—
149C22×C3×C4C2411yescharacteristic
150C22×Dic3—22no—
151C22×Dic3C2×D414yes—
152C2×D12—24no—
153C2×D12—24no—
154C2×D12—24no—
155C2×D12—24no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—24no—
161C2×D12—22no—
162C2×D12—41no—
163C2×D12—41no—
164C2×D12—42no—
165C2×D12—42no—
166C2×D12—42no—
167C2×D12—42no—
168C2×D12—42no—
169C2×D12C2×Q814yes—
170C2×Dic6—28no—
171C3×M16—22no—
172C3×M16—22no—
173C3×M16—42no—
174C2×C3×C8—22no—
175C2×C3×C8—42no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4C2×D414yes—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—42no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186#217—22no—
187#217—22no—
188#217—22no—
189#217—22no—
190(C2×D3)⋊C4—88no—
191C4⋊Dic3—28no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3C2×D414yes—
196Dic3⋊C4—28no—
197Dic3⋊C4—28no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202#208—22no—
203#208—22no—
204#208—42no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—42no—
207C25—61no—
208C22×D4—34no—
209C22×D4—32no—
210C22×D4—61no—
211C22×D4—61no—
212C22×D4—62no—
213C22×D4—62no—
214C22×D4—61no—
215C22×D4—61no—
216C22×D4—61no—
217C22×D4—62no—
218C23×C4C22×D311yescharacteristic
219C2×M16—21no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—32no—
223C2×M16—61no—
224C2×M16—62no—
225C2×M16—62no—
226C2×M16C22×D312yes—
227C22×C8—21no—
228C22×C8—61no—
229C4⋊Q8—64no—
230C4⋊Q8—64no—
231C4⋊Q8—64no—
232#125—64no—
233#125—64no—
234#125—64no—
235C22⋊Q8—68no—
236C22⋊Q8—68no—
237C22⋊Q8—68no—
238C4×Q8—68no—
239#117—21no—
240#117—21no—
241#117—22no—
242#117—22no—
243#117—32no—
244#117—32no—
245#117—61no—
246#117—61no—
247#117—61no—
248#117—61no—
249#117—61no—
250#117—61no—
251#117D1212yes—
252#117C22×D312yes—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4D1212yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—68no—
262C2×C42—21no—
263C2×C42—32no—
264C2×C42—61no—
265C2×C42C22×D312yes—
266#102—68no—
267#102—68no—
268C23×C3—42no—
269C23×C3#12011yescharacteristic
270C23×C3#12011yescharacteristic
271C23×C3C22⋊Q812yes—
272C23×C3#12412yes—
273C23×C3C22×D411yescharacteristic
274C22×D3—24no—
275C22×D3—48no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—44no—
279C22×D3—44no—
280C22×D3—88no—
281C22×D3—84no—
282C22×D3—84no—
283C22×D3—88no—
284C2×C3×C4—24no—
285C2×C3×C4—44no—
286C2×C3×C4—42no—
287C2×C3×C4—82no—
288C2×C3×C4#12011yescharacteristic
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4C22⋊Q812yes—
291C2×C3×C4C22⋊Q812yes—
292C2×C3×C4C22⋊Q812yes—
293C2×C3×C4C22⋊Q812yes—
294C2×C3×C4C22⋊Q812yes—
295C2×C3×C4#12412yes—
296C2×C3×C4#12412yes—
297C2×C3×C4#12412yes—
298C2×C3×C4#12412yes—
299C2×C3×C4#12412yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×Q811yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—44no—
309C2×Dic3—82no—
310D12—24no—
311D12—24no—
312D12—42no—
313D12—42no—
314D12—44no—
315D12—44no—
316Dic6—88no—
317C3×C8—42no—
318C3⋊C8—42no—
319C24—62no—
320C24—64no—
321C24—121no—
322C24—121no—
323C24—122no—
324C24—244no—
325C24D3×D411yescharacteristic
326C2×Q8—68no—
327C2×D4—34no—
328C2×D4—64no—
329C2×D4—64no—
330C2×D4—64no—
331C2×D4—64no—
332C2×D4—64no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—121no—
341C2×D4—121no—
342C2×D4—122no—
343C2×D4—122no—
344C2×D4—122no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—34no—
348C22×C4—62no—
349C22×C4C2×D1211yescharacteristic
350C22×C4C2×D1211yescharacteristic
351C22×C4#23512yes—
352C22×C4#23512yes—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4#23712yes—
357C22×C4D3×Q812yes—
