Finite Group Databasebeta

← #10883523

Subgroups — #10883523

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
1C2×C3×#440C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#78491C211yesmaximal, characteristic, complemented
5#72059C212yesmaximal, complemented
6#71637C211yesmaximal, characteristic, complemented
7#71617C211yesmaximal, characteristic, complemented
8#70647C212yesmaximal, complemented
9#70286C212yesmaximal, complemented
10#70218C212yesmaximal, complemented
11#69841C211yesmaximal, characteristic, complemented
12#28178—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#942C2212yes—
17C2×#932C2212yes—
18C2×#901C2212yes—
19#5477C2211yescharacteristic
20C3×#440—22no—
21C3×#440—22no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C2×#836C2212yes—
27C2×#833C2212yes—
28#5110C2212yes—
29#5031—21no—
30#5028—22no—
31#5027—21no—
32#5019—21no—
33#5016—22no—
34#5015—21no—
35#4966C2212yes—
36#4966C2212yes—
37#4924C2211yescharacteristic
38#4924C2211yescharacteristic
39#4924C2211yescharacteristic
40#4706—22no—
41#4706—22no—
42#4552C2212yes—
43#4552C2212yes—
44#4530C2212yes—
45#4530C2212yes—
46#3037—31no—
47#3036—31no—
48C2×#440D311yescharacteristic, complemented
49#2932—31no—
50#2078—32no—
51#2019—32no—
52#1908—32no—
53#1907—32no—
54#1838—31no—
55#1801—31no—
56#1800—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×#235C2312yes—
61C2×C3×SD16—22no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—21no—
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C3×C4×D4—21no—
73C3×C4×D4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#216—22no—
81#942—24no—
82#942—42no—
83C2×#215—22no—
84C2×#214—22no—
85#932—24no—
86#932—42no—
87C2×#213—22no—
88#930—21no—
89#930—21no—
90C2×#208C2312yes—
91C2×#208C2312yes—
92C2×#208C2312yes—
93#901—42no—
94#901—42no—
95C2×ΓL(1,𝔽25)—22no—
96C2×D3×C8—22no—
97#875D412yes—
98#874D411yescharacteristic
99C4×D12—21no—
100C4×D12—21no—
101#869D411yescharacteristic
102C3×#107—22no—
103C3×#103—22no—
104C3×#102—22no—
105C3×#97D412yes—
106C3×#97D412yes—
107#836—42no—
108#833—42no—
109(C2×D3)⋊C8—24no—
110#810—22no—
111#810—22no—
112#809—22no—
113#809—22no—
114C4⋊(C3⋊C8)—21no—
115C4⋊(C3⋊C8)—21no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118C4⋊(C3⋊C8)—21no—
119C4⋊(C3⋊C8)—21no—
120C2×#137—32no—
121C2×AGL(1,ℤ/8)—32no—
122C22×SD16C2×D311yescharacteristic
123C2×#131—32no—
124#532—31no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×D4C2×D311yescharacteristic
127#464—62no—
128#463—62no—
129#462—61no—
130#461—61no—
131#460—61no—
132#459—61no—
133#440—22no—
134#440—22no—
135#435—62no—
136#435—62no—
137C2×#107C2×D311yescharacteristic
138#423—31no—
139#423—31no—
140#423—31no—
141#417—32no—
142#417—32no—
143C2×#103C2×D311yescharacteristic
144C2×#102C2×D311yescharacteristic
145C2×#101—32no—
146C2×#100—32no—
147#408—32no—
148C2×#97C2×D311yescharacteristic
149#343—32no—
150#343—32no—
151#341—32no—
152#341—32no—
153C24×C3—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—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4#4112yes—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4—82no—
172C22×C3×C4#4111yescharacteristic
173C22×C3×C4#4111yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—22no—
176#235—22no—
177#235—22no—
178#235—22no—
179#235—22no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12C2×D412yes—
183C2×C4×D3—22no—
184C2×Dic6C2×D412yes—
185C3×SD16—28no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8C2×D412yes—
191C2×C3×C8C2×D412yes—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C22⋊C4—41no—
199C3×C22⋊C4—41no—
200C3×C42—41no—
201C3×C42—41no—
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206#216—44no—
207#215—44no—
208#214—44no—
209#213—44no—
210(C2×D3)⋊C4—22no—
211(C2×D3)⋊C4—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214#208—22no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—22no—
219#208—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223ΓL(1,𝔽25)—44no—
224D3×C8—44no—
225C2×#41—32no—
226C22×Q8C22×D311yescharacteristic
227C22×D4C22×D311yescharacteristic
228C23×C4—21no—
229#137—64no—
230#137—122no—
231AGL(1,ℤ/8)—64no—
232AGL(1,ℤ/8)—122no—
233C2×Q16—62no—
234C2×SD16—22no—
235C2×SD16—22no—
236C2×SD16—22no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×D8—62no—
240#131—122no—
241#131—122no—
242C2×M16—32no—
243C2×M16—32no—
244C2×M16—32no—
245C2×M16—62no—
246C22×C8—62no—
247C22×C8C22×D311yescharacteristic
