Finite Group Databasebeta

← #10944295

Subgroups — #10944295

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#131C211yesmaximal, characteristic, complemented
2#84098C211yesmaximal, characteristic
3#84094C211yesmaximal, characteristic, complemented
4C2×#5134C212yesmaximal, complemented
5C2×#5114C211yesmaximal, characteristic, complemented
6#71990C211yesmaximal, characteristic, complemented
7#71989C211yesmaximal, characteristic, complemented
8#71988C212yesmaximal, complemented
9#71973C212yesmaximal, complemented
10#71972C212yesmaximal, complemented
11#71971C211yesmaximal, characteristic, complemented
12#32938—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#131—28no—
15C2×C3×#131—24no—
16C22×C3×M16C2211yescharacteristic
17C22×C3×M16C2212yes—
18C23×C3×C8C2212yes—
19C23×C3×C8C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#930C2212yes—
22C2×#924C2211yescharacteristic
23C2×#924C2212yes—
24C22×#204C2211yescharacteristic
25#5231—21no—
26#5230—21no—
27#5223—21no—
28#5223—22no—
29#5222—21no—
30#5222—22no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2212yes—
34#5212C2212yes—
35#5134—28no—
36#5114—24no—
37C2×#821C2211yescharacteristic
38C2×#821C2212yes—
39C2×#818C2212yes—
40C2×#817C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2212yes—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C22×#131D311yescharacteristic, complemented
47C2×#468—32no—
48C2×#465—31no—
49#2616—31no—
50#2615—31no—
51#1895—31no—
52#1894—31no—
53#1893—32no—
54#1892—32no—
55#1809—32no—
56#1808—31no—
57C2×C3×#41—24no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4D412yes—
63C22×C3×D4D411yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C22×#67—21no—
67C22×#67—22no—
68C22×D12C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C3×#131—232no—
71C2×C3×M16—22no—
72C2×C3×M16—24no—
73C2×C3×M16—24no—
74C2×C3×M16—24no—
75C2×C3×M16D418yescomplemented
76C22×C3×C8—24no—
77C22×C3×C8—24no—
78C22×C3×C8—24no—
79C22×C3×C8—24no—
80C22×C3×C8D414yescomplemented
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2312yes—
83C22×C3×C8C2311yescharacteristic
84C2×#217—21no—
85C2×#217—22no—
86C2×#217—21no—
87C2×#217—22no—
88C2×#217—21no—
89C2×#217—22no—
90#930—24no—
91#930—22no—
92#930—22no—
93#930—24no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—22no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2312yes—
100#924—24no—
101#924—22no—
102#924—22no—
103#924—24no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—22no—
106C2×#204—24no—
107C2×#204—22no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112#821—24no—
113#821—22no—
114#818—24no—
115#817—22no—
116#816—24no—
117#816—22no—
118C22×#41C2×D311yescharacteristic
119C22×SD16—31no—
120C2×#131—28no—
121C2×#131—24no—
122C22×M16C2×D311yescharacteristic
123C22×M16C2×D312yes—
124C23×C8C2×D312yes—
125C23×C8C2×D311yescharacteristic
126C2×C22⋊Q8—32no—
127C2×C22⋊Q8—31no—
128C2×#121—32no—
129C2×#121—31no—
130#468—68no—
131#465—64no—
132C2×#107—32no—
133C2×#106—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—31no—
137C2×#102—32no—
138#395—61no—
139#394—61no—
140#394—62no—
141#393—61no—
142#392—61no—
143#392—62no—
144#380—31no—
145#380—31no—
146#380—32no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3C2×D411yescharacteristic
152C24×C3C2×D412yes—
153C23×D3—21no—
154C3×#41—28no—
155C3×#41—216no—
156C3×#41—28no—
157C2×C3×Q8—22no—
158C2×C3×Q8—24no—
159C2×C3×D4—24no—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4114yes—
176C22×C3×C4C2411yescharacteristic
177C2×#67—44no—
178C2×#67—42no—
179C2×#67—82no—
180C2×#67—82no—
181C2×#67—82no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C2×D12—22no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—22no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—22no—
195C3×M16—28no—
196C3×M16—216no—
197C2×C3×C8—216no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—28no—
202C2×C3×C8C2×D412yes—
203C2×C3×C8C2×D414yes—
204C2×C3×C8C2×D412yes—
205C2×C3×C8C2×D414yes—
206#217—44no—
207#217—42no—
208#217—81no—
209#217—82no—
210#217—81no—
211#217—82no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—42no—
214C4⋊Dic3—21no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—21no—
219C4⋊Dic3—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222Dic3⋊C4—44no—
223Dic3⋊C4—42no—
224#204—44no—
225C2×#41—22no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—22no—
229C22×Q8—31no—
230C22×Q8D1211yescharacteristic
231C22×D4—31no—
232C22×D4—62no—
233C22×D4—61no—
234C22×D4D1211yescharacteristic
235C22×D4D1212yes—
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C2×SD16—62no—
239C2×SD16—64no—
240#131—232no—
241C2×M16—22no—
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C2×M16#6718yescomplemented
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8#6714yescomplemented
251C22×C8C22×D311yescharacteristic
252C22×C8C22×D312yes—
