Finite Group Databasebeta

← #10822293

Subgroups — #10822293

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Q16C211yesmaximal, characteristic, complemented
2C2×#5816C211yesmaximal, characteristic
3C22×#942C211yesmaximal, characteristic
4#84062C211yesmaximal, characteristic, complemented
5C2×#5225C211yesmaximal, characteristic, complemented
6C2×#5113C211yesmaximal, characteristic
7#72073C214yesmaximal, complemented
8#72072C214yesmaximal, complemented
9#71965C211yesmaximal, characteristic, complemented
10#23425—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C22×C3×Q16—24no—
15C22×C3×Q16—24no—
16C22×C3×Q16—24no—
17C22×C3×Q16C2214yes—
18C23×C3×C8C2211yescharacteristic
19#5816—24no—
20#5816—24no—
21C2×#947C2212yes—
22C2×#946—22no—
23C2×#944C2212yes—
24C22×#216C2212yes—
25C2×#942—24no—
26C2×#942—24no—
27C2×#942—24no—
28C22×#215C2212yes—
29C2×#929—22no—
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5229—22no—
34#5229C2214yes—
35#5225—24no—
36C2×#835C2214yes—
37#5211—22no—
38#5113—24no—
39C2×#821C2212yes—
40C2×#816C2212yes—
41#4550C2214yes—
42C22×#137—31no—
43C23×Q16D311yescharacteristic, complemented
44C2×#536—31no—
45C2×#419—31no—
46C2×#416—31no—
47#2310—31no—
48#2016—34no—
49#2015—34no—
50#1729—31no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—22no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8D414yes—
64C22×C3×Q8D414yes—
65C22×C3×Q8C2312yes—
66C22×C3×Q8C2312yes—
67C23×C3×C4—21no—
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235D414yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×Q16—232no—
81C2×C3×Q16—28no—
82C2×C3×Q16—28no—
83C2×C3×Q16—28no—
84C22×C3×C8—22no—
85C22×C3×C8—22no—
86C22×C3×C8D414yes—
87C22×C3×C8C2312yes—
88C2×#217—21no—
89C2×#217D412yes—
90#947—24no—
91#947—24no—
92#946—216no—
93#944—24no—
94#944—24no—
95C2×#216—24no—
96C2×#216—24no—
97C2×#216—24no—
98#942—232no—
99C2×#215—24no—
100C2×#215—24no—
101C2×#215—24no—
102#929—216no—
103C2×(C2×D3)⋊C4—24no—
104#926—22no—
105#926—22no—
106#926—22no—
107#926—22no—
108C2×C4⋊Dic3D412yes—
109C2×C4⋊Dic3C2312yes—
110C2×Dic3⋊C4—24no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2312yes—
113C2×#208—22no—
114C2×#208—22no—
115C2×#208D414yes—
116C22×C3⋊C8C2312yes—
117#835—28no—
118C3⋊#95—28no—
119#821—24no—
120#816—24no—
121C22×#41—31no—
122C23×Q8C2×D311yescharacteristic
123C23×Q8C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—24no—
128C22×Q16—24no—
129C22×Q16—24no—
130C22×Q16—32no—
131C22×Q16C2×D314yes—
132C22×SD16—32no—
133C22×M16—31no—
134C23×C8C2×D311yescharacteristic
135#536—64no—
136#536—64no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—62no—
141C2×#117—31no—
142C2×#117—31no—
143#419—64no—
144#416—64no—
145C2×#103—34no—
146C2×#103—32no—
147C2×#102—32no—
148#391—34no—
149#391—62no—
150#381—62no—
151#336—34no—
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C2×C3×Q8—24no—
155C2×C3×Q8—24no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—28no—
159C2×C3×Q8—28no—
160C2×C3×Q8—28no—
161C2×C3×Q8—28no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—42no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8—44no—
174C2×C3×Q8—42no—
175C2×C3×Q8—44no—
176C2×C3×Q8—42no—
177C2×C3×Q8—42no—
178C2×C3×Q8C2×D414yes—
179C2×C3×Q8C2×D414yes—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—24no—
197C2×#67—42no—
198C2×#67—42no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3C2×D414yes—
202#235—28no—
203#235—28no—
204#235—44no—
205#235—44no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C2×D412yes—
212C2×C4×D3—24no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×D412yes—
221C3×Q16—232no—
222C2×C3×C8—28no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8C2×D414yes—
226#217—24no—
227#217—42no—
228#216—216no—
229#215—216no—
230(C2×D3)⋊C4—216no—
231C4⋊Dic3—24no—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234Dic3⋊C4—216no—
235C4×Dic3—22no—
236C4×Dic3—22no—
237C4×Dic3—42no—
238#208—28no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8C2×D414yes—
242C2×#41—34no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—21no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—31no—
261C22×Q8#6714yes—
262C22×Q8#6714yes—
263C22×Q8C22×D312yes—
264C22×Q8C22×D312yes—
265C22×D4—31no—
266C22×D4—61no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271#137—632no—
272C2×Q16—232no—
273C2×Q16—28no—
274C2×Q16—28no—
275C2×Q16—28no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×Q16—64no—
279C2×SD16—64no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×M16—34no—
283C2×M16—62no—
284C2×M16—62no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—32no—
288C22×C8#6714yes—
289C22×C8C22×D312yes—
290C4⋊Q8—64no—
291C4⋊Q8—64no—
292#124—64no—
293#124—64no—
294#123—616no—
295C22⋊Q8—616no—
296#117—62no—
297#117—62no—
298#117—62no—
