Finite Group Databasebeta

← #10794884

Subgroups — #10794884

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5745C211yesmaximal, characteristic
2#83768C211yesmaximal, characteristic
3C2×#5124C211yesmaximal, characteristic, complemented
4C2×#4966C211yesmaximal, characteristic
5C2×#4964C211yesmaximal, characteristic, complemented
6C2×C3×#343C211yesmaximal, characteristic, complemented
7#71987C211yesmaximal, characteristic, complemented
8#69001C214yesmaximal, complemented
9#69000C214yesmaximal, complemented
10#20679—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5745—24no—
19#5745C418yescomplemented
20C2×#907C2214yes—
21C2×D3×M16—22no—
22C2×#901—22no—
23#5133C2212yes—
24#5133C2212yes—
25#5131—21no—
26#5124—24no—
27C2×#822C2214yescomplemented
28#5120—21no—
29#5113—22no—
30C2×#810—21no—
31#4966—22no—
32#4966C2214yes—
33C2×#809—21no—
34#4964—22no—
35#4964—22no—
36C2×#808C2212yescomplemented
37C2×#807C2212yescomplemented
38C3×#343C2214yescomplemented
39#4554C2212yescomplemented
40#4554C2212yescomplemented
41C2×#568—31no—
42C2×#430—31no—
43C2×#428—31no—
44#2896—31no—
45C2×#417—31no—
46#1816—34no—
47#1815—34no—
48#1742—31no—
49C2×#343D311yescharacteristic, complemented
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C22×#67—21no—
53C2×#235—21no—
54C2×#235—22no—
55C2×#235—21no—
56C2×#235—22no—
57C2×#235C2×C414yes—
58C2×#235D412yes—
59C2×#235Q814yes—
60C2×#235C2312yes—
61C22×D12D411yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6D411yescharacteristic
64C2×C3×M16—21no—
65C2×C3×M16—22no—
66C2×C3×M16—21no—
67C2×C3×M16—22no—
68C2×C3×M16—22no—
69C2×C3×M16C2×C414yes—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×#217—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4×Dic3—21no—
88C2×#208—21no—
89C2×#208—22no—
90C2×#208—22no—
91C2×#208C2×C414yes—
92C2×#208D414yes—
93C2×#208D412yes—
94C2×#208D412yes—
95C2×#208C2312yes—
96C22×C3⋊C8—21no—
97C22×C3⋊C8C2312yes—
98#907—28no—
99#907C2×C4116yescomplemented
100D3×M16—216no—
101#901—216no—
102C2×ΓL(1,𝔽25)—24no—
103C2×D3×C8—24no—
104#822—28no—
105#821—22no—
106#821—22no—
107#816—22no—
108#816—22no—
109#810—24no—
110#809—24no—
111#808—24no—
112#807—24no—
113C22×#41—31no—
114#568—38no—
115#568—64no—
116C2×#131—34no—
117C2×#131—62no—
118C22×M16—31no—
119C22×M16—31no—
120C22×M16—62no—
121C22×M16C2×D311yescharacteristic
122C22×M16C2×D311yescharacteristic
123C2×#117—31no—
124C22×C4⋊C4C2×D311yescharacteristic
125#430—64no—
126C2×#108—34no—
127#428—62no—
128#428—62no—
129C2×#107—32no—
130C2×#106—32no—
131#419—61no—
132#418—61no—
133#417—32no—
134#417—32no—
135#417—34no—
136#417—62no—
137#416—62no—
138C2×#103—61no—
139C2×#102—61no—
140#343—32no—
141#343—32no—
142#343C2×D314yescomplemented
143C24×C3#4111yescharacteristic
144C23×D3—21no—
145C22×C3×C4—22no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—82no—
149C22×C3×C4C22×C412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×Q812yes—
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—22no—
161C2×#67—24no—
162C2×#67—22no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166#235—24no—
167#235—24no—
168#235—24no—
169#235—24no—
170#235—24no—
171#235—22no—
172#235—22no—
173#235C4⋊C418yes—
174#235C22×C414yes—
175#235C2×Q814yes—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12C4⋊C414yes—
179C2×D12C2×D412yes—
180C2×C4×D3—24no—
181C2×C4×D3—22no—
182C2×C4×D3—24no—
183C2×C4×D3—22no—
184C2×Dic6—22no—
185C2×Dic6—21no—
186C2×Dic6C4⋊C414yes—
187C2×Dic6C2×D412yes—
188C3×M16—24no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—42no—
