Finite Group Databasebeta

← #10697530

Subgroups — #10697530

Order 768.

712 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82417C211yesmaximal, characteristic
2C2×#5008C211yesmaximal, characteristic, complemented
3C2×#4842C211yesmaximal, characteristic, complemented
4C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
5#69771C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10235—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938—21no—
11C2×#930—21no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#832—21no—
17#5212—21no—
18C2×#821—21no—
19C2×#817—21no—
20C2×#816—22no—
21#5008—22no—
22#5008—22no—
23#4987—21no—
24C2×#808—21no—
25#4961—22no—
26#4935C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4842—22no—
29#4842—22no—
30C3×D4⋊C8—22no—
31#4459—22no—
32C2×#476—31no—
33C2×#476—31no—
34#2537—31no—
35#1956—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—
57#930—42no—
58#930—42no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D412yes—
67C22×C3⋊C8—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72#869—21no—
73#869—21no—
74#869—21no—
75#869D412yescomplemented
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yescomplemented
78C3×C4×C8—21no—
79C3×C4×C8—21no—
80#832—24no—
81C3⋊#95—22no—
82#821—24no—
83#817—24no—
84#816—44no—
85#808—24no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yescomplemented
88C2×C4⋊Q8—31no—
89C2×#121—61no—
90C2×#121—61no—
91C2×C4×D4—31no—
92C2×C4×D4C2×D311yescharacteristic
93#476—62no—
94#476—62no—
95#476—62no—
96#476—62no—
97C2×#106—61no—
98C2×#106—61no—
99C2×C4⋊C8—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#103—62no—
102C2×#102—61no—
103C2×#102—61no—
104C2×C4×C8C2×D311yescharacteristic
105#394—62no—
106#384—31no—
107#380—61no—
108#380—61no—
109D4⋊C8—22no—
110D4⋊C8—62no—
111C24×C3—21no—
112C23×D3—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4SD1612yes—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C2×#67—44no—
130C2×#67—44no—
131C2×#67—82no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—41no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12SD1612yes—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×Dic6—22no—
148C2×Dic6—42no—
149C2×Dic6—42no—
150C2×Dic6—41no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—41no—
154C2×C3×C8—41no—
155C2×C3×C8—41no—
156C2×C3×C8—41no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42#4111yescharacteristic
165#217—44no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169C4⋊Dic3—22no—
170C4⋊Dic3—21no—
171C4⋊Dic3—22no—
172C4⋊Dic3—21no—
173C4⋊Dic3—41no—
174C4⋊Dic3—41no—
175C4⋊Dic3—41no—
176C4⋊Dic3—41no—
177C4⋊Dic3—41no—
178C4⋊Dic3—41no—
179C4⋊Dic3—42no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C22×Q8—61no—
184C22×D4—31no—
185C22×D4—62no—
186C22×D4D1211yescharacteristic
187C23×C4—21no—
188C23×C4—61no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—61no—
192C4⋊Q8—32no—
193C4⋊Q8—61no—
194C4⋊Q8—61no—
195C4⋊Q8—61no—
196#121—122no—
197#121—122no—
198#121—122no—
199#121—122no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—61no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4—61no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4D1211yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—61no—
219C2×C42C22×D311yescharacteristic
220#106—64no—
221#106—64no—
222C4⋊C8—21no—
223C4⋊C8—32no—
224C4⋊C8—61no—
225C4⋊C8#6712yescomplemented
226#103—124no—
227#102—64no—
228#102—64no—
229C4×C8—21no—
230C4×C8—21no—
231#95—62no—
232C23×C3—22no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3#12011yescharacteristic
240C22×D3—22no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—41no—
250C3×D4—41no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12711yescharacteristic
266C2×C3×C4C2×SD1611yescharacteristic
267C2×C3×C4C2×SD1611yescharacteristic
268C2×C3×C4#13511yescharacteristic
269C2×C3×C4#13511yescharacteristic
270C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
271C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
272C2×C3×C4#13711yescharacteristic
273C2×C3×C4#13711yescharacteristic
274#67—88no—
275C2×Dic3—42no—
276C2×Dic3—41no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—42no—
282C2×Dic3—81no—
283C2×Dic3—81no—
284D12—22no—
285D12—22no—
286D12—41no—
287D12—41no—
288C4×D3—81no—
289C4×D3—81no—
290C4×D3—81no—
291C4×D3—81no—
292Dic6—44no—
293Dic6—82no—
294C3×C8—42no—
295C3×C8—42no—
296C3⋊C8—42no—
297C24—21no—
298C24—61no—
299C2×Q8—62no—
300C2×Q8—122no—
301C2×Q8—122no—
