Finite Group Databasebeta

← #10898415

Subgroups — #10898415

Order 768.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76252C211yesmaximal, characteristic
2#75970C211yesmaximal, characteristic, complemented
3C3×#1941C211yesmaximal, characteristic, complemented
4#70589C211yesmaximal, characteristic, complemented
5#70588C211yesmaximal, characteristic, complemented
6#70585C211yesmaximal, characteristic
7#70569C211yesmaximal, characteristic
8#69774C211yesmaximal, characteristic
9#69755C211yesmaximal, characteristic, complemented
10#69617C211yesmaximal, characteristic
11#69616C211yesmaximal, characteristic, complemented
12#68909C211yesmaximal, characteristic
13#68863C211yesmaximal, characteristic
14#68840C211yesmaximal, characteristic, complemented
15#68818C211yesmaximal, characteristic, complemented
16#29442—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#887C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#817C2211yescharacteristic
26C2×#816C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29#4994—21no—
30#4992—21no—
31#4988C2211yescharacteristic
32#4988C2211yescharacteristic
33#4984C2211yescharacteristic
34#4984C2211yescharacteristic
35#4978—21no—
36#4977—21no—
37Dic3×C4⋊C4C2211yescharacteristic
38#4973C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44#4939C2211yescharacteristic
45#4932C2211yescharacteristic
46#4929C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4675—21no—
49#4674—21no—
50#4655—21no—
51#4651—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2644—31no—
61#2320—31no—
62#2088—31no—
63#2066—31no—
64#2065—31no—
65#2055—31no—
66#1968—31no—
67#1941D311yescharacteristic, complemented
68#1936—31no—
69#1935—31no—
70#1926—31no—
71#1920—31no—
72#1880—31no—
73#1865—31no—
74#1768—31no—
75C22×D12C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
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⋊Dic3—21no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3C2311yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C2×#204—22no—
105C2×#204—22no—
106C2×#204—41no—
107#887—22no—
108#887—22no—
109C3×#106—22no—
110C3×#106—41no—
111C3×C4×C8—41no—
112C3×#95—21no—
113C3×#95—21no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95D411yescharacteristic
123C3⋊#95D411yescharacteristic
124C3⋊#95D411yescharacteristic
125C3⋊#95D411yescharacteristic
126#821—22no—
127#817—22no—
128#816—22no—
129Dic3⋊C8—22no—
130Dic3⋊C8—22no—
131#810—22no—
132#810—41no—
133#809—22no—
134#809—41no—
135#803—41no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×#106—31no—
139C2×#106C2×D311yescharacteristic
140C2×#106C2×D311yescharacteristic
141C2×C4⋊C8—31no—
142C2×C4⋊C8—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#97—31no—
150C2×C4×C8C2×D311yescharacteristic
151#397—61no—
152#397—61no—
153#397—61no—
154#396—61no—
155#391—31no—
156#391—31no—
157#390—31no—
158#390—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389C2×D311yescharacteristic
164#389C2×D311yescharacteristic
165#387—61no—
166#387—61no—
167#383—61no—
168#382—61no—
169C4×C4⋊C4—31no—
170#376—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
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—21no—
204C2×Dic6—21no—
205C2×Dic6#4112yes—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8#4112yes—
211C2×C3×C8#4112yes—
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—81no—
220C3×C4⋊C4—81no—
221C3×C4⋊C4#4112yes—
222C3×C42—21no—
223C3×C42—21no—
224(C2×D3)⋊C4—81no—
225(C2×D3)⋊C4—81no—
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—81no—
235C4⋊Dic3—81no—
236Dic3⋊C4—24no—
237Dic3⋊C4—81no—
238C4×Dic3—24no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—81no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250#204—44no—
251C22×Q8—31no—
252C22×D4—31no—
253C2×SD16—62no—
254C2×SD16—62no—
255C2×SD16—121no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260C4×Q8—62no—
261C4×Q8—62no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#106—22no—
286#106—41no—
