Finite Group Databasebeta

← #10876053

Subgroups — #10876053

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75982C211yesmaximal, characteristic, complemented
2#73695C211yesmaximal, characteristic
3C3×#1920C211yesmaximal, characteristic, complemented
4#70588C211yesmaximal, characteristic, complemented
5#70585C211yesmaximal, characteristic
6#69617C211yesmaximal, characteristic
7#69475C211yesmaximal, characteristic, complemented
8#68909C211yesmaximal, characteristic
9#68909C211yesmaximal, characteristic
10#68905C211yesmaximal, characteristic, complemented
11#68903C211yesmaximal, characteristic
12#68845C211yesmaximal, characteristic, complemented
13#68844C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68656C211yesmaximal, characteristic, complemented
16#27577—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#887C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#816C2211yescharacteristic
26C2×Dic3⋊C8C2211yescharacteristic
27#4988C2211yescharacteristic
28#4988C2211yescharacteristic
29#4984C2211yescharacteristic
30#4977C2211yescharacteristic
31#4977C2211yescharacteristic
32#4976—21no—
33Dic3×C4⋊C4—21no—
34#4972C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39#4940—21no—
40#4940—21no—
41#4939C2211yescharacteristic
42C4×Dic3⋊C4C2211yescharacteristic
43C2×#803C2211yescharacteristic
44#4675—21no—
45#4675—21no—
46#4650—21no—
47#4647—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2513—31no—
61#2351—31no—
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2065—31no—
66#1936—31no—
67#1920D311yescharacteristic, complemented
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1865—31no—
73#1805—31no—
74#1763—31no—
75C22×D12C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×Dic3⋊C4D411yescharacteristic
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C2×#204—22no—
109C2×#204—22no—
110C2×#204—22no—
111#887—22no—
112#887—22no—
113#887—22no—
114#887—41no—
115C4×Dic6—22no—
116C4×Dic6—41no—
117C3×#106—22no—
118C3×#95—21no—
119C3×#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—41no—
129#816—22no—
130Dic3⋊C8—41no—
131#810—22no—
132#810—22no—
133#809—22no—
134#809—41no—
135#803—41no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×Q8—31no—
140C2×#106C2×D311yescharacteristic
141C2×C4⋊C8—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97—31no—
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#389—61no—
157#389—61no—
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#382—31no—
162#382—31no—
163#382—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—61no—
166C4×C4⋊C4C2×D311yescharacteristic
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6#4112yes—
204C2×Dic6#4112yes—
205C2×Dic6#4112yes—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4#4112yes—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—81no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—42no—
226(C2×D3)⋊C4—81no—
227(C2×D3)⋊C4—81no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—42no—
235Dic3⋊C4—24no—
236Dic3⋊C4—41no—
237Dic3⋊C4—41no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—81no—
241C4×Dic3—24no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—42no—
247C4×Dic3—42no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254#204—44no—
255C22×Q8—31no—
256C22×D4—31no—
257C2×SD16—62no—
258C2×SD16—62no—
259C2×SD16—62no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4×Q8—62no—
265C4×Q8—62no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—121no—
269C4×Q8—121no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—21no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297#106—22no—
298C4⋊C8—121no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#102—121no—
304#102—121no—
305#97—121no—
306#95—21no—
307#95—21no—
308#95—31no—
309#95—31no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369D12—41no—
370D12—41no—
371D12—41no—
372D12—41no—
373Dic6—22no—
374Dic6—22no—
375Dic6—22no—
376Dic6#12614yes—
377C3×C8—41no—
378C3×C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—61no—
382C2×Q8—32no—
383C2×Q8—32no—
384C2×Q8—32no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—124no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—64no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—121no—
435C4⋊C4—121no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—241no—
440C4⋊C4#23512yes—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—241no—
444C22⋊C4—241no—
445C22⋊C4—241no—
446C22⋊C4—241no—
447C42—21no—
448C42—21no—
449C42—64no—
450C42—61no—
451C42—61no—
452C42—61no—
453C42—61no—
454C42—81no—
455C42—122no—
456C42—122no—
457C22×C3#52511yescharacteristic
458C22×C3#54611yescharacteristic
459C22×C3#54811yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57511yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57711yescharacteristic
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4C2×#12611yescharacteristic
478C3×C4#53911yescharacteristic
479C3×C4#55211yescharacteristic
480C3×C4#55311yescharacteristic
481Dic3—24no—
482Dic3—41no—
483Dic3—41no—
484Dic3—41no—
485Dic3—41no—
486Dic3—81no—
487Dic3—81no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23C2×D3×D411yescharacteristic
496Q8—34no—
497Q8—62no—
498Q8—62no—
499Q8—62no—
500D4—121no—
501D4—121no—
502D4—121no—
503D4—121no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4C2×#23511yescharacteristic
544C2×C4#100711yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4#100911yescharacteristic
549C8—41no—
550C8—41no—
551C8—121no—
552C8—121no—
553C2×C3#248111yescharacteristic
554C2×C3#311111yescharacteristic
555C2×C3#312611yescharacteristic
556C2×C3#332111yescharacteristic
557C2×C3#332611yescharacteristic
558C2×C3#337111yescharacteristic
559C2×C3#337111yescharacteristic
560D3—161no—
561C22—241no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22#554711yescharacteristic
569C22#563311yescharacteristic
570C22#565411yescharacteristic
571C22#576211yescharacteristic
572C22#577011yescharacteristic
573C22#577311yescharacteristic
574C22#577711yescharacteristic
575C4—41no—
576C4—42no—
577C4—41no—
578C4—64no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—121no—
583C4—121no—
584C4—121no—
585C4—121no—
586C4—241no—
587C4—241no—
588C4#553011yescharacteristic
589C4#558111yescharacteristic
590C4#569211yescharacteristic
591C4D3×#12611yescharacteristic
592C3#2757711yescharacteristic, complemented, Sylow
593C2—481no—
594C2#7501911yescharacteristic
595C2#7966111yescharacteristic
596C2#7978011yescharacteristic
597C2#8105111yescharacteristic
598C2#8121811yescharacteristic
599C2#8138611yescharacteristic
600C2#8143411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1920D3
hypocenterC3#27577
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
1. #75982C2semidirect
3. C3×#1920C2semidirect
4. #70588C2semidirect
7. #69475C2semidirect
10. #68905C2semidirect
12. #68845C2semidirect
13. #68844C2semidirect
15. #68656C2semidirect
67. #1920D3semidirect
592. C3#27577semidirect

Maximal subgroups

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

subgroupindex
#759822
#736952
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