Finite Group Databasebeta

← #10829935

Subgroups — #10829935

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2C2×C3×C4×D8C211yesmaximal, characteristic
3#82376C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic, complemented
5#72045C212yesmaximal
6#71604C211yesmaximal, characteristic, complemented
7#70642C212yesmaximal, complemented
8#70216C212yesmaximal, complemented
9#70213C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69771C211yesmaximal, characteristic, complemented
12#24185—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936C2212yes—
17C22×#214C2212yes—
18C2×#926—21no—
19C2×#924—21no—
20C2×#924—21no—
21#5542—24no—
22#5542—24no—
23C2×#887C2212yes—
24C2×#879—22no—
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×C4×D8—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32#5221C2212yes—
33C2×#832C2212yes—
34#5212—21no—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#5019—24no—
38#4972C2212yes—
39C2×#810C2212yes—
40#4948—22no—
41#4935C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×C4×D8D311yescharacteristic
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1956—31no—
54#1880—32no—
55#1863—32no—
56#1802—31no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2312yes—
63C2×C3×D8—22no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81#936—24no—
82#936—24no—
83C2×#214—24no—
84C2×#214—24no—
85C2×#214—42no—
86#926—28no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93#924—28no—
94#924—44no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4D412yes—
98C2×C4×Dic3—22no—
99C2×C4×Dic3C2312yes—
100C22×C3⋊C8C2312yes—
101#887—24no—
102#887—24no—
103#879—216no—
104#869—22no—
105#869D414yes—
106C4×Dic6—22no—
107C4×Dic6—22no—
108C3×#107—22no—
109C3×#102—22no—
110C3×#102—22no—
111C3×C4×C8—22no—
112#832—42no—
113C3⋊#95—22no—
114#816—24no—
115#816—42no—
116#810—24no—
117C4⋊(C3⋊C8)—22no—
118C22×SD16—32no—
119C22×D8C2×D311yescharacteristic
120#556—64no—
121#556—64no—
122C2×C4⋊Q8—31no—
123C2×#126—62no—
124C2×#124—32no—
125C2×C22⋊Q8—61no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×D4C2×D311yescharacteristic
130C2×C4×D4C2×D311yescharacteristic
131C2×#117—61no—
132#461—64no—
133C4×D8—24no—
134C2×#107C2×D311yescharacteristic
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#102—32no—
140C2×#102C2×D311yescharacteristic
141C2×#102C2×D311yescharacteristic
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C24×C3—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4C2×D412yes—
164C2×C3×D4#4112yes—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—21no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3C2×D412yes—
179C22×Dic3C2×D412yes—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6C2×D414yes—
189C3×D8—28no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8#4112yes—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—42no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#217—28no—
205#217—28no—
206#217—44no—
207#217—82no—
208#214—48no—
209C4⋊Dic3—22no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3—81no—
216Dic3⋊C4—28no—
217Dic3⋊C4—24no—
218Dic3⋊C4—44no—
219C4×Dic3—28no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C22×Q8—31no—
226C22×Q8—32no—
227C22×D4C22×D311yescharacteristic
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—21no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C2×D8—22no—
235C2×D8—22no—
236C2×D8—22no—
237C22×C8—32no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242#126—616no—
243#124—64no—
244#124—64no—
245C22⋊Q8—68no—
246C22⋊Q8—124no—
247C4×Q8—64no—
248C4×Q8—64no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4#6714yes—
255#117—68no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C42—32no—
274C2×C42—62no—
275C2×C42C22×D311yescharacteristic
276#107—22no—
277C4⋊C8—62no—
278#103—64no—
279#103—64no—
280#103—122no—
281#102—22no—
282#102—22no—
283#102—122no—
284C4×C8—22no—
285#95—62no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3C22×D411yescharacteristic
