Finite Group Databasebeta

← #10883500

Subgroups — #10883500

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83966C212yesmaximal
2C2×Dic3×Q16C211yesmaximal, characteristic, complemented
3#78657C211yesmaximal, characteristic
4C3×#2020C211yesmaximal, characteristic, complemented
5#72103C212yesmaximal, complemented
6#72074C211yesmaximal, characteristic, complemented
7#70123C212yesmaximal, complemented
8#70097C212yesmaximal, complemented
9#69994C211yesmaximal, characteristic, complemented
10#68928C211yesmaximal, characteristic, complemented
11#68921C211yesmaximal, characteristic, complemented
12#28176—31nomaximal, Sylow
13C2×C3×#137C2212yes—
14C22×C3×Q16C2211yescharacteristic
15#5832C2211yescharacteristic
16C2×#948C2211yescharacteristic
17C2×Q8×Dic3C2212yes—
18C2×#942C2212yes—
19C2×#907C2211yescharacteristic
20C2×C3×#101C2212yes—
21C3×#408C2211yescharacteristic
22#5232C2212yes—
23C2×#836C2212yes—
24C2×#835C2212yes—
25#5185—22no—
26#5182—22no—
27Dic3×Q16—21no—
28Dic3×Q16—21no—
29Dic3×Q16—21no—
30Dic3×Q16—21no—
31#5166—22no—
32#5163—22no—
33#5133C2212yes—
34#5118C2212yes—
35#5118C2211yescharacteristic
36#5110C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#4729—21no—
40#4729—21no—
41#67⋊C8—21no—
42#67⋊C8—21no—
43C3×#341C2211yescharacteristic
44#4558C2212yes—
45#4554C2212yes—
46#4554C2212yes—
47#4552C2211yescharacteristic
48#3037—32no—
49C2×C4×Q16—31no—
50#2931—31no—
51#2078—31no—
52#2078—31no—
53#2020—31no—
54#2020D311yescharacteristic, complemented
55#1908—32no—
56#1908—32no—
57#1837—31no—
58#1801—32no—
59C2×C3×#41C2311yescharacteristic
60C22×C3×Q8C2312yes—
61C2×#235C2311yescharacteristic
62C3×#137—24no—
63C3×#137—42no—
64C2×C3×Q16—24no—
65C2×C3×Q16—22no—
66C2×C3×Q16—21no—
67C2×C3×Q16—21no—
68C2×C3×SD16—22no—
69C2×C3×M16C2312yes—
70C2×C3×M16C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72#948—42no—
73#947D411yescharacteristic
74Q8×Dic3—22no—
75Q8×Dic3—22no—
76Q8×Dic3—22no—
77Q8×Dic3—22no—
78#944D411yescharacteristic
79C2×#216—22no—
80#942—24no—
81#942—42no—
82C2×#215—22no—
83#940D411yescharacteristic
84#939—21no—
85#939—21no—
86#936D411yescharacteristic
87C4×#67—21no—
88C4×#67—21no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3—22no—
92C2×C4×Dic3C2311yescharacteristic
93C2×#208—21no—
94C2×#208C2311yescharacteristic
95C2×#208C2312yes—
96C22×C3⋊C8—21no—
97C22×C3⋊C8C2311yescharacteristic
98#907—42no—
99C2×ΓL(1,𝔽25)—21no—
100C2×D3×C8—21no—
101C3×#101—42no—
102C3×C22⋊C8—21no—
103C3×C22⋊C8—21no—
104#836—42no—
105#835—22no—
106#835—22no—
107#835—22no—
108#832—22no—
109#821—22no—
110(C2×D3)⋊C8—21no—
111(C2×D3)⋊C8—21no—
112#818—21no—
113#818—21no—
114#816—22no—
115Dic3⋊C8—22no—
116Dic3⋊C8—22no—
117Dic3⋊C8—21no—
118Dic3⋊C8—21no—
119C8×Dic3D411yescharacteristic
120C8×Dic3D411yescharacteristic
121C8×Dic3D411yescharacteristic
122C8×Dic3D411yescharacteristic
123C2×#137—32no—
124C2×#137C2×D312yes—
125C22×Q16C2×D311yescharacteristic
126C2×#131—31no—
127C2×#131—31no—
128#532—31no—
129C2×C4×Q8—32no—
130#464—62no—
131#464—62no—
132#462—62no—
133#460—62no—
134C4×Q16—61no—
135C4×Q16—61no—
136C4×Q16—61no—
137C4×Q16—61no—
138#436—61no—
139#436—61no—
140C8×D4—61no—
141C8×D4—61no—
142C2×#107—31no—
143#423—31no—
144#423—32no—
145#417—32no—
