Finite Group Databasebeta

← #10883526

Subgroups — #10883526

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82441C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic
4#79042C211yesmaximal, characteristic, complemented
5C3×#1801C211yesmaximal, characteristic, complemented
6#70527C211yesmaximal, characteristic, complemented
7#70517C211yesmaximal, characteristic, complemented
8#70123C212yesmaximal, complemented
9#70117C212yesmaximal, complemented
10#69010C212yesmaximal, complemented
11#69006C212yesmaximal, complemented
12#28178—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#942C2212yes—
15C22×#215C2211yescharacteristic
16C2×#908C2212yes—
17C2×#907C2212yes—
18#5477C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×#101C2212yes—
23C2×#835C2211yescharacteristic
24#5133C2212yes—
25C2×#823C2212yes—
26#5129C2212yes—
27#5031—21no—
28#5028—22no—
29#5027—21no—
30#5026—22no—
31#5026—22no—
32#4966C2212yes—
33C2×#809C2211yescharacteristic
34C2×#807C2211yescharacteristic
35#4924C2211yescharacteristic
36C2×#803C2211yescharacteristic
37#4844—22no—
38#4843—21no—
39#4841—21no—
40#4836—22no—
41#4836—22no—
42C3×#343C2212yes—
43#4558C2212yes—
44#4558C2212yes—
45#4554C2212yes—
46#3037—31no—
47#3036—31no—
48C2×#440—31no—
49#2932—31no—
50#2078—32no—
51#2019—32no—
52#1908—32no—
53#1907—32no—
54#1838—31no—
55#1801D311yescharacteristic, complemented
56#1800—31no—
57C22×C3×Q8C2311yescharacteristic
58C2×#235C2312yes—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C2×C3×M16C2312yes—
62C2×C3×M16C2312yes—
63C3×C4×Q8—21no—
64C3×C4×Q8—21no—
65C3×C4×Q8—21no—
66C3×C4×Q8—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#216—22no—
72#942—24no—
73#942—42no—
74C2×#215—22no—
75C2×#215—22no—
76C2×#215—22no—
77C2×#215—22no—
78#930—21no—
79#930—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×#208C2312yes—
83C22×C3⋊C8C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85#908—24no—
86#908—42no—
87#907—42no—
88#907—42no—
89C2×D24—22no—
90C2×#204—22no—
91C2×ΓL(1,𝔽25)—22no—
92C2×D3×C8—22no—
93#875D412yes—
94#874D411yescharacteristic
95C4×D12—21no—
96C4×D12—21no—
97C4×D12—21no—
98C4×D12—21no—
99C4×D12—21no—
100C4×D12—21no—
101#869D411yescharacteristic
102C3×C4⋊C8—21no—
103C3×C4⋊C8—21no—
104C3×C4⋊C8—21no—
105C3×C4⋊C8—21no—
106C3×#101—42no—
107#835—22no—
108#823—42no—
109#821—22no—
110(C2×D3)⋊C8—24no—
111#816—22no—
112#810—22no—
113#809—22no—
114#809—22no—
115#807—22no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118#803D412yes—
119#803D412yes—
120C2×#137—32no—
121C2×AGL(1,ℤ/8)—32no—
122C22×SD16—31no—
123C2×#131—32no—
124#532—31no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×D4—31no—
127#464—62no—
128#463—62no—
129#462—61no—
130#461—61no—
131#460—61no—
132#459—61no—
133#440—62no—
134#440—62no—
135#435—62no—
136#435—62no—
137C2×#107—31no—
138#423—31no—
139#423C2×D311yescharacteristic
140#423C2×D311yescharacteristic
141#417—32no—
142#417—32no—
143C2×#103—31no—
144C2×#102—31no—
145C2×#101C2×D312yes—
146C2×#100—32no—
147#408—32no—
148C2×#97—31no—
149#343—32no—
150#343C2×D312yes—
151#341—32no—
152#341—32no—
153C23×D3—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4#4111yescharacteristic
163C22×C3×C4#4111yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—22no—
166C22×Dic3—21no—
167#235—22no—
168#235—22no—
169#235—22no—
170#235—22no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C2×D412yes—
179C2×D12#4112yes—
180C2×C4×D3—22no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—82no—
184C2×Dic6C2×D412yes—
185C3×M16—22no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C42—41no—
196C3×C42—41no—
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201#216—44no—
202#215—28no—
203#215—44no—
204(C2×D3)⋊C4—22no—
205(C2×D3)⋊C4—22no—
206(C2×D3)⋊C4—41no—
207(C2×D3)⋊C4—41no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212#208—22no—
213#208—22no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8C2×D412yes—
220C2×C3⋊C8C2×D412yes—
221D24—44no—
222#204—44no—
223ΓL(1,𝔽25)—44no—
224D3×C8—44no—
225C2×#41—32no—
226C22×Q8C22×D311yescharacteristic
227C22×D4—31no—
228C23×C4—61no—
229#137—64no—
230#137—122no—
231AGL(1,ℤ/8)—64no—
232AGL(1,ℤ/8)—122no—
233C2×Q16—62no—
234C2×SD16—62no—
235C2×SD16—62no—
236C2×SD16—62no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×D8—62no—
240#131—122no—
241#131—122no—
242C2×M16—32no—
243C2×M16—62no—
244C2×M16C22×D312yes—
245C2×M16C22×D312yes—
246C22×C8—31no—
247C22×C8—31no—
248C22×C8—62no—
249C4⋊Q8—31no—
250#127—31no—
251#124—32no—
252#121—61no—
253#121—61no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—61no—
