Finite Group Databasebeta

← #10836631

Subgroups — #10836631

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×#2629C211yesmaximal, characteristic
2C2×#5063C211yesmaximal, characteristic, complemented
3#82235C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic
5#71738C211yesmaximal, characteristic, complemented
6#69621C211yesmaximal, characteristic, complemented
7#69618C211yesmaximal, characteristic, complemented
8#69616C211yesmaximal, characteristic, complemented
9#69614C211yesmaximal, characteristic, complemented
10#69588C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#69526C211yesmaximal, characteristic
13#69526C211yesmaximal, characteristic
14#69519C211yesmaximal, characteristic
15#69507C211yesmaximal, characteristic
16#24719—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C3×#398—21no—
22C3×#398—21no—
23C3×#396—21no—
24C3×#396—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#384C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#5063—24no—
34#5000—24no—
35#4978—21no—
36#4976—21no—
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4945—21no—
46#4944—21no—
47C4×C4⋊Dic3C2211yescharacteristic
48#4929C2211yescharacteristic
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#2629D311yescharacteristic
65#2323—31no—
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—
77C23×C3×C4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C3×#125—22no—
80C3×#125—22no—
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—22no—
107#924—22no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—22no—
130#810—22no—
131#810—22no—
132#810—22no—
133#810—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×#125C2×D311yescharacteristic
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
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—21no—
163#398—21no—
164#396—21no—
165#396—21no—
166#391—31no—
167#389—31no—
168#389—61no—
169#389C2×D311yescharacteristic
170#389C2×D311yescharacteristic
171#388—61no—
172#388—61no—
173#387—61no—
174#384C2×D311yescharacteristic
175#380—31no—
176#380C2×D311yescharacteristic
177#380C2×D311yescharacteristic
178C4×C4⋊C4—31no—
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
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6#4112yes—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4—81no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—81no—
233C3×C22⋊C4—81no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42C2×D412yes—
237C3×C42#4112yes—
238#217—24no—
239#217—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249C4×Dic3—42no—
250C4×Dic3—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×Q8—31no—
264C23×C4C22×D311yescharacteristic
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—31no—
269#125—22no—
270#125—22no—
271C22⋊Q8—62no—
272C22⋊Q8—62no—
273C4×Q8—62no—
274C4×Q8—62no—
275#117—22no—
276#117—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
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—61no—
302C2×C42—61no—
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—61no—
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
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
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—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
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#12112yes—
366C2×C3×C4#12112yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C42−1+411yescharacteristic
372C2×C3×C42−1+411yescharacteristic
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3⋊C8—22no—
390C3⋊C8—22no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C24C2×#6711yescharacteristic
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
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—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—32no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—64no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—122no—
456C4⋊C4—241no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—64no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—122no—
466C42—241no—
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—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#54711yescharacteristic
493C3×C4#54811yescharacteristic
494C3×C4#55211yescharacteristic
495C3×C4#55611yescharacteristic
496Dic3—42no—
497Dic3—42no—
498Dic3—81no—
499Dic3—81no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C22×#6711yescharacteristic
508Q8—62no—
509Q8—62no—
510Q8—121no—
511Q8—121no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
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—62no—
542C2×C4—62no—
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×D3×D411yescharacteristic
556C2×C4C2×#23711yescharacteristic
557C2×C4#100711yescharacteristic
558C2×C4#100711yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4D3×#4111yescharacteristic
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—
580C22C2×#93911yescharacteristic
581C22#581611yescharacteristic
582C22#582811yescharacteristic
583C22#583611yescharacteristic
584C22#583611yescharacteristic
585C22#583711yescharacteristic
586C22#583811yescharacteristic
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—122no—
597C4—122no—
598C4—241no—
599C4—241no—
600C4#563111yescharacteristic
601C4#563311yescharacteristic
602C4#563911yescharacteristic
603C4#564111yescharacteristic
604C3#2471911yescharacteristic, complemented, Sylow
605C2—81no—
606C2#8281411yescharacteristic
607C2#8342011yescharacteristic
608C2#8342311yescharacteristic
609C2#8343911yescharacteristic
610C2#8344611yescharacteristic
611C2#8347911yescharacteristic
612C2#8348111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2629C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2629D3
hypocenterC3#24719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2629

Complements

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

NH ≅ G/Ntype
2. C2×#5063C2semidirect
5. #71738C2semidirect
6. #69621C2semidirect
7. #69618C2semidirect
8. #69616C2semidirect
9. #69614C2semidirect
10. #69588C2semidirect
11. #69568C2semidirect
604. C3#24719semidirect

Maximal subgroups

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

subgroupindex
C3×#26292
C2×#50632
#822352
C2×#50002
#717382
#696212
#696182
#696162
#696142
#695882
#695682
#695262
#695192
#695072
#247193