358C22×C4C23×D311yescharacteristic
359M16—22no—
360M16—22no—
361M16—42no—
362M16—62no—
363M16—62no—
364M16—122no—
365C2×C8—22no—
366C2×C8—42no—
367C2×C8—62no—
368C2×C8—122no—
369C4⋊C4—22no—
370C4⋊C4—34no—
371C4⋊C4—42no—
372C4⋊C4—68no—
373C4⋊C4—68no—
374C4⋊C4—68no—
375C4⋊C4—62no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4C2×D1214yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—42no—
382C22⋊C4—62no—
383C22⋊C4—62no—
384C22⋊C4—62no—
385C22⋊C4—62no—
386C22⋊C4—248no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—62no—
392C42—62no—
393C42—62no—
394C42—62no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3#39312yes—
401C22×C3#39312yes—
402C22×C3C2×#12011yescharacteristic
403C22×C3C2×C22⋊Q811yescharacteristic
404C22×C3C2×#12411yescharacteristic
405C2×D3—48no—
406C2×D3—44no—
407C2×D3—84no—
408C2×D3—88no—
409C2×D3—88no—
410C2×D3—84no—
411C3×C4—44no—
412C3×C4—82no—
413C3×C4#39312yes—
414C3×C4#39312yes—
415C3×C4C2×C22⋊Q811yescharacteristic
416C3×C4C2×C22⋊Q811yescharacteristic
417C3×C4C2×#12411yescharacteristic
418C3×C4C2×#12411yescharacteristic
419Dic3—44no—
420Dic3—44no—
421C23—42no—
422C23—64no—
423C23—128no—
424C23—124no—
425C23—124no—
426C23—122no—
427C23—122no—
428C23—244no—
429C23—244no—
430C23—248no—
431C23—248no—
432C23#88211yescharacteristic
433C23#88211yescharacteristic
434C23#88512yes—
435C23#89612yes—
436C23C2×D3×D411yescharacteristic
437Q8—248no—
438D4—64no—
439D4—64no—
440D4—124no—
441D4—124no—
442D4—122no—
443D4—122no—
444C2×C4—24no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—64no—
448C2×C4—64no—
449C2×C4—82no—
450C2×C4—124no—
451C2×C4—242no—
452C2×C4#87512yes—
453C2×C4#87512yes—
454C2×C4#88211yescharacteristic
455C2×C4#88211yescharacteristic
456C2×C4#88512yes—
457C2×C4#88512yes—
458C2×C4#88512yes—
459C2×C4#89612yes—
460C2×C4#89612yes—
461C2×C4#89612yes—
462C2×C4#89712yes—
463C2×C4#89712yes—
464C2×C4C22×D1211yescharacteristic
465C2×C4C2×#23511yescharacteristic
466C2×C4C2×#23511yescharacteristic
467C2×C4C2×D3×D411yescharacteristic
468C2×C4C2×#23711yescharacteristic
469C2×C4C2×D3×Q811yescharacteristic
470C8—42no—
471C8—122no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3C2×#39311yescharacteristic
477C2×C3#302112yes—
478D3—84no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—42no—
483C22—42no—
484C22—128no—
485C22—124no—
486C22—244no—
487C22—248no—
488C22—248no—
489C22—244no—
490C22#467312yes—
491C22#467312yes—
492C22C2×#88211yescharacteristic
493C22C2×#88511yescharacteristic
494C22C2×#89611yescharacteristic
495C4—44no—
496C4—82no—
497C4—124no—
498C4—124no—
499C4#467312yes—
500C4#467312yes—
501C4C2×#87511yescharacteristic
502C4C2×#88511yescharacteristic
503C4C2×#89611yescharacteristic
504C4C2×#89711yescharacteristic
505C3#2346311yescharacteristic, complemented, Sylow
506C2—21no—
507C2—21no—
508C2—22no—
509C2—22no—
510C2—244no—
511C2C2×#467311yescharacteristic
512C2#7899412yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1733C2
socleC22×C3C2×#120
prime radicalC23×D12C22
hypercenter#1733D3
hypocenterC3#23463
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. #85725C2semidirect
5. C3×#1733C2semidirect
6. #71987C2semidirect
7. #71578C2semidirect
8. #68763C2semidirect
18. #5519C22semidirect
33. C3×#339C22semidirect
34. #4554C22semidirect
35. #4523C22semidirect
43. #1733D3semidirect
54. C22×D12Q8semidirect
130. #339C2×D3semidirect
505. C3#23463semidirect

Maximal subgroups

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

subgroupindex
#857252
C2×#51312
#822692
C2×#50012
C3×#17332
#719872
#715782
#687632
#234633