253C22×C8C22×D311yescharacteristic
254C22⋊Q8—64no—
255C22⋊Q8—62no—
256C22⋊Q8—62no—
257C22⋊Q8—64no—
258#121—64no—
259#121—62no—
260#121—62no—
261#121—64no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—62no—
276#107—64no—
277#106—62no—
278#103—62no—
279#103—64no—
280#102—62no—
281#102—64no—
282#95—61no—
283#95—62no—
284#95—61no—
285#95—62no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3#12112yes—
297C23×C3#12114yes—
298C23×C3C22×D411yescharacteristic
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—41no—
304C3×Q8—28no—
305C3×D4—216no—
306C3×D4—28no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4#12112yes—
318C2×C3×C4#12114yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D412yes—
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4112yes—
323#67—84no—
324#67—88no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338D12—41no—
339D12—42no—
340D12—41no—
341Dic6—41no—
342Dic6—42no—
343Dic6—41no—
344C3×C8—22no—
345C3×C8—24no—
346C3×C8#12014yes—
347C24—61no—
348C24C2×D1212yes—
349C24C2×D1211yescharacteristic
350#41—28no—
351#41—216no—
352#41—28no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—61no—
356C2×Q8—62no—
357C2×Q8—61no—
358C2×Q8—62no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—61no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4—124no—
370C2×D4—242no—
371C2×D4—242no—
372C2×D4—242no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4C2×D1212yes—
385C22×C4C2×D1211yescharacteristic
386C22×C4#23512yes—
387C22×C4#23514yes—
388C22×C4C2×#6714yes—
389C22×C4C2×#6712yes—
390C22×C4C23×D311yescharacteristic
391SD16—124no—
392M16—28no—
393M16—216no—
394C2×C8—216no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—28no—
399C2×C8C2×#6712yes—
400C2×C8C2×#6714yes—
401C2×C8C2×#6712yes—
402C2×C8C2×#6714yes—
403C4⋊C4—61no—
404C4⋊C4—62no—
405C4⋊C4—61no—
406C4⋊C4—62no—
407C4⋊C4—61no—
408C4⋊C4—62no—
409C4⋊C4—61no—
410C4⋊C4—62no—
411C4⋊C4—122no—
412C4⋊C4—124no—
413C22⋊C4—122no—
414C22⋊C4—124no—
415C22⋊C4—122no—
416C22⋊C4—124no—
417C22⋊C4—242no—
418C22⋊C4—241no—
419C22⋊C4—242no—
420C22⋊C4—241no—
421C22×C3—22no—
422C22×C3—24no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3C2×#12111yescharacteristic
434C22×C3C2×#12112yes—
435C22×C3#57611yescharacteristic
436C22×C3#57712yes—
437C22×C3#57811yescharacteristic
438C2×D3—81no—
439C2×D3—81no—
440C2×D3—81no—
441C2×D3—82no—
442C2×D3—82no—
443C3×C4—24no—
444C3×C4—22no—
445C3×C4C2×#12011yescharacteristic
446C3×C4#52512yes—
447C3×C4#52511yescharacteristic
448Dic3—81no—
449Dic3—81no—
450Dic3—81no—
451Dic3—82no—
452Dic3—82no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—122no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—122no—
468C23#93012yes—
469C23#93014yes—
470C23C22×D1211yescharacteristic
471Q8—28no—
472Q8—121no—
473Q8—122no—
474Q8—121no—
475D4—216no—
476D4—28no—
477D4—121no—
478D4—122no—
479D4—121no—
480D4—244no—
481D4—248no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—122no—
505C2×C4#93012yes—
506C2×C4#93014yes—
507C2×C4C2×#23511yescharacteristic
508C2×C4C2×#23512yes—
509C2×C4C22×#6711yescharacteristic
510C2×C4C22×#6712yes—
511C8—22no—
512C8—24no—
513C8#95314yes—
514C2×C3—42no—
515C2×C3—44no—
516C2×C3#284711yescharacteristic
517C2×C3#305711yescharacteristic
518C2×C3#306111yescharacteristic
519C2×C3#306212yes—
520C2×C3#306312yes—
521D3—161no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—22no—
526C22—24no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—241no—
535C22—241no—
536C22—241no—
537C22—242no—
538C22—242no—
539C22#575112yes—
540C22#575211yescharacteristic
541C22#575311yescharacteristic
542C22C2×#93011yescharacteristic
543C22C2×#93012yes—
544C4—24no—
545C4—22no—
546C4—241no—
547C4—241no—
548C4—241no—
549C4—242no—
550C4—242no—
551C4#579112yes—
552C4#579111yescharacteristic
553C4C2×#95311yescharacteristic
554C3#3293811yescharacteristic, complemented, Sylow
555C2—42no—
556C2—44no—
557C2—481no—
558C2#8379811yescharacteristic
559C2#8380111yescharacteristic
560C2#8380212yes—
561C2#8380812yes—
562C2#8403211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#131C2
socleC23×C3C22×D4
prime radical#84098C2
hypercenterC22×#131D3
hypocenterC3#32938
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C22×#131

Complements

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

NH ≅ G/Ntype
1. C22×C3×#131C2semidirect
3. #84094C2semidirect
4. C2×#5134C2semidirect
5. C2×#5114C2semidirect
6. #71990C2semidirect
7. #71989C2semidirect
8. #71988C2semidirect
9. #71973C2semidirect
10. #71972C2semidirect
11. #71971C2semidirect
46. C22×#131D3semidirect
75. C2×C3×M16D4semidirect
80. C22×C3×C8D4semidirect
245. C2×M16#67semidirect
250. C22×C8#67semidirect
554. C3#32938semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1312
#840982
#840942
C2×#51342
C2×#51142
#719902
#719892
#719882
#719732
#719722
#719712
#329383