Finite Group Databasebeta

← #10883515

Subgroups — #10883515

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82307C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic, complemented
4#79042C211yesmaximal, characteristic, complemented
5C3×#1800C211yesmaximal, characteristic, complemented
6#70527C211yesmaximal, characteristic, complemented
7#70517C211yesmaximal, characteristic, complemented
8#70090C212yesmaximal, complemented
9#70088C212yesmaximal, complemented
10#69011C212yesmaximal, complemented
11#69006C212yesmaximal, complemented
12#28177—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#932C2212yes—
15C22×#213C2211yescharacteristic
16C2×#908C2212yes—
17C2×#907C2212yes—
18#5477C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×#100C2212yes—
23C2×#832C2211yescharacteristic
24#5133C2212yes—
25C2×#823C2212yes—
26#5129C2212yes—
27#5019—21no—
28#5016—22no—
29#5015—21no—
30#5014—22no—
31#5014—22no—
32#4966C2212yes—
33C2×#809C2211yescharacteristic
34C2×#808C2211yescharacteristic
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#3036—31no—
47#3036—31no—
48C2×#442—31no—
49#2932—31no—
50#2079—32no—
51#2018—32no—
52#1907—32no—
53#1907—32no—
54#1838—31no—
55#1800—31no—
56#1800D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×#235C2312yes—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C2×C3×M16C2312yes—
63C2×C3×M16C2312yes—
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#214—22no—
72#932—24no—
73#932—42no—
74C2×#213—22no—
75C2×#213—22no—
76C2×#213—22no—
77C2×#213—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×#100—42no—
107#832—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#808—22no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118#803D412yes—
119#803D412yes—
120C2×AGL(1,ℤ/8)—32no—
121C2×AGL(1,ℤ/8)—32no—
122C22×D8—31no—
123C2×#131—32no—
124#532—31no—
125C2×C4×D4—31no—
126C2×C4×D4C2×D311yescharacteristic
127#463—62no—
128#463—62no—
129#461—61no—
130#461—61no—
131#459—61no—
132#459—61no—
133#442—62no—
134#442—62no—
135#435—62no—
136#435—62no—
137C2×#106—31no—
138#423—31no—
139#423C2×D311yescharacteristic
140#423C2×D311yescharacteristic
141#417—32no—
142#417—32no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#100—32no—
146C2×#100C2×D312yes—
147#408—32no—
148C2×#97—31no—
149#343—32no—
150#343C2×D312yes—
151#341—32no—
152#341—32no—
153C24×C3—21no—
154C23×D3—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4#4112yes—
161C22×C3×C4—21no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4—82no—
165C22×C3×C4#4111yescharacteristic
166C22×C3×C4#4111yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—22no—
169C22×Dic3—21no—
170#235—22no—
171#235—22no—
172#235—22no—
173#235—22no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12C2×D412yes—
182C2×D12#4112yes—
183C2×C4×D3—22no—
184C2×C4×D3—41no—
185C2×C4×D3—41no—
186C2×C4×D3—82no—
187C2×Dic6C2×D412yes—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C22⋊C4—41no—
197C3×C22⋊C4—41no—
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202#214—44no—
203#213—28no—
204#213—44no—
205(C2×D3)⋊C4—22no—
206(C2×D3)⋊C4—22no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213#208—22no—
214#208—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8C2×D412yes—
221C2×C3⋊C8C2×D412yes—
222D24—44no—
223#204—44no—
224ΓL(1,𝔽25)—44no—
225D3×C8—44no—
226C2×#41—32no—
227C22×D4—31no—
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—61no—
231AGL(1,ℤ/8)—64no—
232AGL(1,ℤ/8)—64no—
233AGL(1,ℤ/8)—122no—
234AGL(1,ℤ/8)—122no—
235C2×SD16—62no—
236C2×SD16—62no—
237C2×D8—62no—
238C2×D8—62no—
239C2×D8—62no—
240C2×D8—62no—
241C2×D8—62no—
242#131—122no—
243#131—122no—
244C2×M16—32no—
245C2×M16—62no—
246C2×M16C22×D312yes—
247C2×M16C22×D312yes—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8—62no—
251C4⋊Q8—31no—
252#127—31no—
253#124—32no—
254#121—61no—
255#121—61no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263C4×D4—61no—
264C4×D4—61no—
265C4×D4—61no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—61no—
270C2×C42C22×D311yescharacteristic
271#106—62no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—61no—
277C4⋊C8—61no—
278#103—62no—
279#103—62no—
280#102—62no—
281#102—62no—
282#102—62no—
283#102—62no—
284#100—42no—
285#100—122no—
286C22⋊C8—64no—
287#97—32no—
288#97—32no—
289C23×C3—22no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—81no—
293C23×C3—81no—
294C23×C3C2×#4111yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—81no—