287#106—62no—
288C4⋊C8—62no—
289C4⋊C8—62no—
290#103—62no—
291#103—62no—
292#103—121no—
293#102—62no—
294#102—62no—
295#102—121no—
296#97—121no—
297C4×C8—41no—
298#95—21no—
299#95—21no—
300#95—31no—
301#95—31no—
302#95—31no—
303#95—31no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312C23×C3C22×D411yescharacteristic
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
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×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—22no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365D12—41no—
366D12—41no—
367D12—41no—
368D12—41no—
369Dic6—22no—
370Dic6—22no—
371Dic6—41no—
372Dic6—41no—
373C3×C8—22no—
374C3×C8—22no—
375C3⋊C8—41no—
376C3⋊C8—41no—
377C24—61no—
378C2×Q8—32no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4C23×D311yescharacteristic
404SD16—124no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8#23912yes—
416C2×C8#23912yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—64no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—241no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4#23512yes—
435C22⋊C4—241no—
436C22⋊C4—241no—
437C22⋊C4—241no—
438C22⋊C4—241no—
439C22⋊C4—241no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—64no—
444C42—61no—
445C42—61no—
446C42—61no—
447C42—61no—
448C42—241no—
449C22×C3#52511yescharacteristic
450C22×C3#53211yescharacteristic
451C22×C3#53711yescharacteristic
452C22×C3#57711yescharacteristic
453C22×C3#57711yescharacteristic
454C22×C3#57711yescharacteristic
455C22×C3#57711yescharacteristic
456C2×D3—81no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4#55111yescharacteristic
470C3×C4#55211yescharacteristic
471C3×C4#55311yescharacteristic
472C3×C4#55611yescharacteristic
473Dic3—42no—
474Dic3—41no—
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23C2×D3×D411yescharacteristic
488Q8—62no—
489Q8—62no—
490Q8—121no—
491Q8—121no—
492D4—121no—
493D4—121no—
494D4—121no—
495D4—121no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4C2×#23511yescharacteristic
536C2×C4#100411yescharacteristic
537C2×C4C2×#23911yescharacteristic
538C2×C4C2×#23911yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4#101011yescharacteristic
541C8—22no—
542C8—22no—
543C8—121no—
544C8—121no—
545C2×C3#260111yescharacteristic
546C2×C3#312511yescharacteristic
547C2×C3#312611yescharacteristic
548C2×C3#317311yescharacteristic
549C2×C3#317411yescharacteristic
550C2×C3#326011yescharacteristic
551C2×C3#326311yescharacteristic
552D3—161no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22—241no—
560C22#554711yescharacteristic
561C22#559711yescharacteristic
562C22#572511yescharacteristic
563C22#576211yescharacteristic
564C22#576211yescharacteristic
565C22#577011yescharacteristic
566C22#577011yescharacteristic
567C4—41no—
568C4—41no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—121no—
574C4—121no—
575C4—122no—
576C4—121no—
577C4—121no—
578C4—241no—
579C4—241no—
580C4#551411yescharacteristic
581C4#558411yescharacteristic
582C4#563711yescharacteristic
583C4#569411yescharacteristic
584C3#2944211yescharacteristic, complemented, Sylow
585C2—481no—
586C2#7609511yescharacteristic
587C2#7966011yescharacteristic
588C2#7966111yescharacteristic
589C2#8064611yescharacteristic
590C2#8064711yescharacteristic
591C2#8188711yescharacteristic
592C2#8190811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1941C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1941D3
hypocenterC3#29442
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. #75970C2semidirect
3. C3×#1941C2semidirect
4. #70589C2semidirect
5. #70588C2semidirect
9. #69755C2semidirect
11. #69616C2semidirect
14. #68840C2semidirect
15. #68818C2semidirect
67. #1941D3semidirect
584. C3#29442semidirect

Maximal subgroups

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

subgroupindex
#762522
#759702
C3×#19412
#705892
#705882
#705852
#705692
#697742
#697552
#696172
#696162
#689092
#688632
#688402
#688182
#294423