193C3×M16C22×C414yes—
194C2×C3×C8—24no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—42no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202#217—41no—
203#217—41no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4×Dic3—41no—
209C4×Dic3—41no—
210#208—22no—
211#208—22no—
212#208—42no—
213#208C22⋊C418yes—
214#208C22⋊C418yes—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—42no—
217C2×C3⋊C8C22×C414yes—
218C2×C3⋊C8C2×D414yes—
219C2×C3⋊C8C2×D414yes—
220ΓL(1,𝔽25)—216no—
221D3×C8—216no—
222C2×#41—34no—
223C2×#41—32no—
224C2×#41—34no—
225C2×#41—32no—
226C2×#41—61no—
227C2×#41—62no—
228C2×#41—61no—
229C2×#41—62no—
230C22×Q8—31no—
231C22×D4—31no—
232C22×D4—61no—
233C23×C4—21no—
234C23×C4—61no—
235C23×C4C22×D311yescharacteristic
236#131—316no—
237#131—616no—
238#131—68no—
239C2×M16—21no—
240C2×M16—22no—
241C2×M16—21no—
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—34no—
245C2×M16—32no—
246C2×M16—32no—
247C2×M16—34no—
248C2×M16—32no—
249C2×M16—64no—
250C2×M16—616no—
251C2×M16—61no—
252C2×M16—62no—
253C2×M16—62no—
254C2×M16C4×D314yes—
255C2×M16C22×D312yes—
256C2×M16C22×D312yes—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—32no—
260C22×C8—64no—
261C22×C8—61no—
262#117—61no—
263#117—61no—
264#117—61no—
265#117—61no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—61no—
275C2×C42—61no—
276#108—68no—
277#107—64no—
278#106—64no—
279#103—64no—
280#103—62no—
281#103—62no—
282#102—64no—
283#102—62no—
284#102—62no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3#11712yes—
288C23×C3#12111yescharacteristic
289C23×C3#12111yescharacteristic
290C23×C3C22⋊Q812yes—
291C23×C3C2×#4111yescharacteristic
292C22×D3—22no—
293C22×D3—24no—
294C22×D3—81no—
295C22×D3—81no—
296C2×C3×C4—42no—
297C2×C3×C4—82no—
298C2×C3×C4—82no—
299C2×C3×C4—82no—
300C2×C3×C4C2×C22⋊C412yes—
301C2×C3×C4C2×C22⋊C412yes—
302C2×C3×C4C2×C4⋊C412yes—
303C2×C3×C4C2×C4⋊C412yes—
304C2×C3×C4#11712yes—
305C2×C3×C4#12112yes—
306C2×C3×C4#12112yes—
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12111yescharacteristic
309C2×C3×C4C22⋊Q812yes—
310C2×C3×C4C22⋊Q812yes—
311C2×C3×C4C22⋊Q812yes—
312C2×C3×C4C23×C411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×Q811yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318#67—24no—
319#67—24no—
320#67—28no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—81no—
326C2×Dic3—81no—
327D12—22no—
328D12—24no—
329D12C2×C4⋊C414yes—
330C4×D3—24no—
331C4×D3—24no—
332C4×D3—28no—
333Dic6—22no—
334Dic6—24no—
335Dic6C2×C4⋊C414yes—
336C3×C8—24no—
337C3×C8—42no—
338C3⋊C8—42no—
339C3⋊C8C2×C22⋊C418yes—
340C24—61no—
341C24#23511yescharacteristic
342#41—38no—
343#41—34no—
344#41—34no—
345#41—64no—
346#41—64no—
347#41—64no—
348#41—64no—
349#41—64no—
350#41—62no—
351#41—62no—
352C2×Q8—34no—
353C2×Q8—32no—
354C2×Q8—62no—
355C2×Q8—61no—
356C2×D4—34no—
357C2×D4—32no—
358C2×D4—64no—
359C2×D4—62no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—61no—
363C2×D4—62no—
364C22×C4—22no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—61no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—62no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—82no—
375C22×C4C2×C4×D312yes—
376C22×C4#23511yescharacteristic
377C22×C4#23512yes—
378C22×C4D3×D412yes—
379C22×C4D3×D411yescharacteristic
380C22×C4D3×D411yescharacteristic
381C22×C4D3×D411yescharacteristic
382C22×C4D3×D411yescharacteristic
383C22×C4D3×Q812yes—
384C22×C4C23×D311yescharacteristic
385M16—24no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—38no—
390M16—38no—
