Finite Group Databasebeta

← #10823201

Subgroups — #10823201

Order 768.

1458 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic, complemented
3C2×#5225C211yesmaximal, characteristic, complemented
4C2×#4966C211yesmaximal, characteristic
5#82070C211yesmaximal, characteristic
6C2×#4924C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#69681C214yesmaximal, complemented
9#69680C214yesmaximal, complemented
10#23547—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×#216C2212yes—
16C2×#942—24no—
17C2×#942—24no—
18C2×#942—24no—
19C22×#215C2212yes—
20C2×#926C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C3×#391—21no—
24C3×#391—21no—
25C3×#391C2214yes—
26C2×C3×#95—22no—
27#5225—24no—
28C2×#835C2214yes—
29#5213—22no—
30C2×#810C2212yes—
31#4966—24no—
32C2×#809C2212yes—
33#4939C2212yes—
34C4×(C2×D3)⋊C4—21no—
35C4×(C2×D3)⋊C4—21no—
36#4929C2212yes—
37#4924—24no—
38C2×C4⋊(C3⋊C8)C2214yes—
39#4529C2214yes—
40C22×#137—31no—
41C2×#423—31no—
42C2×#417—31no—
43C2×#419—31no—
44#2297—31no—
45C2×#391D311yescharacteristic, complemented
46#2056—34no—
47#2055—34no—
48#1736—31no—
49C22×C3×Q8—21no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8D414yes—
56C22×C3×Q8C2312yes—
57C23×C3×C4—21no—
58C23×C3×C4—21no—
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235D414yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6C2311yescharacteristic
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2312yes—
80C2×#217—21no—
81C2×#216—24no—
82C2×#216—24no—
83C2×#216—24no—
84#942—232no—
85C2×#215—24no—
86C2×#215—24no—
87C2×#215—24no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4×Dic3—21no—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208D414yes—
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2312yes—
102C3×#95—24no—
103C3×#95—24no—
104#835—28no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#810—24no—
108#809—24no—
109C4⋊(C3⋊C8)—28no—
110C22×#41—31no—
111C23×Q8C2×D311yescharacteristic
112C2×#137—64no—
113C2×#137—64no—
114C2×#137—64no—
115C22×Q16—32no—
116C22×SD16—32no—
117C22×M16—31no—
118C22×M16—31no—
119C2×#117—31no—
120C22×C4⋊C4C2×D311yescharacteristic
121C22×C42C2×D311yescharacteristic
122#423—64no—
123C2×C4⋊C8—34no—
124#419—64no—
125#417—64no—
126C2×#103—34no—
127C2×#103—32no—
128C2×#102—32no—
129#391—21no—
130#391—21no—
131#391—32no—
132#391C2×D314yes—
133#390—32no—
134#386—62no—
135C4×C22⋊C4—61no—
136C4×C22⋊C4—61no—
137C2×#95—22no—
138#340—34no—
139C24×C3C2×D411yescharacteristic
140C23×D3—21no—
141C2×C3×Q8—24no—
142C2×C3×Q8—22no—
143C2×C3×Q8—28no—
144C2×C3×Q8—28no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—42no—
150C2×C3×Q8—42no—
151C2×C3×Q8—44no—
152C2×C3×Q8—44no—
153C2×C3×Q8C2×D414yes—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—44no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4#4112yes—
173C22×C3×C4C2411yescharacteristic
174C2×#67—24no—
175C2×#67—42no—
176C2×#67—42no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179#235—28no—
180#235—28no—
181#235—44no—
182#235—44no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12C2×D412yes—
189C2×C4×D3—24no—
190C2×C4×D3—42no—
191C2×C4×D3—42no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6C2×D412yes—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—42no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—22no—
205#217—42no—
206#216—216no—
207#215—216no—
208(C2×D3)⋊C4—44no—
209(C2×D3)⋊C4—44no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4×Dic3—42no—
213#208—28no—
214#208—44no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8C2×D414yes—
221C2×#41—34no—
222C2×#41—62no—
223C2×#41—62no—
224C2×#41—62no—
225C2×#41—62no—
226C2×#41—62no—
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—31no—
234C22×Q8D1214yes—
235C22×Q8C22×D312yes—
236C22×D4—31no—
237C22×D4—61no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—61no—
241C23×C4#6712yes—
242C23×C4C22×D311yescharacteristic
243#137—632no—
244C2×Q16—64no—
245C2×Q16—64no—
246C2×Q16—64no—
247C2×SD16—64no—
248C2×SD16—64no—
249C2×SD16—64no—
250C2×M16—34no—
251C2×M16—62no—
252C2×M16—62no—
253C2×M16—62no—
254C2×M16—62no—
255C2×M16—62no—
256C22×C8—32no—
257C22×C8—32no—
258#117—62no—
259#117—62no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—61no—
271C2×C42#6711yescharacteristic
272C2×C42#6711yescharacteristic
273C2×C42C22×D312yes—
274C4⋊C8—68no—
275#103—64no—
276#103—68no—
277#102—64no—
278#95—24no—
279#95—24no—
280#95—62no—
281#95—62no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3#12112yes—
285C23×C3#12112yes—
286C23×C3#12112yes—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C3×Q8—24no—
