Finite Group Databasebeta

← #10836629

Subgroups — #10836629

Order 768.

738 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×#2323C211yesmaximal, characteristic
2C2×#5022C211yesmaximal, characteristic, complemented
3#82265C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic
5#71694C211yesmaximal, characteristic
6#71694C211yesmaximal, characteristic
7#71626C211yesmaximal, characteristic, complemented
8#69675C211yesmaximal, characteristic, complemented
9#69674C211yesmaximal, characteristic, complemented
10#69621C211yesmaximal, characteristic, complemented
11#69618C211yesmaximal, characteristic, complemented
12#69520C211yesmaximal, characteristic
13#69519C211yesmaximal, characteristic
14#69491C211yesmaximal, characteristic, complemented
15#69467C211yesmaximal, characteristic, complemented
16#24719—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#870C2211yescharacteristic
21C3×#391C2211yescharacteristic
22C3×#389—21no—
23C3×#389C2211yescharacteristic
24C3×#388—21no—
25C3×#388—21no—
26C3×#387—21no—
27C3×#380C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5022—24no—
38#5000—24no—
39#4982—21no—
40#4982—21no—
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4948—21no—
46#4948—21no—
47#4935C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52C2×#803C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#433—31no—
64#2629—31no—
65#2323D311yescharacteristic
66#2077—31no—
67#2076—31no—
68#2035—31no—
69#2035—31no—
70#1989—31no—
71#1948—31no—
72#1944—31no—
73#1941—31no—
74#1935—31no—
75#1933—31no—
76#1803—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#870—22no—
118#870—22no—
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—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#830—22no—
134#808—22no—
135#808—22no—
136#808—22no—
137#807—22no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)D414yes—
140#803—22no—
141C4×C3⋊C8—22no—
142C2×#125—31no—
143C2×C22⋊Q8C2×D311yescharacteristic
144C2×C4×Q8C2×D311yescharacteristic
145C2×#117C2×D311yescharacteristic
146#479—64no—
147#433—64no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#103—31no—
157C2×#103—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×#97—31no—
161C2×C4×C8—31no—
162#398—61no—
163#398—61no—
164#396—61no—
165#396—61no—
166#391C2×D311yescharacteristic
167#389—21no—
168#389—31no—
169#389—31no—
170#389C2×D311yescharacteristic
171#388—21no—
172#388—21no—
173#387—21no—
174#384—31no—
175#380—31no—
176#380—31no—
177#380C2×D311yescharacteristic
178C4×C4⋊C4C2×D311yescharacteristic
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187C24×C3C2×D411yescharacteristic
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8#4112yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—81no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—42no—
232C3×C42—81no—
233C3×C42C2×D412yes—
234C3×C42#4112yes—
235#217—81no—
236#217—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—81no—
246C4⋊Dic3—81no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8C2×D412yes—
262C2×C3⋊C8C2×D412yes—
263C22×Q8C22×D311yescharacteristic
264C23×C4C22×D311yescharacteristic
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—31no—
269#125—62no—
270#125—62no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275#117—22no—
276#117—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4#6711yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4#6711yescharacteristic
300C2×C22⋊C4#6711yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—62no—
306#106—62no—
307#106—62no—
308#106—62no—
309C4⋊C8—34no—
310C4⋊C8—62no—
311#103—62no—
312#103—62no—
313#103—62no—
314#103—62no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#97—62no—
318C4×C8—62no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3C22×D411yescharacteristic
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×Q8—41no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4#12112yes—
370C2×C3×C4#12112yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376C2×C3×C42−1+411yescharacteristic
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C3⋊C8—22no—
390C3⋊C8—22no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C24C2×#6711yescharacteristic
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8#23512yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C23×D311yescharacteristic
420C2×C8—32no—
421C2×C8—32no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—42no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—81no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4#23512yes—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—81no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—81no—
467C42D3×D412yes—
468C42#23712yes—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3C2×#12111yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3#53611yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#54711yescharacteristic
493C3×C4#54811yescharacteristic
494C3×C4#55211yescharacteristic
495C3×C4#55611yescharacteristic
496Dic3—81no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C22×#6711yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4#93912yes—
554C2×C4#93912yes—
555C2×C4C2×#23511yescharacteristic
556C2×C4C2×#23511yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4#101011yescharacteristic
560C2×C4#101011yescharacteristic
561C8—62no—
562C8—62no—
563C8—121no—
564C8—121no—
565C2×C3—81no—
566C2×C3#241511yescharacteristic
567C2×C3#306011yescharacteristic
568C2×C3#306311yescharacteristic
569C2×C3#312011yescharacteristic
570C2×C3#312611yescharacteristic
571C2×C3#324211yescharacteristic
572C2×C3#324411yescharacteristic
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22#579111yescharacteristic
581C22C2×#93911yescharacteristic
582C22#582511yescharacteristic
583C22#583611yescharacteristic
584C22#583711yescharacteristic
585C22#583711yescharacteristic
586C22#583811yescharacteristic
587C4—22no—
588C4—22no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—241no—
597C4—241no—
598C4—241no—
599C4—241no—
600C4#554111yescharacteristic
601C4#554211yescharacteristic
602C4#555111yescharacteristic
603C4#555211yescharacteristic
604C3#2471911yescharacteristic, complemented, Sylow
605C2—81no—
606C2#8235111yescharacteristic
607C2#8255311yescharacteristic
608C2#8257511yescharacteristic
609C2#8340011yescharacteristic
610C2#8342011yescharacteristic
611C2#8344211yescharacteristic
612C2#8345511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2323D3
hypocenterC3#24719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#5022C2semidirect
7. #71626C2semidirect
8. #69675C2semidirect
9. #69674C2semidirect
10. #69621C2semidirect
11. #69618C2semidirect
14. #69491C2semidirect
15. #69467C2semidirect
604. C3#24719semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×#50222
#822652
C2×#50002
#716942
#716262
#696752
#696742
#696212
#696182
#695202
#695192
#694912
#694672
#247193