Finite Group Databasebeta

← #10844872

Subgroups — #10844872

Order 768.

754 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2568C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic
4C2×#5024C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic
6#69683C211yesmaximal, characteristic
7#69682C211yesmaximal, characteristic
8#69681C211yesmaximal, characteristic
9#69681C211yesmaximal, characteristic
10#69678C211yesmaximal, characteristic
11#69677C211yesmaximal, characteristic
12#69675C211yesmaximal, characteristic
13#69674C211yesmaximal, characteristic
14#69475C211yesmaximal, characteristic
15#69468C211yesmaximal, characteristic
16#25306—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#400—21no—
23C3×#400—21no—
24C3×#391—21no—
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#382—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31#5230—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38#5090—24no—
39#5024—24no—
40#4985—21no—
41#4978—21no—
42#4978—21no—
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4936C2211yescharacteristic
48#4929C2211yescharacteristic
49#4929C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#462—31no—
63C2×#446—31no—
64#2581—31no—
65#2568D311yescharacteristic
66#2068—31no—
67#2066—31no—
68#2056—31no—
69#2056—31no—
70#1971—31no—
71#1969—31no—
72#1960—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1763—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#216—22no—
97C2×#216—22no—
98C2×#216—22no—
99C2×#216—22no—
100C2×#216—41no—
101C2×#216—41no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124#835—22no—
125#835—22no—
126#835—22no—
127#835—22no—
128#835—41no—
129#835—41no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#810—22no—
139#810—22no—
140#808—41no—
141#807—41no—
142C4⋊(C3⋊C8)—22no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)D414yes—
145#803—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×Q16—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#462—64no—
154#446—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#400—21no—
172#400—21no—
173#400—61no—
174#395—61no—
175#391—21no—
176#391—31no—
177#391—31no—
178#391C2×D311yescharacteristic
179#391C2×D311yescharacteristic
180#391C2×D311yescharacteristic
181#391C2×D311yescharacteristic
182#389—31no—
183#389—31no—
184#387—61no—
185#387—61no—
186#382—21no—
187#382—31no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—41no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—42no—
243C3×C42C2×D412yes—
244C3×C42#4112yes—
245#216—44no—
246#216—44no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D412yes—
267C2×C3⋊C8C2×D412yes—
268C2×C3⋊C8#4112yes—
269C22×Q8—31no—
270C22×Q8#6711yescharacteristic
271C22×Q8C22×D311yescharacteristic
272C22×Q8C22×D311yescharacteristic
273C2×Q16—62no—
274C2×Q16—62no—
275C2×Q16—62no—
276C2×Q16—62no—
277C2×Q16—121no—
278C2×Q16—121no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—62no—
288C4×Q8—62no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4#6711yescharacteristic
298C2×C4⋊C4#6711yescharacteristic
299C2×C4⋊C4#6711yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C42—21no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307#107—121no—
308#106—121no—
309C4⋊C8—34no—
310C4⋊C8—62no—
311C4⋊C8—62no—
312#103—62no—
313#103—62no—
314#103—62no—
315#103—62no—
316#103—62no—
317#103—62no—
318#103—121no—
319#103—121no—
320#97—62no—
321#97—62no—
322C4×C8—62no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340C23×C3C22×D411yescharacteristic
341C3×Q8—44no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4#12112yes—
378C2×C3×C4#12112yes—
379C2×C3×C4#13712yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386C2×Dic3—22no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398Dic6—22no—
399Dic6—22no—
400Dic6—41no—
401Dic6—41no—
402C3⋊C8—22no—
403C3⋊C8—22no—
404C3⋊C8—22no—
405C3⋊C8—41no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—32no—
421C2×Q8—41no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C23×D311yescharacteristic
442Q16—124no—
443Q16—124no—
444C2×C8—32no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—42no—
479C42—122no—
480C42D3×D412yes—
481C42#23712yes—
482C22×C3C2×#12111yescharacteristic
483C22×C3#54811yescharacteristic
484C22×C3#55011yescharacteristic
485C22×C3C2×#13711yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57811yescharacteristic
488C22×C3#57811yescharacteristic
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#52911yescharacteristic
500C3×C4#54211yescharacteristic
501C3×C4#54611yescharacteristic
502C3×C4#54811yescharacteristic
503Dic3—42no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507C23C22×#6711yescharacteristic
508Q8—44no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—62no—
518Q8—62no—
519Q8—121no—
520Q8—121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—62no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4#93912yes—
561C2×C4#93912yes—
562C2×C4#94212yes—
563C2×C4C2×D3×D411yescharacteristic
564C2×C4C2×#23711yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4#100911yescharacteristic
568C2×C4#101011yescharacteristic
569C8—62no—
570C8—62no—
571C8—62no—
572C8—121no—
573C2×C3#242611yescharacteristic
574C2×C3#305611yescharacteristic
575C2×C3#305911yescharacteristic
576C2×C3#337411yescharacteristic
577C2×C3#337711yescharacteristic
578C2×C3#339811yescharacteristic
579C2×C3#340211yescharacteristic
580C22C2×#93911yescharacteristic
581C22C2×#94211yescharacteristic
582C22#582211yescharacteristic
583C22#583111yescharacteristic
584C22#583411yescharacteristic
585C22#583811yescharacteristic
586C22#583811yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—122no—
598C4—241no—
599C4—241no—
600C4—241no—
601C4#567611yescharacteristic
602C4#567711yescharacteristic
603C4#568011yescharacteristic
604C4#568211yescharacteristic
605C3#2530611yescharacteristic, complemented, Sylow
606C2#8273211yescharacteristic
607C2#8299511yescharacteristic
608C2#8339911yescharacteristic
609C2#8367811yescharacteristic
610C2#8368111yescharacteristic
611C2#8368611yescharacteristic
612C2#8369911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2568C2
socle, prime radicalC23×C3C22×D4
hypercenter#2568D3
hypocenterC3#25306
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
605. C3#25306semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
C2×#50902
#824802
C2×#50242
#717862
#696832
#696822
#696812
#696782
#696772
#696752
#696742
#694752
#694682
#253063