259C4×D4—61no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263C4×D4—61no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—61no—
268C2×C42—21no—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—61no—
276C4⋊C8—61no—
277#103—62no—
278#103—62no—
279#103—62no—
280#102—62no—
281#102—62no—
282#102—62no—
283#101—42no—
284#100—122no—
285C22⋊C8—64no—
286#97—32no—
287#97—32no—
288C23×C3C2×#4111yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—81no—
294C22×D3—81no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—81no—
306C2×C3×C4#12111yescharacteristic
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12111yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316#67—42no—
317#67—42no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—81no—
321D12—22no—
322D12—22no—
323D12—22no—
324D12—22no—
325D12—41no—
326D12—41no—
327D12—81no—
328D12—81no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—82no—
332C4×D3—82no—
333Dic6—22no—
334Dic6—22no—
335C3×C8—42no—
336C3×C8—42no—
337C3⋊C8—22no—
338C3⋊C8—22no—
339C3⋊C8—42no—
340C24—61no—
341#41—62no—
342#41—62no—
343#41—62no—
344#41—62no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—32no—
351C2×Q8#23912yes—
352C2×D4—32no—
353C2×D4—32no—
354C2×D4—62no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—121no—
367C22×C4—121no—
368C22×C4—242no—
369C22×C4#23511yescharacteristic
370C22×C4#23511yescharacteristic
371C22×C4C23×D311yescharacteristic
372Q16—124no—
373SD16—68no—
374SD16—124no—
375SD16—124no—
376D8—124no—
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—62no—
382M16—62no—
383M16—124no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—62no—
393C2×C8—124no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—121no—
405C22⋊C4—121no—
406C42—41no—
407C42—41no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42D3×D411yescharacteristic
411C42D3×D411yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3C2×#12111yescharacteristic
415C22×C3#55311yescharacteristic
416C22×C3#55311yescharacteristic
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—82no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—42no—
430C3×C4—81no—
431C3×C4#52511yescharacteristic
432C3×C4#53511yescharacteristic
433C3×C4#54611yescharacteristic
434C3×C4#54611yescharacteristic
435Dic3—42no—
436Dic3—81no—
437C23—62no—
438C23—62no—
439C23—121no—
440C23—121no—
441C23—241no—
442C23—241no—
443C23C2×#23511yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448Q8—62no—
449Q8—62no—
450D4—62no—
451D4—62no—
452D4—62no—
453D4—62no—
454D4—122no—
455D4—122no—
456D4—121no—
457D4—121no—
458D4—241no—
459D4—241no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—62no—
467C2×C4—81no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—121no—
471C2×C4—242no—
472C2×C4—242no—
473C2×C4—241no—
474C2×C4#88411yescharacteristic
475C2×C4#88411yescharacteristic
476C2×C4#88411yescharacteristic
477C2×C4#88411yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C2×C4C2×D3×D411yescharacteristic
480C2×C4#100411yescharacteristic
481C2×C4C2×#23911yescharacteristic
482C2×C4D3×#4111yescharacteristic
483C2×C4#100911yescharacteristic
484C8—42no—
485C8—42no—
486C8—62no—
487C8—62no—
488C8—122no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3#253511yescharacteristic
492C2×C3#306811yescharacteristic
493C2×C3#306911yescharacteristic
494D3—82no—
495D3—82no—
496C22—21no—
497C22—21no—
498C22—122no—
499C22—122no—
500C22—122no—
501C22—122no—
502C22—242no—
503C22—242no—
504C22—241no—
505C22—241no—
506C22C2×#88411yescharacteristic
507C22#551411yescharacteristic
508C22#553011yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—81no—
515C4—122no—
516C4—241no—
517C4#560711yescharacteristic
518C4#561211yescharacteristic
519C4#562711yescharacteristic
520C4#563011yescharacteristic
521C3#2817811yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—242no—
525C2—242no—
526C2#7393311yescharacteristic
527C2#7871211yescharacteristic
528C2#7872011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1801C2
socleC22×C3C2×#121
prime radical#79055C2
hypercenter#1801D3
hypocenterC3#28178
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #82441C2semidirect
2. C2×#5026C2semidirect
4. #79042C2semidirect
5. C3×#1801C2semidirect
6. #70527C2semidirect
7. #70517C2semidirect
8. #70123C2semidirect
9. #70117C2semidirect
10. #69010C2semidirect
11. #69006C2semidirect
55. #1801D3semidirect
521. C3#28178semidirect

Maximal subgroups

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

subgroupindex
#824412
C2×#50262
#790552
#790422
C3×#18012
#705272
#705172
#701232
#701172
#690102
#690062
#281783