Finite Group Databasebeta

← #10844932

Subgroups — #10844932

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2548C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic, complemented
4C2×#5024C211yesmaximal, characteristic, complemented
5#71765C211yesmaximal, characteristic
6#69653C211yesmaximal, characteristic, complemented
7#69652C211yesmaximal, characteristic, complemented
8#69651C211yesmaximal, characteristic, complemented
9#69651C211yesmaximal, characteristic, complemented
10#69649C211yesmaximal, characteristic, complemented
11#69646C211yesmaximal, characteristic, complemented
12#69645C211yesmaximal, characteristic, complemented
13#69644C211yesmaximal, characteristic, complemented
14#69475C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#25310—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#394—21no—
24C3×#394—21no—
25C3×#390—21no—
26C3×#390C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31#5230—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5075—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×#460—31no—
63C2×#446—31no—
64#2581—31no—
65#2548D311yescharacteristic
66#2067—31no—
67#2065—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1961—31no—
73#1949—31no—
74#1936—31no—
75#1934—31no—
76#1763—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C3×#127—22no—
84C3×#127—22no—
85C3×#124—44no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×#214—22no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—22no—
109C2×#214—41no—
110C2×#214—41no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×#95—21no—
125#832—22no—
126#832—22no—
127#832—22no—
128#832—22no—
129#832—41no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#810—22no—
140#810—22no—
141#808—41no—
142#807—41no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)D414yes—
146#803—22no—
147#803—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127C2×D311yescharacteristic
152C2×#124—21no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#460—64no—
156#446—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8—31no—
173#400—61no—
174#395—61no—
175#394—21no—
176#394—21no—
177#391—31no—
178#391—31no—
179#390—21no—
180#390C2×D311yescharacteristic
181#390C2×D311yescharacteristic
182#390C2×D311yescharacteristic
183#390C2×D311yescharacteristic
184#389—31no—
185#389—31no—
186#387—61no—
187#387—61no—
188#382—31no—
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—41no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—42no—
212C2×C3×D4—42no—
213C2×C3×D4—42no—
214C2×C3×D4—42no—
215C2×C3×D4—81no—
216C2×C3×D4—81no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6#4112yes—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C22⋊C4—81no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258#214—44no—
259#214—44no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C2×D412yes—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8#4112yes—
282C22×Q8—31no—
283C22×Q8#6711yescharacteristic
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C22×D311yescharacteristic
287C22×D4C22×D311yescharacteristic
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—121no—
293C2×SD16—121no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—31no—
298#127—22no—
299#127—22no—
300#124—44no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—62no—
304C4×Q8—62no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4#6711yescharacteristic
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C42—21no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—121no—
331C4⋊C8—34no—
332C4⋊C8—62no—
333C4⋊C8—62no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—62no—
343#97—62no—
344C4×C8—62no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3C22×D411yescharacteristic
369C3×Q8—44no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—84no—
379C3×D4—84no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4#12112yes—
402C2×C3×C4#12112yes—
403C2×C3×C4AGL(1,ℤ/8)12yes—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4H(5,𝔽2)11yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×Dic3—22no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422Dic6—22no—
423Dic6—22no—
424Dic6—41no—
425Dic6—41no—
426C3⋊C8—22no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C3⋊C8—41no—
430C24—21no—
431C24—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—32no—
435C2×Q8—41no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—124no—
473SD16—124no—
474C2×C8—32no—
475C2×C8—32no—
476C2×C8—32no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—42no—
515C42—122no—
516C42D3×D412yes—
517C42#23712yes—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3C2×#12111yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3#55011yescharacteristic
535C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
536C22×C3#57511yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57711yescharacteristic
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—81no—
546C3×C4—81no—
547C3×C4#52911yescharacteristic
548C3×C4#54211yescharacteristic
549C3×C4#54611yescharacteristic
550C3×C4#54811yescharacteristic
551Dic3—42no—
552Dic3—81no—
553Dic3—81no—
554Dic3—81no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C22×#6711yescharacteristic
570Q8—44no—
571Q8—62no—
572Q8—62no—
573Q8—121no—
574Q8—121no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—84no—
584D4—84no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—62no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4#93212yes—
619C2×C4#93912yes—
620C2×C4#93912yes—
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4#100911yescharacteristic
626C2×C4#101011yescharacteristic
627C8—62no—
628C8—62no—
629C8—62no—
630C8—121no—
631C2×C3—161no—
632C2×C3—161no—
633C2×C3#242611yescharacteristic
634C2×C3#305511yescharacteristic
635C2×C3#305711yescharacteristic
636C2×C3#336811yescharacteristic
637C2×C3#337111yescharacteristic
638C2×C3#339711yescharacteristic
639C2×C3#340111yescharacteristic
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22C2×#93211yescharacteristic
655C22C2×#93911yescharacteristic
656C22#582211yescharacteristic
657C22#583111yescharacteristic
658C22#583411yescharacteristic
659C22#583611yescharacteristic
660C22#583611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—81no—
668C4—81no—
669C4—122no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#567611yescharacteristic
674C4#567711yescharacteristic
675C4#568011yescharacteristic
676C4#568211yescharacteristic
677C3#2531011yescharacteristic, complemented, Sylow
678C2—161no—
679C2—161no—
680C2#8262111yescharacteristic
681C2#8299511yescharacteristic
682C2#8337711yescharacteristic
683C2#8357611yescharacteristic
684C2#8358711yescharacteristic
685C2#8371211yescharacteristic
686C2#8371811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2548C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2548D3
hypocenterC3#25310
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
3. #82480C2semidirect
4. C2×#5024C2semidirect
6. #69653C2semidirect
7. #69652C2semidirect
8. #69651C2semidirect
9. #69651C2semidirect
10. #69649C2semidirect
11. #69646C2semidirect
12. #69645C2semidirect
13. #69644C2semidirect
14. #69475C2semidirect
15. #69468C2semidirect
677. C3#25310semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
C2×#50752
#824802
C2×#50242
#717652
#696532
#696522
#696512
#696492
#696462
#696452
#696442
#694752
#694682
#253103