302C2×Q8—121no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—32no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—61no—
312C2×D4—61no—
313C2×D4—61no—
314C2×D4—124no—
315C2×D4—124no—
316C2×D4—242no—
317C2×D4#20412yes—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—61no—
326C22×C4—62no—
327C22×C4—121no—
328C22×C4—121no—
329C22×C4—121no—
330C22×C4—121no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4C2×D1211yescharacteristic
335C22×C4D3×D411yescharacteristic
336C22×C4D3×D411yescharacteristic
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—62no—
344C2×C8—121no—
345C2×C8—121no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—62no—
351C4⋊C4—61no—
352C4⋊C4—122no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C4⋊C4—121no—
356C4⋊C4—121no—
357C4⋊C4—121no—
358C4⋊C4—121no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C42#23511yescharacteristic
366C42D3×D411yescharacteristic
367C42D3×D411yescharacteristic
368C42C2×#6711yescharacteristic
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#39211yescharacteristic
377C22×C3#45011yescharacteristic
378C22×C3#45011yescharacteristic
379C22×C3#45211yescharacteristic
380C22×C3#45211yescharacteristic
381C22×C3#45311yescharacteristic
382C22×C3#45411yescharacteristic
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C2×D3—81no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—81no—
393C3×C4#44811yescharacteristic
394C3×C4#44811yescharacteristic
395C3×C4#44911yescharacteristic
396C3×C4#44911yescharacteristic
397C3×C4#46511yescharacteristic
398C3×C4#46511yescharacteristic
399C3×C4#46811yescharacteristic
400C3×C4#46811yescharacteristic
401Dic3—82no—
402Dic3—81no—
403Dic3—82no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—62no—
412C23—121no—
413C23—121no—
414C23—121no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23#88211yescharacteristic
419Q8—124no—
420Q8—242no—
421D4—22no—
422D4—22no—
423D4—41no—
424D4—41no—
425D4—62no—
426D4—62no—
427D4—121no—
428D4—121no—
429D4—248no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—121no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—122no—
443C2×C4—121no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4—241no—
452C2×C4#88211yescharacteristic
453C2×C4#88411yescharacteristic
454C2×C4#88411yescharacteristic
455C2×C4C2×#20411yescharacteristic
456C2×C4#90411yescharacteristic
457C2×C4#90811yescharacteristic
458C2×C4#90911yescharacteristic
459C2×C4#91111yescharacteristic
460C2×C4#91211yescharacteristic
461C2×C4D3×SD1611yescharacteristic
462C2×C4#91511yescharacteristic
463C2×C4#93011yescharacteristic
464C2×C4#93711yescharacteristic
465C2×C4#94011yescharacteristic
466C8—42no—
467C8—42no—
468C8—122no—
469C2×C3—82no—
470C2×C3#162911yescharacteristic
471C2×C3#163411yescharacteristic
472C2×C3#163511yescharacteristic
473C2×C3#163511yescharacteristic
474C2×C3#200011yescharacteristic
475C2×C3#200011yescharacteristic
476C2×C3#200611yescharacteristic
477D3—82no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—122no—
486C22—122no—
487C22—122no—
488C22—241no—
489C22—241no—
490C22—241no—
491C22—241no—
492C22#473111yescharacteristic
493C22#473311yescharacteristic
494C22#477611yescharacteristic
495C22#478311yescharacteristic
496C22#482211yescharacteristic
497C22#482711yescharacteristic
498C22#496011yescharacteristic
499C4—21no—
500C4—21no—
501C4—81no—
502C4—241no—
503C4—242no—
504C4—242no—
505C4#473411yescharacteristic
506C4#473811yescharacteristic
507C4#477811yescharacteristic
508C4#478011yescharacteristic
509C4#511411yescharacteristic
510C4#513411yescharacteristic
511C4#515811yescharacteristic
512C4#517011yescharacteristic
513C3#1023511yescharacteristic, complemented, Sylow
514C2—82no—
515C2—242no—
516C2#6818111yescharacteristic
517C2#6819511yescharacteristic
518C2#6820311yescharacteristic
519C2#6820411yescharacteristic
520C2#6996211yescharacteristic
521C2#7020511yescharacteristic
522C2#7039211yescharacteristic
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#10235
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×#5008C2semidirect
3. C2×#4842C2semidirect
4. C2×C3×D4⋊C8C2semidirect
5. #69771C2semidirect
6. C2×#4459C2semidirect
7. #67702C2semidirect
37. C2×D4⋊C8D3semidirect
75. #869D4semidirect
77. C3×C4⋊C8D4semidirect
87. C4⋊(C3⋊C8)D4semidirect
225. C4⋊C8#67semidirect
513. C3#10235semidirect

Maximal subgroups

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

subgroupindex
#824172
C2×#50082
C2×#48422
C2×C3×D4⋊C82
#697712
C2×#44592
#677022
#102353