Finite Group Databasebeta

← #10853178

Subgroups — #10853178

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73494C211yesmaximal, characteristic
2#73456C211yesmaximal, characteristic
3C3×#1851C211yesmaximal, characteristic
4#71988C211yesmaximal, characteristic
5#71985C211yesmaximal, characteristic
6#70464C211yesmaximal, characteristic, complemented
7#70462C211yesmaximal, characteristic, complemented
8#70461C211yesmaximal, characteristic, complemented
9#70454C211yesmaximal, characteristic, complemented
10#69929C211yesmaximal, characteristic
11#69925C211yesmaximal, characteristic
12#68643C211yesmaximal, characteristic, complemented
13#68634C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#26014—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#877—21no—
21C2×#870C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#817C2211yescharacteristic
32#4949—21no—
33#4948—21no—
34#4943—21no—
35#4935C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4928C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4654—21no—
43#4653—21no—
44#4653—21no—
45#4647—21no—
46C3×#336C2211yescharacteristic
47C3×#336C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2441—31no—
61#2344—31no—
62#2100—31no—
63#2088—31no—
64#2086—31no—
65#2059—31no—
66#1989—31no—
67#1948—31no—
68#1921—31no—
69#1918—31no—
70#1854—31no—
71#1851D311yescharacteristic
72#1824—31no—
73#1809—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91#926—22no—
92#926—41no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3Q814yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115#877—44no—
116#870—22no—
117#870—41no—
118C3×C4⋊C8—22no—
119C3×C4⋊C8—41no—
120C3×C22⋊C8—22no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#818—22no—
132#818—22no—
133#817—41no—
134#817—41no—
135C22×M16C2×D311yescharacteristic
136C2×#125—31no—
137C2×#117—31no—
138C2×#117—61no—
139C22×C4⋊C4—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×C4⋊C8C2×D311yescharacteristic
145C2×C4⋊C8C2×D311yescharacteristic
146C2×C22⋊C8C2×D311yescharacteristic
147#399—61no—
148#396—61no—
149#389—31no—
150#389—31no—
151#389—31no—
152#389—31no—
153#389—61no—
154#387—61no—
155#387—61no—
156#387—61no—
157#384—31no—
158#381—61no—
159#380—31no—
160#380—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4—31no—
163C4×C22⋊C4C2×D311yescharacteristic
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336C2×D311yescharacteristic
177#336C2×D311yescharacteristic
178C24×C3C2×D411yescharacteristic
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×Q812yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—24no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C3×M16—44no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8#4112yes—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—41no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—21no—
221#217—42no—
222#217—42no—
223#217—81no—
224C4⋊Dic3—24no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—42no—
239C4⋊Dic3—81no—
240C4⋊Dic3C2×Q812yes—
241C4⋊Dic3C2×Q812yes—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249C4×Dic3—42no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C23×C4—61no—
253C23×C4C22×D311yescharacteristic
254C2×M16—22no—
255C2×M16—22no—
256C2×M16—41no—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#125—62no—
262#125—121no—
263#117—62no—
264#117—124no—
265#117—121no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4D1211yescharacteristic
289C2×C22⋊C4D1211yescharacteristic
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#107—62no—
297#106—121no—
298#106—121no—
299C4⋊C8—22no—
300C4⋊C8—41no—
301C22⋊C8—22no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95D1211yescharacteristic
311#95D1211yescharacteristic
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3C22⋊Q812yes—
319C23×C3C22×D411yescharacteristic
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4C22⋊Q812yes—
338C2×C3×C4AGL(1,ℤ/8)12yes—
339C2×C3×C4C22×Q811yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4H(5,𝔽2)11yescharacteristic
344C2×C3×C42−1+411yescharacteristic
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—48no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C3×C8—22no—
371C3×C8—41no—
372C3×C8—41no—
373C3×C8—41no—
374C24C2×D1211yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—64no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4C2×D1211yescharacteristic
391C22×C4C2×D1211yescharacteristic
392C22×C4C2×D1211yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4C2×D1211yescharacteristic
395C22×C4#23712yes—
396C22×C4D3×Q812yes—
397C22×C4C23×D311yescharacteristic
398M16—44no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8#23712yes—
411C4⋊C4—32no—
412C4⋊C4—32no—
413C4⋊C4—32no—
414C4⋊C4—64no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4—241no—
435C4⋊C4—241no—
436C22⋊C4—22no—
437C22⋊C4—41no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—122no—
446C42—122no—
447C42—122no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3C2×C22⋊Q811yescharacteristic
456C22×C3#54611yescharacteristic
457C22×C3#54711yescharacteristic
458C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57711yescharacteristic
461C22×C3#57811yescharacteristic
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4#53311yescharacteristic
468C3×C4#54411yescharacteristic
469C3×C4#55111yescharacteristic
470C3×C4#55211yescharacteristic
471Dic3—42no—
472Dic3—42no—
473Dic3—42no—
474Dic3—81no—
475Dic3—81no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23#89712yes—
486C23C22×D1211yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—128no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4#89712yes—
530C2×C4#90812yes—
531C2×C4C2×#23711yescharacteristic
532C2×C4C2×#23711yescharacteristic
533C2×C4#100411yescharacteristic
534C2×C4C2×D3×Q811yescharacteristic
535C2×C4D3×#4111yescharacteristic
536C2×C4#101011yescharacteristic
537C8—22no—
538C8—41no—
539C8—41no—
540C8—41no—
541C2×C3—42no—
542C2×C3#243811yescharacteristic
543C2×C3#307711yescharacteristic
544C2×C3#308211yescharacteristic
545C2×C3#331711yescharacteristic
546C2×C3#333011yescharacteristic
547C2×C3#334511yescharacteristic
548C2×C3#335211yescharacteristic
549C22—22no—
550C22—22no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22C2×#89711yescharacteristic
557C22#555411yescharacteristic
558C22#555511yescharacteristic
559C22C2×#90811yescharacteristic
560C22#574811yescharacteristic
561C22#575111yescharacteristic
562C22#575311yescharacteristic
563C4—22no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—122no—
569C4—122no—
570C4—122no—
571C4—241no—
572C4—241no—
573C4—241no—
574C4—241no—
575C4—241no—
576C4#558711yescharacteristic
577C4#562211yescharacteristic
578C4#562311yescharacteristic
579C4#565011yescharacteristic
580C3#2601411yescharacteristic, complemented, Sylow
581C2—42no—
582C2#7441511yescharacteristic
583C2#7905911yescharacteristic
584C2#7910111yescharacteristic
585C2#7947111yescharacteristic
586C2#7951411yescharacteristic
587C2#7953011yescharacteristic
588C2#7955711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1851C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1851D3
hypocenterC3#26014
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
6. #70464C2semidirect
7. #70462C2semidirect
8. #70461C2semidirect
9. #70454C2semidirect
12. #68643C2semidirect
13. #68634C2semidirect
14. #68624C2semidirect
15. #68621C2semidirect
580. C3#26014semidirect

Maximal subgroups

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

subgroupindex
#734942
#734562
C3×#18512
#719882
#719852
#704642
#704622
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