Finite Group Databasebeta

← #10883494

Subgroups — #10883494

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3037C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic
3#83275C211yesmaximal, characteristic, complemented
4C2×C4×#215C211yesmaximal, characteristic, complemented
5#72107C212yesmaximal, complemented
6#72103C212yesmaximal, complemented
7#71637C211yesmaximal, characteristic, complemented
8#70527C211yesmaximal, characteristic, complemented
9#70516C211yesmaximal, characteristic, complemented
10#70303C212yesmaximal, complemented
11#70302C212yesmaximal, complemented
12#28175—31nomaximal, Sylow
13C2×C3×#137C2212yes—
14C3×#532C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#949C2212yes—
17C2×#948C2212yes—
18C22×#215C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#464—22no—
21C3×#462—21no—
22C3×#460—21no—
23C3×#423C2211yescharacteristic
24C3×#417C2212yes—
25#5234C2212yes—
26#5232C2212yes—
27C2×#836C2212yes—
28C2×#835C2211yescharacteristic
29C2×#823C2212yes—
30#5083—21no—
31#5072—21no—
32#5057—22no—
33C4×#215—21no—
34C4×#215—21no—
35C4×#215—21no—
36C4×#215—21no—
37#5012—21no—
38#5012—21no—
39D4×C3⋊C8—21no—
40D4×C3⋊C8—21no—
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4924C2211yescharacteristic
44#4924C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4558C2212yes—
47#4558C2212yes—
48#4554C2212yes—
49#4530C2212yes—
50#3037D311yescharacteristic, complemented
51#3036—31no—
52C2×C4×SD16—31no—
53#2931—31no—
54#2079—32no—
55#2019—32no—
56#1908—32no—
57#1907—32no—
58#1837—31no—
59#1801—31no—
60#1800—31no—
61C2×C3×#41C2312yes—
62C22×C3×Q8C2311yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C3×#137—24no—
66C3×#137—42no—
67C2×C3×Q16—22no—
68C2×C3×SD16—22no—
69C2×C3×M16C2312yes—
70C3×C4⋊Q8D411yescharacteristic
71C3×#127D411yescharacteristic
72C3×#124D412yes—
73C3×#121—21no—
74C3×#121—21no—
75C3×C4×Q8—21no—
76C3×C4×Q8—21no—
77C3×C4×Q8—21no—
78C3×C4×Q8—21no—
79C3×C4×D4—21no—
80C3×C4×D4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C42—21no—
84C2×C3×C42C2311yescharacteristic
85#949—24no—
86#949—42no—
87#948—42no—
88#948—42no—
89C2×#215—22no—
90C2×#215—22no—
91C2×#215—22no—
92C2×#215—21no—
93C2×#215—21no—
94C2×#213—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×#208—22no—
98C2×#208C2312yes—
99C2×#208C2312yes—
100C22×C3⋊C8—22no—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C4×D12—21no—
104C4×D12—21no—
105C4×D12—21no—
106C4×D12—21no—
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×#103—22no—
110C3×#102—22no—
111#836—42no—
112#835—22no—
113#835—21no—
114#835—21no—
115#832—22no—
116#830—22no—
117#830—22no—
118#823—42no—
119#809—21no—
120#809—21no—
121#808—21no—
122#808—21no—
123C4⋊(C3⋊C8)—21no—
124C4⋊(C3⋊C8)—21no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127C4×C3⋊C8D411yescharacteristic
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D411yescharacteristic
131C2×#137C2×D312yes—
132C2×AGL(1,ℤ/8)—32no—
133C22×SD16—31no—
134C2×#131—32no—
135#532C2×D311yescharacteristic
136C2×C4×Q8C2×D311yescharacteristic
137C2×C4×D4—31no—
138#464—22no—
139#463—62no—
140#462—21no—
141#461—61no—
142#460—21no—
143#459—61no—
144C4×SD16—61no—
145C4×SD16—61no—
146C4×SD16—61no—
147C4×SD16—61no—
148#436—61no—
149#436—61no—
150C8×D4—61no—
151C8×D4—61no—
152C2×#106—31no—
153#423—31no—
154#423—31no—
155#423C2×D311yescharacteristic
156#417—32no—
157#417C2×D312yes—
158C2×#103—31no—
159C2×#102—31no—
160C2×#101—32no—
161C2×#100—32no—
162#408—32no—
163C2×C4×C8—31no—
164#343—32no—
165#343—32no—
166#341—32no—
167#341—32no—
168C23×D3—21no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8#4112yes—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4#4111yescharacteristic
188C22×C3×C4#4111yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12#4112yes—
197C2×C4×D3—41no—
198C2×C4×D3—41no—
199C2×C4×D3—82no—
200C3×Q16—44no—
201C3×SD16—44no—
202C3×M16—22no—
203C3×M16—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4—41no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C42—41no—
214C3×C42—41no—
215C3×C42C2×D411yescharacteristic
216C3×C42C2×D411yescharacteristic
217C3×C42C2×D411yescharacteristic
218C3×C42C2×D411yescharacteristic
219#215—24no—
220#215—24no—
221#215—44no—
222#213—44no—
223(C2×D3)⋊C4—41no—
224(C2×D3)⋊C4—41no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227#208—22no—
228#208—22no—
229#208—22no—
230#208—22no—
231#208—44no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—44no—
239C2×C3⋊C8C2×D411yescharacteristic
240C2×C3⋊C8C2×D411yescharacteristic
241C2×C3⋊C8C2×D411yescharacteristic
242C2×C3⋊C8C2×D411yescharacteristic
243C2×#41C22×D312yes—
244C22×Q8C22×D311yescharacteristic
245C22×D4—31no—
246C23×C4—61no—
247#137—24no—
248#137—42no—
249AGL(1,ℤ/8)—64no—
250AGL(1,ℤ/8)—122no—
251C2×Q16—22no—
252C2×SD16—22no—
253C2×SD16—61no—
254C2×SD16—61no—
255C2×SD16—62no—
256C2×SD16—62no—
257C2×SD16—62no—
