Finite Group Databasebeta

← #10823005

Subgroups — #10823005

Order 768.

1326 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
1C22×C3×#137C211yesmaximal, characteristic
2C2×#5237C211yesmaximal, characteristic
3#84072C211yesmaximal, characteristic
4C22×#836C211yesmaximal, characteristic, complemented
5C2×#5225C211yesmaximal, characteristic, complemented
6#83750C211yesmaximal, characteristic, complemented
7#72070C214yesmaximal, complemented
8#72053C214yesmaximal, complemented
9#71998C211yesmaximal, characteristic, complemented
10#23522—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C2×C3×#137—24no—
14C2×C3×#137—24no—
15C2×C3×#137—22no—
16C2×C3×#137—22no—
17C22×C3×Q16C2212yes—
18C22×C3×SD16C2212yes—
19C22×C3×M16C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5237—24no—
25#5233—22no—
26#5232—22no—
27C2×#836—24no—
28C2×#836C2214yes—
29C2×#836C2212yes—
30#5225—22no—
31#5225C2214yes—
32#5225C2212yes—
33C2×#835—22no—
34C2×#833C2212yes—
35#5215C2212yes—
36#5119—22no—
37#5118—22no—
38#5102C2214yes—
39#4552C2214yes—
40C22×#137D311yescharacteristic
41#2905—31no—
42#2896—31no—
43C2×#419—31no—
44C2×#413—31no—
45C22×#101—31no—
46#2047—34no—
47#2046—34no—
48#1738—31no—
49C2×C3×#41—22no—
50C2×C3×#41—21no—
51C2×C3×#41—21no—
52C2×C3×#41—22no—
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C2×C3×#41—22no—
56C2×C3×#41D414yes—
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8—22no—
62C22×C3×Q8—21no—
63C22×C3×Q8—21no—
64C22×C3×Q8—22no—
65C22×C3×Q8D414yes—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C3×#137—216no—
74C3×#137—216no—
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C2×C3×M16D412yes—
84C2×C3×M16D412yes—
85C22×C3×C8C2312yes—
86C2×#217—21no—
87#926—21no—
88#926—21no—
89#926D412yes—
90#926C2312yes—
91C2×C4⋊Dic3D412yes—
92C2×C4×Dic3—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—22no—
101C2×#208—22no—
102C2×#208C2312yes—
103C2×#208C2312yes—
104C2×#208C2312yes—
105C22×C3⋊C8—21no—
106C22×C3⋊C8—21no—
107C22×C3⋊C8—21no—
108#836—28no—
109#836—24no—
110#835—28no—
111#835—24no—
112#833—24no—
113#832—24no—
114Dic3⋊C8—28no—
115C22×#41C2×D311yescharacteristic
116C23×Q8C2×D311yescharacteristic
117C2×#137—24no—
118C2×#137—24no—
119C2×#137—22no—
120C2×#137—22no—
121C22×Q16C2×D312yes—
122C22×SD16C2×D312yes—
123C22×M16—31no—
124C22×M16—31no—
125C22×M16—31no—
126C22×M16C2×D311yescharacteristic
127C2×#117—31no—
128#424—34no—
129#424—62no—
130#423—62no—
131#419—34no—
132#419—32no—
133#419—62no—
134#418—32no—
135#417—62no—
136#416—62no—
137C2×#103—62no—
138#413—64no—
139C2×#101—34no—
140C2×#101—32no—
141C2×#101—64no—
142C2×#100—32no—
143#341—34no—
144C24×C3—21no—
145C24×C3#4111yescharacteristic
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41—42no—
151C3×#41—42no—
152C3×#41—42no—
153C3×#41—42no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—24no—
158C2×C3×Q8—24no—
159C2×C3×Q8—24no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—41no—
170C2×C3×Q8—41no—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×Q8—42no—
175C2×C3×Q8—42no—
176C2×C3×Q8C2×D414yes—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—41no—
185C2×C3×D4—41no—
186C2×C3×D4—42no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—41no—
193C22×C3×C4—41no—
194C22×C3×C4—42no—
195C22×C3×C4—41no—
196C22×C3×C4—41no—
197C22×C3×C4—42no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4#4111yescharacteristic
205C22×C3×C4#4112yes—
206C22×C3×C4#4112yes—
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—22no—
209C3×Q16—216no—
210C3×SD16—216no—
211C3×M16—24no—
212C3×M16—24no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8C2×D414yes—
216#217—22no—
217#217—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3C2×D414yes—
220C4×Dic3—22no—
221C4×Dic3—22no—
222#208—22no—
223#208—22no—
224#208—22no—
225#208—22no—
226#208—22no—
227#208—22no—
228#208—42no—
229#208—42no—
230#208—42no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—42no—
235C2×C3⋊C8—42no—
236C2×C3⋊C8—42no—
237C2×#41—22no—
238C2×#41—21no—
239C2×#41—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—21no—
243C2×#41—21no—
244C2×#41#6714yes—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8#6714yes—
254C22×Q8C22×D312yes—
255C22×Q8C22×D311yescharacteristic
256C22×D4—21no—
257C22×D4C22×D311yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C22×D311yescharacteristic
261#137—216no—
262#137—216no—
263C2×Q16—24no—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—32no—
272C2×M16—32no—
273C2×M16—32no—
274C2×M16—61no—
275C2×M16—61no—
276C2×M16—61no—
277C2×M16—62no—
278C2×M16—62no—
279C2×M16—62no—
280C2×M16—62no—
281C2×M16—62no—
282C2×M16—62no—
283C2×M16#6712yes—
284C2×M16#6712yes—
285C22×C8—61no—