299#117—62no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—64no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—64no—
305C2×C22⋊C4—61no—
306C2×C42—32no—
307C2×C42—61no—
308#103—68no—
309#103—64no—
310#102—64no—
311#95—68no—
312C23×C3—24no—
313C23×C3#12012yes—
314C23×C3#12112yes—
315C23×C3#12712yes—
316C23×C3C22×D411yescharacteristic
317C22×D3—22no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—44no—
321C3×Q8—24no—
322C3×Q8—24no—
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—48no—
330C3×Q8—48no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—48no—
338C2×C3×C4—48no—
339C2×C3×C4#12014yes—
340C2×C3×C4#12012yes—
341C2×C3×C4#12014yes—
342C2×C3×C4#12114yes—
343C2×C3×C4#12114yes—
344C2×C3×C4#12112yes—
345C2×C3×C4#12712yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352#67—48no—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—28no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—44no—
360D12—22no—
361D12—22no—
362D12—41no—
363D12—41no—
364C4×D3—48no—
365Dic6—22no—
366Dic6—22no—
367Dic6—41no—
368Dic6—41no—
369C3×C8—24no—
370C3⋊C8—24no—
371C24—61no—
372C24D3×D411yescharacteristic
373#41—68no—
374#41—68no—
375#41—124no—
376#41—124no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—32no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8C2×#6714yes—
408C2×Q8C2×#6714yes—
409C2×D4—32no—
410C2×D4—64no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—122no—
417C2×D4—122no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—34no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—61no—
428C22×C4—64no—
429C22×C4—62no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4D3×D412yes—
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D412yes—
435C22×C4#23712yes—
436C22×C4C2×#6712yes—
437C22×C4C2×#6712yes—
438C22×C4C2×#6712yes—
439C22×C4C23×D311yescharacteristic
440Q16—232no—
441Q16—616no—
442SD16—616no—
443M16—68no—
444C2×C8—28no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—34no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8C2×#6714yes—
451C4⋊C4—616no—
452C4⋊C4—64no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C22⋊C4—616no—
456C22⋊C4—64no—
457C22⋊C4—122no—
458C42—62no—
459C42—62no—
460C42—122no—
461C22×C3—24no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3#39214yes—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×#12111yescharacteristic
467C22×C3C2×#12711yescharacteristic
468C22×C3#57814yes—
469C2×D3—42no—
470C2×D3—44no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—44no—
474C3×C4—44no—
475C3×C4#39214yes—
476C3×C4C2×#12011yescharacteristic
477C3×C4C2×#12011yescharacteristic
478C3×C4C2×#12111yescharacteristic
479C3×C4C2×#12111yescharacteristic
480Dic3—42no—
481Dic3—44no—
482Dic3—82no—
483C23—24no—
484C23—62no—
485C23—124no—
486C23—121no—
487C23—121no—
488C23#93712yes—
489C23#93912yes—
490C23#94012yes—
491C23C2×D3×D411yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—24no—
496Q8—48no—
497Q8—48no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—62no—
503Q8—62no—
504Q8—121no—
505Q8—121no—
506D4—62no—
507D4—62no—
508D4—128no—
509D4—121no—
510D4—121no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—68no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—128no—
525C2×C4—124no—
526C2×C4—122no—
527C2×C4#93712yes—
528C2×C4#93714yes—
529C2×C4#93814yes—
530C2×C4#93912yes—
531C2×C4#93914yes—
532C2×C4#94012yes—
533C2×C4#95314yes—
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×#23711yescharacteristic
537C2×C4C22×#6711yescharacteristic
538C2×C4C22×#6711yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C8—24no—
541C8—64no—
542C2×C3—24no—
543C2×C3C2×#39211yescharacteristic
544C2×C3#301112yes—
545C2×C3#304412yes—
546C2×C3#315612yes—
547D3—82no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—122no—
552C22—124no—
553C22—242no—
554C22—242no—
555C22#522314yes—
556C22#576614yes—
557C22C2×#93711yescharacteristic
558C22C2×#93911yescharacteristic
559C22C2×#94011yescharacteristic
560C4—44no—
561C4—44no—
562C4—122no—
563C4—124no—
564C4—242no—
565C4#522314yes—
566C4C2×#93711yescharacteristic
567C4C2×#93811yescharacteristic
568C4C2×#93911yescharacteristic
569C4C2×#95311yescharacteristic
570C3#2342511yescharacteristic, complemented, Sylow
571C2—24no—
572C2—242no—
573C2#8387712yes—
574C2#8388112yes—
575C2#8388412yes—
576C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×Q16C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC23×Q16D3
hypocenterC3#23425
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
1. C23×C3×Q16C2semidirect
4. #84062C2semidirect
5. C2×#5225C2semidirect
7. #72073C2semidirect
8. #72072C2semidirect
9. #71965C2semidirect
43. C23×Q16D3semidirect
570. C3#23425semidirect

Maximal subgroups

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

subgroupindex
C23×C3×Q162
C2×#58162
C22×#9422
#840622
C2×#52252
C2×#51132
#720732
#720722
#719652
#234253