294C3×Q8—24no—
295C3×Q8—42no—
296C3×Q8—42no—
297C3×Q8—48no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—48no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—48no—
308C2×C3×C4—42no—
309C2×C3×C4—44no—
310C2×C3×C4—84no—
311C2×C3×C4#12014yes—
312C2×C3×C4#12112yes—
313C2×C3×C4#12112yes—
314C2×C3×C4#12112yes—
315C2×C3×C4#12314yes—
316C2×C3×C4#12314yes—
317C2×C3×C4#12414yes—
318C2×C3×C4#13714yescomplemented
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325#67—48no—
326C2×Dic3—22no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332D12—22no—
333D12—22no—
334D12—41no—
335D12—41no—
336C4×D3—48no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341C3⋊C8—24no—
342C3⋊C8—42no—
343C24—61no—
344C24C2×#6711yescharacteristic
345#41—68no—
346#41—68no—
347#41—124no—
348#41—124no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—32no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—61no—
366C2×Q8—61no—
367C2×Q8C2×D1214yes—
368C2×D4—32no—
369C2×D4—64no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—122no—
376C2×D4—122no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—61no—
388C22×C4—64no—
389C22×C4—62no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4C2×D1212yes—
393C22×C4#23512yes—
394C22×C4D3×D412yes—
395C22×C4D3×D412yes—
396C22×C4#23712yes—
397C22×C4#23712yes—
398C22×C4C2×#6712yes—
399C22×C4C2×#6712yes—
400C22×C4C2×#6711yescharacteristic
401C22×C4C23×D311yescharacteristic
402Q16—616no—
403SD16—616no—
404M16—68no—
405M16—124no—
406C2×C8—34no—
407C2×C8—62no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—62no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C22⋊C4—124no—
418C22⋊C4—122no—
419C22⋊C4—124no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—122no—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3#39414yes—
430C22×C3C2×#12111yescharacteristic
431C22×C3C2×#12111yescharacteristic
432C22×C3C2×#12111yescharacteristic
433C22×C3C2×#13712yes—
434C22×C3#57512yes—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C3×C4—24no—
441C3×C4—44no—
442C3×C4—42no—
443C3×C4—82no—
444C3×C4#39414yes—
445C3×C4C2×#12011yescharacteristic
446C3×C4C2×#12311yescharacteristic
447C3×C4C2×#12311yescharacteristic
448C3×C4C2×#12411yescharacteristic
449Dic3—42no—
450Dic3—82no—
451C23—22no—
452C23—22no—
453C23—62no—
454C23—121no—
455C23—121no—
456C23—122no—
457C23—122no—
458C23#93012yes—
459C23#93812yes—
460C23#93912yes—
461C23C22×#6711yescharacteristic
462Q8—24no—
463Q8—24no—
464Q8—48no—
465Q8—42no—
466Q8—42no—
467Q8—62no—
468Q8—62no—
469Q8—121no—
470Q8—121no—
471D4—62no—
472D4—62no—
473D4—128no—
474D4—121no—
475D4—121no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—62no—
489C2×C4—84no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—128no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4#88214yes—
497C2×C4#88314yes—
498C2×C4#88514yes—
499C2×C4#88614yes—
500C2×C4#93012yes—
501C2×C4#93812yes—
502C2×C4#93912yes—
503C2×C4#94214yescomplemented
504C2×C4C22×D1211yescharacteristic
505C2×C4C2×#23511yescharacteristic
506C2×C4C2×D3×D411yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×#23711yescharacteristic
509C2×C4C2×#23711yescharacteristic
510C8—64no—
511C8—122no—
512C2×C3—24no—
513C2×C3C2×#39411yescharacteristic
514C2×C3#303712yes—
515C2×C3#303911yescharacteristic
516C2×C3#304011yescharacteristic
517C2×C3#305612yes—
518D3—82no—
519C22—22no—
520C22—24no—
521C22—22no—
522C22—24no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—242no—
527C22—242no—
528C22#496114yes—
529C22C2×#93011yescharacteristic
530C22C2×#93811yescharacteristic
531C22C2×#93911yescharacteristic
532C22C2×#94212yes—
533C22#582212yes—
534C4—24no—
535C4—42no—
536C4—44no—
537C4—82no—
538C4—122no—
539C4—242no—
540C4#496114yes—
541C4C2×#88211yescharacteristic
542C4C2×#88311yescharacteristic
543C4C2×#88511yescharacteristic
544C4C2×#88611yescharacteristic
545C3#2354711yescharacteristic, complemented, Sylow
546C2—24no—
547C2—242no—
548C2C2×#496111yescharacteristic
549C2#8244112yes—
550C2#8273212yes—
551C2#8294511yescharacteristic
552C2#8320911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#391C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#391D3
hypocenterC3#23547
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×C3×#391C2semidirect
3. C2×#5225C2semidirect
6. C2×#4924C2semidirect
7. #71558C2semidirect
8. #69681C2semidirect
9. #69680C2semidirect
45. C2×#391D3semidirect
318. C2×C3×C4#137semidirect
503. C2×C4#942semidirect
545. C3#23547semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C2×C3×#3912
C2×#52252
C2×#49662
#820702
C2×#49242
#715582
#696812
#696802
#235473