Finite Group Databasebeta

← #10944298

Subgroups — #10944298

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×#948C211yesmaximal, characteristic, complemented
2C3×#2616C211yesmaximal, characteristic, complemented
3#84098C211yesmaximal, characteristic
4C2×#5049C212yesmaximal, complemented
5C2×#5048C211yesmaximal, characteristic, complemented
6#71697C211yesmaximal, characteristic, complemented
7#71696C212yesmaximal, complemented
8#69517C211yesmaximal, characteristic, complemented
9#69515C212yesmaximal, complemented
10#69513C212yesmaximal, complemented
11#69512C211yesmaximal, characteristic, complemented
12#32938—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#948—28no—
17C2×#948—24no—
18C22×#215C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×#930C2212yes—
21C22×#208C2211yescharacteristic
22C22×#208C2212yes—
23C23×C3⋊C8C2212yes—
24C23×C3⋊C8C2211yescharacteristic
25C3×#395—21no—
26C3×#394—21no—
27C3×#394—22no—
28C3×#380C2211yescharacteristic
29C3×#380C2212yes—
30#5231—21no—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33#5223—22no—
34#5223—21no—
35#5212C2212yes—
36#5212C2211yescharacteristic
37#5049—28no—
38#5048—24no—
39C2×#809C2211yescharacteristic
40C2×#809C2212yes—
41C2×#808C2211yescharacteristic
42C2×#807C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#468—32no—
48C2×#465—31no—
49#2616D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
63C22×C3×D4D412yes—
64C22×C3×D4D411yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C22×#67—22no—
69C22×D12C2311yescharacteristic
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×#217—22no—
83C2×#217—21no—
84#948—232no—
85C2×#215—22no—
86C2×#215—24no—
87#930—22no—
88#930—24no—
89#930—22no—
90#930—24no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×#208—22no—
96C2×#208—24no—
97C2×#208—24no—
98C2×#208—24no—
99C2×#208D418yescomplemented
100C22×C3⋊C8—24no—
101C22×C3⋊C8—24no—
102C22×C3⋊C8—24no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8D414yescomplemented
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C3×#95—21no—
109C3×#95—22no—
110#835—22no—
111#835—24no—
112C3⋊#95—22no—
113C3⋊#95—21no—
114#809—22no—
115#809—24no—
116#808—22no—
117#807—24no—
118C22×#41C2×D311yescharacteristic
119C22×SD16—31no—
120C2×#131—64no—
121C2×#131—68no—
122C22×M16—32no—
123C22×M16—31no—
124C23×C8—31no—
125C23×C8—32no—
126C2×C22⋊Q8C2×D312yes—
127C2×C22⋊Q8C2×D311yescharacteristic
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—21no—
139#394—21no—
140#394—22no—
141#393—61no—
142#392—61no—
143#392—62no—
144#380—31no—
145#380—32no—
146#380C2×D311yescharacteristic
147#380C2×D312yes—
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×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4#4112yes—
182C22×C3×C4#4114yes—
183C22×C3×C4C2411yescharacteristic
184C2×#67—42no—
185C2×#67—44no—
186C2×#67—82no—
187C2×#67—82no—
188C2×#67—82no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C2×D12—21no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—44no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—81no—
204C3×C22⋊C4—82no—
205#217—81no—
206#217—82no—
207#215—44no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—44no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214#208—28no—
215#208—216no—
216C2×C3⋊C8—216no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—28no—
221C2×C3⋊C8C2×D412yes—
222C2×C3⋊C8C2×D414yes—
223C2×C3⋊C8C2×D414yes—
224C2×C3⋊C8C2×D412yes—
225C2×#41—22no—
226C2×#41—24no—
227C2×#41—22no—
228C2×#41—24no—
229C22×Q8#6711yescharacteristic
230C22×Q8C22×D311yescharacteristic
231C22×D4—31no—
232C22×D4—62no—
233C22×D4—61no—
234C22×D4#6712yes—
235C22×D4#6711yescharacteristic
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D312yes—
238C2×SD16—62no—
239C2×SD16—64no—
240#131—632no—
241C2×M16—38no—
242C2×M16—64no—
243C2×M16—64no—
244C2×M16—64no—
245C2×M16—62no—
246C22×C8—31no—
247C22×C8—34no—
248C22×C8—31no—