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—41no—
306C3×D4—41no—
307C3×D4—41no—
308C3×D4—41no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—48no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—84no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12012yes—
321C2×C3×C4#13512yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C42−1+411yescharacteristic
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341Dic6—24no—
342Dic6—24no—
343Dic6—42no—
344Dic6—42no—
345Dic6—44no—
346C3×C8—22no—
347C3×C8—41no—
348C3⋊C8—42no—
349C24—21no—
350C24—21no—
351C2×Q8—34no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4#23512yes—
371C2×D4C2×#6712yes—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386C22×C4C23×D311yescharacteristic
387SD16—128no—
388D8—28no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8#23512yes—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—68no—
402C4⋊C4—62no—
403C4⋊C4—64no—
404C4⋊C4—64no—
405C4⋊C4—64no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—124no—
409C4⋊C4—122no—
410C4⋊C4—241no—
411C4⋊C4C2×#6712yes—
412C22⋊C4—24no—
413C22⋊C4—42no—
414C22⋊C4—68no—
415C22⋊C4—68no—
416C22⋊C4—124no—
417C22⋊C4—242no—
418C42—68no—
419C42—62no—
420C42—62no—
421C42D3×D412yes—
422C42C2×#6712yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54811yescharacteristic
439C22×C3#55011yescharacteristic
440C22×C3C2×#13511yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3#57711yescharacteristic
443C22×C3#57811yescharacteristic
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—81no—
447C3×C4#44914yes—
448C3×C4#52511yescharacteristic
449C3×C4#53711yescharacteristic
450C3×C4#54611yescharacteristic
451C3×C4#54711yescharacteristic
452Dic3—44no—
453Dic3—42no—
454Dic3—42no—
455Dic3—82no—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C2×D3×D411yescharacteristic
471Q8—64no—
472Q8—64no—
473Q8—124no—
474Q8—122no—
475Q8—122no—
476D4—22no—
477D4—22no—
478D4—22no—
479D4—22no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—48no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—84no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4#91112yes—
507C2×C4#91212yes—
508C2×C4#93712yes—
509C2×C4#93712yes—
510C2×C4C2×#23511yescharacteristic
511C2×C4C2×#23511yescharacteristic
512C2×C4C2×D3×D411yescharacteristic
513C2×C4#100911yescharacteristic
514C2×C4#101011yescharacteristic
515C2×C4C22×#6711yescharacteristic
516C8—22no—
517C8—41no—
518C8—122no—
519C2×C3—82no—
520C2×C3—82no—
521C2×C3#240711yescharacteristic
522C2×C3C2×#44911yescharacteristic
523C2×C3#301311yescharacteristic
524C2×C3#336011yescharacteristic
525C2×C3#336711yescharacteristic
526C2×C3#339311yescharacteristic
527C2×C3#339711yescharacteristic
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#555211yescharacteristic
543C22#555311yescharacteristic
544C22C2×#91111yescharacteristic
545C22C2×#91211yescharacteristic
546C22#577011yescharacteristic
547C22#577111yescharacteristic
548C22C2×#93711yescharacteristic
549C4—22no—
550C4—42no—
551C4—81no—
552C4—122no—
553C4—122no—
554C4—124no—
555C4—242no—
556C4#515814yes—
557C4#555011yescharacteristic
558C4#555111yescharacteristic
559C4#579111yescharacteristic
560C4#583311yescharacteristic
561C3#2418511yescharacteristic, complemented, Sylow
562C2—82no—
563C2—82no—
564C2#7956811yescharacteristic
565C2#7957111yescharacteristic
566C2#7957911yescharacteristic
567C2#7958111yescharacteristic
568C2#8240211yescharacteristic
569C2C2×#515811yescharacteristic
570C2#8391311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×C4×D8D3
hypocenterC3#24185
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
3. #82376C2semidirect
4. C2×#5019C2semidirect
6. #71604C2semidirect
7. #70642C2semidirect
8. #70216C2semidirect
9. #70213C2semidirect
10. #69788C2semidirect
11. #69771C2semidirect
561. C3#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C3×C4×D82
#823762
C2×#50192
#720452
#716042
#706422
#702162
#702132
#697882
#697712
#241853