300C22×D3—81no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—41no—
304C3×D4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—82no—
309C2×C3×C4—82no—
310C2×C3×C4—81no—
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321#67—42no—
322#67—42no—
323C2×Dic3—22no—
324C2×Dic3—41no—
325C2×Dic3—81no—
326D12—22no—
327D12—22no—
328D12—22no—
329D12—22no—
330D12—41no—
331D12—41no—
332D12—81no—
333D12—81no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—82no—
337C4×D3—82no—
338Dic6—22no—
339Dic6—22no—
340C3×C8—42no—
341C3×C8—42no—
342C3⋊C8—22no—
343C3⋊C8—22no—
344C3⋊C8—42no—
345C24—21no—
346C24—61no—
347#41—62no—
348#41—62no—
349#41—62no—
350#41—62no—
351C2×Q8—32no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—32no—
358C2×D4—32no—
359C2×D4—62no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4—61no—
367C2×D4#23912yes—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—62no—
372C22×C4—61no—
373C22×C4—82no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4—242no—
377C22×C4#23511yescharacteristic
378C22×C4#23511yescharacteristic
379C22×C4C23×D311yescharacteristic
380SD16—124no—
381SD16—124no—
382D8—68no—
383D8—124no—
384D8—124no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—62no—
390M16—62no—
391M16—124no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—32no—
395C2×C8—32no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—62no—
401C2×C8—124no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—62no—
411C22⋊C4—62no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C42D3×D411yescharacteristic
415C42D3×D411yescharacteristic
416C42D3×D411yescharacteristic
417C42D3×D411yescharacteristic
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—42no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C2×#12111yescharacteristic
426C22×C3#55311yescharacteristic
427C22×C3#55311yescharacteristic
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—82no—
433C2×D3—81no—
434C2×D3—81no—
435C2×D3—82no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—81no—
441C3×C4#52511yescharacteristic
442C3×C4#53511yescharacteristic
443C3×C4#54611yescharacteristic
444C3×C4#54611yescharacteristic
445Dic3—42no—
446Dic3—81no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—62no—
451C23—62no—
452C23—81no—
453C23—81no—
454C23—121no—
455C23—121no—
456C23—241no—
457C23—241no—
458C23C2×#23511yescharacteristic
459Q8—62no—
460Q8—62no—
461D4—22no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—62no—
466D4—62no—
467D4—62no—
468D4—62no—
469D4—122no—
470D4—122no—
471D4—121no—
472D4—121no—
473D4—241no—
474D4—241no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—62no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—242no—
486C2×C4—242no—
487C2×C4—241no—
488C2×C4#88411yescharacteristic
489C2×C4#88411yescharacteristic
490C2×C4#88411yescharacteristic
491C2×C4#88411yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4#100411yescharacteristic
495C2×C4C2×#23911yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#100911yescharacteristic
498C8—42no—
499C8—42no—
500C8—62no—
501C8—62no—
502C8—122no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—82no—
506C2×C3#253511yescharacteristic
507C2×C3#306811yescharacteristic
508C2×C3#306811yescharacteristic
509D3—82no—
510D3—82no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—242no—
523C22—242no—
524C22—241no—
525C22—241no—
526C22C2×#88411yescharacteristic
527C22#551411yescharacteristic
528C22#553011yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—81no—
534C4—122no—
535C4—241no—
536C4#560711yescharacteristic
537C4#561211yescharacteristic
538C4#562711yescharacteristic
539C4#563011yescharacteristic
540C3#2817711yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—82no—
544C2—242no—
545C2—242no—
546C2#7393311yescharacteristic
547C2#7870311yescharacteristic
548C2#7872011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1800C2
socleC22×C3C2×#121
hypercenter#1800D3
hypocenterC3#28177
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #82307C2semidirect
2. C2×#5014C2semidirect
3. #79055C2semidirect
4. #79042C2semidirect
5. C3×#1800C2semidirect
6. #70527C2semidirect
7. #70517C2semidirect
8. #70090C2semidirect
9. #70088C2semidirect
10. #69011C2semidirect
11. #69006C2semidirect
56. #1800D3semidirect
540. C3#28177semidirect

Maximal subgroups

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

subgroupindex
#823072
C2×#50142
#790552
#790422
C3×#18002
#705272
#705172
#700902
#700882
#690112
#690062
#281773