249C22×C8—32no—
250C22×C8—64no—
251C22×C8—64no—
252C22×C8—64no—
253C22×C8—64no—
254C22⋊Q8—22no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258#121—62no—
259#121—64no—
260#121—64no—
261#121—62no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—61no—
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—21no—
283#95—22no—
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×Q8—41no—
306C3×Q8—41no—
307C3×Q8—42no—
308C3×D4—216no—
309C3×D4—28no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4#12112yes—
329C2×C3×C4#12114yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D412yes—
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4112yes—
334#67—84no—
335#67—88no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341D12—41no—
342D12—41no—
343D12—42no—
344C3⋊C8—22no—
345C3⋊C8—24no—
346C3⋊C8#12014yes—
347C24—61no—
348C24C2×#6711yescharacteristic
349C24C2×#6712yes—
350#41—28no—
351#41—28no—
352#41—216no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—22no—
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—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4D3×D412yes—
385C22×C4D3×D414yes—
386C22×C4#23712yes—
387C22×C4#23714yes—
388C22×C4C2×#6712yes—
389C22×C4C2×#6711yescharacteristic
390C22×C4C23×D311yescharacteristic
391SD16—124no—
392M16—616no—
393M16—68no—
394C2×C8—32no—
395C2×C8—34no—
396C2×C8—32no—
397C2×C8—34no—
398C2×C8—68no—
399C2×C8—616no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—61no—
410C4⋊C4—62no—
411C4⋊C4—61no—
412C4⋊C4—62no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—81no—
416C22⋊C4—82no—
417C22⋊C4—122no—
418C22⋊C4—124no—
419C22⋊C4—241no—
420C22⋊C4—242no—
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×C4—81no—
446C3×C4—81no—
447C3×C4—82no—
448C3×C4C2×#12011yescharacteristic
449C3×C4#52512yes—
450C3×C4#52511yescharacteristic
451Dic3—81no—
452Dic3—82no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—121no—
464C23—122no—
465C23—121no—
466C23—122no—
467C23—121no—
468C23#93912yes—
469C23#93914yes—
470C23C22×#6711yescharacteristic
471Q8—28no—
472Q8—41no—
473Q8—42no—
474Q8—41no—
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—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4#93912yes—
506C2×C4#93914yes—
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×D3×D412yes—
509C2×C4C2×#23711yescharacteristic
510C2×C4C2×#23712yes—
511C8—34no—
512C8—62no—
513C8—64no—
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—24no—
526C22—22no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—241no—
535C22—241no—
536C22—242no—
537C22—241no—
538C22—242no—
539C22C2×#93911yescharacteristic
540C22C2×#93912yes—
541C22#583511yescharacteristic
542C22#583712yes—
543C22#583811yescharacteristic
544C4—24no—
545C4—22no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4—241no—
550C4—242no—
551C4#558611yescharacteristic
552C4#558612yes—
553C4D3×#12011yescharacteristic
554C3#3293811yescharacteristic, complemented, Sylow
555C2—42no—
556C2—44no—
557C2—481no—
558C2#8260111yescharacteristic
559C2#8339811yescharacteristic
560C2#8340012yes—
561C2#8342111yescharacteristic
562C2#8342212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2616C2
socleC23×C3C22×D4
prime radical#84098C2
hypercenter#2616D3
hypocenterC3#32938
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2616

Complements

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

NH ≅ G/Ntype
1. C22×#948C2semidirect
2. C3×#2616C2semidirect
4. C2×#5049C2semidirect
5. C2×#5048C2semidirect
6. #71697C2semidirect
7. #71696C2semidirect
8. #69517C2semidirect
9. #69515C2semidirect
10. #69513C2semidirect
11. #69512C2semidirect
49. #2616D3semidirect
99. C2×#208D4semidirect
104. C22×C3⋊C8D4semidirect
554. C3#32938semidirect

Maximal subgroups

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

subgroupindex
C22×#9482
C3×#26162
#840982
C2×#50492
C2×#50482
#716972
#716962
#695172
#695152
#695132
#695122
#329383