Finite Group Databasebeta

← #10697503

Subgroups — #10697503

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82417C211yesmaximal, characteristic
2C2×#5019C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
5#69777C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10231—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938—21no—
11C22×#214—21no—
12C2×#930—21no—
13C2×C4×D12C2211yescharacteristic
14C2×#869C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×C4×C8C2211yescharacteristic
17#5212—21no—
18C2×#821—21no—
19C2×#818—21no—
20#5019—22no—
21#5019—22no—
22#4987—21no—
23C2×#810—21no—
24#4961—22no—
25#4935C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4846—22no—
28#4846—22no—
29C3×D4⋊C8—22no—
30#4551—22no—
31#4459—22no—
32C2×#474—31no—
33C2×#461—31no—
34#2537—31no—
35#1990—31no—
36C2×D4⋊C8—31no—
37C2×D4⋊C8D311yescharacteristic, complemented
38#1296—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—22no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×Dic6—21no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—22no—
55#938—42no—
56#938—42no—
57C2×#214—24no—
58C2×#214—42no—
59C2×#214—42no—
60#930—42no—
61#930—42no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D412yes—
70C22×C3⋊C8—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75#869—21no—
76#869—21no—
77#869—21no—
78#869D412yescomplemented
79C3×C4⋊C8—21no—
80C3×C4⋊C8D412yescomplemented
81C3×C4×C8—21no—
82C3×C4×C8—21no—
83C3⋊#95—22no—
84#821—24no—
85#818—24no—
86#810—24no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yescomplemented
89C22×SD16—61no—
90C2×C4⋊Q8—31no—
91C2×#121—61no—
92C2×#121—61no—
93C2×C4×D4—31no—
94C2×C4×D4C2×D311yescharacteristic
95#474—62no—
96#474—62no—
97#461—62no—
98#461—62no—
99C2×#107—61no—
100C2×C4⋊C8—31no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#103—61no—
103C2×#102—61no—
104C2×C4×C8C2×D311yescharacteristic
105#394—62no—
106#384—31no—
107#380—61no—
108#380—61no—
109#340—62no—
110D4⋊C8—22no—
111D4⋊C8—62no—
112C24×C3—21no—
113C23×D3—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4D812yes—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C2×#67—44no—
131C2×#67—44no—
132C2×#67—82no—
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3—41no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12SD1612yescomplemented
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—41no—
148C2×Dic6—22no—
149C2×Dic6—41no—
150C2×Dic6—41no—
151C2×Dic6—41no—
152C2×Dic6—41no—
153C2×Dic6—41no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—41no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42#4111yescharacteristic
168#217—44no—
169#214—48no—
170(C2×D3)⋊C4—44no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173C4⋊Dic3—21no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—21no—
177C4⋊Dic3—41no—
178C4⋊Dic3—41no—
179C4⋊Dic3—41no—
180C4⋊Dic3—41no—
181C4⋊Dic3—41no—
182C4⋊Dic3—41no—
183C4⋊Dic3—42no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C22×Q8—61no—
188C22×D4—31no—
189C22×D4—62no—
190C22×D4D1211yescharacteristic
191C23×C4—21no—
192C23×C4—61no—
193C2×SD16—64no—
194C2×SD16—122no—
195C2×SD16—122no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—61no—
199C4⋊Q8—32no—
200C4⋊Q8—61no—
201C4⋊Q8—61no—
202C4⋊Q8—61no—
203#121—122no—
204#121—122no—
205#121—122no—
206#121—122no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—32no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4D1211yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—61no—
226C2×C42C22×D311yescharacteristic
227#107—64no—
228C4⋊C8—21no—
229C4⋊C8—32no—
230C4⋊C8—61no—
231C4⋊C8#6712yescomplemented
232#103—64no—
233#102—64no—
234C4×C8—21no—
235C4×C8—21no—
236#95—62no—
237C23×C3—22no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3#12011yescharacteristic
245C22×D3—22no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C3×D4—22no—
253C3×D4—22no—
254C3×D4—41no—
255C3×D4—41no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4#12011yescharacteristic
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12711yescharacteristic
271C2×C3×C4C2×D811yescharacteristic
272C2×C3×C4C2×SD1611yescharacteristic
273C2×C3×C4#13511yescharacteristic
274C2×C3×C4#13511yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
277C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
278C2×C3×C4#13711yescharacteristic
279#67—88no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—42no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—81no—
288C2×Dic3—81no—
289D12—22no—
290D12—22no—
291D12—41no—
292D12—41no—
293C4×D3—81no—
294C4×D3—81no—
295C4×D3—81no—