248C22×C8C22×D311yescharacteristic
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—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—61no—
263C4×D4—61no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42C22×D311yescharacteristic
270#107—22no—
271C4⋊C8—61no—
272C4⋊C8—61no—
273C4⋊C8—61no—
274C4⋊C8—61no—
275C4⋊C8—61no—
276C4⋊C8—61no—
277#103—22no—
278#103—62no—
279#103—62no—
280#102—22no—
281#102—62no—
282#102—62no—
283#101—122no—
284#100—122no—
285C22⋊C8—64no—
286#97#6712yes—
287#97#6712yes—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—81no—
292C23×C3—81no—
293C23×C3C2×#4111yescharacteristic
294C22×D3—22no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×D4—22no—
300C3×D4—22no—
301C3×D4—41no—
302C3×D4—41no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—82no—
311C2×C3×C4—82no—
312C2×C3×C4—81no—
313C2×C3×C4—81no—
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324#67—42no—
325#67—42no—
326C2×Dic3—22no—
327D12—22no—
328D12—22no—
329D12—81no—
330D12—81no—
331C4×D3—42no—
332C4×D3—42no—
333Dic6—22no—
334Dic6—22no—
335C3×C8—22no—
336C3×C8—22no—
337C3⋊C8—42no—
338C3⋊C8—42no—
339C3⋊C8—42no—
340C24—21no—
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#23512yes—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—32no—
358C2×D4—62no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4#23512yes—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—62no—
368C22×C4—82no—
369C22×C4#23711yescharacteristic
370C22×C4#23711yescharacteristic
371C22×C4C23×D311yescharacteristic
372Q16—124no—
373SD16—28no—
374SD16—124no—
375SD16—124no—
376D8—124no—
377M16—62no—
378M16—62no—
379M16—62no—
380M16—62no—
381M16—62no—
382M16—62no—
383M16—124no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—124no—
392C2×C8C2×#6712yes—
393C2×C8C2×#6712yes—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—62no—
405C22⋊C4—62no—
406C42—41no—
407C42—41no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3C2×#12111yescharacteristic
420C22×C3#55311yescharacteristic
421C22×C3#55311yescharacteristic
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—42no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4#52511yescharacteristic
433C3×C4#53511yescharacteristic
434C3×C4#54611yescharacteristic
435C3×C4#54611yescharacteristic
436Dic3—42no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—62no—
441C23—81no—
442C23—81no—
443C23C2×#23711yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448Q8—62no—
449Q8—62no—
450D4—22no—
451D4—22no—
452D4—41no—
453D4—41no—
454D4—62no—
455D4—62no—
456D4—122no—
457D4—122no—
458D4—241no—
459D4—241no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—62no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4#93811yescharacteristic
475C2×C4#93811yescharacteristic
476C2×C4#93811yescharacteristic
477C2×C4#93811yescharacteristic
478C2×C4C2×#23511yescharacteristic
479C2×C4C2×#23511yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C2×C4#100911yescharacteristic
482C2×C4#100911yescharacteristic
483C2×C4C22×#6711yescharacteristic
484C8—22no—
485C8—22no—
486C8—122no—
487C8—122no—
488C8—122no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—82no—
492C2×C3#253511yescharacteristic
493C2×C3#306811yescharacteristic
494C2×C3#306911yescharacteristic
495D3—82no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22—81no—
502C22—81no—
503C22—122no—
504C22—122no—
505C22—122no—
506C22#556511yescharacteristic
507C22#556511yescharacteristic
508C22C2×#93811yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—81no—
515C4—81no—
516C4—122no—
517C4#555011yescharacteristic
518C4#555011yescharacteristic
519C4#579111yescharacteristic
520C4#583011yescharacteristic
521C3#2817811yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—82no—
525C2—242no—
526C2#8241511yescharacteristic
527C2#8396311yescharacteristic
528C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC22×C3C2×#121
prime radical#82307C2
hypercenterC2×#440D3
hypocenterC3#28178
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×C3×#440C2semidirect
2. #82441C2semidirect
4. #78491C2semidirect
5. #72059C2semidirect
6. #71637C2semidirect
7. #71617C2semidirect
8. #70647C2semidirect
9. #70286C2semidirect
10. #70218C2semidirect
11. #69841C2semidirect
48. C2×#440D3semidirect
521. C3#28178semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4402
#824412
#823072
#784912
#720592
#716372
#716172
#706472
#702862
#702182
#698412
#281783