146#417—32no—
147C2×#103—32no—
148C2×#101—32no—
149C2×#101C2×D312yes—
150#408—31no—
151#408C2×D311yescharacteristic
152C2×C4×C8—31no—
153#343—32no—
154#343—32no—
155#341—32no—
156#341—31no—
157#341C2×D311yescharacteristic
158C3×#41—22no—
159C3×#41—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×Q8#4114yes—
167C2×C3×D4—21no—
168C2×C3×D4C2×D411yescharacteristic
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4C2411yescharacteristic
172C2×#67—21no—
173C2×#67—21no—
174C2×#67—21no—
175C22×Dic3—22no—
176C22×Dic3#4111yescharacteristic
177C22×Dic3#4111yescharacteristic
178#235—22no—
179#235—22no—
180C2×D12C2×D411yescharacteristic
181C2×C4×D3—21no—
182C2×Dic6C2×D411yescharacteristic
183C3×Q16—24no—
184C3×Q16—24no—
185C3×Q16—44no—
186C3×SD16—44no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195#217—21no—
196#217—21no—
197#216—44no—
198#215—44no—
199(C2×D3)⋊C4—21no—
200(C2×D3)⋊C4—21no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—42no—
204C4⋊Dic3—42no—
205Dic3⋊C4—21no—
206Dic3⋊C4—21no—
207C4×Dic3—42no—
208C4×Dic3—42no—
209C4×Dic3C2×D411yescharacteristic
210C4×Dic3C2×D411yescharacteristic
211C4×Dic3C2×D411yescharacteristic
212C4×Dic3C2×D411yescharacteristic
213#208—22no—
214#208—22no—
215#208—22no—
216#208—42no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—42no—
221C2×C3⋊C8C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223ΓL(1,𝔽25)—42no—
224D3×C8—42no—
225C2×#41—31no—
226C2×#41C22×D311yescharacteristic
227C22×Q8C22×D312yes—
228#137—24no—
229#137—42no—
230#137—64no—
231#137—122no—
232C2×Q16—24no—
233C2×Q16—22no—
234C2×Q16—21no—
235C2×Q16—21no—
236C2×Q16—62no—
237C2×SD16—22no—
238C2×SD16—62no—
239#131—122no—
240#131—122no—
241C2×M16—31no—
242C2×M16—32no—
243C2×M16—61no—
244C2×M16—61no—
245C2×M16C22×D311yescharacteristic
246C2×M16C22×D312yes—
247C22×C8—31no—
248C22×C8—61no—
249C22×C8—61no—
250C22×C8C22×D311yescharacteristic
251C4⋊Q8—31no—
252#127—31no—
253#124—31no—
254#124—31no—
255#121—61no—
256#121—61no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×Q8—62no—
260C4×Q8—62no—
261C4×D4—61no—
262C4×D4—61no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—62no—
265C2×C42—31no—
266C2×C42—62no—
267#107—61no—
268#107—61no—
269C4⋊C8—61no—
270C4⋊C8—61no—
271C4⋊C8—62no—
272C4⋊C8—62no—
273#103—62no—
274#103—62no—
275#103—62no—
276#103—62no—
277#102—62no—
278#102—62no—
279#101—42no—
280#101—122no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—61no—
284C22⋊C8—61no—
285C4×C8—31no—
286C4×C8—31no—
287C4×C8—31no—
288C4×C8—31no—
289C23×C3—21no—
290C23×C3C2×#4111yescharacteristic
291C22×D3—21no—
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—22no—
295C3×Q8—42no—
296C3×Q8—42no—
297C3×D4—22no—
298C3×D4—42no—
299C2×C3×C4—21no—
300C2×C3×C4—24no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4H(5,𝔽2)12yes—
309#67—42no—
310#67—42no—
311#67—42no—
312C2×Dic3—22no—
313C2×Dic3—21no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3—82no—
317C2×Dic3#12111yescharacteristic
318C2×Dic3#12111yescharacteristic
319C2×Dic3#12111yescharacteristic
320C2×Dic3#12111yescharacteristic
321D12—22no—
322C4×D3—41no—
323C4×D3—41no—
324Dic6—22no—
325C3×C8—21no—
326C3×C8—21no—
327C3×C8—41no—
328C3×C8—42no—
329C3⋊C8—21no—