296C4×D3—81no—
297Dic6—42no—
298Dic6—42no—
299Dic6—81no—
300Dic6—81no—
301C3×C8—42no—
302C3×C8—42no—
303C3⋊C8—42no—
304C24—21no—
305C24—61no—
306C2×Q8—62no—
307C2×Q8—121no—
308C2×Q8—121no—
309C2×Q8—121no—
310C2×Q8—121no—
311C2×Q8—121no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—32no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—124no—
324C2×D4—124no—
325C2×D4—242no—
326C2×D4D2412yes—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—61no—
335C22×C4—62no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4C2×D1211yescharacteristic
344C22×C4D3×D411yescharacteristic
345C22×C4D3×D411yescharacteristic
346SD16—128no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—62no—
354C2×C8—121no—
355C2×C8—121no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—61no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—61no—
362C4⋊C4—122no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—124no—
372C22⋊C4—124no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C42#23511yescharacteristic
376C42D3×D411yescharacteristic
377C42D3×D411yescharacteristic
378C42C2×#6711yescharacteristic
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3#39211yescharacteristic
387C22×C3#44711yescharacteristic
388C22×C3#44911yescharacteristic
389C22×C3#45011yescharacteristic
390C22×C3#45211yescharacteristic
391C22×C3#45511yescharacteristic
392C22×C3#45611yescharacteristic
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—81no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—81no—
403C3×C4#44711yescharacteristic
404C3×C4#44811yescharacteristic
405C3×C4#44911yescharacteristic
406C3×C4#44911yescharacteristic
407C3×C4#46511yescharacteristic
408C3×C4#46611yescharacteristic
409C3×C4#46811yescharacteristic
410C3×C4#46911yescharacteristic
411Dic3—82no—
412Dic3—82no—
413Dic3—81no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—62no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23#88211yescharacteristic
429Q8—122no—
430Q8—122no—
431Q8—241no—
432Q8—241no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437D4—62no—
438D4—62no—
439D4—121no—
440D4—121no—
441D4—248no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—241no—
459C2×C4—241no—
460C2×C4—241no—
461C2×C4—241no—
462C2×C4—241no—
463C2×C4—241no—
464C2×C4#88211yescharacteristic
465C2×C4#88411yescharacteristic
466C2×C4#88411yescharacteristic
467C2×C4C2×D2411yescharacteristic
468C2×C4#90411yescharacteristic
469C2×C4#90811yescharacteristic
470C2×C4#90811yescharacteristic
471C2×C4#91111yescharacteristic
472C2×C4#91211yescharacteristic
473C2×C4D3×SD1611yescharacteristic
474C2×C4#91511yescharacteristic
475C2×C4#93011yescharacteristic
476C2×C4#93711yescharacteristic
477C2×C4#94011yescharacteristic
478C8—42no—
479C8—42no—
480C8—122no—
481C2×C3—82no—
482C2×C3#163911yescharacteristic
483C2×C3#164611yescharacteristic
484C2×C3#164711yescharacteristic
485C2×C3#165411yescharacteristic
486C2×C3#199911yescharacteristic
487C2×C3#200011yescharacteristic
488C2×C3#200711yescharacteristic
489D3—82no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22—241no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22#473211yescharacteristic
505C22#473411yescharacteristic
506C22#477611yescharacteristic
507C22#478311yescharacteristic
508C22#482511yescharacteristic
509C22#482611yescharacteristic
510C22#496011yescharacteristic
511C4—21no—
512C4—21no—
513C4—81no—
514C4—241no—
515C4—242no—
516C4—242no—
517C4#473211yescharacteristic
518C4#473411yescharacteristic
519C4#477811yescharacteristic
520C4#478011yescharacteristic
521C4#511511yescharacteristic
522C4#513511yescharacteristic
523C4#515811yescharacteristic
524C4#517011yescharacteristic
525C3#1023111yescharacteristic, complemented, Sylow
526C2—82no—
527C2—242no—
528C2#6819611yescharacteristic
529C2#6820711yescharacteristic
530C2#6820811yescharacteristic
531C2#6821111yescharacteristic
532C2#6996111yescharacteristic
533C2#7020511yescharacteristic
534C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC23×C3#120
prime radical#82417C2
hypercenterC2×D4⋊C8D3
hypocenterC3#10231
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#5019C2semidirect
3. C2×#4846C2semidirect
4. C2×C3×D4⋊C8C2semidirect
5. #69777C2semidirect
6. C2×#4459C2semidirect
7. #67702C2semidirect
37. C2×D4⋊C8D3semidirect
78. #869D4semidirect
80. C3×C4⋊C8D4semidirect
88. C4⋊(C3⋊C8)D4semidirect
141. C2×D12SD16semidirect
231. C4⋊C8#67semidirect
525. C3#10231semidirect

Maximal subgroups

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

subgroupindex
#824172
C2×#50192
C2×#48462
C2×C3×D4⋊C82
#697772
C2×#44592
#677022
#102313