258C2×D8—62no—
259#131—122no—
260#131—122no—
261C2×M16—32no—
262C2×M16—32no—
263C2×M16—62no—
264C2×M16C22×D312yes—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—62no—
268C4⋊Q8#6711yescharacteristic
269#127#6711yescharacteristic
270#124#6712yes—
271#121—21no—
272#121—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C22⋊C4—61no—
287C2×C42—21no—
288C2×C42C22×D311yescharacteristic
289#106—61no—
290#106—61no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—61no—
294C4⋊C8—61no—
295C4⋊C8—61no—
296C4⋊C8—61no—
297#103—22no—
298#103—61no—
299#103—61no—
300#103—62no—
301#102—22no—
302#102—61no—
303#102—61no—
304#102—62no—
305#101—122no—
306#100—122no—
307C22⋊C8—62no—
308C22⋊C8—62no—
309C4×C8—31no—
310C4×C8—31no—
311C4×C8—31no—
312C4×C8—31no—
313C23×C3—22no—
314C23×C3C2×#4111yescharacteristic
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—81no—
319C22×D3—81no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—81no—
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C4H(5,𝔽2)11yescharacteristic
352C2×Dic3—41no—
353C2×Dic3—81no—
354D12—22no—
355D12—22no—
356D12—41no—
357D12—41no—
358C4×D3—82no—
359C4×D3—82no—
360C3×C8—42no—
361C3⋊C8—21no—
362C3⋊C8—21no—
363C3⋊C8—21no—
364C3⋊C8—21no—
365C3⋊C8—42no—
366C3⋊C8—42no—
367C24—61no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8#23712yes—
378C2×Q8C2×#6712yes—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—32no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4C2×#6712yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—61no—
393C22×C4—121no—
394C22×C4—121no—
395C22×C4—242no—
396C22×C4#23711yescharacteristic
397C22×C4#23711yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—44no—
400SD16—44no—
401SD16—64no—
402SD16—64no—
403SD16—124no—
404D8—124no—
405M16—22no—
406M16—22no—
407M16—62no—
408M16—62no—
409M16—62no—
410M16—62no—
411M16—124no—
412C2×C8—22no—
413C2×C8—31no—
414C2×C8—31no—
415C2×C8—31no—
416C2×C8—31no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—124no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C42—41no—
437C42—41no—
438C42D3×D411yescharacteristic
439C42D3×D411yescharacteristic
440C42C2×#6711yescharacteristic
441C42C2×#6711yescharacteristic
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3C2×#12111yescharacteristic
448C22×C3#55311yescharacteristic
449C22×C3#55311yescharacteristic
450C2×D3—42no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—82no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—81no—
461C3×C4#47112yes—
462C3×C4#52511yescharacteristic
463C3×C4#53511yescharacteristic
464C3×C4#54611yescharacteristic
465C3×C4#54611yescharacteristic
466Dic3—81no—
467C23—22no—
468C23—62no—
469C23—121no—
470C23—121no—
471C23—241no—
472C23—241no—
473C23C2×#23711yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—22no—
477Q8—22no—
478Q8—41no—
479Q8—41no—
480D4—22no—
481D4—22no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—62no—
487D4—62no—
488D4—121no—
489D4—121no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—81no—
500C2×C4—121no—
501C2×C4—242no—
502C2×C4—242no—
503C2×C4—241no—
504C2×C4#93911yescharacteristic
505C2×C4#93911yescharacteristic
506C2×C4#93911yescharacteristic
507C2×C4#93911yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×#23711yescharacteristic
510C2×C4#100411yescharacteristic
511C2×C4D3×#4111yescharacteristic
512C2×C4#100911yescharacteristic
513C2×C4C22×#6711yescharacteristic
514C8—42no—
515C8—61no—
516C8—61no—
517C8—61no—
518C8—61no—
519C8—122no—
520C8—122no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—82no—
524C2×C3#253511yescharacteristic
525C2×C3C2×#47111yescharacteristic
526C2×C3#306711yescharacteristic
527D3—82no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—122no—
534C22—242no—
535C22—242no—
536C22—241no—
537C22—241no—
538C22#561411yescharacteristic
539C22#561811yescharacteristic
540C22C2×#93911yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—42no—
546C4—81no—
547C4—241no—
548C4#520312yes—
549C4#560611yescharacteristic
550C4#560711yescharacteristic
551C4#580611yescharacteristic
552C4#582911yescharacteristic
553C3#2817511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—82no—
557C2—242no—
558C2#8266611yescharacteristic
559C2C2×#520311yescharacteristic
560C2#8400111yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3037C2
socleC22×C3C2×#121
prime radical#83297C2
hypercenter#3037D3
hypocenterC3#28175
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
1. C3×#3037C2semidirect
3. #83275C2semidirect
4. C2×C4×#215C2semidirect
5. #72107C2semidirect
6. #72103C2semidirect
7. #71637C2semidirect
8. #70527C2semidirect
9. #70516C2semidirect
10. #70303C2semidirect
11. #70302C2semidirect
50. #3037D3semidirect
553. C3#28175semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#832972
#832752
C2×C4×#2152
#721072
#721032
#716372
#705272
#705162
#703032
#703022
#281753