Finite Group Databasebeta

← #10867916

Subgroups — #10867916

Order 768.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82162C211yesmaximal, characteristic
2#73921C211yesmaximal, characteristic
3C3×#1852C211yesmaximal, characteristic
4#71553C211yesmaximal, characteristic
5#70643C211yesmaximal, characteristic, complemented
6#70642C211yesmaximal, characteristic, complemented
7#70611C211yesmaximal, characteristic, complemented
8#70609C211yesmaximal, characteristic, complemented
9#69937C211yesmaximal, characteristic
10#69913C211yesmaximal, characteristic
11#69526C211yesmaximal, characteristic
12#68915C211yesmaximal, characteristic, complemented
13#68798C211yesmaximal, characteristic, complemented
14#68789C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#27035—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24#5212C2211yescharacteristic
25#5210—21no—
26C4×#217—21no—
27C4×#217C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4984C2211yescharacteristic
35#4978—21no—
36#4977C2211yescharacteristic
37#4976—21no—
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4952—21no—
47#4951—21no—
48#4659—21no—
49#4651—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2495—31no—
61#2360—31no—
62#2074—31no—
63#2068—31no—
64#2036—31no—
65#2035—31no—
66#1949—31no—
67#1942—31no—
68#1923—31no—
69#1920—31no—
70#1918—31no—
71#1863—31no—
72#1852D311yescharacteristic
73#1823—31no—
74C4×#103—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—41no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217D411yescharacteristic
91C2×#217D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—22no—
97#924—22no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#887—22no—
110#887—41no—
111C3×#107—22no—
112C3×#107—41no—
113C3×C22⋊C8—22no—
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⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#830—41no—
128#818—22no—
129#816—22no—
130#816—22no—
131Dic3⋊C8—22no—
132C8×Dic3—41no—
133#810—22no—
134#810—41no—
135#808—22no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8—31no—
138C22×C4⋊C4C2×D311yescharacteristic
139C2×#107—31no—
140C2×#107C2×D311yescharacteristic
141C2×#107C2×D311yescharacteristic
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8C2×D311yescharacteristic
150C2×C4×C8—31no—
151#400—61no—
152#397—61no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#389—61no—
158#388—61no—
159#387—61no—
160#385—61no—
161#383—61no—
162#382—31no—
163#380—31no—
164#380C2×D311yescharacteristic
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—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—
179C24×C3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—81no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6#4112yes—
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×C4⋊C4—21no—
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—42no—
223C3×C4⋊C4#4112yes—
224C3×C22⋊C4—42no—
225#217—24no—
226#217—22no—
227#217—22no—
228#217—81no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—81no—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—42no—
245C4×Dic3—42no—
246C4×Dic3—81no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C22×Q8—31no—
254C23×C4—21no—
255C23×C4C22×D311yescharacteristic
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262C4×Q8—62no—
263C4×Q8—121no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—31no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42—61no—
289#107—22no—
290#107—41no—
291#107—62no—
292#106—62no—
293C4⋊C8—62no—
294#103—62no—
295#103—62no—
296#103—62no—
297#103—121no—
298C22⋊C8—22no—
299C22⋊C8—121no—
300C4×C8—121no—
301#95—21no—
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—
312#95—61no—
313#95—61no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3#13512yes—
321C23×C3C22×D411yescharacteristic
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
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×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—84no—
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—21no—
349C2×Dic3—21no—
350C2×Dic3—22no—
351C2×Dic3—22no—
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—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—
370C2×Dic3—41no—
371C2×Dic3#13712yes—
372Dic6—22no—
373Dic6—22no—
374Dic6—41no—
375Dic6—41no—
376C3×C8—22no—
377C3×C8—41no—
378C3⋊C8—41no—
379C3⋊C8—41no—
380C24D3×D411yescharacteristic
381C2×Q8—32no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—81no—
408C22×C4D3×D411yescharacteristic
409C22×C4C23×D311yescharacteristic
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—
421C2×C8#23512yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—32no—
430C4⋊C4—42no—
431C4⋊C4—64no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—122no—
438C4⋊C4—241no—
439C4⋊C4—241no—
440C4⋊C4—241no—
441C4⋊C4#23712yes—
442C22⋊C4—42no—
443C22⋊C4—62no—
444C22⋊C4—64no—
445C22⋊C4—62no—
446C22⋊C4—241no—
447C42—61no—
448C42—61no—
449C42—61no—
450C42—61no—
451C42—122no—
452C42—122no—
453C42—241no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3#52511yescharacteristic
462C22×C3#54611yescharacteristic
463C22×C3#54711yescharacteristic
464C22×C3C2×#13511yescharacteristic
465C22×C3C2×#13711yescharacteristic
466C22×C3#57611yescharacteristic
467C22×C3#57811yescharacteristic
468C3×C4—41no—
469C3×C4—42no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4#52911yescharacteristic
474C3×C4#53911yescharacteristic
475C3×C4#55311yescharacteristic
476C3×C4#55611yescharacteristic
477Dic3—22no—
478Dic3—41no—
479Dic3—41no—
480Dic3—42no—
481Dic3—42no—
482Dic3—41no—
483Dic3—81no—
484Dic3—81no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#91212yes—
492C23C2×D3×D411yescharacteristic
493Q8—62no—
494Q8—62no—
495Q8—121no—
496Q8—121no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—32no—
502C2×C4—41no—
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—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—84no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4C2×#23511yescharacteristic
542C2×C4C2×#23711yescharacteristic
543C2×C4#100411yescharacteristic
544C2×C4#100711yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C8—22no—
548C8—41no—
549C8—121no—
550C8—121no—
551C2×C3—42no—
552C2×C3#246611yescharacteristic
553C2×C3#310811yescharacteristic
554C2×C3#312411yescharacteristic
555C2×C3#329711yescharacteristic
556C2×C3#331211yescharacteristic
557C2×C3#333611yescharacteristic
558C2×C3#335711yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22#550711yescharacteristic
567C22#559111yescharacteristic
568C22#565611yescharacteristic
569C22C2×#91211yescharacteristic
570C22#576611yescharacteristic
571C22#577411yescharacteristic
572C22D3×#13711yescharacteristic
573C4—42no—
574C4—41no—
575C4—62no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—122no—
580C4—121no—
581C4—121no—
582C4—121no—
583C4—122no—
584C4—241no—
585C4—241no—
586C4#554011yescharacteristic
587C4#555711yescharacteristic
588C4#563811yescharacteristic
589C4#564111yescharacteristic
590C3#2703511yescharacteristic, complemented, Sylow
591C2—42no—
592C2#7443511yescharacteristic
593C2#7915511yescharacteristic
594C2#7923811yescharacteristic
595C2#8001511yescharacteristic
596C2#8019611yescharacteristic
597C2#8126911yescharacteristic
598C2#8145711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1852D3
hypocenterC3#27035
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
5. #70643C2semidirect
6. #70642C2semidirect
7. #70611C2semidirect
8. #70609C2semidirect
12. #68915C2semidirect
13. #68798C2semidirect
14. #68789C2semidirect
15. #68781C2semidirect
590. C3#27035semidirect

Maximal subgroups

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

subgroupindex
#821622
#739212
C3×#18522
#715532
#706432
#706422
#706112
#706092
#699372
#699132
#695262
#689152
#687982
#687892
#687812
#270353