391M16—42no—
392M16—616no—
393M16—62no—
394M16—62no—
395M16—122no—
396M16C2×C4×D314yes—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—34no—
401C2×C8—34no—
402C2×C8—34no—
403C2×C8—42no—
404C2×C8—616no—
405C2×C8—62no—
406C2×C8—122no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C22⋊C4—121no—
416C22⋊C4—121no—
417C42—121no—
418C42—121no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3#38012yes—
428C22×C3#38012yes—
429C22×C3C2×#11711yescharacteristic
430C22×C3C2×#12111yescharacteristic
431C22×C3C2×C22⋊Q811yescharacteristic
432C2×D3—24no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—84no—
436C3×C4—82no—
437C3×C4#38012yes—
438C3×C4#38012yes—
439C3×C4C22×C22⋊C411yescharacteristic
440C3×C4C22×C4⋊C411yescharacteristic
441C3×C4C2×#12111yescharacteristic
442C3×C4C2×C22⋊Q811yescharacteristic
443Dic3—24no—
444Dic3—81no—
445Dic3—81no—
446C23—41no—
447C23—41no—
448C23—62no—
449C23—64no—
450C23—241no—
451C23—241no—
452C23#87212yes—
453C23#88411yescharacteristic
454C23#88411yescharacteristic
455C23#89612yes—
456C23C2×#23511yescharacteristic
457Q8—34no—
458Q8—64no—
459Q8—62no—
460D4—34no—
461D4—68no—
462D4—64no—
463D4—64no—
464D4—64no—
465D4—62no—
466C2×C4—42no—
467C2×C4—64no—
468C2×C4—62no—
469C2×C4—68no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4#87212yes—
480C2×C4D3×C22⋊C412yes—
481C2×C4D3×C22⋊C412yes—
482C2×C4#88411yescharacteristic
483C2×C4#88411yescharacteristic
484C2×C4#88412yes—
485C2×C4#88412yes—
486C2×C4D3×C4⋊C412yes—
487C2×C4D3×C4⋊C412yes—
488C2×C4#89612yes—
489C2×C4#89612yes—
490C2×C4#89612yes—
491C2×C4C22×C4×D311yescharacteristic
492C2×C4C2×#23511yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×D3×D411yescharacteristic
495C2×C4C2×D3×D411yescharacteristic
496C2×C4C2×D3×Q811yescharacteristic
497C8—24no—
498C8—38no—
499C8—42no—
500C8—122no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—22no—
505C2×C3C2×#38011yescharacteristic
506C2×C3#306811yescharacteristic
507C2×C3#306911yescharacteristic
508D3—44no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—42no—
516C22—42no—
517C22—64no—
518C22—122no—
519C22—122no—
520C22—244no—
521C22#466712yes—
522C22#466712yes—
523C22C2×#87211yescharacteristic
524C22C2×#88411yescharacteristic
525C22C2×#89611yescharacteristic
526C4—64no—
527C4—82no—
528C4—241no—
529C4—241no—
530C4#466712yes—
531C4#466712yes—
532C4C2×D3×C22⋊C411yescharacteristic
533C4C2×#88411yescharacteristic
534C4C2×D3×C4⋊C411yescharacteristic
535C4C2×#89611yescharacteristic
536C3#2067911yescharacteristic, complemented, Sylow
537C2—22no—
538C2—22no—
539C2—21no—
540C2—21no—
541C2—124no—
542C2C2×#466711yescharacteristic
543C2#7870311yescharacteristic
544C2#7871211yescharacteristic
subgroupG/H
centerC22C2×#884
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#343C2
socleC22×C3C2×#121
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#343D3
hypocenterC3#20679
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. C2×#5124C2semidirect
5. C2×#4964C2semidirect
6. C2×C3×#343C2semidirect
7. #71987C2semidirect
8. #69001C2semidirect
9. #69000C2semidirect
19. #5745C4semidirect
27. C2×#822C22semidirect
36. C2×#808C22semidirect
37. C2×#807C22semidirect
38. C3×#343C22semidirect
39. #4554C22semidirect
40. #4554C22semidirect
49. C2×#343D3semidirect
99. #907C2×C4semidirect
142. #343C2×D3semidirect
536. C3#20679semidirect

Maximal subgroups

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

subgroupindex
C2×#57452
#837682
C2×#51242
C2×#49662
C2×#49642
C2×C3×#3432
#719872
#690012
#690002
#206793