330C3⋊C8—21no—
331C3⋊C8—41no—
332C3⋊C8—42no—
333#41—22no—
334#41—22no—
335#41—62no—
336#41—62no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—31no—
343C2×Q8D3×D411yescharacteristic
344C2×Q8#23714yes—
345C2×D4—21no—
346C2×D4—31no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4D3×D411yescharacteristic
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4C23×D311yescharacteristic
358Q16—24no—
359Q16—24no—
360Q16—44no—
361Q16—124no—
362SD16—44no—
363SD16—124no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—62no—
368M16—62no—
369M16—62no—
370M16—122no—
371M16—122no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—31no—
376C2×C8—31no—
377C2×C8—62no—
378C2×C8—61no—
379C2×C8—62no—
380C2×C8—122no—
381C2×C8—122no—
382C2×C8D3×D411yescharacteristic
383C2×C8D3×D411yescharacteristic
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—122no—
389C4⋊C4—122no—
390C22⋊C4—61no—
391C22⋊C4—61no—
392C22⋊C4—61no—
393C22⋊C4—61no—
394C42—31no—
395C42—31no—
396C42—31no—
397C42—31no—
398C42—122no—
399C42—122no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3C2×#12111yescharacteristic
406C22×C3#55312yes—
407C2×D3—41no—
408C2×D3—41no—
409C2×D3—41no—
410C3×C4—41no—
411C3×C4—44no—
412C3×C4#52511yescharacteristic
413C3×C4#53511yescharacteristic
414C3×C4#54612yes—
415Dic3—21no—
416Dic3—21no—
417Dic3—21no—
418Dic3—41no—
419Dic3—82no—
420Dic3#47312yes—
421C23—21no—
422C23—61no—
423C23D3×#4111yescharacteristic
424Q8—24no—
425Q8—24no—
426Q8—22no—
427Q8—42no—
428Q8—42no—
429Q8—62no—
430D4—22no—
431D4—42no—
432D4—62no—
433D4—122no—
434D4—122no—
435D4—122no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—62no—
447C2×C4—61no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—242no—
453C2×C4C2×D3×D411yescharacteristic
454C2×C4C2×D3×D411yescharacteristic
455C2×C4C2×#23712yes—
456C2×C4#100412yes—
457C8—21no—
458C8—21no—
459C8—41no—
460C8—42no—
461C8—61no—
462C8—61no—
463C8—121no—
464C8—122no—
465C2×C3—22no—
466C2×C3—81no—
467C2×C3#253511yescharacteristic
468C2×C3C2×#47311yescharacteristic
469C2×C3#306711yescharacteristic
470D3—81no—
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22#560112yes—
480C22D3×#12111yescharacteristic
481C4—32no—
482C4—41no—
483C4—44no—
484C4—61no—
485C4—61no—
486C4—61no—
487C4—121no—
488C4—242no—
489C4#558611yescharacteristic
490C4#559112yes—
491C4#571511yescharacteristic
492C3#2817611yescharacteristic, complemented, Sylow
493C2—22no—
494C2—81no—
495C2—241no—
496C2#7668611yescharacteristic
497C2D3×#47311yescharacteristic
498C2#8094311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2020C2
socleC22×C3C2×#121
prime radical#5832C22
hypercenter#2020D3
hypocenterC3#28176
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2020

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×Q16C2semidirect
4. C3×#2020C2semidirect
5. #72103C2semidirect
6. #72074C2semidirect
7. #70123C2semidirect
8. #70097C2semidirect
9. #69994C2semidirect
10. #68928C2semidirect
11. #68921C2semidirect
54. #2020D3semidirect
492. C3#28176semidirect

Maximal subgroups

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

subgroupindex
#839662
C2×Dic3×Q162
#786572
C3×#20202
#721032
#720742
#701232
#700972
#699942
#689282
#689212
#281763