286C22×C8—61no—
287C22×C8—61no—
288C22×C8C22×D312yes—
289#117—32no—
290#117—32no—
291#117—61no—
292#117—61no—
293C2×C4⋊C4—32no—
294C2×C22⋊C4—61no—
295C2×C42—61no—
296C4⋊C8—68no—
297#103—68no—
298#103—64no—
299#102—64no—
300#101—68no—
301#101—64no—
302#100—64no—
303C23×C3—22no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—82no—
308C23×C3#12111yescharacteristic
309C23×C3#12111yescharacteristic
310C23×C3#12312yes—
311C23×C3#12312yes—
312C23×C3C2×#4111yescharacteristic
313C3×Q8—24no—
314C3×Q8—22no—
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8—42no—
318C3×Q8—41no—
319C3×Q8—42no—
320C3×Q8—41no—
321C3×Q8—44no—
322C3×Q8—44no—
323C3×D4—22no—
324C3×D4—22no—
325C3×D4—44no—
326C3×D4—44no—
327C3×D4—41no—
328C3×D4—41no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4#12012yes—
340C2×C3×C4#12012yes—
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12112yes—
344C2×C3×C4#12112yes—
345C2×C3×C4#12112yes—
346C2×C3×C4#12112yes—
347C2×C3×C4#12312yes—
348C2×C3×C4#12312yes—
349C2×C3×C4#12412yes—
350C2×C3×C4#12412yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×Dic3—24no—
358C2×Dic3—82no—
359C3×C8—24no—
360C3⋊C8—42no—
361C3⋊C8—42no—
362C3⋊C8—42no—
363C24—21no—
364C24#23711yescharacteristic
365#41—24no—
366#41—24no—
367#41—24no—
368#41—24no—
369#41—42no—
370#41—42no—
371#41—42no—
372#41—42no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8C2×#6714yes—
396C2×Q8C2×#6712yes—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4C2×#6712yes—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—62no—
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4#23711yescharacteristic
421C22×C4#23712yes—
422C22×C4#23712yes—
423C22×C4C2×#6712yes—
424C22×C4C2×#6712yes—
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C23×D311yescharacteristic
428Q16—216no—
429SD16—216no—
430M16—24no—
431M16—24no—
432M16—62no—
433M16—62no—
434M16—62no—
435M16—62no—
436M16—62no—
437M16—62no—
438M16—122no—
439M16—122no—
440M16—122no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—122no—
447C2×C8—122no—
448C2×C8—122no—
449C2×C8C2×#6714yes—
450C4⋊C4—34no—
451C4⋊C4—62no—
452C22⋊C4—62no—
453C22⋊C4—62no—
454C42—62no—
455C42—62no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3#39412yes—
466C22×C3#39412yes—
467C22×C3C2×#12111yescharacteristic
468C22×C3C2×#12311yescharacteristic
469C22×C3C2×#12311yescharacteristic
470C3×C4—42no—
471C3×C4—44no—
472C3×C4#39412yes—
473C3×C4#39412yes—
474C3×C4C2×#12011yescharacteristic
475C3×C4C2×#12111yescharacteristic
476C3×C4C2×#12111yescharacteristic
477C3×C4C2×#12411yescharacteristic
478Dic3—44no—
479C23—22no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—82no—
484C23#93512yes—
485C23#93512yes—
486C23#93811yescharacteristic
487C23#93811yescharacteristic
488C23C2×#23711yescharacteristic
489Q8—24no—
490Q8—22no—
491Q8—24no—
492Q8—22no—
493Q8—44no—
494Q8—44no—
495Q8—42no—
496Q8—41no—
497Q8—42no—
498Q8—41no—
499D4—22no—
500D4—22no—
501D4—44no—
502D4—44no—
503D4—41no—
504D4—41no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—64no—
516C2×C4—242no—
517C2×C4#93512yes—
518C2×C4#93512yes—
519C2×C4#93612yes—
520C2×C4#93612yes—
521C2×C4#93811yescharacteristic
522C2×C4#93811yescharacteristic
523C2×C4#93912yes—
524C2×C4#93912yes—
525C2×C4#93912yes—
526C2×C4#93912yes—
527C2×C4#95312yes—
528C2×C4#95312yes—
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4C2×#23711yescharacteristic
532C2×C4C22×#6711yescharacteristic
533C2×C4C22×#6711yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C8—24no—
536C8—122no—
537C8—122no—
538C8—122no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—82no—
544C2×C3C2×#39411yescharacteristic
545C2×C3#306912yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22#522212yes—
556C22#522212yes—
557C22C2×#93511yescharacteristic
558C22C2×#93511yescharacteristic
559C22C2×#93811yescharacteristic
560C4—42no—
561C4—44no—
562C4—124no—
563C4#522212yes—
564C4#522212yes—
565C4C2×#93611yescharacteristic
566C4C2×#93911yescharacteristic
567C4C2×#93911yescharacteristic
568C4C2×#95311yescharacteristic
569C3#2352211yescharacteristic, complemented, Sylow
570C2—21no—
571C2—21no—
572C2—22no—
573C2—22no—
574C2—82no—
575C2#8399812yes—
576C2C2×#522211yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#137C2
socleC22×C3C2×#121
prime radicalC22×C3×#41C22
hypercenterC22×#137D3
hypocenterC3#23522
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
4. C22×#836C2semidirect
5. C2×#5225C2semidirect
6. #83750C2semidirect
7. #72070C2semidirect
8. #72053C2semidirect
9. #71998C2semidirect
569. C3#23522semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1372
C2×#52372
#840722
C22×#8362
C2×#52252
#837502
#720702
#720532
#719982
#235223